Microstructural Characterization of Materials

Microstructural Characterization of Materials 2nd Edition DAVID BRANDON AND WAYNE D. KAPLAN Technion, Israel Institute of Technology, Israel

Microstructural Characterization of Materials 2nd Edition

Microstructural Characterization of Materials 2nd Edition DAVID BRANDON AND WAYNE D. KAPLAN Technion, Israel Institute of Technology, Israel

Copyright Ó 2008

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) 1243 779777

Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (þ44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The Publisher and the Author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the Publisher nor the Author shall be liable for any damages arising herefrom. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Ltd, 6045 Freemont Blvd, Mississauga, Ontario L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Brandon, D. G. Microstructural Characterization of Materials / David Brandon and Wayne D. Kaplan. – 2nd ed. p. cm. – (Quantitative software engineering series) Includes bibliographical references and index. ISBN 978-0-470-02784-4 (cloth) – ISBN 978-0-470-02785-1 (pbk.) 1. Materials–Microscopy. 2. Microstructure. I. Kaplan, Wayne D. II. Title. TA417.23.B73 2008 620.1’1299–dc22 2007041704 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978 0 470 02784 4 (cloth) ISBN 978 0 470 02785 1 (paper) Typeset in 10/12 pt Times by Thomson Digital, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

Contents Preface to the Second Edition Preface to the First Edition

xi xiii

1 The Concept of Microstructure 1.1 Microstructural Features 1.1.1 Structure–Property Relationships 1.1.2 Microstructural Scale 1.1.3 Microstructural Parameters 1.2 Crystallography and Crystal Structure 1.2.1 Interatomic Bonding in Solids 1.2.2 Crystalline and Amorphous Phases 1.2.3 The Crystal Lattice Summary Bibliography Worked Examples Problems

1 7 7 10 19 24 25 30 30 42 46 46 51

2 Diffraction Analysis of Crystal Structure 2.1 Scattering of Radiation by Crystals 2.1.1 The Laue Equations and Bragg’s Law 2.1.2 Allowed and Forbidden Reflections 2.2 Reciprocal Space 2.2.1 The Limiting Sphere Construction 2.2.2 Vector Representation of Bragg’s Law 2.2.3 The Reciprocal Lattice 2.3 X-Ray Diffraction Methods 2.3.1 The X-Ray Diffractometer 2.3.2 Powder Diffraction–Particles and Polycrystals 2.3.3 Single Crystal Laue Diffraction 2.3.4 Rotating Single Crystal Methods 2.4 Diffraction Analysis 2.4.1 Atomic Scattering Factors 2.4.2 Scattering by the Unit Cell 2.4.3 The Structure Factor in the Complex Plane 2.4.4 Interpretation of Diffracted Intensities 2.4.5 Errors and Assumptions 2.5 Electron Diffraction 2.5.1 Wave Properties of Electrons

55 56 56 59 60 60 61 61 63 67 73 76 78 79 80 81 83 84 85 90 91

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Contents

2.5.2 Ring Patterns, Spot Patterns and Laue Zones 2.5.3 Kikuchi Patterns and Their Interpretation Summary Bibliography Worked Examples Problems 3 Optical Microscopy 3.1 Geometrical Optics 3.1.1 Optical Image Formation 3.1.2 Resolution in the Optical Microscope 3.1.3 Depth of Field and Depth of Focus 3.2 Construction of The Microscope 3.2.1 Light Sources and Condenser Systems 3.2.2 The Specimen Stage 3.2.3 Selection of Objective Lenses 3.2.4 Image Observation and Recording 3.3 Specimen Preparation 3.3.1 Sampling and Sectioning 3.3.2 Mounting and Grinding 3.3.3 Polishing and Etching Methods 3.4 Image Contrast 3.4.1 Reflection and Absorption of Light 3.4.2 Bright-Field and Dark-Field Image Contrast 3.4.3 Confocal Microscopy 3.4.4 Interference Contrast and Interference Microscopy 3.4.5 Optical Anisotropy and Polarized Light 3.4.6 Phase Contrast Microscopy 3.5 Working with Digital Images 3.5.1 Data Collection and The Optical System 3.5.2 Data Processing and Analysis 3.5.3 Data Storage and Presentation 3.5.4 Dynamic Range and Digital Storage 3.6 Resolution, Contrast and Image Interpretation Summary Bibliography Worked Examples Problems 4 Transmission Electron Microscopy 4.1 Basic Principles 4.1.1 Wave Properties of Electrons 4.1.2 Resolution Limitations and Lens Aberrations 4.1.3 Comparative Performance of Transmission and Scanning Electron Microscopy

94 96 98 103 103 114 123 125 125 130 133 134 134 136 136 139 143 143 144 145 148 149 150 152 152 157 163 165 165 165 166 167 170 171 173 173 176 179 185 185 187 192

Contents vii

4.2

4.3

4.4

4.5

4.6

4.7

Specimen Preparation 4.2.1 Mechanical Thinning 4.2.2 Electrochemical Thinning 4.2.3 Ion Milling 4.2.4 Sputter Coating and Carbon Coating 4.2.5 Replica Methods The Origin of Contrast 4.3.1 Mass–Thickness Contrast 4.3.2 Diffraction Contrast and Crystal Lattice Defects 4.3.3 Phase Contrast and Lattice Imaging Kinematic Interpretation of Diffraction Contrast 4.4.1 Kinematic Theory of Electron Diffraction 4.4.2 The Amplitude–Phase Diagram 4.4.3 Contrast From Lattice Defects 4.4.4 Stacking Faults and Anti-Phase Boundaries 4.4.5 Edge and Screw Dislocations 4.4.6 Point Dilatations and Coherency Strains Dynamic Diffraction and Absorption Effects 4.5.1 Stacking Faults Revisited 4.5.2 Quantitative Analysis of Contrast Lattice Imaging at High Resolution 4.6.1 The Lattice Image and the Contrast Transfer Function 4.6.2 Computer Simulation of Lattice Images 4.6.3 Lattice Image Interpretation Scanning Transmission Electron Microscopy Summary Bibliography Worked Examples Problems

5 Scanning Electron Microscopy 5.1 Components of The Scanning Electron Microscope 5.2 Electron Beam–Specimen Interactions 5.2.1 Beam-Focusing Conditions 5.2.2 Inelastic Scattering and Energy Losses 5.3 Electron Excitation of X-Rays 5.3.1 Characteristic X-Ray Images 5.4 Backscattered Electrons 5.4.1 Image Contrast in Backscattered Electron Images 5.5 Secondary Electron Emission 5.5.1 Factors Affecting Secondary Electron Emission 5.5.2 Secondary Electron Image Contrast 5.6 Alternative Imaging Modes 5.6.1 Cathodoluminescence 5.6.2 Electron Beam Induced Current 5.6.3 Orientation Imaging Microscopy

194 195 198 199 201 202 203 205 205 207 213 213 213 215 216 218 219 221 227 230 230 230 231 232 234 236 238 238 247 261 262 264 265 266 269 271 277 279 280 283 286 288 288 288 289

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Contents

5.7

5.8

5.6.4 Electron Backscattered Diffraction Patterns 5.6.5 OIM Resolution and Sensitivity 5.6.6 Localized Preferred Orientation and Residual Stress Specimen Preparation and Topology 5.7.1 Sputter Coating and Contrast Enhancement 5.7.2 Fractography and Failure Analysis 5.7.3 Stereoscopic Imaging 5.7.4 Parallax Measurements Focused Ion Beam Microscopy 5.8.1 Principles of Operation and Microscope Construction 5.8.2 Ion Beam–Specimen Interactions 5.8.3 Dual-Beam FIB Systems 5.8.4 Machining and Deposition 5.8.5 TEM Specimen Preparation 5.8.6 Serial Sectioning Summary Bibliography Worked Examples Problems

289 291 292 294 295 295 298 298 301 302 304 306 306 310 314 315 318 318 326

6 Microanalysis in Electron Microscopy 6.1 X-Ray Microanalysis 6.1.1 Excitation of Characteristic X-Rays 6.1.2 Detection of Characteristic X-Rays 6.1.3 Quantitative Analysis of Composition 6.2 Electron Energy Loss Spectroscopy 6.2.1 The Electron Energy-Loss Spectrum 6.2.2 Limits of Detection and Resolution in EELS 6.2.3 Quantitative Electron Energy Loss Analysis 6.2.4 Near-Edge Fine Structure Information 6.2.5 Far-Edge Fine Structure Information 6.2.6 Energy-Filtered Transmission Electron Microscopy Summary Bibliography Worked Examples Problems

333 334 334 338 343 357 360 361 364 365 366 367 370 375 375 386

7 Scanning Probe Microscopy and Related Techniques 7.1 Surface Forces and Surface Morphology 7.1.1 Surface Forces and Their Origin 7.1.2 Surface Force Measurements 7.1.3 Surface Morphology: Atomic and Lattice Resolution 7.2 Scanning Probe Microscopes 7.2.1 Atomic Force Microscopy 7.2.2 Scanning Tunnelling Microscopy 7.3 Field-Ion Microscopy and Atom Probe Tomography

391 392 392 396 397 400 403 410 413

Contents

7.3.1 Identifying Atoms by Field Evaporation 7.3.2 The Atom Probe and Atom Probe Tomography Summary Bibliography Problems

ix

414 416 417 420 420

8 Chemical Analysis of Surface Composition 8.1 X-Ray Photoelectron Spectroscopy 8.1.1 Depth Discrimination 8.1.2 Chemical Binding States 8.1.3 Instrumental Requirements 8.1.4 Applications 8.2 Auger Electron Spectroscopy 8.2.1 Spatial Resolution and Depth Discrimination 8.2.2 Recording and Presentation of Spectra 8.2.3 Identification of Chemical Binding States 8.2.4 Quantitative Auger Analysis 8.2.5 Depth Profiling 8.2.6 Auger Imaging 8.3 Secondary-Ion Mass Spectrometry 8.3.1 Sensitivity and Resolution 8.3.2 Calibration and Quantitative Analysis 8.3.3 SIMS Imaging Summary Bibliography Worked Examples Problems

423 424 426 428 429 431 431 433 434 435 436 437 438 440 442 444 445 446 448 448 453

9 Quantitative and Tomographic Analysis of Microstructure 9.1 Basic Stereological Concepts 9.1.1 Isotropy and Anisotropy 9.1.2 Homogeneity and Inhomogeneity 9.1.3 Sampling and Sectioning 9.1.4 Statistics and Probability 9.2 Accessible and Inaccessible Parameters 9.2.1 Accessible Parameters 9.2.2 Inaccessible Parameters 9.3 Optimizing Accuracy 9.3.1 Sample Size and Counting Time 9.3.2 Resolution and Detection Errors 9.3.3 Sample Thickness Corrections 9.3.4 Observer Bias 9.3.5 Dislocation Density Revisited 9.4 Automated Image Analysis 9.4.1 Digital Image Recording 9.4.2 Statistical Significance and Microstructural Relevance

457 458 459 461 463 466 467 468 476 481 483 485 487 489 490 491 494 495

x

Contents

9.5

Tomography and Three-Dimensional Reconstruction 9.5.1 Presentation of Tomographic Data 9.5.2 Methods of Serial Sectioning 9.5.3 Three-Dimensional Reconstruction Summary Bibliography Worked Examples Problems

495 496 498 499 500 503 503 514

Appendices Appendix 1: Useful Equations Interplanar Spacings Unit Cell Volumes Interplanar Angles Direction Perpendicular to a Crystal Plane Hexagonal Unit Cells The Zone Axis of Two Planes in the Hexagonal System Appendix 2: Wavelengths Relativistic Electron Wavelengths X-Ray Wavelengths for Typical X-Ray Sources

517 517 517 518 518 519 520 521 521 521 521

Index

523

Preface to the Second Edition The last decade has seen several major changes in the armoury of tools that are routinely available to the materials scientist and engineer for microstructural characterization. Some of these changes reflect continuous technological improvements in the collection, processing and recording of image data. Several other innovations have been both dramatic and unexpected, not least in the rapid acceptance these tools have gained in both the research and industrial development communities. The present text follows the guidelines laid down for the first edition, exploring the methodology of materials characterization under the three headings of crystal structure, microstructural morphology and microanalysis. One additional chapter has been added, on Scanning Probe Microscopy, a topic that, at the time that the first edition was written, was very much a subject for active research, but a long way from being commonly accessible in university and industrial laboratories. Today, atomic force and scanning tunnelling microscopy have found applications in fields as diverse as optronics and catalysis, friction and cosmetics. It has proved necessary to split the chapter on Electron Microscopy into two chapters, one on Transmission techniques, and the other on Scanning methods. These two expanded chapters reflect the dramatic improvements in the resolution available for lattice imaging in transmission, and the revolution in sampling and micro-machining associated with the introduction of the focused ion beam in scanning technology. The final chapter, on Quantitative Analysis, has also been expanded, to accommodate the rapid advances in three-dimensional reconstruction that now enable massive data sets to be assembled which include both chemical and crystallographic data embedded in a frame of reference given by microstructural morphology. Not least among the new innovations are orientation imaging microscopy, which allows the relative crystallographic orientations of the grains of a polycrystalline sample to be individually mapped, and atom probe tomography, in which the ions extracted from the surface of a sharp metallic needle are chemically identified and recorded in three dimensions. This last instrument is a long way from being widely available, but a number of laboratories do offer their services commercially, bringing three-dimensional analysis and characterization well below the nanometre range, surely the ultimate in microstructural characterization. It only remains to note the greatest difference between the present text and its predecessor: digital recording methods have all but replaced photography in every application that we have considered, and we have therefore included sections on digital recording, processing and analysis. This ‘digital revolution’ has crept up on us slowly, following the on-going improvements in the storage capacity and processing speed for computer hardware and software. Today, massive amounts of digital image data can be handled rapidly and reliably.

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Preface to the Second Edition

At the same time, it is still up to the microscopist and the engineer to make the critical decisions associated with the selection of samples for characterization, the preparation of suitable sections and the choice of characterization methods. This task is just as difficult today as it always was in the past. Hopefully, this new text will help rather than confuse! Most of the data in this book are taken from work conducted in collaboration with our colleagues and students at the Technion. We wish to thank the following for their contributions: David Seidman, Rik Brydson, Igor Levin, Moshe Eizenberg, Arnon Siegmann, Menachem Bamberger, Michael Silverstein, Yaron Kauffmann, Christina Scheu, Gerhard Dehm, Ming Wei, Ludmilla Shepelev, Michal Avinun, George Levi, Amir Avishai, Tzipi Cohen, Mike Lieberthal, Oren Aharon, Hila Sadan, Mor Baram, Lior Miller, Adi Karpel, Miri Drozdov, Gali Gluzer, Mike Lieberthal, and Thangadurai Paramasivam. D.B. W.D.K.

Preface to the First Edition Most logical decisions rely on providing acceptable answers to precise questions, e.g. what, why and how? In the realm of scientific and technical investigation, the first question is typically what is the problem or what is the objective? This is then followed by a why question which attempts to pinpoint priorities, i.e. the urgency and importance of finding an acceptable answer. The third type of question, how is usually concerned with identifying means and methods, and the answers constitute an assessment of the available resources for resolving a problem or achieving an objective. The spectrum of problems arising in materials science and technology very often depends critically on providing adequate answers to these last two questions. The answers may take many forms, but when materials expertise is involved, they frequently include a need to characterize the internal microstructure of an engineering material. This book is an introduction to the expertise involved in assessing the microstructure of engineering materials and to the experimental methods which are available for this purpose. For this text to be meaningful, the reader should understand why the investigation of the internal structure of an engineering material is of interest and appreciate why the microstructural features of the material are so often of critical engineering importance, This text is intended to provide a basic grasp of both the methodology and the means available for deriving qualitative and quantitative microstructural information from a suitable sample. There are two ways of approaching materials characterization. The first of these is in terms of the engineering properties of materials, and reflects the need to know the physical, chemical and mechanical properties of the material before we can design an engineering system or manufacture its components. The second form of characterization is that which concerns us in this book, namely the microstructural characterization of the material. In specifying the internal microstructure of an engineering material we include the chemistry, the crystallography and the structural morphology, with the term materials characterization being commonly taken to mean just this specification. Characterization in terms of the chemistry involves an identification of the chemical constituents of the material and an analysis of their relative abundance, that is a determination of the chemical composition and the distribution of the chemical elements within the material. In this present text, we consider methods which are available for investigating the chemistry on the microscopic scale, both within the bulk of the material and at the surface. Crystallography is the study of atomic order in the crystal structure. A crystallographic analysis serves to identify the phases which are present in the structure, and to describe the atomic packing of the various chemical elements within these phases. Most phases are highly ordered, so that they are crystalline phases in which the atoms are packed together in a well-ordered, regularly repeated array. Many solid phases possess no such long-range order, and their structure is said to be amorphous or glassy. Several

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Preface to the First Edition

quasicrystalline phases have also been discovered in which classical long-range order is absent, but the material nevertheless possesses well-defined rotational symmetry. The microstructure of the material also includes those morphological features which are revealed by a microscopic examination of a suitably prepared specimen sample. A study of the microstructure may take place on many levels, and will be affected by various parameters associated with specimen preparation and the operation of the microscope, as well as by the methods of data reduction used to interpret results. Nevertheless, all microstructural studies have some features in common. They provide an image of the internal structure of the material in which the image contrast depends upon the interaction between the specimen and some incident radiation used to probe the sample morphology. The image is usually magnified, so that the region of the specimen being studied is small compared with the size of the specimen. Care must be exercised in interpreting results as being ‘typical’ of the bulk material. While the specimen is a threedimensional object, the image is (with few exceptions) a two-dimensional projection. Even a qualitative interpretation of the image requires an understanding of the spatial relationship between the two-dimensional imaged features and the three-dimensional morphology of the bulk specimen. Throughout this book we are concerned with the interpretation of the interaction between the probe and a sample prepared from a given material, and we limit the text to probes of X-rays, visible light or energetic electrons. In all cases, we include three stages of investigation, namely specimen preparation, image observation and recording, and the analysis and interpretation of recorded data. We will see that these three aspects of materials characterization interact: the microstructural morphology defines the phase boundaries, and the shape and dimensions of the grains or particles, the crystallography determines the phases present and the nature of the atomic packing within these phases, while the microchemistry correlates with both the crystallography of the phases and the microstructural morphology. This text is intended to demonstrate the versatility and the limitations of the more important laboratory tools available for microstructural characterization. It is not a compendium of all of the possible methods, but rather a teaching outline of the most useful methods commonly found in student laboratories, university research departments and industrial development divisions. Most of the data in this book are taken from work conducted in collaboration with our colleagues and students at the Technion. We wish to thank the following for their contributions: Moshe Eizenberg, Arnon Siegmann, Menachem Bamberger, Christina Scheu, Gerhard Dehm, Ming Wei, Ludmilla Shepelev, Michal Avinun, George Levi, Mike Lieberthal, and Oren Aharon. D.B. W.D.K.

1 The Concept of Microstructure This text provides a basic introduction to the most commonly used methods of microstructural characterization. It is intended for students of science and engineering whose course requirements (or curiosity) lead them to explore beyond the accepted causal connection between the engineering properties of materials and their microstructural features, and prompt them to ask how the microstructures of diverse materials are characterized in the laboratory. Most introductory textbooks for materials science and engineering emphasize that the processing route used to manufacture a component (shaping processes, thermal treatment, mechanical working, etc.) effectively determines the microstructural features (Figure 1.1). They note the interrelation between the microstructure and the chemical, physical, and/or mechanical properties of materials, developing expressions for the dependence of these properties on such microstructural concepts as grain size or precipitate volume fraction. What they do not usually do is to give details of either the methods used to identify microstructural features, or the analysis required to convert a microstructural observation into a parameter with some useful engineering significance. This book covers three aspects of microstructural characterization (Table 1.1). First, the different crystallographic phases which are present in the sample are identified. Secondly, the morphology of these phases (their size, shape and spatial distribution) are characterized. Finally, the local chemical composition and variations in chemical composition are determined. In all three cases we will explore the characterization of the microstructure at both the qualitative and the quantitative level. Thus, in terms of crystallography, we will be concerned not only with qualitative phase identification, but also with the elementary aspects of applied crystallography used to determine crystal structure, as well as with the quantitative determination of the volume fraction of each phase. As for the microstructure, we will introduce stereological relationships which are needed to convert a qualitative observation of morphological features, such as the individual grains seen in a cross-section, into a clearly defined microstructural parameter, the grain size. Similarly, we shall not be Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

2

Microstructural Characterization of Materials

Figure 1.1 The microstructure of an engineering material is a result of its chemical composition and processing history. The microstructure determines the chemical, physical and mechanical properties of the material, and hence limits its engineering performance.

satisfied with the microanalytical identification of the chemical elements present in a specific microstructural feature, but rather we shall seek to determine the local chemical composition through microanalysis. Throughout the text we shall attempt to determine both the sensitivity of the methods described (the limits of detection) and their accuracy (the spatial or spectral resolution, or the concentration errors). In general terms, microstructural characterization is achieved by allowing some form of probe to interact with a carefully prepared specimen sample. The most commonly used probes are visible light, X-ray radiation and a high energy electron beam. These three types of probe, taken in the same order, form the basis for optical microscopy, X-ray diffraction and electron microscopy. Once the probe has interacted with the sample, the scattered or excited signal is collected and processed into a form where it can be interpreted, either qualitatively or quantitatively. Thus, in microscopy, a two-dimensional image of the specimen is obtained, while in microanalysis a spectrum is collected in which the signal intensity is recorded as a function of either its energy or wavelength. In diffraction the signal may be displayed as either a diffraction pattern or a diffraction spectrum. All the instrumentation that is used to characterize materials microstructure includes components that have five distinct functions (Figure 1.2). First, the probe is generated by a Table 1.1 On the qualitative level, microstructural characterization is concerned with the identification of the phases present, their morphology (size and shape), and the identification of the chemical constituents in each phase. At the quantitative level, it is possible to determine the atomic arrangements (applied crystallography), the spatial relationships between microstructural features (stereology), and the microchemical composition (microanalysis). Qualitative analysis Quantitative analysis

Phase identification

Microstructural morphology

Microchemical identification

Applied crystallography

Stereology

Microanalysis

The Concept of Microstructure

3

Figure 1.2 Microstructural characterization relies on the interaction of a material sample with a probe. The probe is usually visible light, X-rays or a beam of high energy electrons. The resultant signal must be collected and interpreted. If the signal is elastically scattered an image can be formed by an optical system. If the signal is inelastically scattered, or generated by secondary emission the image is formed by a scanning raster (as in a television monitor).

source that is filtered and collimated to provide a well-defined beam of known energy or wavelength. This probe beam then interacts with a prepared sample mounted on a suitable object stage. The signal generated by the interaction between the probe and the sample then passes through an optical system to reach the image plane, where the signal data are collected and stored. Finally, the stored data are read out, processed and recorded, either as a final image, or as diffraction data, or as a chemical record (for example, a composition map). The results then have to be interpreted! In all the methods of characterization which we shall discuss, two forms of interaction between the probe and the specimen occur (Figure 1.3): 1. Elastic scattering, which is responsible for the intensity peaks in X-ray diffraction spectra that are characteristic of the phases present and their orientation in the sample. Elastic scattering also leads to diffraction contrast in transmission electron microscopy (TEM), where it is directly related to the nature of the crystal lattice defects present in the sample (grain boundaries, dislocations and other microstructural features). 2. Inelastic scattering, in which the energy in the probe is degraded and partially converted into other forms of signal. In optical microscopy, microstructural features may be

4

Microstructural Characterization of Materials

Figure 1.3 An elastically scattered signal may be optically focused, to form an image in real space (the spatial distribution of microstructural features), or the scattering angles can be analysed from a diffraction pattern in reciprocal space (the angular distribution of the scattered signal). Inelastic scattering processes generate both an energy loss spectra, and secondary, excited signals, especially secondary electrons and characteristic X-rays.

revealed because they partially absorb some wavelengths of the ‘visible’ light that illuminates the specimen. Gold and copper have a characteristic colour because they absorb some of the shorter wavelengths (blue and green light) but reflect the longer wavelengths (red and yellow). The reflection is an elastic process while absorption is an inelastic process. In electron microscopy, the high energy electrons interacting with a specimen often lose energy in well-defined increments. These inelastic energy losses are then characteristic of the electron energy levels of the atoms in the sample, and the energy loss spectra can be analysed to identify the chemical composition of a region of the sample beneath the electron beam (the ‘probe’). Certain electron energy losses are accompanied by the emission of characteristic X-rays. These X-rays can also be analysed, by either energy dispersive or wavelength dispersive spectroscopy, to yield accurate information on the distribution of the chemical elements in the sample. Elastic scattering processes are characteristic of optical or electro-optical systems which form an image in real space (the three dimensions in which we live), but elastic scattering is also a characteristic of diffraction phenomena, which are commonly analysed in reciprocal space. Reciprocal space is used to represent the scattering angles that we record in real space (see below). In real space we are primarily concerned with the size, shape and spatial distribution of the features observed, but in reciprocal space it is the angle through which the signal is scattered by the sample and the intensity of this signal that are significant. These angles are inversely related to the size or separation of the features responsible for the characteristic intensity peaks observed in diffraction. The elastically scattered signals collected in optical imaging and diffraction are compared in Figure 1.4. In optical imaging we study the spatial distribution of features in the image plane, while in a diffraction pattern or diffraction spectrum we study the angular distribution of the signal scattered from the sample.

The Concept of Microstructure

5

(a) Object

Focused Image

Optical System Scattered Radiation (b)

Incident Radiation

Scattering Angle

Object

Diffracted Radiation

Figure 1.4 Schematic representations of an optical image (a) and a diffraction pattern (b). In the former distances in the image are directly proportional to distances in the object, and the constant of proportionality is equal to the magnification. In the latter the scattering angle for the diffracted radiation is inversely proportional to the scale of the features in the object, so that distances in a diffraction pattern are inversely proportional to the separation of features in the object.

Inelastic scattering processes dominate the contrast in scanning electron imaging systems (as in a scanning electron microscope; Figure 1.5). In principle it is possible to detect either the loss spectra (the energy distribution in the original probe after it has interacted with the sample) or a secondary signal (the excited particles or photons generated by the probe as a result of inelastic interaction). Large numbers of secondary electrons are emitted when an energetic electron beam strikes a solid sample. It is the detection of this secondary electron signal that makes possible the very striking high resolution images of surface features that are characteristic of scanning electron microscopy (SEM). In what follows we will assume that the student is familiar with those aspects of microstructure and crystallography that are commonly included in introductory courses in materials science and engineering: some knowledge of the Bravais lattices and the concept of crystal symmetry; microstructural concepts associated with polycrystals and polyphase materials (including the significance of microstructural inhomogeneity and anisotropy); and, finally, the thermodynamic basis of phase stability and phase equilibrium in polyphase materials. Throughout this book each chapter will conclude with results obtained from samples of three materials that are representative of a wide range of common engineering materials.

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Microstructural Characterization of Materials

Figure 1.5 A scanning image is formed by scanning a focused probe over the specimen and collecting a data signal from the sample. The signal is processed and displayed on a fluorescent screen with the same time-base as that used to scan the probe. The magnification is the ratio of the monitor screen size to the amplitude of the probe scan on the specimen. The signal may be secondary electrons, characteristic X-rays, or a wide variety of other excitation phenomena.

We will explore the information that can be obtained for these materials from each of the methods of microstructural characterization that we discuss. The materials we have selected are: . . .

a low alloy steel containing approximately 0.4% C; a dense, glass-containing alumina; a thin-film microelectronic device based on the Al/TiN/Ti system.

An engineering polymer or a structural composite could equally well have been selected for these examples. The principles of characterization would have been the same, even though the details of interpretation differed. Our choice of the methods of microstructural characterization that we describe is as arbitrary as our selection of these ‘typical’ materials. Any number of methods of investigation are used to characterize the microstructure of engineering materials, but this text is not a compendium of all known techniques. Instead we have chosen to limit ourselves to those established methods that are commonly found in a well-equipped industrial development department or university teaching laboratory. The methods selected include optical and electron microscopy (both scanning and transmission), X-ray and electron diffraction, and the commoner techniques of microanalysis (energy dispersive and wavelength dispersive X-ray analysis, Auger electron spectroscopy, X-ray photospectroscopy and electron energy loss spectroscopy). We also discuss surface probe microscopy (SPM), including the atomic force microscope and the scanning tunnelling microscope, since one or more versions of

The Concept of Microstructure

7

this instrumentation are now commonly available. We also include a brief account of the remarkable chemical and spatial resolution that can be achieved by atom probe tomography, even though this equipment is certainly not commonly available at the time of writing. In each case a serious attempt is made to describe the physical principles of the method, clarify the experimental limitations, and explore the extent to which each technique can be used to yield quantitative information

1.1

Microstructural Features

When sectioned, polished and suitably etched, nearly all engineering materials will be found to exhibit structural features that are characteristic of the material. These features may be visible to the unaided eye, or they may require a low-powered optical microscope to reveal the detail. The finest sub-structure will only be visible in the electron microscope. Many of the properties of engineering solids are directly and sensitively related to the microstructural features of the material. Such properties are said to be structure sensitive. In such cases, the study of microstructure can reveal a direct causal relationship between a particular microstructural feature and a specific physical, chemical or engineering property. In what follows we shall explore some of these structure–property relationships and attempt to clarify further the meaning of the term microstructure. 1.1.1

Structure–Property Relationships

It is not enough to state that ‘materials characterization is important’, since it is usual to distinguish between those properties of a material that are structure-sensitive and those that are structure-insensitive. Examples of structure-insensitive properties are the elastic constants, which vary only slowly with composition or grain size. For example, there is little error involved in assuming that all steels have the same tensile (Young’s) modulus, irrespective of their composition. In fact the variation in the elastic modulus of structural materials with temperature (typically less than 10 %) exceeds that associated with alloy chemistry, grain size or degree of cold work. The thermal expansion coefficient is another example of a property which is less affected by variations in microstructural morphology than it is by composition, temperature or pressure. The same is true of the specific gravity (or density) of a solid material. In contrast, the yield strength, which is the stress that marks the onset of plastic flow in engineering alloys, is a sensitive function of several microstructural parameters: the grain size of the material, the dislocation density, and the distribution and volume fraction of second-phase particles. Thermal conductivity and electrical resistivity are also structuresensitive properties, and heat treating an alloy may have a large affect on its thermal and electrical conductivity. This is often because both the thermal and the electrical conductivity are drastically reduced by the presence of alloying elements in solid solution in the matrix. Perhaps the most striking example of a structure-sensitive property is the fracture toughness of an engineering material, which measures the ability of a structural material to inhibit crack propagation and prevent catastrophic brittle failure. Very small changes in

8

Microstructural Characterization of Materials

chemistry and highly localized grain boundary segregation (the migration of an impurity to the boundary, driven by a reduction in the boundary energy), may cause a catastrophic loss of ductility, reducing the fracture toughness by an order of magnitude. Although such segregation effects are indeed an example of extreme structure-sensitivity, they are also extremely difficult to detect, since the bulk impurity levels associated with segregation need only be of the order of 10
5 [10 parts per million (ppm)]. A classic example of a structure-sensitive property relation is the Petch equation, which relates an engineering property, the yield strength of a steel sy, to a microstructural feature, its grain size D, in terms of two material constants, s0 and ky: sy ¼ s0 þk y D
1=2

ð1:1Þ

This relation presupposes that we are able to determine the grain size of the material quantitatively and unambiguously. The meaning of the term grain size, is explored in more detail in Section 1.1.3.1. The fracture surfaces of engineering components that have failed in service, as well as those of standard specimens that have failed in a tensile, creep or mechanical fatigue test, are frequently subjected to microscopic examination in order to characterize the microstructural mechanisms responsible for the fracture (a procedure which is termed fractography). In brittle, polycrystalline samples much of the fracture path often lies along specific lowindex crystallographic planes within the crystal lattices of the individual grains. Such fractures are said to be transgranular or cleavage failures. Since neighbouring grains have different orientations in space, the cleavage surfaces are forced to change direction at the grain boundaries. The line of intersection of the cleavage plane with the grain boundary in one grain is very unlikely to lie in an allowed cleavage plane within the second grain, so that cleavage failures in polycrystalline materials must either propagate on unfavourable crystal lattice planes, or else link up by intergranular grain boundary failure, which takes place at the grain boundaries between the cleavage cracks. A fractographic examination of the failure surface reveals the relative extent of intergranular and transgranular failure (Figure 1.6). By determining the three-dimensional nature of brittle crack propagation and its dependence on grain size or grain boundary chemistry we are able to explore critical aspects of the failure mechanism. Ductile failures are also three-dimensional. A tensile crack in a ductile material typically propagates by the nucleation of small cavities in a region of hydrostatic tensile stress that is generated ahead of the crack tip. The nucleation sites are often small, hard inclusions, either within the grains or at the grain boundaries, and the distribution of the cavities depends on the spatial distribution of these nucleating sites. The cavities grow by plastic shear at the root of the crack, until they join up to form a cellular, ridged surface, termed a dimpled fracture (Figure 1.7). The complex topology of this dimpled, ductile failure may not be immediately obvious from a two-dimensional micrograph of the fracture surface, and in fractography it is common practice to image the failure twice, tilting the sample between the recording of the two images. This process is equivalent to viewing the surface from two different points of view, and allows a rough surface to be viewed and analysed stereoscopically, in threedimensions. The third dimension is deduced from the changes in horizontal displacement for any two points that lie at different heights with respect to the plane of the primary image, a

The Concept of Microstructure

9

Figure 1.6 A scanning electron microscope image showing transgranular and intergranular brittle failure in a partially porous ceramic, aluminium oxynitride (AlON). In three dimensions some intergranular failure is always present, since transgranular failure occurs by cleavage on specific crystallographic planes.

Figure 1.7 Intergranular dimple rupture in a steel specimen resulting from microvoid coalescence at grain boundaries. From Victor Kerlins, Modes of Fracture, Metals Handbook Desk Edition, 2nd Edition, ASM International, 1998, in ASM Handbooks Online (http://www. asmmaterials.info), ASM International, 2004. Reprinted with permission of ASM International. All rights reserved.

10

Microstructural Characterization of Materials

Figure 1.8 The principle of stereoscopic imaging. The left and right eyes see the world from two different positions, so that two points at different heights subtend different angular separations when viewed by the two eyes.

phenomenon termed parallax (Figure 1.8, see Section 4.3.6.3). Our two eyes give us the same impression of depth when the brain superimposes the two views of the world which we receive from each eye separately. The scale of the microstructure determines many other mechanical properties, just as the grain size of a steel is related to its yield strength. The fracture strength of a brittle structural material sf is related to the fracture toughness Kc by the size c ofpthe ffiffiffi processing defects present in the material (cracks, inclusions or porosity), i.e. sf / Kc / c. The contribution of workhardening to the plastic flow stress (the stress required to continue plastic flow after plastic strain due to a stress increment above the yield stress, Dsy) depends on both the dislocation pffiffiffi density r and the elastic shear modulus G, i.e. Dsy / G r. Similarly, the effectiveness of precipitation hardening by a second phase (the increase in yield stress associated with the nucleation and growth of small second-phase particles, Dsp) is often determined by the average separation of the second phase precipitates L, through the relationship Dsp / G/L. 1.1.2

Microstructural Scale

Microstructure is a very general term used to cover a wide range of structural features, from those visible to the naked eye down to those corresponding to the interatomic distances in the crystal lattice. It is good practice to distinguish between macrostructure, mesostructure, microstructure and nanostructure.

The Concept of Microstructure 11

Macrostructure refers to those features which approach the scale of the engineering component and are either visible to the naked eye, or detectable by common methods of nondestructive evaluation (dye-penetrant testing, X-ray radiography, or ultrasonic testing). Examples include major processing defects such as large pores, foreign inclusions, or shrinkage cracks. Nondestructive evaluation and nondestructive testing are beyond the scope of this text. Mesostructure is a less common term, but is useful to describe those features that are on the borderline of the visible. This is particularly the case with the properties of composite materials, which are dominated by the size, shape, spatial distribution and volume fraction of the reinforcement, as well as by any cracking present at the reinforcement interface or within the matrix, or other forms of defect (gas bubbles or dewetting defects). The mesoscale is also important in adhesive bonding and other joining processes: the lateral dimensions of an adhesive or a brazed joint, for example, or the heat-affected zone (HAZ) adjacent to a fusion weld. Microstructure covers the scale of structural phenomena most commonly of concern to the materials scientist and engineers, namely grain and particle sizes, dislocation densities and particle volume fractions, microcracking and microporosity. Finally, the term nanostructure is restricted to sub-micrometre features: the width of grain boundaries, grain-boundary films and interfaces, the early nucleation stages of precipitation, regions of local ordering in amorphous (glassy) solids, and very small, nanoparticles whose properties are dominated by the atoms positioned at the particle surface. Quantum dots come into this category, as do the stable thin films often formed at boundaries, interfaces and free surfaces. Table 1.2 summarizes these different microstructural scales in terms of the magnification range required to observe the features concerned. 1.1.2.1 The Visually Observable. The human eye is a remarkably sensitive data collection and imaging system, but it is limited in four respects: . . . .

the the the the

range of electromagnetic wavelengths that the eye can detect; signal intensity needed to trigger image ‘recognition’ in the brain; angular separation of image details that the eye can resolve; integration time over which an image is recorded by the eye.

The eye is sensitive to wavelengths ranging from about 0.4 to 0.7 mm, corresponding to a colour scale from dark red to violet. The peak sensitivity of the eye is in the green, and is usually quoted as 0.56 mm, a characteristic emission peak in the spectrum from a mercury vapour lamp. As a consequence, optical microscopes are commonly focused using a green filter, while the phosphors used for the screens of transmission electron microscopes and in monitors for image scanning systems often fluoresce in the green. The integration time of the eye is about 0.1 s, after which the signal on the retina decays. Sufficient photons have to be captured by the retina within this time in order to form an image. In absolute darkness, the eye ‘sees’ isolated flashes of light, that, at low intensities, constitute a background of random noise. At low light levels the eye also requires several minutes to achieve its maximum sensitivity (a process termed dark adaptation). Nevertheless, when properly dark-adapted, the eye detects of the order of 50 % of the incident ‘green’ photons, and a statistically significant image will be formed if of the order of 100 photons

12

Microstructural Characterization of Materials

Table 1.2 Scale of microstructural features, the magnification required to reveal these features, and some common techniques available for studying their morphology. Scale

Macrostructure

Typical ·1 magnification Common Visual techniques inspection X-ray radiography

Mesostructure

Microstructure Nanostructure ·104

·106 X-ray diffraction

Scanning electron microscopy

Scanning and transmission electron microscopy Atomic force microscopy

Grain and particle sizes

Dislocation substructure

·10

2

Optical microscopy

Ultrasonic inspection

Characteristic features

Production defects

Porosity, cracks Phase morphology and inclusions and anisotropy

Scanning tunnelling microscopy High resolution transmission electron microscopy

Crystal and interface structure Grain and phase Point defects and boundaries point defect clusters Precipitation phenomena

can contribute to each picture element (or pixel). This is as good as the best available military night-viewing systems, but these systems can integrate the image signal over a much longer period of time than the 0.1 s available to the eye, so that they can operate effectively at much lower light levels. The ability to identify two separate features that subtend a small angle at the eye is termed the resolution of the eye, and is a function of the pupil diameter (the aperture of the eye) and the distance at which the features are viewed. The concept of resolution was defined by Raleigh in terms of the apparent width of a point source. If the point source subtends an angle 2a at the lens, then Abbe showed that its apparent width d in the plane of the source was given by d ¼ 1.2l/m sin a, where l is the wavelength of the radiation from the source and m is the index of refraction of the intervening medium. Raleigh assumed that two point sources could be distinguished when the peak intensity collected from one point source coincided with the first minimum in the intensity collected from the other (Figure 1.9) That is, the resolution, defined by this Raleigh criterion, is exactly equal to the apparent diameter of a point source d viewed with the aid of a lens subtending an angle a. Larger objects are blurred in the image, so that an object having a dimension d appears to have a dimension d þ d. Objects smaller than d can be detected, but have a reduced intensity – and appear to have a size still equal to d. The limit of detection is usually determined by background noise levels, but is always less than the resolution limit. In general, intensity signals that exceed the background noise by more than 10 % can be detected. The diameter of the fully dilated pupil (the aperture that controls the amount of light entering the eye) is about 6 mm, while it is impossible to focus on an object if it is too close

The Concept of Microstructure 13 Image Intensity

Optical System

2α

Two Point Sources in the Object Plane Image Plane

Figure 1.9 The Raleigh criterion defines the resolution in terms of the separation of two identical point sources that results in the centre of the image of one source falling on the first minimum in the image of the second source.

(termed the near point, typically about 150 mm). It follows that sin a for the eye is of the order of 0.04. Using green light at 0.56 mm and taking m¼ 1 (for air), we arrive at an estimate for deye of just under 0.2 mm. That is, the unaided eye can resolve features which are a few tenths of a millimetre apart. The eye records of the order of 106 image features in the field of view at any one time, corresponding to an object some 20 cm across at the near point (roughly the size of this page!). 1.1.2.2 ‘With The Aid of The Optical Microscope’. An image which has been magnified by a factor M will contain resolvable features whose size ranges down to the limit dictated by the resolving power d of the objective lens in the microscope. In the image these ‘just resolved’ features will have a separation Md. If Md<deye then the unaided eye will not be able to resolve all the features recorded in the magnified image. On the other hand, if Md>deye then the image will appear blurred and fewer resolvable features will be present. That is, less information will be available to the observer. It follows that there is an ‘optimum’ magnification, corresponding to the ratio deye/d, at which the eye is just able to resolve all the features present in the magnified image and the density of resolvable image points (pixels) in the field of view is a maximum. Lower magnifications will image a larger area of the specimen, but at the cost of restricting the observable resolution. In some cases, for example in high resolution electron microscopy, this may actually be desirable. A hand lens is then used to identify regions of particular interest, and these regions are then enlarged, usually electronically. Higher magnifications than the optimum are seldom justified, since the image features then appear blurred and no additional information is gained. The optical microscope uses visible electromagnetic radiation as the probe, and the best optical lens systems have values of m sin a of the order of unity (employing a high-refractive index, inert oil as the medium between the objective lens and the specimen). It follows that

14

Microstructural Characterization of Materials

the best possible resolution is of the order of the wavelength, that is, approximately 0.5 mm. Assuming 0.2 mm for the resolution of the eye, this implies that, at a magnification of ·400, the optical microscope should reveal all the detail that it is capable of resolving. Higher magnifications are often employed (why strain your eyes?), but there is no point in seeking magnifications for the optical microscope greater than ·1000. Modern, digitized imaging systems are capable of recording image intensity levels at a rate of better than 106 pixels s
1, allowing for real-time digital image recording, not only in the optical microscope, but also in any other form of spatially resolved signal collection and processing (see Section 3.5). 1.1.2.3 Electron Microscopy. Attempts to improve the resolution of the optical microscope by reducing the wavelength of the electromagnetic radiation used to form the image have been marginally successful. Ultraviolet (UV) radiation is invisible to the eye, so that the image must be viewed on a fluorescent screen, and special lenses transparent to UV are required. The shorter wavelength radiation is also strongly absorbed by many engineering materials, severely limiting the potential applications for a ‘UV’ microscope. Far more success has been achieved by reducing the size of the source to the sub-micrometre range and scanning a light probe over the sample in an x–y raster while recording a scattered or excited photon signal. Such near-field microscopes have found applications, especially in biology where they are used for the study of living cells. Once again, however, such instruments fall outside the scope of this text. Attempts have also been made to develop an X-ray microscope, focusing a subnanometre wavelength X-ray beam by using curved crystals, but it is difficult to find practical solutions to the immense technical problems. More successful has been the use of synchrotron X-radiation at energies as high as 400 keV, using X-ray microtomography to generate three-dimensional image information at sub-micrometre resolutions. Such facilities are not generally available. Electrons are the only feasible alternative. An electron beam of fixed energy will exhibit wavelike properties, the wavelength l being derived, to a good approximation, from the de Broglie relationship: l ¼ h/(2 meV)1/2, where h is Planck’s constant, m is the mass of the electron, e is the electron charge and V is the accelerating voltage. With V in kilovolts and l in nanometres, the constant h/(2me)1/2 is approximately equal to 0.037. For an accelerating voltage of only 1 kV this wavelength is much less than the interplanar spacing in a crystalline solid. However, as we shall see in Section 4.1.2, it is not that easy to focus an electron beam. Electromagnetic lenses are needed, and various lens aberrations limit the acceptable values of the collection angle a for the scattered electrons to between 10
2 and 10
3 rad (360 ¼ 2 p rad). At these small angles a  sin a, and the Raleigh resolution criterion reduces to d ¼ 1.2l/a. (In the electron microscope m ¼ 1, since the electron beam will only propagate without energy loss in a vacuum.) Typical interatomic distances in solids are of the order of 0.2 to 0.5 nm, so that, in principle, atomic resolution in the electron microscope ought to be achievable at 100 kV. This is indeed the case, but transmission electron microscopes require thin samples which may be difficult to prepare, and in practice the optimum operating voltage for achieving consistent resolution of the atomic arrays in a crystal lattice is between 200 kV and 400 kV. Commercial transmission electron microscopes guarantee sub-nanometre resolutions and are capable of detecting the microstructural and nanostructural features present in engineering materials.

The Concept of Microstructure 15

Scanning electron microscopes, in which the resolution depends on focusing the electron beam into a fine probe, have an additional, statistical resolution limit. The beam current available in the probe Ip decreases rapidly as the beam diameter d is reduced, according to the relationship I p / d 8=3 , and the intensity of the signal that can be detected is proportional to the beam current. For some types of signal (most notably, X-rays, whose characteristic wavelengths are dependent on the chemical species), the statistical limit is reached at probe diameters which are much larger than those set by the potential electromagnetic performance of the probe lens. For thick samples in the scanning electron microscope, the resolution in characteristic X-ray maps of chemical concentration is usually limited, both by the statistics of X-ray generation and by spreading of the electron beam as it loses energy to the sample. In general, the best spatial resolution for X-ray maps is of the order of 1 mm. A secondary electron image, obtained in the same scanning electron microscope, may have a resolution that is only limited by the aperture of the probe-forming lens and the wavelength of the electron beam. Combining the Abbe equation, the Raleigh criterion and the de Broglie relationship (linking wavelength to accelerating voltage), we can estimate the approximate minimum size of an electron beam probe for secondary electron imaging in a scanning electron microscope. Since the image is formed from a secondary signal, we also need to consider the volume of material beneath the impinging beam which generates the signal. As the accelerating voltage is increased, the electrons will penetrate deeper into the sample and will be scattered (both elastically and inelastically) over a wider angle. It follows that lower accelerating voltages are desirable in order to limit this spread of the beam within the sample. However, a better signal-to-background ratio will be obtained at higher accelerating voltages. Most scanning electron microscopes are designed to operate in the range 1–30 kV, the lower accelerating voltages being preferred for low-density (lower atomic number) specimens. Assuming the probe convergence angle a to be 10
2 rad and taking 4 kV as a typical lower limit to the accelerating voltage, this yields a potential resolution of about 2 nm for the secondary electron image in the scanning electron microscope. A great deal will be said in later chapters about the factors limiting the spatial resolution in the different methods of microscopy used to characterize microstructural morphology. For the time being, it will suffice to note that five factors should be considered: . . . . .

the physical characteristics of the probe source; the optical properties of the imaging system; the nature of the specimen–probe interaction; the statistics of imaging data collection and storage; image processing, and the display and recording of the final image.

Every technology has its limitations, and methods of microstructural characterization should be chosen according to the information required. Modern equipment for microstructural characterization often combines several techniques on one platform, so the limitations of a specific technique do not always mean that we have to prepare different samples, or search for another piece of equipment. 1.1.2.4 ‘Seeing Atoms’. A common, and not in the least foolish, question is ‘can we see atoms?’ The answer is, in a certain sense, ‘yes’, although the detailed science and technology behind the imaging of atomic and molecular structure is beyond the scope of this text.

16

Microstructural Characterization of Materials

The story starts in Berlin, in 1933, when the electron microscope was developed by Ernst .. Ruska. Shortly afterwards, in 1937, Erwin Muller, also in Berlin, demonstrated that a polished tungsten needle, mounted in a vacuum chamber, would emit electrons from its tip when a positive voltage was applied to the needle. This field emission process was used to image differences in the work function, the energy required to extract an electron from the surface of the metal. A field emission image of the dependence of the work function on the crystallographic orientation was formed by radial projection of the electrons onto a .. fluorescent screen. Some 15 years later Muller and Bahadur admitted a small quantity of gas into the chamber, reversed the voltage on the tungsten needle, and observed ionization of the gas at the tip surface. When the low pressure gas was cooled to cryogenic temperatures, the radially projected image on a fluorescent screen showed regular arrays of bright spots, and the field-ion microscope was born (Figure 1.10). The intensity of these bright spots reflects the electric field enhancement over individual atoms protruding from the surface of the tip. By pulsing the tip voltage or by using pulsed laser excitation, the protruding surface atoms can be ionized and accelerated radially away from the tip. These field-evaporated atoms can be detected with high efficiency using a time-of-flight mass spectrometer, so that not only can the surface of a metal tip be imaged at ‘atomic’ resolution in a field-ion microscope, but in many cases a high proportion of the individual atoms can be identified with reasonable certainty in a field-ion atom probe. In recent years further technical developments have improved this instrumentation to the point where positional time-offlight detectors can record millions of field-evaporated individual ions and display each of

Figure 1.10 The field-ion microscope was the first successful attempt to image ‘atoms’: (a) A schematic diagram of the instrument; (b) a tungsten tip imaged by field-ion microscopy.

The Concept of Microstructure 17

the atomic species present in a three-dimensional array, clearly resolving initial stages in the nucleation of phase precipitation and the chemistry of interface segregation, all at resolutions on the atomic scale (Section 7.3.2). Unfortunately, only engineering materials which possess some electrical conductivity can be studied, and many of these materials lack the mechanical strength to withstand the high electric field strengths needed to obtain controlled field-evaporation of atoms from the very sharp, sub-micrometre sample tip. A rather more useful instrument for observing ‘atoms’ is the scanning tunnelling microscope. In this technique a sharp needle is used as a probe rather than as a specimen. The probe needle, mounted on a compliant cantilever in vacuum, is brought to within a few atomic distances of the sample surface and the vertical distance z of the tip of the probe from the surface is precisely controlled by using a laser sensor and piezoelectric drives. The tip is usually scanned at constant z across the x–y plane, while monitoring either the tip current at constant applied voltage, or the tip voltage at constant applied current (Section 7.2.2). The contrast periodicity observed in a scanning tunnelling image reflects periodicity in the atomic structure of the surface, and some of the first images published demonstrated unequivocally and dramatically that the equilibrium {111} surface of a silicon single crystal is actually restructured to form a 7 · 7 rhombohedral array (Figure 1.11). By varying the voltage of the tip with respect to the specimen, the electron density of states in the sample can be determined at varying distances beneath the surface of the solid. The resolution of the scanning tunnelling microscope depends primarily on the mechanical stability of the system, and all the commercial instruments available guarantee ‘atomic’ resolution. In principle, there is no reason why the same needle should not be used to monitor the force between the needle and a specimen surface over which the needle is scanned. As the needle approaches the surface it first experiences a van der Waals attraction (due to

Figure 1.11 The scanning tunnelling microscope provides data on the spatial distribution of the density of states in the electron energy levels at and beneath the surface. In this example the 7 · 7 rhombohedral unit cell of the restructured {111} surface of a silicon crystal is clearly resolved. Reprinted from I.H. Wilson, p. 162 in Walls and Smith (eds) Surface Science Techniques, Copyright 1994, with permission from Elsevier Science.

18

Microstructural Characterization of Materials

polarization forces), which at shorter distances is replaced by a repulsion force, as the needle makes physical contact with the specimen. Scanning the probe needle over the surface at constant displacement (constant z) and monitoring the changes in the van der Waals force yields a scanning image in which, under suitable conditions, ‘atomic’ resolution may also be observed (Figure 1.12). No vacuum is now necessary, so the atomic force microscope (Section 7.2.1) is particularly useful for studying solid surfaces in gaseous or liquid media (something that the field-ion microscope and the scanning tunnelling microscope cannot do). The atomic force microscope is a powerful tool for imaging both organic membranes and polymers. By vibrating the tip (using a piezoelectric transducer), it is possible to monitor the elastic compliance of the substrate at atomic resolution, providing information on the spatial distribution of the atomic bonding at the surface. Surface force microscopy, developed earlier, primarily by Israelachvili, uses much the same principle, but with a resolution of the order of micrometres. In this case curved cylindrical surfaces of cleaved mica are brought together at right angles, eventually generating a circular area of contact. The surface forces are monitored as a function of both the separation of the cylinders and the environment, and this instrument has proved to be a particularly powerful tool for fundamental studies of surface-active and lubricating films. For these studies the mica surfaces can be coated, for example by physical vapour or chemical vapour deposition, to control the nature of the lubricant substrate. All of these techniques provide spatially resolved information on the surface structure of a solid and have been grouped together under the heading surface probe microscopy (SPM). In all cases, these techniques reveal one or another aspect of the surface atomic morphology and the nature of the interatomic bonding at the surface. In a very real sense, today we are indeed able to ‘see atoms’. The limitations of SPM will be explored in Chapter 7, while the extent to

Figure 1.12 AFM scan of Ca-rich particles on the basal surface of sapphire: (a) a top view showing the faceting of the particle: (b) an inclined view of the same area showing the three dimensional morphology of the particle. Reprinted with permission from A. Avishai, PhD, Thesis, Thin-Equilibrium Amorphous Films at Model Metal–Ceramic Interfaces, 2004, Technion, Israel Institute of Technology; Haifa, p. 75.

The Concept of Microstructure 19

which transmission electron microscopy and scanning electron microscopy also allow us to ‘see atoms’ is discussed in the appropriate sections of Chapter 4. 1.1.3

Microstructural Parameters

Microstructural features are commonly described in qualitative terms: for example, the structure may be reported to be equiaxed, when the structure appears similar in all directions, or the particles of a second phase may be described as acicular or plate-like. Theories of material properties, on the other hand, frequently attempt to define a quantitative dependence of a measured property on the microstructure, through the introduction of microstructural parameters. These parameters, for example grain size, porosity or dislocation density, must also be measured quantitatively if they are to have any predictive value within the framework of a useful theory. A classic example is the Petch relationship which links the yield strength of a steel sy to its grain size D [see Equation (1.1)]. In many cases there is a chasm of uncertainty between the qualitative microstructural observations and their association with a predicted or measured material property. To bridge this chasm we need to build two support piers. To construct the first we note that many of the engineering properties of interest are stochastic in nature, rather than deterministic. That is, the property is not single-valued, but is best described by a probability function. We know that not all men are 180 cm tall, nor do they all weigh 70 kg. Rather, there are empirical functions describing the probability that an individual taken from any well-defined population will have a height or weight that falls within a set interval. Clearly, if an engineering property is stochastic in nature, then it is likely that the material parameters which determine that property will also be stochastic, so that we need to determine the statistical distribution of the parameters used to define these material properties. The second support pier needed to build our bridge is on the microstructural side of the chasm. It is constructed from our understanding and knowledge of the mechanisms of image formation and the origins of image contrast, in combination with a quantitative analysis of the spatial relationship between microstructural image observations and the bulk structure of the material. This geometry-based, quantitative analysis of microstructure is termed stereology, the science of spatial relationships, and is an important component of this book (see Chapter 9). For the time being, we will only consider the significance of some terms which are used to define a few common microstructural parameters. 1.1.3.1 Grain Size. Most engineering materials, especially structural materials, are polycrystalline, that is, they consist of a three-dimensional assembly of individual grains, each of which is a single crystal whose crystal lattice orientation in space differs from that of its neighbours. The size and shape of these individual grains are as varied as the grains of sand on the seashore. If we imagine the polycrystalline aggregate separated out into these individual grains, we might legitimately choose to define the grain size as the average separation of two parallel tangent planes that touch the surfaces of any randomly oriented grain. This definition is termed the caliper diameter DC, and it is rather difficult to measure (Figure 1.13). We could also imagine counting the number of grains extracted from a unit volume of the sample, NV, and then defining an average grain size as DV ¼ NV
1/3. This definition is unambiguous and independent of any anisotropy or inhomogeneity in the material. However, most samples for microstructural observation are prepared by taking a

20

Microstructural Characterization of Materials

Figure 1.13 Grain size may be defined in several ways that are not directly related to one another, for example the mean caliper diameter, the average section diameter on a planar section, or the average intercept length along a random line.

planar section through the solid, three-dimensional microstructure, so it is more practical to count the number of grains that have been revealed by etching the polished surface of a twodimensional section through the sample, and then transform the number of grains intercepted per unit area of the section NA into an average ‘grain size’ by writing DA ¼ NA
1/2. This is a very common procedure for determining grain size, and may well be the preferred measure. However, we could also lay down a set of lines on our ‘random’ polished-andetched section, and count the number of intercepts which these ‘test’ lines make with the grain boundary traces seen on the section. In general, samples are usually sectioned on planes selected in accordance with the specimen geometry, for example, parallel or perpendicular to a rolling plane, and test lines are also usually drawn parallel or perpendicular to specific directions, such as the rolling

The Concept of Microstructure 21

direction of a metal sheet. However, if both the test line on the surface and the sample section itself are truly random, then the test line is a random intercept of the boundary array present in the bulk material. The number of intercepts per unit length of a test line NL gives us yet another measure of the grain size, DL the mean linear intercept, where DL ¼ NL
1, and is also a commonly accepted definition for grain size (with or without some factor of proportionality). However, this is not quite the end of the story. In industrial quality control a totally different, qualitative ‘measure’ of grain size is commonly used. The sample microstructure is compared with a set of standard microstructures (ASTM Grain-size Charts) and assigned an ASTM (American Society for Testing Materials) grain size DASTM on the basis of the observer’s visual judgement of the ‘best-fit’ with the ASTM chart. This is of course a rather subjective measure of grain size that depends heavily on the experience of the observer. What happens if the grains are not of uniform size? Well, any section through the sample will cut through an individual grain at a position determined by the distance of the centre of gravity of the grain from the plane of the section, so that grains of identical volume may have different intercept areas on the plane of the section, depending on how far this plane lies from their centre of gravity. This distribution of intercept areas will be convoluted with the ‘true’ grain size distribution, and it is difficult (but not impossible) to derive the grain size distribution in the bulk material from an observed distribution of the areas intercepted on the polished section. If the grains are also elongated (as may happen if the grain boundary energy depends on orientation) or if elongated grains are partially aligned (as will be the case for a plastically deformed ductile metal or for a sample that has been cast from the melt in a temperature gradient), then we shall have to think very carefully indeed about the significance of any grain size parameter that has been measured from a planar surface section. These problems are treated in more detail in Chapter 9. 1.1.3.2 Dislocations and Dislocation Density. Dislocations control many of the mechanical properties of engineering materials. A dislocation is a line defect in the crystal lattice which generates a local elastic strain field. Dislocations may interact with the free surface, dispersed particles and internal interfaces (such as grain boundaries), as well as with each other. Each dislocation line is characterized by a displacement vector, the Burgers vector, which defines the magnitude of the elastic strain field around the dislocation. The angle between the Burgers vector and the dislocation line in large part determines the nature of the dislocation strain field, that is the shear, tensile and compressive displacements of the atoms from their equilibrium positions. A Burgers vector parallel to the dislocation line defines a screw dislocation, while a Burgers vector perpendicular to the dislocation line defines an edge dislocation. All other configurations are referred to as mixed dislocations. In a thin-film specimen imaged by TEM, the Burgers vector and line-sense of the dislocations can usually be determined unambiguously from the diffraction contrast generated in the region of the dislocation by the interaction of the dislocation strain field with the electron beam (Section 4.4.5). However, the determination of the dislocation density is often ambiguous. A good theoretical definition of the dislocation density is ‘the total line length of the dislocations per unit volume of the sample’. This definition is unambiguous and independent of the dislocation distribution, and the dislocations may interact or be aligned along particular crystallographic directions. However, this definition does not make any allowance for families of dislocations that have different Burgers

22

Microstructural Characterization of Materials

vectors. This may not matter if the dislocation population is dominated by one specific type of Burgers vector, as is commonly the case in cold-worked metals, in which the slip dislocations are usually of only one type. However, it may matter a great deal when assessing the residual dislocation content in semiconductor single crystals. Moreover, since it is the strain fields of the dislocations that are imaged, and contrast is only observed if the strain field has a component perpendicular to the diffracting planes, dislocations may ‘disappear’ from the image under certain diffraction imaging conditions. A further problem arises when we seek to extend our definition of the term dislocation. For example, when dislocations interact they may form low energy dislocation networks (Figure 1.14), that separate regions of the crystal having slightly different orientations in space. The dislocation network is thus a sub-grain boundary. Are we to include the array of dislocations in the sub-boundary in our count of dislocation density or not? Plastic deformation frequently leads to the formation of dislocation tangles which form cell structures, and within the cell walls it is usually impossible to resolve the individual dislocations (Figure 1.15). In addition, small dislocation loops may be formed by the collapse of point defect clusters that often result from plastic deformation, quenching the sample from high temperatures or radiation damage in a nuclear reactor (Figure 1.16). Should these small loops also be counted as dislocations? A common alternative definition of dislocation density is ‘the number of dislocation intersections per unit area of a planar section’. In an anisotropic sample this definition would be expected to give dislocation densities that show a dependence on the plane of the section, so that the two definitions are not equivalent. Furthermore, a dislocation density determined from observations made using one counting method need not agree with that derived from other measurements, no matter how carefully the dislocation density is defined. The spatial resolution may differ, and the method of specimen preparation could also affect the results, for example, by permitting dislocations to glide out to the surface and annihilate.

Figure 1.14 A low energy array of dislocations forms a dislocation network which constitutes a sub-boundary in the crystal that can interact with slip dislocations, as shown here. Such boundaries separate the crystal into sub-grains of slightly different orientation.

Figure 1.15 Plastic deformation of a ductile metal often results in poorly resolved dislocation tangles which form cells within the grains.

Figure 1.16 Dark-field transmission electron micrograph taken with a g ¼ (200) reflection showing small defects due to 100 kV Kr ion irradiation of Cu at 294 K. Reproduced with permission from R.C. Birtcher, M.A. Kirk, K. Furuya, G.R. Lumpkin and M.O. Ruault, in situ Transmission Electron Microscopy Investigation of Radiation Effects, Journal of Materials Research, 20(7), 1654–1683, 2005. Copyright (2005), with permission from Materials Research Society.

24

Microstructural Characterization of Materials

1.1.3.3 Phase Volume Fraction. Many engineering materials contain more than one phase, and the size, shape and distribution of a second phase are often dominant factors in determining the effect of the second phase on the properties. As with grain size, when the second phase is present as individual particles there are a number of nonequivalent options for defining the particle size, shape and spatial distribution. These definitions are for the most part analogous to those for grain size, discussed above. However, in many cases the second phase forms a continuous, interpenetrating network within the primary phase, in which both interphase boundaries and grain boundaries are present. If the secondary phase particles have shapes that are in any way convoluted (with regions of both positive and negative curvature at the boundaries), then the same particle may intersect the surface of a section more than once, making it very difficult to estimate just how many second phase particles are present. However, there is one microstructural parameter that is independent of both the scale, shape and the distribution of the second phase, and that is the phase volume fraction fV. Since this is both a shape and scale-independent parameter, for crystalline phases fV can be determined both conveniently and quickly from diffraction data (see the Worked Examples of Chapter 2), while local values of the phase volume fraction can be extracted from images of a planar section (Figure 1.17). On a random section, the volume fraction of the second phase can be estimated from the areal fraction of the second phase intercepted by the section, A/A0. At the dawn of metallography (well over 100 years ago!) it was even accepted practice to cut out the regions of interest from a photographic image and then weigh them relative to the ‘weight’ of the total area sampled. This areal estimate is actually equivalent to a lineal estimate, determined from a random line placed across the plane of the section. Providing the line and section are both ‘random’, the length of the line traversing the second phase relative to the total length of test line L/L0 is also an estimate of the phase volume fraction. Finally, a random grid of test points on the sample section can also provide the same information: the number of points falling on regions of the second phase divided by the total number of test points P/P0 again estimates the volume of the second phase relative to the total volume of the sample V/V0. Thus, for the case of truly random sampling: fV ¼ A / A0 ¼ L / L0 ¼ P/P0 ¼ V / V0. For anisotropic samples, these relations do not hold, but useful information on the extent of the microstructural anisotropy can still be determined by comparing results obtained from different sample sections and directions in the material. Porous materials can also be analysed as though they contained a ‘second phase’, although precautions have to be taken when sectioning and polishing a porous sample if artifacts associated with loss of solid material and rounding of the pore edges is to be avoided. It is always a good idea to compare microstructural observations of porous materials with data on pore size and volume fraction determined from physical models using measurements based on density, porisometry or gas adsorption.

1.2

Crystallography and Crystal Structure

The arrangements of the atoms in engineering materials are determined by the chemical bonding forces. Some degree of order at the atomic level is always present in solids, even in what appears to be a featureless, structureless glass or polymer. In what follows we will briefly review the nature of the chemical forces and outline the ways in which these chemical forces are related to the engineering properties. We will then discuss some of the

The Concept of Microstructure 25

Figure 1.17 The volume fraction of a second phase can be determined from the areal fraction of the phase, seen on a random planar section, or from the fractional length of a random test line which intercepts the second phase particles in the section, or from the fraction of points in a test array which falls within the regions of the second phase.

crystallographic tools needed to describe and understand the commonly observed atomic arrangements in ordered, crystalline solids. The body of knowledge that describes and characterizes the structure of crystals is termed crystallography. 1.2.1

Interatomic Bonding in Solids

It is a convenient assumption that atoms in solids are packed together much as one would pack table tennis balls into a box. The atoms (or, if they carry an electrical charge, the ions) are assumed to be spherical, and to have a diameter which depends on their atomic number

26

Microstructural Characterization of Materials

(the number of electrons surrounding the nucleus), their electrical charge (positive, if electrons have been removed to form a cation, or negative if additional electrons have been captured to form an anion), and, to a much lesser extent, the number of neighbouring atoms surrounding the atom being considered (the coordination number of the atom or ion). 1.2.1.1 Ionic Bonding. In an ionically bonded solid the outer, valency electron shells of the atoms are completed or emptied by either accepting or donating electrons. In cooking salt, NaCl, the sodium atom donates an electron to the chlorine atom to form an ion pair (a positively charged sodium cation and a negatively charged chlorine anion). Both cations and anions now have stable outer electron shells, the chlorine ion having a full complement of eight electrons in the outer shell, and the sodium ion having donated its lone excess electron to the chlorine atom. To minimize the electrostatic energy, the negative electrical charge on the cations must be surrounded by positively charged anion neighbours and vice versa. At the same time, the outer electron shell of the anions will contract towards the positively charged nucleus as a result of the excess positive charge on the nucleus, while the excess negative charge on the anions will cause a net expansion of the outer electron shell of the anions. For the case of NaCl, and many other ionic crystals, the larger anions form an ordered (and closely packed) array, while the smaller cations occupy the interstices. Two opposing factors will determine the number of neighbours of opposite charge that surrounds a given ion (the coordination number). First, the electrostatic (Coulombic) attraction between ions of opposite charge tends to maximize the density of the ionic array. At the same time, in order to keep neighbouring ions of similar charge separated, the smaller ion must be larger than the interstices which it occupies in the packing of the larger ion. The smallest possible number of neighbours of opposite charge is 3, and boron (Z ¼ 5) is a small, highly charged cation which often has this low coordination number. Coordination numbers of 4, 6 and 8 are found for steadily increasing ratios of the two ionic radii (Figure 1.18), while the maximum

3

4

8

6

12

Figure 1.18 The number of neighbours of an ion, its coordination number, is primarily determined by the ratio of the radii of the smaller to the larger ion. The regular coordination polyhedra allow for 3, 4, 6, 8 or 12 nearest neighbours.

The Concept of Microstructure 27

coordination number, 12, corresponds to cations and anions having approximately the same size. The anion is not always the larger of the two ions, and when the cation has a sufficiently large atomic number, it may be the anions that occupy the interstices in a cation array. Zirconia, ZrO2, is a good example of packing anions into a cation array. Silicate structures and glasses are also dominated by ionic bonding, but in this case the tightly coordinated cations form a molecular ion, most notably the SiO4 silicate tetrahedron. These tetrahedra may carry a negative charge (as in magnesium silicate, Mg2SiO4), or they may be covalently linked to form a poly-ion, in which the coordination tetrahedra share their corner oxygen atoms (as in quartz) to form oxygen bridges, –(Si–O–Si)–. Borates, phosphates and sulfates can form similar structures, but it is the silicates that dominate in engineering importance. Since only the corners of a silicate tetrahedron may be shared, and not the edges or faces, the silicates form very open, low density structures which can accommodate a wide variety of other cations. In addition, the oxygen corner linkage is quite flexible, and allows any two linked tetrahedra considerable freedom to change their relative orientation. The average negative charge on the silicate ion in a glass varies inversely with the number of oxygen bridges, and this charge is neutralized by the presence of additional cations (known as modifiers) that occupy the interstices between the tetrahedra. The tetrahedra themselves do not readily change their dimensions, although some substitution of the Si4þ ion can occur (most notably by B3þ or Al3þ). Since the oxygen bridges constrain the distance between neighbouring tetrahedra, the glasses possess well-defined short-range order that may extend up to 2 nm from the centres of the silicate tetrahedra. 1.2.1.2 Covalent Bonding. Many important engineering materials are based on chemical bonding in which neighbouring atoms share electrons that occupy molecular orbitals. In diamond. (Figure 1.19), the carbon atoms all have four valency electrons, and by sharing

Figure 1.19 In covalently bonded diamond the carbon atoms are tetrahedrally coordinated to each of their nearest neighbours by shared molecular orbitals.

28

Microstructural Characterization of Materials

each of these with four neighbouring carbon atoms, each atom acquires a full complement of eight electrons for the outer valency shell. It is the strong C
C covalent bond which ensures the chemical stability, not only of diamond, but also of most polymer molecules, which are constructed from covalently linked chains of carbon atoms. The oxygen bridges in silicate glasses are also, to a large extent, covalent bonds, and the same oxygen bridges provide the chain linkage in silicone polymers,
(O
SiHR)
. It is not always possible to describe a bond as simply covalent or ionic. Consider the series NaCl, MgO, AlN, SiC, in which the number of electrons participating in the bond between individual cations and anions increases from one to four. The first two compounds in the series are commonly described as ionic solids. The third, aluminium nitride, could also be described as ionically bonded, although the effective charge on the ions is appreciably less than that predicted by their trivalent nature. The fourth compound, silicon carbide is tetrahedrally coordinated, as is diamond. Since both constituents are in the same group of the periodic table, one might guess that it is covalently bonded. However, this is not entirely correct, and in fact the physical properties are best simulated by assuming that the silicon atoms still carry some positive charge, while the carbon atoms carry a corresponding negative charge. In fact, as the valency increases, so does the contribution to the bond strength from covalent bonding. 1.2.1.3 Metals and Semiconductors. Valence electrons may be shared, not only with a nearest neighbour atom, but quite generally, throughout the solid. That is, the molecular orbitals of the electrons may not be localized to a specific pair of atoms. Electrons which are free to move throughout the solid are said to occupy a conduction band, and to be free electrons. The chemical bonding in such a solid is termed metallic bonding, and is characterized by a balance between two opposing forces: the coulombic attraction between the free electrons and the array of positively charged, metallic cations, and the repulsive forces between the closed shells of these cations. The properties typical of the metallic bond are associated with the mobility of the free electrons (especially the high thermal and electrical conductivity, and the optical reflectivity), and with the nondirectionality of this bond (for example, mechanical plasticity or ductility). In some cases, only small numbers of electrons may be present in the conduction band of a solid, either as a result of thermal excitation or due to the presence of impurities. Such materials are termed semiconductors, and they play a key role in the manufacture of electronic devices for the electronics industry. If electrons are thermally excited to occupy a conduction band, then they leave behind vacant holes, which may also be mobile. Semiconductors in which the negatively charged electrons are the dominant electrical current carriers are termed n-type, while those in which the positively charged holes are responsible for the electronic properties are termed p-type. If certain impurities are present in very low concentrations, then the electrons may not be free to move throughout the solid, and cannot confer electrical conductivity, but they will nevertheless occupy localized states. Many cation impurities in ceramics give rise to localized states that are easily excited and strongly absorb visible light. They are then said to form colour centres. In precious and semi-precious jewels small quantities of cation impurities are dissolved in the single crystal jewel stone and impart the characteristic colour tones, for example, chromium in ruby. Such cation additions are also of major importance in the ceramics industry, and the effects may be either deleterious (discoloration) or advantageous (a variety of attractive enamels and glazes, Figure 1.20). Irradiation of transparent,

The Concept of Microstructure 29

Figure 1.20 The colour and texture of a porcelain glaze depends both on the presence of dispersed pigment particles and controlled amounts of impurity cations that introduce colour centres into the silicate glass. These colour centres may also nucleate localized crystallization. The illustration shows a glass jar from the time of the Roman Empire. Reproduced with permission of the Corning Museum of Glass. (See colour plate section)

nonconducting solids creates large concentrations of point defects in the material. Such radiation damage is also often a cause of colour centres. 1.2.1.4 Polarization Forces. In addition to the three types of chemical bonding discussed above, many of the properties of engineering solids are determined by secondary, or van der Waals bonding, that is associated with molecular polarization forces. In its weakest form, the polarization force arises from the polarizability of an electron orbital. Rare gas atoms will liquefy (and solidify) at cryogenic temperatures as a consequence of the small reduction in potential energy achieved by polarization of an otherwise symmetrical electron orbital. Many molecular gases (H2, N2, O2, CH4) behave similarly. The properties of a number of engineering polymers are dominated by the polarizability of the molecular chain, for example polyethylene,
(CH2
CH2)
. The ductility of the polymer, as well as its softening point and glass transition temperature, are determined by a combination of the molecular weight of the

30

Microstructural Characterization of Materials

polymer chains and their polarizability. This weak bonding is quite sufficient to ensure that engineering components manufactured from polymers are mechanically stable, and, under suitable circumstances, these high molecular weight polymers may partially crystallize. Stronger polarization forces exist when the molecular species has a lower symmetry and possesses a permanent dipole moment. A good example is carbon dioxide (CO2) but similar molecular groupings are also often present in high performance engineering polymers. Organic tissues are largely constructed from giant polar molecules with properties dictated by a combination of the molecular configuration and the position of the polar groups within the molecule. The strongest polarization forces are associated with a dipole moment due to hydrogen, namely the hydrogen bond. Hydrogen in its ionized form is a proton, with no electrons to screen the nucleus. The ionic radius of hydrogen is therefore the smallest possible, and asymmetric molecular groupings which contain hydrogen can have very high dipole moments. The two compounds that demonstrate this best are water and ammonia, (H2O and NH3) and the corresponding molecular groups found in engineering polymers are
(OH) and
(NH2). These groups raise both the tensile strength and the softening point of the polymer. The families of polyamides and polyamines (which include the ‘nylons’, actually a commercial trade name) depend for their strength and stiffness on the strong hydrogen bonding between the polymer chains. 1.2.2

Crystalline and Amorphous Phases

We have outlined the nature of the chemical bonding found in engineering solids and liquids, and the atomic coordination requirements associated with this bonding. In some engineering solids the local atomic packing results in long-range order and the material is termed crystalline, while in others short-range atomic order has no long-range consequences, and the (‘liquid-like’) material is termed amorphous or glassy. Single phase polycrystalline materials are made up of many small crystals or grains. Each grain has identical atomic packing to that of its neighbours, although the neighbouring grains are not in the same relative crystal orientation. In polyphase materials all the grains of each individual phase have the same atomic packing, but this packing generally differs from that of the other phases present in the material. In thermodynamic equilibrium, the grains of each phase also have a unique and fixed composition that depends on the temperature and composition of the material, and can usually be determined from the appropriate phase diagram. In general, the solid phases in engineering materials may be either crystalline or amorphous. Amorphous phases may form by several quite distinct routes: rapid cooling from the liquid phase, condensation from the gaseous phase, or as the result of a chemical reaction. A good example of amorphous phases produced by chemical reaction are the highly protective oxide films formed on aluminium alloy and stainless steel components by surface reactions in air at room temperature. 1.2.3

The Crystal Lattice

The well-developed facets readily observed on many naturally occurring crystals, as well as on many ionic crystals grown from aqueous solution, prompted the development of the science of crystallography in the latter part of the nineteenth century. An analysis of the

The Concept of Microstructure 31

angles between crystal facets permitted an exact description of the symmetry elements of a crystal, and led to speculation that crystal symmetry was a property of the bulk material. This was finally confirmed with the discovery, at the beginning of the twentieth century, that small, single crystals would strongly diffract X-rays at very specific, fixed angles, to give a sharp diffraction pattern that was characteristic of any crystal of the same material when it was oriented in the same relation to the incident X-ray beam, irrespective of the crystal size and shape. The interpretation of these sharp X-ray diffraction maxima in terms of a completely ordered and regular atomic array of the chemical constituents of the crystal followed almost immediately, being pioneered by the father and son team of Lawrence and William Bragg. The concept of the crystal lattice was an integral part of this interpretation. The atoms in a crystal were centred at discrete, essentially fixed distances from one another, and these interatomic separations constituted an array of lattice vectors that could be defined in terms of an elementary unit of volume, the unit cell for the crystal, that displayed all the symmetry elements characteristic of the bulk crystal. 1.2.3.1 Unit Cells and Point Lattices. This section introduces basic crystallography, and describes how a crystalline structure is interpreted and how crystallographic data can be retrieved from the literature. To understand the structure of crystals, it is convenient initially to ignore the positions of the atoms, and just concentrate on the periodicity, using a threedimensional periodic scaffold within which the atoms can be positioned. Such a scaffold is termed a crystal lattice, and is defined as a set of periodic points in space. A single lattice cell (the unit cell) is a parallelepiped, and the unit cells can be packed periodically by integer displacements of the unit cell parameters. The unit cell parameters are the three coordinate lengths (a, b, c) determined by placing the origin of the coordinate system at a lattice point, and the three angles (a, b, g) subtended by the lattice cell axes (Figure 1.21). Thus a unit cell with a ¼ b ¼ c and a ¼ b ¼ g ¼ 90 is a cube. The various unit cells are generated from the different values of a, b, c and a, b, g. An analysis of how to fill space periodically with lattice points shows that only seven different unit cells are required to describe all the possible

c

α β

b

γ a

Figure 1.21 Schematic representation of the lattice parameters of a unit cell.

32

Microstructural Characterization of Materials

point lattices, and these are termed the seven crystal systems. These crystal systems are, in order of increasing crystal symmetry: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. These seven crystal systems are each defined using primitive unit cells, in which each primitive cell only contains a single lattice point placed at the origin of the unit cell. However, more complicated point lattice symmetries are possible, each requiring that every lattice point should have identical surroundings. These permutations were first analysed by the French crystallographer Bravais in 1848, who described the possible (14) point lattices (the Bravais lattices), which are shown in Figure 1.22. 1.2.3.2 Space Groups. In real crystals each individual lattice point actually represents either a group of atoms or a single atom, and it is these atoms that are packed into the crystal. Various degrees of symmetry are possible in this periodic packing of the atoms and atom groups. For example, if a crystal is built of individual atoms located only at the lattice points

Figure 1.22 The 14 Bravais lattices derived from the seven crystal systems.

The Concept of Microstructure 33

of a unit cell, then it will have the highest possible symmetry within that particular crystal system. This symmetry will then be retained if there are symmetrical groups of atoms associated with each lattice point. However, the atomic groups around a lattice point might also pack with a lower symmetry, reducing the symmetry of the crystal, so that it belongs to a different symmetry group within the same crystal class. Combining the symmetries of the atom groups associated with a lattice point with that of the Bravais lattices leads to the definition of space groups, which provide criteria for filling the Bravais point lattices with atoms and groups of atoms in a periodic array. It has been found that there are a total of 230 different periodic space groups, and that the structure of a crystal can always be described by one (or more) of these space groups. This has been found to be the most convenient way to visualize any complex crystal structure. Let us examine the use of space groups to define a crystal structure using a simple example. Assume we wish to know the positions of all the atoms in a copper (Cu) crystal. First we need a literature source which contains the crystallographic data. For materials science we use Pearson’s Handbook of Crystallographic Data for Intermetallic Phases (the name is misleading, since pure metals and ceramics are also included). The principle data listed in the handbook for Cu appear in the format shown in Table 1.3. Following the name of the phase, a structure type is given. This is the name of a real material that serves as an example for this particular crystallographic structure, and in this example copper is its own structure type. Next the Pearson symbol and space group are listed, which refer to the type of lattice cell and the symmetry of the structure. Here, the Pearson symbol, cF4, means a cubic (c) face-centred lattice (F) with four atoms per unit cell (that is, one atom for each lattice point, in this case). The symmetry description Fm
3m is in this case also the name of the space group and is followed by the lattice parameters. For our example, Cu, a ¼ b ¼ c ¼ 0.36148 nm, and a ¼ b ¼ g ¼ 90 . Since a cubic structure clearly has a ¼ b ¼ g ¼ 90 , these values are not listed. Finally the Wyckoff generating sites are given. These are the sites of specific atoms within the crystal structure, upon which the space group symmetry operators act. When combined with the space group, the symmetry operators will generate the positions of all atoms within the unit cell and specify the ‘occupancy’ or occupation factor. For Cu, x ¼ 000, y ¼ 000, z ¼ 000 and Occ ¼ 100. Hence, x ¼ y ¼ z ¼ 0.0 and the occupancy is 1.00. The last term, the occupancy, indicates the probability that a site is occupied by a particular atom species. In the case of Cu, all the sites (neglecting vacancy point defects) are occupied by Cu, so the occupancy is 1.00. How do we generate the crystal structure? We first need details of the symmetry operators for the space group Fm
3m. These can be found in the International Tables For

Table 1.3 Crystallographic data for Cu

Phase

Structure type

Pearson symbol space and group

Cu

Cu

cF4 Fm
3m

a, b, c (nm) 0.36148

a, b, g ( )

Atoms

Point set

x

Cu

4a

000 000 000 100

y

z

Occ

34

Microstructural Characterization of Materials

Figure 1.23 The space group Fm
3m which defines the symmetry for Cu. Reproduced by permission of Kluwer Academic Publishers from International Tables for Crystallography, Volume A, Space Group Symmetry, T. Hahn, ed. (1992).

The Concept of Microstructure 35

Crystallography, Volume A, Space Group Symmetry. An example is given in Figure 1.23. The data in the tables provide all the symmetry operators for any specific space group. In order to generate atomic positions from the generating site data listed for Cu, we add the (x, y, z) values of the generating site to the values listed in the tables. Returning to our example, Cu has a generating site of type 4a with (x,y,z) equal to (0,0,0). In the tables, the Wyckoff generating site for 4a has an operator of (0,0,0). This is our first atom site. Now we activate the general operators which are also listed. The addition of (0,0,0) to our initial generating site of (0,0,0) leaves one atom at the origin, while the addition of (0,1/2,1/2), (1/2,0,1/2) and (1/2,1/2,0) to (0,0,0) places three more atoms, one at the centre of each of the faces of the unit cell adjoining the origin. There are therefore four Cu atoms in our unit cell, as shown schematically in Figure 1.24(a). Figure 1.24(a) does not look like the ‘complete’ FCC structure sketched in most elementary texts, since only those atoms which ‘belong’ to a single unit cell are shown. Additional atoms at sites (1,0,0), (1,1,0), (0,1,0), (1,0,1), (0,1,1), (0,0,1), (1,1,1), (1,1/2,1/2), (1/2,1,1/2) and (1/2,1/2,1) actually belong to the neighboring unit cells in the crystal lattice.

Figure 1.24 Schematic drawing of (a) the unit cell of Cu and (b) the same unit cell, but with additional atoms from neighbouring unit cells in order to demonstrate the face-centred cubic (FCC) packing.

36

Microstructural Characterization of Materials

A more commonly accepted (although no more correct) drawing of the unit cell for a copper crystal is given in Figure 1.24(b). We need a clear definition to decide if an atom ‘belongs’ in a specific unit cell: if 0  x < 1, 0  y < 1, and 0  z < 1, then we define the atom as ‘belonging’ to the unit cell. If not, then the atom belongs to a neighbouring cell. This provides a very easy method to count the number of atoms per cell. Why all this effort just to define four atoms in an FCC configuration? For this simple structure, the formalism is not strictly necessary, but for a structure with 92 atoms per unit cell, use of these tables is the easiest and least error-prone way to define the atomic positions. It is also convenient when computer-simulating crystal structures or diffraction spectra: instead of having to type in the positions of all the atoms in the unit cell, you can just use the space group operations. Now consider an example in which the use of occupation factors is important. A Cu
Ni solid solution (an alloy) with 50 atom % Ni. Both Cu and Ni have the simple FCC structure, with complete solid solubility over the entire composition range. The description of the structure according to Pearson would be very similar to that of Cu, but instead of one generating site there are now two, one for nickel and one for copper but both with the same x, y, and z values, and each having an occupation factor of 0.5, corresponding to the bulk alloy concentration of 50 atom% Ni. Thus each of the four sites in the cell is occupied by both Cu and Ni each with the same probability of occupancy, 0.5. Other than some slight changes in the lattice parameters, this is the only difference between pure Cu and the Cu
Ni random solid solution. 1.2.3.3 Miller Indices and Unit Vectors. Crystal planes and crystal directions in the lattice are described by a vector notation based on a coordinate system defined by the axes and dimensions of the unit cell. A direction is defined by a vector whose origin lies at the origin of the coordinate system and whose length is sufficient to ensure that the x, y and z coordinates of the tip of the vector all correspond to an integer number of unit cell coordinates (Figure 1.25). The length of such a rational vector always corresponds to an

z nc=w

nb=v na=u

x Figure 1.25 The definition of direction indices in a crystal lattice.

y

The Concept of Microstructure 37

interatomic repeat distance in the lattice. In a crystal, two or more lattice directions may be geometrically equivalent, and it is sometimes useful to distinguish a family of crystal directions. For example, in a cubic crystal the x-axis is defined by the direction [100] (in square brackets), but the y and z directions, [010] and [001] are, by symmetry, geometrically equivalent. Angular brackets are used as a shorthand for the family of h100i directions. Of course, in lower symmetry crystals the two directions [100] and [010] may not be equivalent (that is a 6¼ b in the unit cell), and these two directions do not then belong to the same family. Note that all directions which are parallel in the crystal lattice are considered equivalent, regardless of their point of origin, and are denoted by the same direction indices [uvw]. However, a negative index is perfectly legitimate, ½
u vw 6¼ ½uvw and indicates that the u coordinate is negative. Since the direction indices define the shortest repeat distance in the lattice along the line of the vector, they cannot possess a common factor. It follows that the direction indices [422] and [330] should be written [211] and [110], respectively. Finally, the direction indices all have the dimension of length, the unit of length being defined by the dimensions of the unit cell. Crystal planes are described in terms of the reciprocal of the intercepts of the plane with the axes of a coordinate system that is defined by the unit cell (Figure 1.26). If the unit cell parameters are a, b and c, and the crystal plane makes intercepts x*, y* and z* along the axes defined by this unit cell, then the planar indices, termed Miller indices, are (h ¼ na/x*, k ¼ nb/y*, l ¼ nc/z*), where the integer n is chosen to clear the indices (hkl) of fractions. For example, a cube plane only intersects one of the axes of a cubic crystal, so two of the values x*, y* and z* must be 1 and the Miller indices must be one of the three possibilities (100),

z z*=nc/l

y y*=nb/k x*=na/h x Figure 1.26 The definition of Miller indices describing the orientation of a crystal plane within the unit cell of the crystal.

38

Microstructural Characterization of Materials

(010) and (001), (always given in round brackets). All planes which are parallel in the lattice are described by the same indices, irrespective of the intercepts they make with the coordinate axes (although different values of n will be required to clear the fractions). Reversing the sign of all three Miller indices does not define a new plane. That is


lÞ, and these indices refer to three crystallo6¼ ðhk ðhklÞ  ðhklÞ. However, ðhklÞ 6¼ ðhklÞ graphically distinct, nonparallel planes in the lattice. While the letters [uvw], with square brackets, are used to define a set of parallel direction indices, the letters (hkl), with round brackets, are used to define the Miller indices of a parallel set of lattice planes. If a family of geometrically equivalent (but nonparallel) planes is intended, then this can be indicated by curly brackets; that is {hkl}. Note that the dimensions of the Miller indices are those of inverse length, and that the units are the inverse dimensions of the unit cell. Unlike the direction indices, Miller indices having a common factor do have a specific meaning: they refer to fractional values of the interplanar spacing in the unit cell. Thus the indices (422) are divisible by 2, and correspond to planes which are parallel to, but have just half the spacing of the (211) planes. In cubic crystals, but only in cubic crystals, any set of direction indices is always normal to the crystal planes having the same set of Miller indices. That is, the [123] direction in a crystal of cubic symmetry is normal to the (123) plane. This is not generally true in less symmetric crystals, even though it may be true for some symmetry directions. If a number of crystallographically distinct crystal planes intersect along a common direction [uvw], then that shared direction is said to be the zone axis of these planes, and the planes are said to lie on a common zone. There is a simple way of finding out whether or not a particular plane (hkl) lies on a given zone [uvw]. If it does, then hu þ kv þ lw ¼ 0. This is true for all crystal symmetries. 1.2.3.4 The Stereographic Projection. It is a great convenience to be able to plot the prominent crystal planes and directions in a crystal on a two-dimensional projection, similar to the projections familiar to us from geographical mapping. By far the most useful of these is the stereographic projection. In the stereographic projection the crystal is imagined to be positioned at the centre of a sphere, the projection sphere, and the crystal directions and normals to the prominent crystal planes are projected from the centre of this sphere (the centre of the crystal) to intersect its surface (Figure 1.27). Straight lines are then drawn from the south pole of the projection sphere, through the points of intersection of these crystallographic directions and crystal plane normals with the sphere surface, until the lines intersect a plane placed tangential to the sphere at its north pole. All points around the equator of the sphere now project onto the tangent plane as a circle of radius equal to the diameter of the projection sphere. All points on the projection sphere that lie in the northern hemisphere will project within this circle, which is termed the stereogram. Points lying in the southern hemisphere will project outside the circle of the stereogram, but by reversing the direction of projection (from the north pole to a plane tangential to the south pole) we can also represent points lying in the southern hemisphere (but using an open circle in order to distinguish the southern hemisphere points). This avoids having to plot any points outside the area of the stereogram. Any such two-dimensional plot of the crystal plane normals and crystal directions constitutes a stereographic projection. It is conventional to choose a prominent symmetry plane (usually one face of the unit cell) for the plane of these stereographic projections.

The Concept of Microstructure 39

Figure 1.27 Derivation of the stereographic projection, a two-dimensional representation of the angular relationships between crystal planes and directions.

Figure 1.28 shows the stereographic projection for the least symmetrical, triclinic crystal system, with the axes of the unit cell [100], [010] and [001] and the faces of the unit cell (100), (010) and (001) plotted on an (001) projection. The (001) plane contains the [100] and [010] directions, while the [001] zone contains the normals to the (100) and (010) planes. However, as noted above, the crystal directions for this very low symmetry crystal do not coincide with the plane normals having the same indices. A stereogram (stereographic projection) for a cubic crystal, with the plane of the projection parallel to a cube plane is shown in Figure 1.29. Plane normals and crystal directions with the same indices now coincide, as do the plots of crystal planes and the corresponding zones. The high symmetry of the cubic system divides the stereogram into 24 geometrically equivalent unit spherical triangles that are projected onto the stereogram from the surface of the projection sphere (we ignore the southern hemisphere, since reversing the sign of a plane normal does not change the crystal plane). Each of these unit

40

Microstructural Characterization of Materials [100]

[110]

[111]

[110]

[101]

[111]

[010]

(111)

(110)

[011] (101) (111) [001]

(011)

(001)

(011)

[011] [111]

(010)

α

(111) (111)

β

(101)

[101] (110)

[010]

[111] [110] (100)

[110]

γ

[100]

Figure 1.28 An (001) stereographic projection of the lattice of a triclinic crystal showing the (100), (010) and (001) planes and the [100], [010] and [001] directions. The angular unit cell parameters, a, b and g, that define the angles between the axes of the unit cell, are also marked on the stereogram. Note that the normals to the faces of the unit cell do not coincide with the axes of the unit cell.

triangles possesses all the symmetry elements of the cubic crystal, and is bounded by the traces of one plane from each of the families {100}, {110} and {111}. The zones defined by coplanar directions and plane normals pass through the centre of the projection sphere (by definition) and intercept this sphere along circles which have the diameter of the projection sphere. These circles then project onto the stereogram as traces of larger circles, termed great circles, whose maximum curvature is equal to that of the stereogram. The minimum curvature of a great circle is zero (that is, it projects as a straight line), and so straight lines on the stereogram correspond to planes whose traces pass through the centre of the stereographic projection. The bounding edges of any unit spherical triangle that defines the symmetry elements of a crystal are always great circles. Another property of the stereographic projection is that the cone of directions that make a fixed angle to any given crystallographic direction or crystal plane normal also projects as a circle on the plane of the stereogram. Such circles are termed small circles (but beware,

The Concept of Microstructure 41 (100)

(110)

(110)

(111)

(101)

(011)

(010)

(111)

(011)

(001)

(111)

(101)

(010)

(111)

(110)

(110)

(100)

Figure 1.29 A stereographic projection of a cubic crystal with a cube plane parallel to the plane of the projection. The projection consists of 24 unit spherical triangles bounded by great circles. Each triangle contains all the symmetry elements of the crystal.

the axis of the cone of angles does not project to the centre of the circle on the stereogram; Figure 1.30). It follows that the angular scale of a stereographic projection is strongly distorted, as can be seen from a standard Wulff net (Figure 1.31), that is used to define the angular scale of the stereogram, both in small circles of latitude and in great circles of longitude. This Wulff net scale is identical to the angular scale commonly used to map the surface of the globe. Finally, some spherical triangles, defined by the intersection of three great circles, have geometrical properties which are very useful in applied crystallography. The sum of the angles at the intersections that form the corners of the spherical triangle always exceeds 2p, while the sides of the spherical triangles, on a stereographic projection also define angles. Great circles that intersect at 90 must each pass through the pole of the other. If more than one of the six angular elements of a spherical triangle is a right angle, then at least four elements of the triangle are right angles and if only two of the angles are given, then all the remaining four angles can be derived. If the angles represented by the sides of the spherical triangle are denoted by a, b and c, while the opposing angles are A, B and C, then the relation sin a/sin A ¼ sin b/sin B ¼ sin c/sin C always holds (Figure 1.32). (Compare the unit triangle for the triclinic unit cell shown in Figure 1.28.) Hexagonal crystals present a special problem, since the usual Miller indices and direction indices do not reflect the hexagonal symmetry of the crystal. It is common practice to introduce an additional, redundant axis into the basal plane of the hexagonal unit cell. The three axes a1, a2 and a3 then lie at 120 to one another, with the c-axis mutually perpendicular

42

Microstructural Characterization of Materials 100

001

010

010

hkl

100 100 010

001

010

hkl 100

Figure 1.30 A cone defining a constant angle with a direction in the crystal projects as a small circle on the stereogram. Note that the generating pole of the small circle is not at the centre of the projected circle on the stereogram.

(Figure 1.33). A redundant t-axis, drawn parallel to a3, results in a fourth Miller index i when defining a plane: (hkil), but the sum h þ k þ i ¼ 0. A basal plane stereogram for a hexagonal crystal (zinc) is given in Figure 1.34. The angular distance between the poles lying within the stereogram and its centre, the c-axis [0001], depends on the axial ratio of the unit cell, c/a. Families of crystal planes in a hexagonal lattice have similar indices in the four Miller index notation. That is ð10
10Þ and ð1
100Þ are clearly from the same family, while in the three index notation this is not obvious: (100) and ð1
10Þ.

Summary The term microstructure is taken to mean those features of a material, not visible to the eye, that can be revealed by examining a selected sample with a suitable probe. Microstructural information includes the identification of the phases present (crystalline or glassy),

The Concept of Microstructure 43

Figure 1.31 The Wulff net gives the angular scale of the stereogram in terms of small circles of latitude and great circles of longitude, as in a map of the globe.

the determination of their morphology (the grain or particle sizes and their distribution), and the chemical composition of these phases. Microstructural characterization may be either qualitative (‘what does the microstructure look like?’) or quantitative (‘what is the grain size?’). The two commonest forms of probe used to characterize microstructure are electromagnetic radiation and energetic electrons. In the case of electromagnetic radiation, the optical microscope and the X-ray diffractometer are the two most important tools. The optical microscope uses radiation in the visible range of wavelengths (0.4–0.7 mm) to form an image of an object in either reflected or transmitted light. In X-ray diffraction the wavelengths used

c

B a

A C b

Figure 1.32 All six elements of a spherical triangle represent angles. There is a simple trigonometrical relationship linking all six angles (see text).

44

Microstructural Characterization of Materials

v a1=a2=a3

γ =120°

a

c

γ a

u

a2 γ a1

t

a3 Figure 1.33 The hexagonal unit cell showing the use of a four-axis coordinate system, u(a1), v (a2), t(a3) and w(c).

to probe the microstructure are of the order of the interatomic and interplanar spacings in crystals (0.5–0.05 nm). The electron microscope uses a wide range of electron beam energies to probe the microstructure of a specimen. In TEM energies of several hundred kilovolts are common, while in SEM the beam energy may be as low as 1 kV. The interaction of a probe beam with the sample may be either elastic or inelastic. Elastic interaction involves the scattering of the beam without loss of energy, and is the basis of diffraction analysis, either using X-rays or high energy electrons. Inelastic interactions may (1010) (1120)

(2110) (2021) (1121) (1122)

(0110)

(2111)

(1011)

(2112)

(1100)

(1012)

(2201)

(0221) (1101)

(0111) (0112)

(1211) (1212)

(1210)

(1102)

(1212) (1211)

(0001)

(1102)

(1210)

(0112) (0111)

(1101) (2201)

(0221)

(1012) (2112)

(1122)

(1100)

(0110) (2111)

(1011)

(1121)

(2021) (1120)

(2110) (1010)

Figure 1.34 A basal plane stereogram for zinc [hexagonal close-packed (HCP) structure], illustrating the use of the four Miller index system (hkil) for hexagonal crystal symmetry.

The Concept of Microstructure 45

result in contrast in an image formed from elastically scattered radiation (as when one phase absorbs light while another reflects or transmits the light). Inelastic interactions can also be responsible for the generation of a secondary signal. In the scanning electron microscope the primary, high-energy electron beam generates low energy, secondary electrons that are collected from a scanned raster to form the image. Inelastic scattering and energy adsorption is also the basis of many microanalytical techniques for the determination of local chemical composition. Both the energy lost by the primary beam and that generated in the secondary signal may be characteristic of the atomic number of the chemical elements present in the sample beneath the probe. The energy dependence of this signal (the energy spectrum) provides information that identifies the chemical constituents that are present, while the intensity of the signal can be related to the chemical composition. Many engineering properties of materials are sensitive to the microstructure, which in turn depends on the processing conditions. That is, the microstructure is affected by the processing route, while the structure-sensitive properties of a material (not just the mechanical properties) are, in their turn, determined by the microstructure. This includes the microstructural features noted above (grains and particles) as well as various defects in the microstructure, for example porosity, microcracks and unwanted inclusions (phases associated with contamination). The ability of any experimental technique to distinguish closely-spaced features is termed the resolution of the method and is usually limited by the wavelength of the probe radiation, the characteristics of the probe interaction with the specimen and the nature of the image-forming system. In general, the shorter the wavelength and the wider the acceptance angle of the imaging system for the signal, then the better will be the resolution. Magnifications of the order of ·1000 are more than enough to reveal all the microstructural features accessible to the optical microscope. On the other hand, the wavelength associated with energetic electrons is very much less than the interplanar spacings in crystals, so that the transmission electron microscope is potentially able to resolve the crystal lattice itself. The resolution of the scanning electron microscope is usually limited by inelastic scattering events that occur in the sample. This resolution is of the order of a few nanometres for secondary electrons, but only of the order of 1 mm for the characteristic X-rays emitted by the different chemical species Some microscopic methods of materials characterization are capable of resolving individual atoms, in the sense that the images observed reflect a physical effect associated with these atoms. Scanning probe microscopy includes scanning tunnelling and atomic force microscopy, both of which can probe the nanostructure on the atomic scale. Many microstructural features may be quantitatively described by microstructural parameters. Two important examples are the volume fraction of a second phase and the grain or particle size, both of which usually have a major effect on mechanical properties. In many cases the microstructure within any given sample varies, either with respect to direction (anisotropy), or with respect to position (inhomogeneity). Crystal structure (or the lack of it, in an amorphous or glassy material) reflects the nature of the chemical bonding, and the four types of chemical bond, covalent, ionic, metallic and polar (or van der Waals), are responsible for the major properties of the common classes of engineering materials, namely metals and alloys (metallic bonding), ceramics and glasses (covalent and ionic bonding), polymers and plastics (polar and covalent bonding), and semiconductors (primarily covalent bonding).

46

Microstructural Characterization of Materials

In a crystal structure the arrangement of the atoms, ions or molecules is regularly repeated in a characteristic spatial array. The unit cell of this crystal lattice is the smallest unit that contains all the symmetry elements of the bulk crystal, and each cluster of identically arranged atoms in this unit cell can be represented by a single lattice point. There are 14 possible ways of arranging these lattice points to give distinctly different lattice symmetries, the 14 Bravais lattices. Characteristic directions in the crystal lattice correspond to a particular atomic sequence and can be defined by direction indices, while any set of parallel atomic planes can be defined by the normal to these planes, given as Miller indices. Both the direction and the Miller indices are conveniently plotted in two dimensions by mapping them onto an imaginary plane using a stereographic projection. This projection has proved to be the most useful of the geometrically possible mapping options.

Bibliography 1. Villars, P. and Calvert, L.D. (1985) Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, Volumes 1–3, American Society for Metals, Metals Park, OH. 2. Hahn, T. (ed.), (1992) International Tables for Crystallography, Volume A, Space Group Symmetry, Kluwer Academic, London. 3. Barrett, C. and Massalski, T.B. (1980) Structure of Metals, Pergamon Press, Oxford. 4. Callister, W.D. (2006) Materials Science and Engineering: An Introduction, 7th Edition, John Wiley & Sons, Ltd, Chichester.

Worked Examples To demonstrate the type of information that can be obtained, we conclude each chapter with examples of microstructural characterization for three different material systems. In this first chapter we examine the crystallographic structure of the phases to be encountered in future chapters and examine the basic use of the stereographic projection. Having seen in this chapter how the literature data can be used to understand the crystal structure of copper, we now look at two slightly more complicated crystal structures that will be considered later. The first is Fe3C, iron carbide or cementite, which exists in equilibrium with a-Fe in most steels (see the Fe–C equilibrium phase diagram). Pearsons Handbook of Crystallographic Data for Intermetallic Phases lists the data according to the format presented in Table 1.4. Table 1.4 Crystallographic data for Fe3C, iron carbide or cementite, which exists in equilibrium with a-Fe in most steels

Phase

Structure type

Pearson symbol space group

CFe3

CFe3

oP16 Pnma

a, b, c (nm) 0.50890 0.67433 0.45235

a, b, g ( )

Atoms

Point set

x

C Fe1 Fe2

4c 4c 8d

890 250 450 100 036 250 852 100 186 063 328 100

y

z

Occ

The Concept of Microstructure 47

Figure 1.35 Schematic drawing of the structure of Fe3C (cementite) for (a) a single unit cell and (b) eight unit cells. The red spheres represent ions atoms, and the black spheres represent carbon atoms.

Note that the name of the phase is listed as CFe3, not Fe3C, the more accepted chemical designation for cementite. This is because Pearson’s handbook lists the chemical constituents in alphabetical order. We also note that CFe3 is listed as the structure type for this structure, and that it has a primitive (P) orthorhombic (o) structure with 16 (16) atoms per unit cell. It belongs to the space group Pnma, and has three generating sites; one for carbon, which generates a total of four carbon atoms per unit cell, and two for iron (Fe1 and Fe2). The first of these iron atoms (Fe1) generates a total of four atoms while the second (Fe2) generates a total of eight atoms, giving a total of 12 iron atoms per unit cell. Since cementite is a stoichiometric phase, the occupation of each site is constant (there is no significant solid solubility), and the occupation factor for each generating site is therefore 100 %. A schematic drawing of the unit cell, with 16 atoms is given in Figure 1.35(a), and eight unit cells with a more conventional representation (showing the atoms belonging to the neighbouring cells and reflecting the full symmetry of the structure) is shown in Figure 1.35(b). Our second example is a-Al2O3, known as sapphire in its single crystal form (the same name as the gem stone) and alumina or corundum in the polycrystalline form. This is the thermodynamically stable form of alumina. The data given in ‘Pearson’s handbook are presented in Table 1.5. The crystallographic data for a-Al2O3 can be quite confusing. We note that the structure has rhombohedral symmetry, with 10 atoms per unit cell (hR10), and a space group of R
3c. However, the lattice parameters are listed for a hexagonal unit cell (a ¼ 0.4754 nm and c ¼ 1.299 nm). Checking the International Tables for Crystallography under the space group Table 1.5 Crystallographic data for a-Al2O3 (hexagonal unit cell).

Phase

Pearson Symbol Structure and space type group

Al2O3 Al2O3

a, b, c (nm)

hR10 R
3c 0.4754 1.299

Point a, b, g Atoms set x ( ) Al 0

12c 18e

0000 3064

y

z

Occ

0000 0000

3523 2500

100 100

48

Microstructural Characterization of Materials

Figure 1.36 Schematic drawing of the hexagonal (a) and rhombohedral (b) unit cells of aAl2O3. The red spheres represent oxygen anions, and the black spheres represent aluminium cations.

R
3c, you will note that there are two alternative ways to describe this unit cell, the first based on a rhombohedral unit cell, with 10 atoms per unit cell, and the second based on a hexagonal representation, with 30 atoms per unit cell. The symmetry of the structure is not changed by using the hexagonal representation, and it is often more convenient to use the hexagonal unit cell, even though not all of the symmetry operations for the higher symmetry hexagonal structure are correct for a-Al2O3. The generating sites listed in Pearson’s handbook are for the hexagonal unit cell, and includes 12 aluminium cations and 18 oxygen anions, giving a total of 30 atoms per hexagonal unit cell. The structure is stoichiometric, so the occupation factors for both cations and anions are 100 %. A schematic drawing of the two alternative rhombohedral and hexagonal unit cells is given in Figure 1.36. The crystallographic data for a rhombohedral unit cell are presented in Table 1.6. Throughout the rest of this book we will use the hexagonal unit cell to describe a-Al2O3. Now let us examine some of the basic uses of the stereographic projection. As described earlier, a stereographic projection is a map that plots the angles between different crystallographic directions and plane normals. Of course, we could also calculate these angles, and today this is done with computer programs, which use the equations listed in Table 1.6 Crystallographic data for a-Al2O3 (rhombohedral unit cell). Pearson symbol Structure and Space Phase type group Al2O3 Al2O3

a, b, c (nm)

a, b, g Point ( ) Atoms set x

hR10 R
3c 0.51284 55.28 Al 0

4c 6e

y

z

Occ

3520 3520 3520 100 5560
0560 2500 100

The Concept of Microstructure 49 (1010) (2110)

(1120)

(2111)

(1121) (1011)

(1100)

(0110)

(2112)

(1122) (1012)

(1101)

(0111) (1102)

(0112)

(1211)

(1210)

(0001)

(1212)

(1212) (1211)

(1210)

(0112)

(1102)

(0111)

(1101) (1012) (2112) (1100)

(1122) (1011)

(2111)

(0110)

(1121)

(1120)

(2110) (1010)

Figure 1.37 The (0001) stereographic projection for a-Ti (hexagonal).

Appendix I. Nevertheless, the stereographic projection is a useful visual representation of the angular relationships between different crystallographic planes and directions. Titanium metal (Ti) is our example for a stereographic projection. Both body-centred cubic and hexagonal phases exist but the phase a-Ti has the hexagonal structure, with lattice parameters a ¼ 0.29504 nm and c ¼ 0.46833 nm. The stereographic projection (the angular relationships for crystal planes and directions) of any cubic unit cell is independent of the lattice parameter, but this is not true for other structures (compare the equations at the end of the chapter), and for a-Ti we need the ratio of the lattice parameters c/a in order to plot the stereographic projection accurately. Figure 1.37 shows the (0001) basal plane stereographic projection for a-Ti. In principle, we could centre the projection on any other convenient crystallographic direction or plane normal, or even use a coordinate system based on the sample geometry, if this were to prove more convenient. Assume we are interested in finding which planes lie at an angle of 90 to the basal (0001) plane. We draw a great circle (using a Wulff Net or our computer program), and for this simple case the great circle is just the perimeter of the stereogram. We see that all planes having indices (hki0) are at 90 from (0001). Simple! Now let us find the planes which are 90 from the direction½11
22. Again we select a great circle, but this time with its centre on the direction ½11
22 (Figure 1.38). It is important to note that the direction ½11
22 and the pole of the plane ð11
22Þ do not coincide on the projection in Figure 1.38, since ½11
22 is not normal to ð11
22Þ in the hexagonal lattice. This new great circle passes through all the planes which are perpendicular to ½11
22, and hence contain the ½11
22 direction. Of course the stereogram also shows planes whose normals are at angles other than 90 to a chosen direction or plane normal. An example is given in Figure 1.39: the planes

50

Microstructural Characterization of Materials (1010) (2110)

(1120)

(2111)

(1121) (1011)

(1100)

(0110)

(2112)

(1122) (1012)

(1101)

(0111) (1102)

(0112)

(1211)

(1210)

(0001)

(1212)

(1212) (1211)

(1210)

(0112)

(1102)

(0111)

(1101) (1012) (2112) (1100)

[1122] (1122)

(1011) (2111)

(0110)

(1121)

(1120)

(2110) (1010)

Figure 1.38 pole.

The stereographic projection for a-Ti, showing a great circle at 90 to the ½11
22

(1010) (2110)

(1120)

(2111)

(1121) (1011)

(1100)

(0110)

(2112)

(1122) (1012)

(1101)

(0111) (1102)

(0112)

(1211)

(1210)

(0001)

(1212)

(1212) (1211)

(1210)

(0112)

(1102)

(0111)

(1101) (1012) (2112) (1100)

(1122) (1011)

(2111)

(0110)

(1121)

(1120)

(2110) (1010)

Figure 1.39 The (0001) stereographic projection for a-Ti, with a small circle construction showing planes at an angle of 60 to the ð11
20Þ plane.

The Concept of Microstructure 51

whose poles make an angle of 60 with the normal to the ð11
20Þ plane lie on a small circle centred about the normal to ð11
20Þ. Only an arc of the small circle appears within the stereographic projection circle, and this arc corresponds to the angles lying in the northern hemisphere of the projection sphere. The remainder of the small circle can be back-projected from the north pole of the projection sphere. (Note again that the centre of any small circle on the stereographic projection is not at the geometric centre of that circle.)

Problems 1.1. Give three examples of common microstructural features in a polycrystalline, polyphase material. In each case give one example of a physical or mechanical property sensitive to the presence of the feature. 1.2. Give three examples of processing defects which might be present in a bulk material. 1.3. Distinguish between elastic and inelastic scattering of a probe beam of radiation incident on a solid sample. 1.4. What is meant by the term diffraction spectrum? 1.5. Give three examples of structure-sensitive and three examples of structure-insensitive properties of engineering solids. 1.6. What magnification would be needed to make the following features visible to the eye: (a) a 1 mm blowhole in a weld bead; (b) the 10 mm diameter grains in a copper alloy; (c) lattice planes separated by 0.15 nm in a ceramic crystal? 1.7. Why is the resolution attainable in the electron microscope so much better than that of the optical microscope? 1.8. To what extent can we claim to see the ‘real’ features of any microstructure? 1.9. Give three examples of microstructural parameters and, in each case, suggest one way in which these parameters are linked quantitatively to material properties. 1.10. Define the terms symmetry, crystal lattice and lattice point. 1.11. What features of a crystal lattice are described by the directional indices [uvw] and the Miller indices (hkl)? (Note, be very careful and very specific!) 1.12. Using literature data, define the unit cell and give the atomic positions for the following materials: (a) Al; (b) a-Ti;

52

Microstructural Characterization of Materials

(c) a-Fe; (d) TiN. 1.13. From the crystallographic data for aluminium metal, obtained in the previous question, calculate the minimum distance between neighbouring atoms in an aluminium crystal. Compare this value with the diameter of the aluminium atom listed in the periodic table. Which crystallographic direction (or plane) did you use for your calculation and why? 1.14. Find the total number of equivalent planes in the cubic structure (multiplicity factor) belonging to the following families: (a) {100}; (b) {110}; (c) {111}; (d) {210}; (e) {321}. 1.15. Calculate the separation of the ð
1012Þ crystal planes in the a-Al2O3 structure (their d-spacing). Do the same for the ð10
12Þ planes. Are these planes crystallographically equivalent? (Hint: remember the symmetry is actually rhombohedral, not hexagonal.) 1.16. Compare the planar density (number of atoms per unit area) for the following planes in an FCC structure containing one atom per lattice point: (a) {100}; (b) {110}; (c) {111}. Which plane has the highest planar density? 1.17. Repeat the previous question for the BCC structure, also with one atom per lattice point. 1.18. Retrieve the crystallographic data for lead telluride (PbTe). List the positions of the atoms in the unit cell. How many atoms belong to each unit cell? What is the Bravais lattice for this material? 1.19. The compound Mg2Si has a cubic structure. Determine the positions of the atoms and calculate the density of this phase. What is the Bravais lattice? Sketch the unit cell and the positions of the Mg and Si atoms. 1.20. Assume that the initial oxidation rate of a reactive metal depends on the planar density of atoms at the surface. What would then be the relative rate of oxidation expected for the ð1
100Þ and ð
12
10Þ polished surfaces of a Mg crystal? 1.21. The phase a-U has an orthorhombic structure. How many atoms are in each unit cell? What are their locations? Determine the d-spacing (distance between planes) for ð1
11Þ, ð1
10Þ, ð101Þ, and ð210Þ in this phase. 1.22. Show that a FCC unit cell could also be described as a primitive rhombohedral unit cell. What is the relationship between the rhombohedral lattice parameters and the cubic lattice parameter? 1.23. Sketch the stacking sequence of the (111) atomic planes for FCC cobalt (one atom per lattice point). Do the same for (0002) planes in HCP cobalt. (Note, there are two atoms per lattice point in the HCP structure.) Compare the planar densities of the two planes.

The Concept of Microstructure 53

1.24. The surface of a cubic single crystal is parallel to the (001) plane. (a) Determine the angle between the (115) plane and the surface. Do the same for the (224) plane. (b) Determine the angle between the (115) and (224) planes. (c) What is the zone-axis that contains both the (115) and the (224) planes? (d) What is the angle and axis of rotation (the rotation vector) required to bring the (115) plane parallel to what was originally the (224) plane?

2 Diffraction Analysis of Crystal Structure Radiation that strikes an object may be scattered or absorbed. When the scattering is entirely elastic, no energy is lost in the process, and the wavelength (energy) of the scattered radiation remains unchanged. The regular arrays of atoms in a crystal lattice interact elastically with radiation of sufficiently short wavelength, to yield a diffraction spectrum in which the intensity of the radiation that is scattered out of the incident beam is plotted as a function of the scattering angle (Figure 2.1). As we shall see below, the scattering angle is twice the angle of diffraction y. Both the diffraction angles and the intensities in the various diffracted beams are a sensitive function of the crystal structure. The diffraction angles depend on the Bravais point lattice and the unit cell dimensions, while the diffracted intensities depend on the atomic numbers of the constituent atoms (the chemical species) and their geometrical relation with respect to the lattice points. A diffraction pattern or spectrum may be analysed at two levels. A crystalline material may be identified from its diffraction spectrum by comparing the diffraction angles that correspond to the peaks in the spectrum and their relative intensities with a diffraction standard (for example, the JCPDS file). In this procedure the diffraction spectrum is treated as a ‘fingerprint’ of the crystal structure in order to identify the crystalline phases as unambiguously as possible. Alternatively, the diffraction spectrum may be compared with a calculated spectrum, derived from some hypothetical model of the crystal structure. The extent to which the predicted spectrum fits the measured data, the degree of fit, then determines the confidence with which the model chosen is judged to represent the crystal structure. In general, any measured spectrum is first compared with existing data, but if there are serious discrepancies with the known standard spectra then it may be necessary to search for a new model of the crystal lattice in order to explain the results. In recent years, computer procedures have been developed to aid in interpreting crystallographic data, and much of the uncertainty and tedium of earlier procedures has been eliminated. Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

56

Microstructural Characterization of Materials

Intensity (a.u.)

2000

α-Fe

1500

1000

Fe3C

500

α -Fe

α -Fe 0 30

40

50

60

70

80

α -Fe 90

100

2θ (deg) Figure 2.1 Diffraction spectrum from a 0.4 %C steel (Cu Ka radiation, 0.154 nm). Most of the peaks are due to BCC a-Fe, but the asymmetry of the major peak is associated with an overlapping Fe3C peak.

2.1

Scattering of Radiation by Crystals

The condition for a crystalline material to yield a discrete diffraction pattern is that the wavelength of the radiation should be comparable with, or less than the interatomic spacing in the lattice. In practice this means that either X-rays, high energy electrons or neutrons may be used to extract structural information on the crystal lattice. Although suitable sources of neutron radiation are now more readily accessible, they are not generally available. The present text is therefore limited to a discussion of the elastic scattering of X-ray and electron beams, although much of the theory is independent of the nature of the radiation. The required specimen dimensions are dictated by the nature of the radiation employed to obtain the diffraction pattern. All materials are highly transparent to neutrons, and it is quite common for neutron diffraction specimens to be several centimetres thick. X-rays, however, especially at the wavelengths normally used (
0.1 nm), are strongly absorbed by engineering materials and X-ray diffraction data are limited to submillimetre surface layers, fine powders or small crystals. Electron beams used in transmission electron microscopy may have energies of up to a few hundred kilovolts (kV), and at these energies inelastic scattering dominates when the specimen thickness exceeds a tenth of a micrometre. Electron diffraction data are therefore limited to submicrometre specimen thicknesses. Thus, even though neutrons, X-rays and electrons may be diffracted by the same crystal structure, the data collected will refer to a very different sample volume, with important implications for the specimen geometry, data interpretation and the procedures used to select and prepare specimens. 2.1.1

The Laue Equations and Bragg’s Law

A one-dimensional array of atoms interacting with a parallel beam of radiation of wavelength l, incident at an angle a0 will scatter the beam to an angle a and generate a path difference D between the incident and scattered beams (Figure 2.2):

Diffraction Analysis of Crystal Structure 57

Figure 2.2 When the path difference between the incident and the scattered beams from a row of equidistant point scatterers is equal to an integral number of wavelengths, then the scattered beams are in phase, and the amplitudes scattered by each atom will reinforce each other (see text).

D ¼ (y  x) ¼ a(cos a  cos a0), where a is the interatomic spacing. The two beams will be in phase, and hence reinforce each other, if D ¼ hl, where h is an integer. Now consider a crystal lattice made up of a three-dimensional array of atoms represented by regularly spaced lattice points that are set at the corners of a primitive unit cell with lattice parameters a, b, and c. The condition that the scattered (diffracted) beam will be in phase with the incident beam for this three-dimensional array of lattice points is now given by a set of three equations, known as the Laue equations: D ¼ aðcosacosa0 Þ ¼ hl

ð2:1aÞ

D ¼ bðcosbcosb0 Þ ¼ kl

ð2:1bÞ

D ¼ cðcosgcosg0 Þ ¼ ll

ð2:1cÞ

The cosines of the angles a, b and g, and a0, b0 and g0 define the directions of the incident and the diffracted beams with respect to the unit cell of this crystal lattice. The choice of the integers hkl, which are identical to the notation used for Miller indices (Section 1.2.3.3), is, as we shall see below, by no means fortuitous. A more convenient, and completely equivalent, form of the geometrical relation determining the angular distribution of the peak intensities in the diffraction spectrum from a regular crystal lattice is the Bragg equation: nl ¼ 2 dsin


ð2:2Þ

where n is an integer, l is the wavelength of the radiation, d is the spacing of the crystal lattice planes responsible for a particular diffracted beam, and y is the diffraction angle, the angle the incident beam makes with the planes of lattice points (Figure 2.3). The assumption made in deriving the Bragg equation is that the planes of atoms responsible for a diffraction peak behave as a specula mirror, so that the angle of incidence y is equal to the angle of reflection. The path difference between the incident beam and the beams reflected from two consecutive planes is then (x  y) in Figure 2.3. The scattering angle between the incident and the reflected beams is 2y, and y ¼ x(cos2y). But cos(2y) ¼ 1  2 sin2y, while x(siny)¼ d, the interplanar spacing, so that (x  y) ¼ 2d siny. The distance d between the lattice

58

Microstructural Characterization of Materials

y

d

x

Figure 2.3 If each plane of atoms in the crystal behaves as a mirror, so that the angle of incidence is equal to the angle of reflection, then the condition for the beams reflected from successive planes to be in phase, and hence reinforce each other, is given by Bragg’s law (see text).

planes is a function of the Miller indices of the planes and the lattice parameters of the crystal lattice. The general equations (for all Bravais lattices) are as follows: 1 1 ¼ ðS11 h2 þ S22 k2 þ S33 l2 þ 2S12 hk þ 2S23 kl þ 2S31 lhÞ d2 V 2 where V is the volume of the unit cell: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ abc 1cos2 acos2 bcos2 g þ 2cosacosbcosg

ð2:3Þ

ð2:4Þ

and the constants Sij are given by: S11 ¼ b2 c2 sin2 a S22 ¼ c2 a2 sin2 b

S33 ¼ a2 b2 sin2 g

ð2:5aÞ ð2:5bÞ

S12 ¼ abc2 ðcosacosbcosgÞ

ð2:5cÞ

S23 ¼ a2 bcðcosbcosgcosaÞ

ð2:5dÞ

S31 ¼ ab2 cðcosgcosacosbÞ

ð2:5eÞ

For an orthorhombic lattice, for which a ¼ b ¼ g ¼ 90 , these equations reduce to:  2  2  2 1 h k l ¼ þ þ ð2:6Þ d2 a b c with V ¼abc. In the Bragg equation, nl ¼ 2d siny, the integer n is referred to as the order of reflection. A first-order hkl reflection (n ¼ 1) corresponds to a path difference of a single wavelength between the incident and the reflected beams from the (hkl) planes, while a second-order reflection corresponds to a path difference of two wavelengths. However, from the Bragg equation, this path difference of two wavelengths for the second-order dhkl reflection is exactly equivalent to a single wavelength path difference from planes of atoms at one half the dhkl spacing, and therefore corresponds to planes with Miller indices (2h 2k 2l). It is therefore common practice to label the nth order reflection as coming from planes having a

Diffraction Analysis of Crystal Structure 59

110

110

200

200

fcc

bcc

Figure 2.4 The FCC (a) and BCC (b) Bravais lattices contain additional planes of lattice points which lead to some forbidden reflections that are characteristic of these crystal structures. The darker lattice points are to the front of the unit cells.

spacing dhkl/n, and with Miller indices (nh nk nl). The Bragg equation is then written l ¼ 2dhkl siny, where the subscript hkl is now understood to refer to a specific order n of the hkl reflection. For example d110, d220 and d440 would be the first-, second- and fourth-order reflecting planes for the 110 reflections. 2.1.2

Allowed and Forbidden Reflections

Body-centred or face-centred Bravais lattices have planes of lattice points that give rise to destructive (out-of-phase) interference for some orders of reflection. In the BCC lattice (Figure 2.4), the lattice point at 1/2 1/2 1/2 scatters in phase for all orders of the 110 reflections, but will give rise to destructive interference for odd orders of the 100 reflections, that is 100, 300, 500, etc.It isinstructiveto list the allowedreflectionsforprimitive,BCCandFCCBravais lattices as a function of the integer (h2 þ k2 þ l2), (Table 2.1). For cubic symmetry the Bragg equation can be rearranged to giveðh2 þ k2 þ l2 Þ ¼ 2a sin
=l, leading to a regular array of diffracted beams. As can be seen, some values of h2 þ k2 þ l2 are always absent, and 7 is the first of these. Other integers may correspond to more than one reflection, and both the 221 and the 300 planes, corresponding to h2 þ k2 þ l2 ¼ 9, will diffract at the same angle y. Those reflections which are disallowed for a particular lattice are referred to as forbidden reflections, and there are simple rules to determine which reflections are forbidden. In the FCC lattice the Miller indices must be either all odd or all even for a reflection to be allowed, and it is the reflecting planes with mixed odd and even indices that are forbidden. In the BCC lattice the sum h þ k þ l must be even for an allowed reflection, and if the sum of the Miller indices is odd, then the reflection is forbidden. In some diffraction spectra the sequence of the diffraction peaks may be recognized immediately as due to a specific Bravais lattice. For example, the two sets of paired reflections (111 and 200, and then 311 and 222) are characteristic of FCC symmetry. More often, careful measurements and calculations are needed to identify the crystal symmetry responsible for a given diffraction pattern. In most crystals the crystal lattice points correspond to groups of atoms, rather than individual atoms, and the different atomic species will scatter more or less strongly, depending on their atomic number (the number of electrons attached to each atom). In the

60

Microstructural Characterization of Materials

Table 2.1 Allowed reflections in crystals of cubic symmetry with one atom per lattice site, listed in order of the sum of the squares of the Miller indices. h2 þ k2 þ l2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Primitive cubic

Face-centred cubic

Body-centred cubic

100 110 111 200 210 211 – 220 221/300 310 311 222 320 321 – 400

– – 111 200 – – – 220 – – 311 222 – – – 400

– 110 – 200 – 211 – 220 – 310 – 222 – 321 – 400

FCC NaCl structure, (Figure 2.5) the sodium cations are located at the lattice points, while the chlorine anions are displaced from the lattice points by a constant lattice vector 1/2 0 0. Two types of lattice plane now exist, those that contain both cations and anions (mixed planes), such as {200} and {220}, and those that consist of an equi-spaced, alternating sequence of pure anion and cation planes, such as {111} and {311}. Since the cations and anions in the mixed planes are coplanar, the two species always scatter in-phase, and the intensities of the diffraction peaks are enhanced by the additional scattering of the second atomic species. However, the alternating planes of cations and anions scatter out of phase for all the odd-order reflecting planes, and hence reduce the diffracted intensity, while the same alternating cation and anion planes will scatter in-phase for all even-order reflections, thus enhancing the scattered intensity. The extent to which the second atomic species will enhance or reduce the diffracted intensity will depend on the difference in scattering power associated with the difference in the atomic numbers of the two atomic species.

2.2

Reciprocal Space

Bragg’s law indicates that the angles of diffraction are inversely proportional to the spacing of the reflecting planes in the crystal lattice. In order to analyse a diffraction pattern it is therefore helpful to establish a three-dimensional coordinate system in which the axes have the dimensions of inverse length (nm1). Such a system of coordinates is referred to as reciprocal space. 2.2.1

The Limiting Sphere Construction

The value of siny is constrained to lie between 1, so that, from Bragg’s law, the value of 1/d must fall in the range between 0 and 2/l if the parallel planes of atoms are to give rise to a diffracted beam. If the beam of radiation is incident along the x-axis, and a diffracting

Diffraction Analysis of Crystal Structure 61

Figure 2.5 In the NaCl structure each lattice point corresponds to one cation and one anion. These different ions are coplanar for the 200 planes, and therefore scatter in-phase, enhancing the 200 diffraction peak. By contrast, the 111 planes of cations and anions are interleaved, and lead to out of phase interference, reducing the intensity of the 111 diffraction peak. The larger anions (red) are distinguished from the smaller cations (grey).

crystal is located at the origin of the coordinate system, then a sphere of radius 2/l, termed the limiting sphere, will enclose all the allowed values of 1/d in reciprocal space and hence define all the planes in the crystal that have the potential to diffract at the wavelength l. Now imagine a smaller sphere of radius 1/l that lies within this limiting sphere and is placed so that it just touches the limiting sphere on the x-axis, and hence also touches the position of the crystal (Figure 2.6). A line passing through the centre of this second sphere, called the reflecting sphere, which is parallel to the diffracted beam and makes an angle 2y with the x-axis, will intersect the periphery of the reflecting sphere at the point P. 2.2.2

Vector Representation of Bragg’s Law

The two vectors k0 and k in Figure 2.6 define the wave vectors of the incident and diffracted beams, |k0| ¼ |k| ¼ 1/l, and, if the reciprocal lattice vector OP is equal to g, then |g| ¼ 1/d, and Bragg’s law l ¼ 2d siny, can be written as a vector equation: k0 þ g ¼ k. Note that the reciprocal lattice vector g is perpendicular to the diffracting planes, while the wave vectors k0 and k are parallel to the incident and diffracted beams, respectively. As we shall see later, this vector form of the Bragg equation can be very useful indeed. 2.2.3

The Reciprocal Lattice

We can now define a lattice in reciprocal space that is in some ways analogous to the Bravais lattice in real space (Figure 2.7). The origin of coordinates in reciprocal space is the point

62

Microstructural Characterization of Materials

Diffracted Beam

P k

g

2θ

Incident Beam

k0

2θ O Limiting Sphere

Reflecting Sphere

Figure 2.6 The limiting sphere and reflecting sphere constructions (see text for details).

(000) and lies at the position of the crystal, in the centre of the limiting sphere construction (Figure 2.6). Any reciprocal lattice vector ghkl, drawn from this origin to the reciprocal lattice point (hkl), will be normal to the reflecting planes that have the Miller indices (hkl). The distance from 000 to hkl, |ghkl|, is equal to 1/dhkl, so that the successive orders of reflection, 1 to n, are represented by a series of equidistant reciprocal lattice points that lie along a straight line in reciprocal space whose origin is at (000). Note that negative values of n are perfectly legitimate in this representation. Lattice planes in real space

1/d 100

1/d 200

000

100

200

Reciprocal lattice points in reciprocal space

d 100

d 200

Figure 2.7 The definition of reciprocal lattice points in terms of the lattice planes of a crystal, defined by their Miller indices. The example shown here is for the cube planes (h00).

Diffraction Analysis of Crystal Structure 63

The condition for Bragg’s law to be obeyed is that the reciprocal lattice vector g should lie on the reflecting sphere. This can be achieved by varying either l or y, as we shall see later. 2.2.3.1 The Reciprocal Lattice Unit Cell. The dimensions of the reciprocal lattice unit cell can be defined in terms of the corresponding Bravais lattice unit cell. The general equations are: a* ¼

bc sina; V

b* ¼

ca sinb; V

c* ¼

ab sing V

ð2:7Þ

where V is the volume of the unit cell. It is important to note that the reciprocal lattice axes a*, b* and c* of the unit cell in reciprocal space are, in general, not parallel to the axes of the unit cell a, b and c in real space. This is consistent with the previous treatment of direction indices and Miller indices. The axes of the unit cell in real space correspond to the direction indices [100], [010] and [001], while the axes of the reciprocal unit cell correspond to the Miller indices (100), (010) and (001) and are aligned normal to the planes (100), (010) and (001). It should now begin to be clear just why we define Miller indices in terms of reciprocal lengths, and a comparison of the reciprocal lattices for FCC and BCC Bravais lattices may help to clarify this even more. Figure 2.8 compares the two reciprocal lattices. In this figure all forbidden reflections have been excluded. Both reciprocal lattice unit cell dimensions are defined by g vectors of type 200. The FCC reciprocal lattice unit cell contains a bodycentred allowed reciprocal lattice point at 111. This reciprocal lattice cell is therefore BCC, while the BCC reciprocal lattice unit cell contains allowed face-centred reciprocal lattice points of the type 110, and is therefore FCC. Reciprocal lattice points in any direction can be derived by simple vector addition and subtraction. The series 020, 121 and 222 thus constitute a series of reflections that lie along a line in BCC reciprocal space. Each reflection is separated from the next in the series by the reciprocal lattice vector 101. Similarly, the sequence of reflections 200, 111, 022 in FCC reciprocal space are separated by the vector
111. Any two nonparallel reflections g1 and g2 define a plane in reciprocal space. Their common zone n, and any other reflection also lying in this zone g3 must obey the rules of vector geometry, that is g1 · g2 ¼ n, and n·g3 ¼ 0.

2.3

X-Ray Diffraction Methods

We now take a closer look at the limiting sphere and reflecting sphere, this time with a reciprocal lattice superimposed on this construction (Figure 2.9). Three factors determine whether or not a particular crystal plane will give a diffraction peak: 1. The wavelength of the X-rays in the incident beam. Reducing the wavelength of the Xrays increases the diameter of the limiting sphere, and therefore places longer reciprocal lattice vectors, corresponding to smaller interplanar spacings, within the limiting sphere. 2. The angle of the incident beam with respect to the crystal. As the crystal is rotated about its centre (or, equivalently, as the X-ray beam is rotated about the crystal), the reflecting sphere sweeps through the reciprocal lattice points, allowing the different diffracting planes to obey the Bragg law in turn. By rotating about two axes at right angles all the

64

Microstructural Characterization of Materials

Z*

(a)

002

202

022

222

fcc 111 Y

020

*

000

200 220

* X Z*

(b)

002

022

112 222

202 101

bcc

011 121

211

020

Y*

000 110 200 * X

220

Figure 2.8 Allowed reflections of the FCC (a) and BCC (b) Bravais lattices plotted in reciprocal space. The FCC reciprocal lattice has BCC symmetry, while the BCC reciprocal lattice has FCC symmetry.

Diffraction Analysis of Crystal Structure 65

. . Incident beam

. .

. . . . .

.

. .

k

. .

Limiting sphere

. g

.

2θ

. .

.

k0

. .

Reflecting sphere

.

.

. . . .

. . 000

. .

. . . . .

. . . .

. . . . .

. . . .

Figure 2.9 Superimposing the reciprocal lattice on the reflecting sphere construction demonstrates the effects of some experimental variables on diffraction (see text).

reciprocal lattice points lying within the limiting sphere can be made to diffract, each at the appropriate Bragg angle yhkl. 3. The effective size of the reciprocal lattice points. If the diffraction condition had to be obeyed exactly, then diffraction would only be observed at the precise Bragg angle. In practice, the reciprocal lattice points have a finite size determined by the size and perfection of the crystal. In addition, inelastic absorption of the incident beam energy limits the path length of the radiation over which elastic scattering dominates, and hence limits the crystal dimensions that can contribute effectively to coherent diffraction Absorption effects are determined by the value of the absorption coefficient. Typical absorption coefficients for X-rays result in path lengths of the order of tens of micrometres before the coherent elastic scattering of the beam becomes blurred by inelastic scattering events. Since interplanar spacings are usually in the range of tenths of a nanometre, it follows that, for large, otherwise perfect crystals, the effective diameter of a reciprocal lattice point (the uncertainty in the value of g) is of the order of 105|g|. As a consequence, the lattice parameters of crystalline phases can be measured with an error that is typically of the order of 10 ppm, quite sufficient to allow for accurate determination of changes in crystal dimensions due to temperature (thermal expansion), alloying additions or applied stress (in particular, residual stresses associated with processing and assembly). For sufficiently small crystallites and heavily deformed crystals it is possible to measure the range of y over which diffraction from a particular hkl plane is observed, and to derive quantitative information on the crystal size or degree of perfection. So far we have assumed that the incident beam is monochromatic and accurately parallel. Again, the effect of these assumptions is best understood from the reflecting sphere construction, now modified as in Figure 2.10. If there is a spread of wavelengths in the

66

Microstructural Characterization of Materials (a)

λ1 > λ2 Wavelength Dispersive

Reflecting Sphere

Limiting Sphere

(b)

Beam Convergence or Divergence Angle

Reflecting Sphere

Limiting Sphere

Figure 2.10 The effect of variations in X-ray wavelength (a) or inadequate collimation of the beam (b) are readily understood from the reflecting sphere construction.

incident beam, then the limiting sphere becomes a shell, and the reflecting sphere generates a ‘new moon’ crescent, within which reciprocal lattice points satisfy the Bragg law. If the incident beam is not strictly parallel, then the reflecting sphere is rotated about the centre of the limiting sphere, by an angle equal to the divergence or, equivalently, the convergence

Diffraction Analysis of Crystal Structure 67

angle of the incident beam, generating two crescent volumes within which the Bragg law is satisfied. Both these effects introduce errors into the determination of lattice spacing by X-ray diffraction. These errors depend on the value of |g| and the angle between g and the incident beam. 2.3.1

The X-Ray Diffractometer

An X-ray diffractometer comprises a source of X-rays, the X-ray generator, a diffractometer assembly, a detector assembly and X-ray data collection and processing systems. The diffractometer assembly controls the alignment of the beam, as well as the position and orientation of both the specimen and the X-ray detector. The X-rays are generated by accelerating a beam of electrons onto a pure metal target contained in a vacuum tube. The high energy electrons eject ground-state electrons from the atoms of the target material, creating holes, and X-rays are emitted during the refilling of these ground states. If all the electron energy, usually measured in eV, were to be converted into an X-ray quantum, then the frequency n would be given by the quantum relation eV ¼ hn, where h is Planck’s constant. The X-ray wavelength l is proportional to the reciprocal of this frequency, l ¼ c/n, where c is the velocity of light in the medium through which the X-rays propagate. The condition that all the energy of the exciting electron is used to create a photon sets an upper limit on the frequency of the X-rays generated, and hence a lower limit on the X-ray wavelength. The above relations lead to an inverse dependence of this minimum wavelength on the accelerating voltage of the X-ray tube, which is given (in vacuum) by: lmin ¼ 1.243/V, where l is in nanometres and V is in kilovolts. Above this minimum wavelength, there is a continuous distribution of X-ray wavelengths generated by the incident electron beam, whose intensity increases with both incident electron energy and beam current, as well as with the atomic number of the target (that is, the density of electrons in the target material). This continuous distribution of photon energies and wavelengths in the X-rays emitted from the target is referred to as white radiation or Bremsstrahlung (German for ‘radiation braking’ – the slowing down of the electrons by the emission of photons). Superimposed on the continuous spectrum of white radiation are a series of very narrow and intense peaks, the characteristic radiation of the chemical elements (Figure 2.11). A characteristic peak corresponds to the energy released when the hole in an inner electron shell, created by a collision event, is filled by an electron which originates in a higher energy shell of the same atom. Thus ejection of an electron from the K-shell excites the atom to an energy state EK, and if the hole in the K-shell is then filled by an electron from the L-shell, then the energy of the atom will decay to EL, while the decrease in the energy of the excited atom, (EK  EL), will appear as an X-ray photon of fixed wavelength that contributes to the Ka line of the characteristic target spectrum (Figure 2.12). Filling the hole in the K-shell with an electron from the M-shell would have reduced the energy state of the atom even further, to EM, leading to a Kb photon of shorter wavelength than Ka, and a second line in the K-shell spectrum. That is, the residual energy of the excited atom is lower in the EM state than it is in the EL state. Further decay of the energy of the excited atom from the EL and EM energy states will result in the generation of L and M characteristic radiation of much longer wavelength. There are many alternative options for the origin of a donor electron to fill a hole in the L- or

68

Microstructural Characterization of Materials

Kα

Relative intensity

3

Characteristic X-rays 2

Kβ

1

X-rays from a molybdenum target at 35 kV Brehmsstrahlung continuum

0.02

0.04

0.06

0.08

0.1

0.12

Wavelength (nm) Figure 2.11 An energetic electron beam striking a solid molybdenum target generates a continuous spectrum of white X-radiation with a sharp cut off at a minimum wavelength, corresponding to the incident electron energy, together with discontinuous, narrow intensity peaks, the characteristic X-radiation from the molybdenum K-shell. Lα Kβ

Kα

Nucleus

K-shell L-shell M-shell

Figure 2.12 Characteristic X-radiation is generated by electron transitions involving the inner shells, and the wavelengths are specific to the atomic species present in the target material.

Diffraction Analysis of Crystal Structure 69 106

EK,L,M (eV)

105 104 103

K 102

L

101

M

10

100

Atomic Number [Z] Figure 2.13 The excitation energy required to eject an electron from an inner shell increases with atomic number.

M-shells, so that the characteristic L and M spectra consist of several closely spaced lines. Clearly, a characteristic line can only be generated in the target by the incident beam if the electron energy exceeds the excitation energy of the atom for that line. The excitation energy for the ejection of an electron from a given inner shell of the atom increases with the atomic number of the target material (Figure 2.13), since the electrons in any given shell are more tightly bound to a higher atomic number nucleus. The low atomic number elements in the first row of the periodic table only contain electrons in the K-shell, and hence can only give K-lines, while only the heaviest elements (of high atomic number) have M- and N-lines in their spectra. These spectra can then be very complex (Figure 2.14). K

K series α2 β3 β 2 α1 β

L series LI

LII

{ {

LI LII LIII

LIII

{

1

4

log energy

M series MII MIII MIV

MI

MV NI NII NIII N IV NV N NVII VI OI OII OIII

MI MII MIII

3

MIV MV

2

1

O O IV V

Figure 2.14 The atomic energy levels and characteristic X-ray spectrum for a uranium atom. After Barratt and Massalski, Structure of Metals, 3rd revised edition, with permission from Pergamon Press.

70

Microstructural Characterization of Materials

If elastic scattering of the X-rays is to dominate their interaction with a sample, then we need to ensure that intensity losses due to inelastic scattering processes are minimized. A monochromatic X-ray beam traversing a thin sample in the x direction loses intensity I at a rate given by dI/dx ¼ mI, where m is the linear absorption coefficient for the X-rays. It is the mass of material traversed by the beam which is important, rather than the sample thickness, so that the values tabulated in the literature are generally for the mass absorption coefficient m/r, where r is the density, rather than the linear absorption coefficient. The transmitted intensity is then given by:   m ð2:8Þ I=I0 ¼ exp  rx r Two plots of the mass absorption coefficient as a function of the X-ray wavelength are shown in Figure 2.15, one for the case of constructional steel and the other for an aluminium alloy. The lower density aluminium alloy has the lower linear absorption coefficient at any given wavelength. All materials show a general increase in the mass absorption coefficient with wavelength, but with a sequence of step discontinuities. These are referred to as absorption edges, and correspond to the wavelengths at which the incident X-ray photon possesses sufficient energy to ionize the atom by ejecting an inner-shell electron from an atom in the specimen, similar to the ejection of an electron by an energetic incident electron. It follows that the absorption edges are the X-ray equivalents of the minimum excitation energies, involved in the generation of characteristic X-rays, as discussed above. Similar to electron excitation, short wavelength, high energy X-rays can generate secondary characteristic X-rays of longer wavelength in the specimen target, a process termed X-ray fluorescence. Note that in X-ray fluorescent excitation there is no background, and white radiation is not generated. In order to avoid fluorescent radiation and minimize absorption of the incident beam, it is important to select radiation for X-ray diffraction measurements that has a wavelength close

Mass absorption coefficient

10 4

Fe

10 3

Al

10 2 10 1

10 0 10 -1

0.2

0.4

0.6

0.8

1.0

1.2

Wavelength (nm) Figure 2.15 The expected dependence of the X-ray mass absorption coefficient on wavelength for iron and aluminium. Note that the linear absorption of the much lighter aluminium will be less than that of iron at all wavelengths.

Diffraction Analysis of Crystal Structure 71

to an absorption minimum, and on the long wavelength side of the absorption edge for the specimen. Thus Cu Ka radiation (l ¼ 0.154 nm) is a poor choice for diffraction measurements on steels and other iron alloys (EFeK ¼ 7.109 keV, l ¼ 0.17433 nm). However, Co Ka radiation (l ¼ 0.1789 nm) lies just to the long wavelength side of the KFe edge and will therefore give sharp diffraction patterns from steel, free of background fluorescence. Assuming that Co Ka radiation is to be used, then the values of m/r for iron and aluminium are 46 and 67.8, respectively. Inserting the densities of the two metals, 7.88 and 2.70 g.cm3, we can derive the thickness of the sample that will reduce the intensity of the incident beam to 1/e of its initial intensity, namely 27.6 mm for iron and 54.6 mm for aluminium. These values effectively define the thickness of the sample which provides the reflection diffraction signal from solid samples of each metal using CoKa radiation. X-ray diffraction experiments require either monochromatic or white radiation. Monochromatic radiation is generated by exciting K-radiation from a pure metal target and first filtering the beam by interposing a foil that strongly absorbs the b component of the Kradiation without appreciable reduction of the intensity of the a component. This can be accomplished by choosing a filter which has an absorption edge that falls exactly between the Ka and Kb wavelengths. A good example is the use of a nickel filter (ENi K ¼ 0.1488 nm) with a copper target (ECu K ¼ 0.138 nm), transmitting the Cu Ka. beam (0.154 nm) but not the Kb, Figure 2.16(a). More effective selection of a monochromatic beam can be achieved by interposing a single crystal monochromator which is oriented to diffract at the characteristic Ka peak. This monochromatic diffracted beam can then be used either as the source of radiation for the actual sample or to filter the diffracted signal [Figure 2.16(b)]. The monochromator crystal can also be bent into an arc of a circle, so that radiation from a line source striking any point on the arc of the crystal will satisfy the Bragg condition, focusing a diffracted beam from the monochromator to a line at the specimen position [Figure 2.16(c)]. The same effect can be achieved at the detector when the monochromator is placed in the path of the beam diffracted from the specimen. An X-ray spectrum is usually recorded by rotating an X-ray detector about the sample, mounted on the diffractometer goniometer stage. The goniometer allows the sample to be rotated about one or more axes (Figure 2.17). In order to make full use of the potential resolution of the method (determined by the sharpness of the diffraction peaks), the diffractometer must be accurately aligned and calibrated, typically to better than 0.01 . The accurate positioning of the sample is very important, especially when using a bent monochromator in a focusing diffractometer. Any displacement of the plane of the sample will result in a shift in the apparent Bragg angle (Section 2.4.5). Note that if a given plane in the sample is to remain perpendicular to a radius of the focusing circle, then the detector must rotate around the focusing circle at a rate which is twice that of the sample. A number of X-ray detectors can be used (including photographic film), but the commonest is the proportional counter, in which an incident photon ionizes a low pressure gas, generating a cloud of charged ions which are then collected as a current pulse. In the proportional counter the charge carried by the current pulse is proportional to the photon energy, and electronic discrimination can be used to eliminate stray photons whose energy does not correspond to that of the required signal. The present generation of proportional counters have an energy resolution better than 150 eV, and can be used to eliminate most of the background noise associated with white radiation (although they are not able to separate

72

Microstructural Characterization of Materials

(a)

(b) Multi-wavelength noncollimated source

Intensity mass absorption coefficient

Nickel filter

Monochromatic beam at θBragg of the monochromator crystal

θ Bragg

Monochromator Kα Bent monochromator

(c)

Kβ 0.12

0.14

0.16

0.18

Source

Specimen

λ (nm) Figure 2.16 Some important spectrometer features: (a) Cu K radiation filtered by a nickel foil to remove Kb; (b) a monochromator crystal allows a specific wavelength to be selected from an X-ray source; (c) a fully focusing spectrometer maximizes the diffracted intensity collected at the detector.

the Ka and Kb peaks). There is a dead-time associated with the current pulse generated in a proportional counter, and a second photon arriving at the counter within a microsecond or so of the first will not be counted. This sets an upper limit to the counting rate and means that peak intensities recorded at high counting rates may be underestimated. While the maximum thickness of the sample that can be studied by X-ray diffraction is dictated by the mass absorption coefficient for the incident radiation (see above), the lateral dimensions are a function of the diffractometer geometry. For an automated powder diffractometer with a ‘Bragg–Brentano’ geometry the width irradiated perpendicular to the incident beam is typically of the order of 10 mm, while the length of the illuminated patch depends on the angle of incidence, and is typically in the range 1–7 mm (Figure 2.18). The size and spacing of Soller slits (used to collimate the beam and also called divergence slits) determine the area illuminated by the incident beam. For fixed slits, the total illuminated area decreases as the diffraction angle increases (2y), but for sufficiently thick samples, the total irradiated volume is almost independent of 2y. Some Bragg–Brentano diffractometers include an automatic (compensating) divergence slit, which increases the width of the incident beam as the diffraction angle increases. The irradiated volume of the sample then increases with increasing diffraction angle and the calculated integrated

Diffraction Analysis of Crystal Structure 73

Focal circle

Detector Source

θ rotation of specimen

θ

2θ Sample

2θ rotation of detector

Figure 2.17 A sample mounted on a goniometer which can be rotated about one or more axis, and a detector which travels along the focusing circle in the Bragg–Brentano geometry.

intensities must then take into account the dependence of integrated intensity on diffracting volume. 2.3.2

Powder Diffraction–Particles and Polycrystals

The grain size of crystalline engineering materials is generally less than the thickness of material contributing to an X-ray diffraction signal. This is also true of many powder samples, whether compacted or dispersed. The general term powder diffraction is used to describe both the nature of the diffraction pattern formed and the subsequent analysis used to interpret diffraction results obtained from these polycrystalline samples. If we assume that the individual grains are both randomly oriented and much smaller than the incident beam cross-section, then this assumption of random orientation is equivalent to allowing the wave vector of the incident beam k0 to take all possible directions in reciprocal space. That is, the reciprocal lattice is rotated freely about its origin (Figure 2.19). All grains that are oriented for Bragg reflection must have g vectors which touch the surface of the reflecting sphere, and it is these grains that then generate diffraction cones that subtend fixed Bragg angles 2y with the incident beam. The innermost cone corresponds to the lattice planes with the largest d-spacing, corresponding to the minimum observed value of y for diffraction. The minimum d-spacing that can be detected is determined by the radius of the limiting sphere, dmin ¼ l/2. As the detector is rotated about an axis normal to the incident

74

Microstructural Characterization of Materials To detector Source

Soller slits

θ Sample w

θ1

w1

θ1<θ2

θ2

w2

Figure 2.18 Influence of y on the exposed surface area for a powder diffractometer using the Bragg–Brentano geometry.

beam and passing through the sample, the diffracted intensities are recorded as a function of 2y, to give the diffraction spectrum for the sample. If the specimen is rotated about an axis normal to the plane of diffraction (the plane containing the incident and diffracted beams) at a constant rate dy/dt, while the detector is rotated about the same axis at twice this rate, d(2y)/dt, then the normal to the diffracting planes in the crystals which are contributing to the spectrum will remain parallel. This is useful if the grains in the polycrystalline sample are not randomly oriented. Mechanical

Diffraction Analysis of Crystal Structure 75

Figure 2.19 In powder diffraction, a random, polycrystalline sample is equivalent to free rotation of the reciprocal lattice about the centre of the limiting sphere. Each reciprocal lattice vector within the limiting sphere then generates a spherical surface which intersects the reflecting sphere on a cone of allowed reflections from the individual crystals which subtend an angle 2y with the incident beam.

working (plastic deformation) and directional solidification are two processes which tend to align the grains along specific directions or in certain planes. Similar alignment effects are common in thin film electronic devices prepared by chemical vapour deposition (CVD). Such samples are said to possess crystalline texture, and the grains are said to be preferentially oriented. If the normal to the diffracting planes in a diffraction experiment coincides with specific directions in the bulk material (parallel or perpendicular to the direction of mechanical work, for example), then the diffracted intensities of specific crystallographic reflections for these sample orientations may be markedly enhanced or reduced with respect to those intensities calculated for a randomly oriented polycrystal. Preferred orientation plays an important role in many material applications, and is associated with anisotropy of the physical, chemical or mechanical properties. A classic case is that of the magnetic hysteresis of silicon iron, which is markedly different in the h100i and h111i directions. Transformer steels are therefore processed to ensure a strong (and favourable) texture, exhibiting low hysteresis losses. In many mechanical applications texture is considered undesirable, and structural steel sheet is usually cross-rolled (rolled in two directions at right angles) to limit the tendency to align the orientation of the grains. The lattice spacings in polycrystalline samples are dependent on the state of stress of the sample. Stresses may be due to the conditions under which the component performs in service (operating stresses), but they may also result from the way in which the component is assembled into a system (for example, a bolt is put under tension when a nut is tightened). However, residual stresses also result from the processing history: gradients of

76

Microstructural Characterization of Materials

plastic work in the component, variations in cooling rate, and some special surface treatments (ion implantation, chemical surface changes, or mechanical bombardment with hard particles, known as ‘shot-peening’). In all cases the residual stresses present in the material must be in a state of mechanical equilibrium, even though no external forces are being applied. One way of determining residual stress is by accurate X-ray measurement of lattice spacings. It is, of course, the lattice strains within the individual grains which are being sampled, and these must be converted to stresses through a knowledge of the elastic constants of the phases present. It is useful to distinguish two types of residual stress, namely macrostresses and microstresses. Macrostresses are present when large numbers of neighbouring crystals of the same phase experience similar stress levels, and the stresses vary smoothly throughout the sample in order to generate an equilibrium stress state (for example, compressive stresses in the surface layers which are balanced by tensile stresses in the bulk). On the other hand, microstresses may also exist, in which the stresses in the individual grains within any volume element may be widely different and of opposite sign, while the average stress in the component sums to zero. A good example would be stresses due to anisotropy of thermal expansion in a noncubic polycrystal, leading to constraints on the contraction of crystals exerted by their neighbours during heating or cooling. Macrostresses result in a displacement of the diffraction maxima from their equilibrium positions, while microstresses result in a broadening of the diffraction peaks. Given that lattice spacings are measurable by X-ray diffraction to one part in 105, it follows that lattice strains are detectable to approximately 105. For an aluminium alloy with an elastic modulus of 60 GPa, this corresponds to a stress of less than 1 MPa. It follows that accurate determination of residual stress levels that are only a few per cent of the bulk yield stress should be possible. Unfortunately, measurements of residual stress using X-ray diffraction are confined to the surface layers of the sample. They may nevertheless be extremely helpful, for example in controlling the quality of surface coatings. 2.3.3

Single Crystal Laue Diffraction

The powder method depends on measuring the intensity diffracted from a monochromatic incident beam as a function of the Bragg angle, given by the relationship l ¼ 2d sin
, and identifying the lattice planes responsible for each of the diffraction peaks in the spectrum. An alternative would be to use a beam of white radiation and determine the spatial distribution of the intensity diffracted by a rigidly mounted sample. A particular set of diffracting planes in a crystal will then select a wavelength from the incident beam that satisfies the Bragg criterion for the angle at which the crystal is oriented. In a polycrystalline sample this will result in jumbled sets of reflections for the different crystals, many of which will overlap with those from other crystals. The result would be a confused pattern that could not be interpreted. However, if the sample is a single crystal, then the reflections will form a very distinctive Laue pattern which can be used to determine the orientation of the crystal with respect to the incident beam. Initially a photographic film but now, more usually, an areal detector array (a chargecoupled device, CCD) is used to record the single crystal diffraction pattern in a Laue camera. Two camera configurations are possible (Figure 2.20). If the specimen is thin enough, a Laue pattern may be recorded in transmission on a plane perpendicular to the

Diffraction Analysis of Crystal Structure 77

λ4 < λ3 < λ2 < λ1

000

Reflecting sphere Back-reflection laue pattern

Transmission laue pattern

Incident beam Single crystal

Figure 2.20 Reflecting sphere construction for single crystal diffraction using white X-radiation and the experimental configurations used to record Laue diffraction patterns. Each reciprocal lattice point diffracts that wavelength which satisfies the Bragg relation.

incident beam. The Laue reflections from a set of crystal planes that lie on the same symmetry zone will intersect the plane of the film along an ellipse. More commonly, Laue diffraction patterns are recorded in reflection, since there is then no limitation on the thickness of the diffracting crystal. The beam passes through a hole in the centre of the recording plane, which is again normal to the incident beam. The symmetry zones of the reflections from the diffracting planes now intersect the detection plane as arcs of hyperbolae. Examples of Laue patterns taken in both transmission and reflection are given in Figure 2.21. The interpretation of a Laue pattern depends on identifying the symmetry axes of the reflecting zones in order to determine the orientation of the single crystal with respect to an external coordinate system. Information on the perfection of the single crystal can also be derived from a Laue image, since the presence of sub-grains or twinned regions gives rise to

78

Microstructural Characterization of Materials

Figure 2.21 Laue diffraction patterns recorded from a single crystal sample in a symmetrical orientation: (a) in transmission, when the symmetry zones intersect the recording plane on ellipses; (b)in reflection, in which case the symmetry zones intersect on hyperbolae.

additional sets of diffracted beams which are displaced with respect to those due to the main crystal. 2.3.4

Rotating Single Crystal Methods

While much crystallographic structure analysis can be achieved using randomly oriented, powder or polycrystalline samples, single crystals are often required to confirm a crystal lattice model unambiguously. Monochromatic radiation is used and the crystal is mounted at the exact centre of the spectrometer on a goniometer stage. The goniometer allows the crystal axes to be oriented accurately with respect to the spectrometer and permits

Diffraction Analysis of Crystal Structure 79

Layer lines

Incident beam

000

Figure 2.22 Rotating a single crystal about an axis perpendicular to the plane of the spectrometer and using monochromatic radiation brings each lattice point in turn into the diffracting condition, generating layer lines of reflections.

continuous rotation of the crystal about an axis normal to the plane containing the incident beam and the detector. Rotation brings reciprocal lattice points lying on planes parallel to the plane of the spectrometer into the reflecting condition, generating layer lines of reflections (Figure 2.22). The single crystal must be large enough to ensure sufficient resolution for structure analysis in reciprocal space, but not so large as to result in geometrical blurring of the reflections in real space. Using X-rays, crystals between 0.1 mm and 1 mm in size are suitable. The range of the rotation angle is restricted in order to reduce overlap from multiple reflections, and several spectra must be recorded by rotating the crystal about the prominent symmetry axes.

2.4

Diffraction Analysis

So far we have only discussed the geometry of diffraction and shown how a determination of the angular distribution of the diffracted beams can be used to identify the crystal symmetry and determine the lattice parameters to a high degree of accuracy. This information is usually sufficient to identify the crystalline phases present in a solid sample unambiguously, but there is a great deal of information present in the relative intensities of the diffracted beams that we have not yet utilized. To make use of this information we examine the factors which determine the scattered amplitude that we have measured. These factors include the scattering by individual atoms, the summation of the atomic scattering by the unit cells of the crystal lattice, the summation over the individual grains of a polycrystal, and the detection of the diffracted radiation by the diffractometer assembly.

80

Microstructural Characterization of Materials Scattered photon

Incident photon

r

α

Electron

Figure 2.23

2.4.1

Scattering of an X-ray beam by an electron (see text).

Atomic Scattering Factors

We confine our attention to incident X-rays and electrons, which are scattered by electrons in the solid, and we ignore diffraction of incident neutrons, which are scattered by the atomic nuclei. If a is the angle between the scattering direction of the incident beam and the direction in which an interacting electron is accelerated, then J.J. Thomson showed that the scattered intensity is given by: e4 ð2:9Þ I ¼ I0 2 2 4 sin2 a r m c where e and m are the charge and mass of the electron, c is the velocity of electromagnetic radiation and r is the distance of the accelerated electron from the incident beam (Figure 2.23). For an unpolarized X-ray beam we need to average the effect of the electric field components that act on the electromagnetic wave (Figure 2.24). The electric field acts

Figure 2.24 If scattering occurs at an angle 2y in the x–z plane, then the applied electric field is in the y–z plane and the average values of the components Ey and Ez must be equal.

Diffraction Analysis of Crystal Structure 81

80.0 70.0

W

60.0 50.0

f (θ)

40.0 30.0

Fe

20.0 10.0 0.0 0

Al 0.2

0.4

0.6

0.8

1.0

1.2

Sin(θ)/λ [Å -1] Figure 2.25 The atomic scattering factor as a function of Z and y for aluminium, iron and tungsten.

perpendicular to the plane of scattering, x  z, and lies in the y  z plane, with components Ey and Ez. For an unpolarized beam, these two components are, on average, equal, while the values of a, the angle between the components of E and the scattering direction, are given by ay ¼ p/2 and az ¼ (p/2  2y). Assuming that each component contributes half the total intensity, the factor sin2 a should be replaced by ðsin2 ay þ sin2 az Þ=2. Substituting for ay and az leads to the relationship I ¼ I0

e4 ð1 þ cos2 2
Þ : r 2 m 2 c4 2

ð2:10Þ

where the term ð1 þ cos2 2
Þ=2 is called the Lorentz polarization factor. Each atom in the sample contains Z electrons, where Z is the atomic number. In the direction of the incident beam all the electrons in the atom will scatter in phase. If the atomic scattering factor f(y) is defined as the amplitude scattered by a single atom divided by the amplitude scattered by an electron, then it follows that f(0) ¼ Z, while for y > 0, f(y) < Z, since at larger scattering angles the electrons around an atom will scatter increasingly out of phase. The y dependence of the atomic scattering factors for iron (Fe), aluminium (Al) and tungsten (W) are shown in Figure 2.25. 2.4.2

Scattering by the Unit Cell

The next step is to derive the amplitude scattered by a unit cell of the crystal structure. Any path difference d between the X-ray beam scattered from an atom at the origin and from another atom elsewhere in the unit cell will correspond to a phase difference between the two scattered beams which is given by j ¼ 2pd=l. If the position of the second atom is defined by the vector r in the direction [uvw], such that the coordinates of the atom in the unit cell are (x, y, z) with u ¼ x/a, v ¼ y/b and w ¼ z/c, and the atom lies on the plane (hkl), defined

82

Microstructural Characterization of Materials

Figure 2.26 Geometry of atomic positions and scattering planes in the unit cell.

by the reciprocal lattice vector g (Figure 2.26), then the phase difference for radiation scattered by these two atoms into the hkl reflection is just: jhkl ¼ 2pðhu þ kv þ lwÞ ¼ 2pg · r

ð2:11Þ

Each atom will scatter an amplitude A, which depends on the atomic scattering factor fZy for that atom, and we can represent both the phase and the amplitude of the scattered wave from each atom by a vector A. Using complex notation, Aeij ¼ Aðcosj þ isinjÞ, and the contribution to the amplitude scattered into the diffracted beam hkl by an atom at uvw in the unit cell will be given by (Figure 2.27):

Resultant amplitude A3

A

φ3

Aisin( φ)

Σ A n isin(φ n)

Aeij / f exp½2piðhu þ kv þ lwÞ ¼ f expð2pig · rÞ

ð2:12Þ

A

A2 A1

φ2 φ1

ΣA ncos(φn)

Acos( φ)

Figure 2.27 The amplitude–phase diagram and its use to sum the scattered amplitudes contributing to a particular reflection by all the atoms in the unit cell.

Diffraction Analysis of Crystal Structure 83

2.4.3

The Structure Factor in the Complex Plane

By ignoring the constant of proportionality, which corresponds to the scattering due to a single electron, we can define a normalized scattering factor due to a complete unit cell of the crystal for an hkl reflection by summing the contributions to this reflection from all the N atoms in the unit cell. This new parameter, the structure factor for the hkl reflection, is then given by: Fhkl ¼

N X 1

fn exp½2piðhun þ kvn þ lwn Þ ¼

N X

fn expð2pig · rÞ

ð2:13Þ

1

Before proceeding further, we note that, in the complex plane, the following relationships hold: 1. eij ¼ cosj þ i sinj, so that the real component of the amplitude is resolved along the x-axis and the imaginary component along the y-axis of phase space. 2. I ¼ jAj·jA* j, that is the intensity scattered by any combination of atoms is derived from the phase space vector amplitude by multiplying the real component of the amplitude by its complex conjugate. 3. enpi ¼ ð1Þn , so that phase angles corresponding to even and odd multiples of p have no imaginary component of the amplitude, and simply add or subtract from the total scattered amplitude. 4. eix þ eix ¼ 2cosx, which is a second condition for no imaginary component. Copper has an FCC unit cell containing four atoms, each atom situated at a Bravais lattice point, so that the values of [uvw] are [0 0 0], [1/2 1/2 0], [1/2 0 1/2] and [0 1/2 1/2]. It follows that the structure factors for copper are given by: Fhkl ¼ f ½1 þ epiðh þ kÞ þ epiðh þ lÞ þ epiðk þ lÞ . If h, k and l are all odd or all even, then Fhkl ¼ 4f, but if h, k and l are mixed integers, then Fhkl ¼ f(1 þ 1  2) ¼ 0, the same result we have noted previously. In the BCC unit cell, characteristic of a-Fe, the atoms are at the two Bravais lattice points [0 0 0] and [1/2 1/2 1/2], and the structure factors are given by Fhkl ¼ f ½1 þ epiðh þ k þ lÞ . For h þ k þ l even it follows that Fhkl ¼ 2f, while if h þ k þ l is odd, then Fhkl ¼ f(1  1) ¼ 0, again as we noted previously. Cubic diamond has an FCC unit cell in which each lattice point corresponds to two atoms, one at the site of the lattice point and the other displaced by a vector [1/4 1/4 1/4]. This is equivalent to two interpenetrating FCC lattices related to one another by this same displacement vector. It follows that the structure factors are given by Fhkl ¼ f ½1 þ epiðh þ kÞ þ epiðh þ lÞ þ epiðk þ lÞ ½1 þ epi=2ðh þ k þ lÞ . There are now three possibilities. For h þ k þ l odd, the amplitude vector for the set of four ‘second’ atoms has no real component and this vector points either vertically up or verticallypdown. In both cases the ffiffiffi structure factor for the allowed FCC reflections is increased by 2 over that for a single atom at the origin. A resultant phase angle of p/4 is introduced. For h þ k þ l even, only the real component exists, and may be either negative, reducing the structure factor of an allowed FCC reflection to zero, or positive, doubling the structure factor to twice the value for a single atom at each lattice point. Finally, consider the case of common salt (NaCl, with an FCC structure). In this case the cations sit on the Bravais lattice points while the anions occupy a second FCC lattice

84

Microstructural Characterization of Materials

displaced by [1/2 1/2 1/2]. The scattering factors of the cations and anions are different, and the structure factors for the different reflections are now given by the relationship Fhkl ¼ ½1 þ epiðh þ kÞ þ epiðh þ lÞ þ epiðk þ lÞ ½ fNa þ fCl epiðh þ k þ lÞ . Even values of h þ k þ l now result in reinforcement of the intensity for the allowed FCC reflections: Fhkl ¼ 4 (fNa þ fCl), while odd values of h þ k þ l reduce the intensity Fhkl ¼ 4(fNa  fCl).

2.4.4

Interpretation of Diffracted Intensities

We are now in a position to summarize all the physical factors that determine the intensity of an observed diffraction peak in a recorded spectrum: 1. The Lorentz polarization factor is associated with scattering of unpolarized electromagnetic radiation:   1 þ cos2 2
ð2:14Þ 2 The exact form of the polarization factor is dependent on the geometry of the diffractometer (see 4 below). 2. The structure factors for the different reflecting planes in the crystal lattice, which include the effect of the atomic scattering factors for all the atoms present in the material: Fhkl ¼

N X

fn exp½2piðhun þ kvn þ lwn Þ

ð2:15Þ

1

3. The multiplicity of the reflecting planes P, which gives the number of planes belonging to a particular family of Miller indices (determined by the symmetry of the crystal). For example, in cubic crystals, planes whose poles fall within the unit triangle have a multiplicity of 24, since there are 24 unit triangles, while those whose poles that lie along the edges of a unit triangle (and are therefore common to two triangles) have a multiplicity of 12. However, there are four {111} planes in the stereogram, which correspond to the apices of the unit triangles (shared by six triangles) and yield a multiplicity of 4, while the {100} reflections correspond to poles on the coordinate axes, shared by eight triangles, and have a multiplicity of 3. 4. The sampling geometry. In the powder method a collector of finite cross section only samples that proportion of the cone of radiation which is diffracted at the Bragg angle. The fraction of radiation collected is given by the Lorentz-polarization factor: L¼

1 þ cos2 2acos2 2
sin2
cos
ð1 þ cos2 2aÞ

ð2:16Þ

where y is the diffracting angle determined from Bragg’s law, and a is the diffracting angle of any monochromator in the diffractometer. 5. Absorption effects that depend on the size of the sample and its geometry. In general, absorption can be expected to increase at large values of the diffraction angle. The absorption correction can be written as A0 (y), and can be estimated for given geometries and sample densities, using standard tables of mass absorption coefficients For a thick

Diffraction Analysis of Crystal Structure 85

specimen in the Bragg–Brentano diffractometer the absorption factor will not be a function of the diffraction angle and is simply A0 ¼ 1/2 m. Thus this factor cancels when calculated integrated intensities are normalized. 6. Temperature is also an important factor, since at high temperatures random atomic vibration will reduce the coherence of the scattering from the more closely spaced crystal planes, corresponding to the higher values of y. This effect increases with siny/l and is more significant at small d values and larger values of hkl. Assuming an isotropic behaviour, its influence on the overall integrated intensity can be expressed by the factor: e2Bsin

2


=l2

ð2:17Þ

where B ¼ 8p2 U


2

ð2:18Þ


2

and U is the mean square displacement of each atom. Summing all the above effects yields a general relationship for the diffracted integrated intensity:   1 þ cos2 2acos2 2
2Bsin2
2 I ¼ kjF j 2 ð2:19Þ PAð
Þ·exp  l sin
cos
ð1 þ cos2 2aÞ where k represents a scaling factor which includes I0. Equation 2.19 can be used to calculate simulated integrated intensities for any given structure from an appropriate computer program. The net integrated intensity is first calculated for each hkl reflection, and then all the calculated intensities are normalized with respect to the maximum calculated integrated peak intensity, which is assigned a value of 100%: n ¼ Ihkl

2.4.5

Ihkl max ·100 Ihkl

ð2:20Þ

Errors and Assumptions

In the present treatment there is little justification for an exact analysis of the errors involved in X-ray diffraction measurements, but some semi-qualitative discussion is necessary. It is important to distinguish between errors in the measurement of peak positions and errors in the determination of peak intensities. It is especially important to recognize the need for accurate calibration and alignment if small changes in lattice parameter (of the order of 105) are to be resolved. X-ray diffraction may only be able to sample a small volume of material, but this can be an advantage in the analysis of thin films, coatings and solid state devices. Such applications are critical for a wide range of systems where the engineering properties are associated with the surface and nearsurface regions, such as microelectronic components, optronic devices, wear parts and machine tools. The monitoring of surface stress or chemical change at the surface can often be accomplished by X-ray diffraction, and in situ commercial X-ray systems are available for extracting measurements at high temperatures or in a controlled environment.

86

Microstructural Characterization of Materials

As an example, we summarize the errors in measuring lattice parameters by using a diffractometer in the Bragg–Brentano geometry. The Bragg–Brentano diffractometer is the most common automated diffractometer and has a well-defined optical focusing system. The focal circle (Figure 2.28), is tangential to both the source and the specimen surface. Since the specimen is rotated in increments of y, and the detector in increments of 2y, the radius of the focal circle decreases as y increases, and the detector must remain on the (changing) focal circle in order to minimize the spread of the signal beam. These operating conditions control the three major errors in peak measurement that limit the accuracy when determining the lattice spacings and lattice parameters. The first is peak broadening, and is due either to incorrect alignment of the diffractometer, or to misplacement of the specimen within the diffractometer goniometer. This error can

Focal Circle Source

θ1

θ1 2θ1

Sample

Source

θ1

θ1

le

Samp

Figure 2.28

The Bragg–Brentano geometry at two different Bragg angles.

Diffraction Analysis of Crystal Structure 87 R ∆2 θ ∆2 θ

θ h

Surface

θ

θ

Figure 2.29 A diffraction signal from below the focusing plane (sample surface) resulting in an error in 2y.

be reduced, either by better alignment or by the use of smaller receiving slits in front of the detector, but residual broadening will always lead to some error in the measurement of the diffraction angle. The second and third sources of error are both directly related to the specimen geometry. The diffraction signal originates from a region beneath the specimen surface, not from the surface itself. This results in an error in the apparent diffraction angle, as illustrated in Figure 2.29. A similar error will result if the specimen is placed either above or below the focal point on the goniometer axis. Using Figure 2.29 we derive the resulting defocus error in y from the relationship: sinD2
sinð1802
Þ ¼ h=sin
R

ð2:21Þ

and thus: D2
ffi

2hcos
R

ð2:22Þ

Finally, Dd D2
cos
2hcos
cos2
Da ¼ ¼ ¼k ¼ d tan
sin
R sin
a

ð2:23Þ

Errors in lattice parameter measurements can be reduced by extrapolating the values calculated from each observed diffraction peak to a (hypothetical) value for y ¼ 90 .That is, by plotting the lattice parameters as a function of cos2y/siny and extrapolating to 0. An alternative is to spread some powder from a diffraction standard (with known lattice parameters) on the surface of the specimen. The peak positions due to diffraction from the standard powder can then be used to correct for systematic errors.

88

Microstructural Characterization of Materials

Some additional factors should be considered in the measurement of peak intensities, especially those associated with efficient data collection. There are two considerations: the geometry of the sample in the spectrometer, and the response of the detector to the incident photons. Both have been mentioned previously. Some geometrical peak broadening is always associated with the finite diffracting volume in the sample, and while the detector response should remain constant over long periods it will always be wavelength-dependent. Measurement of the comparative intensities from different diffraction peaks (expressed as a percentage of the intensity of the strongest diffracted peak) is an important diagnostic tool, both for identifying unknown phases in the sample, and for refining crystal structure models. Such measurements are sensitive to the presence of preferred orientation. The measured height of a diffraction peak in an experimental spectrum is a rather inaccurate estimate of the relative peak intensity, and it is important to determine the integrated peak intensities, that is the total area under the peak, after subtracting background noise, (Figure 2.30). In addition to a dependence on preferred orientation, the integrated intensity of a diffraction peak may vary with temperature (see above), or as a result of transmission losses (if the specimen thickness is below the characteristic absorption thickness). The reduction in integrated intensity associated with a thin sample can be corrected by multiplying the integrated intensity using a factor that takes into account the pathlength in the sample: ½1expð2mt cosec
Þ

ð2:24Þ

For thick specimens, this factor reduces to 1. If the absorption coefficient m is known, then in principle the thickness of a thin sample t can be measured (not very accurately) by comparing the intensities with those from a bulk sample. For random samples, mt is best determined from relative integrated intensities by a refinement procedure based on the entire diffraction spectrum, but we can also compare the intensity ratio of a single pair of

600

Intensity (a.u.)

500

Measured Intensity Background

400 300 200 100 0 50

55

60

65

70

2θ Figure 2.30 A weak diffraction peak, illustrating the calculation of integrated peak intensity from the total area beneath the peak after subtracting background noise.

Diffraction Analysis of Crystal Structure 89

reflections recorded at two Bragg angles y1 and y2, I(t) ¼ I1/I2 to I(1) for the bulk sample, using the expression: Y¼

I ðt Þ 1expð2mtcosec
1 Þ ¼ I ð1Þ 1expð2mtcosec
2 Þ

ð2:25Þ

This method of determining thickness is practicable when mt has values in the range 0.01–0.5. Preferred orientation changes the relative values of the integrated peak intensities observed in the diffraction pattern, and reflects the presence of microstructural features that may strongly affect the material properties. Accurate methods for determining preferred orientation, especially the probability of finding a crystal having a specific orientation (the crystallite orientation distribution function) are beyond the scope of this text. However, there is a fairly simple method which is often used to determine qualitative values of texture. This method (termed the Harris method) is based on a relation for the volume fraction of a phase having a crystal orientation lying within a small solid angle dO about an angle (a, b, g) in an inverse pole figure (see below): Pða; b; gÞdO=4p

ð2:26Þ

where P(a, b, g) depends only on a, b, and g. Since every crystal must have some orientation: ZZ 1 Pða; b; gÞdO ¼ 1 ð2:27Þ 4p For randomly oriented materials P is independent of a, b, and g: ZZ P dO ¼ 1 4p

ð2:28Þ

so that P ¼ 1 for a random polycrystal. Values of P greater than one then indicate that the corresponding crystallographic direction has a higher probability than would be found in a random polycrystal, while values of P less than one indicate that this direction is less likely to be found than in a random polycrystal. P can be experimentally determined in a Bragg– Brentano diffractometer using: P 0 I ðhklÞ IðhklÞ P ð2:29Þ Pða; b; gÞ ¼ IðhklÞ I 0 ðhklÞ where I(hkl) is the measured integrated intensity from the plane (hkl) of the sample, and I0 (hkl) is the measured or calculated integrated intensity from the same plane in a randomly oriented standard of the same material. Values of P are plotted on an inverse pole figure. This is a stereographic projection whose coordinate system corresponds to the geometry of the sample and on which contours of relative peak intensity are plotted. The contours of constant P then define the degree of texture relative to the specimen geometry for the hkl reflecting plane used in the measurements. The measured integrated intensities not only indicate the degree of texture for any selected diffracting plane, they can also determine the amount of each phase present in a multiphase material. Quantitative phase analysis is important for determining the effect of different processing parameters on the phase content. To determine the amount of the phase

90

Microstructural Characterization of Materials

a in a mixture of a and b, we first redefine the integrated intensity in Equation 2.19, isolating the constants and focusing on the significant variables: Ia ¼

K1 ca mm

ð2:30Þ

where ca is the concentration of phase a, mm is the linear absorption coefficient for the mixture of phases, and K1 is a constant. The linear absorption coefficient will depend on the relative amount of each phase present in the mixture: !   mb mm ma þ ob ¼ oa ð2:31Þ rm ra rb where o is the weight-fraction of each phase and r is its density. Rearranging Equation (2.31) and combining with Equation (2.30) yields: Ia ¼

K1 ca ca ðma mb Þ þ mb

ð2:32Þ

Comparing Ia from the phase mixture with Ia from a pure sample of the a phase, we obtain:   ma o a r Ia a  ¼  ð2:33Þ Ia;p oa ma  mb þ mb ra

rb

rb

If we know the mass absorption coefficients for each phase, Equation (2.33) is sufficient for quantitative phase analysis, but if the mass absorption coefficients are unknown then we need to prepare a set of standard specimens (usually mixed powders) and construct a calibration curve. In any case, experimental calibration is usually a good idea, since the variation of Equation (2.33) with oa is nonlinear, as a result of the very different absorption coefficients of the different phases. Many other X-ray diffraction techniques are beyond the scope of this book. These include the X-ray determination of particle size, X-ray residual stress analysis, structure refinement by spectrum fitting (Rietveld analysis) and thin-film X-ray techniques (especially important for semiconductor device technology). References for some of these methods can be found in the Bibliography for this chapter.

2.5

Electron Diffraction

The dual wave-particle nature of electrons is expressed by the de Broglie relationship for the momentum of an electron, p ¼ mv ¼ h/l, where m is the mass and v the velocity of the electron, and l is the electron wavelength. Substituting for the electron energy, in electronvolts, eV ¼ 0.5 mv2, yields an equation for the nonrelativistic electron wavelength: h l ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2meV

ð2:34Þ

At an accelerating voltage of 100 keV, the electron wavelength is 0.0037 nm, much less than the interplanar spacing in crystals, so that the Bragg angles for electron diffraction are

Diffraction Analysis of Crystal Structure 91

always very small when compared with those for X-ray diffraction. That is, the elastic scattering of electrons occurs at very small angles. Electron diffraction is, as we shall see, both a major source of contrast in thin film electron microscopy, and an important analytical tool in its own right. 2.5.1

Wave Properties of Electrons

The de Broglie relationship is not sufficient to define the wavelength of an electron at high energies, since relativistic effects become important at these energies. If the rest mass of the electron is m0, then the relativistic mass is given by m ¼ m0 þ eV=c2, where c is the velocity of electromagnetic radiation (the velocity of light). Substituting in the de Broglie relation and rearranging leads us to the relativistic equation for the wavelength of the electron l in terms of the accelerating voltage V (compare the nonrelativistic equation, above): l¼

h h ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2m0 eVð1 þ eV=m0 c2 Þ

ð2:35Þ

The relativistic correction is significant at the accelerating voltages used in transmission electron microscopy, typically 100–400 keV. An electron beam can be focused with the help of electromagnetic lenses, but the focusing mechanism is quite different from that used in the optical microscope which, as we shall see in Chapter 3, relies on the refractive index of glass and geometrical optics to achieve a sharp focus. A magnetic field will deflect any electron which has a component of velocity perpendicular to the magnetic field vector, and the deflecting force acts in a direction that is perpendicular to the plane containing both the velocity and the magnetic field vectors. As a consequence, the electron follows a helical path when passing through a uniform magnetic field. The magnetic field generated by an electromagnetic lens is cylindrically symmetric, and a divergent electron beam passing through such a lens will be brought to a focus, providing the angular divergence is small. We are fortunate that the elastic scattering of electrons is limited to small angles and permits both electron diffraction patterns and electron microscope images to be brought sharply into focus in the electron microscope. Assuming a wavelength of 0.0037 nm (the wavelength associated with 100 keV electrons) and interplanar spacings of the order of 0.2 nm, we expect Bragg scattering angles of less than 1 . At these small angles it is common to quote the Bragg angle in radians and to use the approximation siny
y. The Bragg relationship can then be written l ¼ 2dy, and we will use this form of Bragg’s equation to describe the elastic scattering of electrons passing through a thin-film specimen in the transmission electron microscope. Finally, the influence of inelastic scattering is often important, since electrons which have lost some energy by inelastic scattering will have a longer wavelength. In such a case, the electron beam will no longer be monochromatic, and cannot therefore be brought to a sharp focus. The stopping cross-section for electrons s is defined as s ¼ ð1=NÞðdE=dxÞ, where N is the number of atoms per unit volume and dE/dx is the rate of energy loss per unit distance travelled by the electrons. As the energy of the electrons is increased (a higher accelerating voltage in the microscope), s decreases, although at very low electron energies the innershell electrons will no longer contribute to inelastic scattering of the incident beam

Microstructural Characterization of Materials I/N (dE/dX) Inelastic scattering cross-section (a.u.)

92

6 5 4 3 2 1 0 103

104

105

Incident electron beam energy (eV)

Figure 2.31

Inelastic scattering cross-section as a function of incident electron beam energy.

and the scattering cross-section will actually decrease. The general shape of the scattering cross-section curve is shown in Figure 2.31. The inelastic scattering cross-section increases with Z, so that higher atomic number materials can only give sharp electron diffraction patterns if the sample is very thin. For a 200 keV incident electron beam, tungsten or gold films thicker than about 100 nm will absorb the incident beam energy and can only give electron diffraction patterns in reflection, when the electron beam is at glancing incidence. The maximum thickness for transmission electron diffraction from steel at 200 kV is of the order of 120 nm, while silicon and aluminium specimens must be less than about 150 nm in thickness. These values are two orders of magnitude less than the maximum sample thickness for a transmission X-ray diffraction experiment. 2.5.1.1 The Limiting Sphere for an Electron Beam. The limiting and reflecting sphere constructions that we have used to analyse X-ray diffraction phenomena in reciprocal space are equally valid for electron diffraction patterns obtained in transmission electron microscopy (TEM), although there are two significant differences. The first concerns the very short wavelength of the electrons when compared with the interplanar spacings in crystals, that is |k0| ¼ |k||g| (Figure 2.32). As a consequence, as noted above, siny
y. In Figure 2.32 the reflecting sphere construction has been rotated by 90 , a symbolic nod to the engineering design of the modern electron microscope, in which the beam source (the electron gun) is almost invariably mounted vertically, generating a beam of electrons which penetrates a thin-film specimen mounted in the horizontal plane. This may be compared with the standard design for X-ray diffraction units, in which the X-ray beam is usually generated in the horizontal plane in order to ensure maximum mechanical stability. The second modification concerns the effective size of the reciprocal lattice points in electron diffraction. The elastic scattering cross-section for electrons is much greater than that for X-rays, so that the intensity scattered into the diffracted beam increases rapidly with sample thickness, to the point at which all the energy in the incident beam may be transferred into the diffracted beam and the diffracted beam starts to be rediffracted back into the incident beam (Figure 2.33). As we shall see, this process, termed double

Diffraction Analysis of Crystal Structure 93

Figure 2.32 The wave vectors in electron diffraction k are very large when compared with the reciprocal lattice vectors g, allowing for some simple geometrical approximations.

Figure 2.33 Double diffraction of the electron beam leads to diffracted intensities which oscillate with film thickness.

94

Microstructural Characterization of Materials

diffraction, leads to oscillations in the diffracted intensity with increasing specimen thickness. These oscillations have a periodicity t0, the extinction thickness, which is characteristic of the electron energy and the structure factor for the actively diffracting planes: t0 ¼

pVc ljFðhklÞj

ð2:36Þ

where Vc is the volume of the lattice unit cell. Typical values for t0 at the accelerating voltages used in TEM are less than 100 nm. In electron diffraction, this extinction thickness limits the effective size of the reciprocal lattice points in reciprocal space to approximately 1/t0, of the order of 102d (compared with values of 104d or less in X-ray diffraction). An additional ‘small crystallite’ effect may also dominate the Bragg condition in electron diffraction, increasing the size of the reciprocal lattice points even further in very thin films and for very small nanocrystals (see Chapter 4). In electron diffraction both the radius of the reflecting sphere and the size of the reciprocal lattice points are large when compared with the conditions obtained in X-ray diffraction, relaxing the diffracting condition set by Bragg’s law so that several diffracted beams of electrons may be scattered simultaneously from a thin sample that has a symmetry axis in the plane of the film (Figure 2.34). Such multi-beam diffraction would not be possible in an X-ray diffraction experiment. 2.5.2

Ring Patterns, Spot Patterns and Laue Zones

The electron diffraction pattern obtained in transmission from a thin film of a single crystal which is oriented with a major zone axis (a symmetry axis) parallel to the electron beam will contain all the reciprocal lattice points which are intersected by the reflecting sphere. This

Figure 2.34 Since the radius of curvature of the reflecting sphere is large and the reciprocal lattice points have a finite diameter, Bragg diffraction occurs even though the Bragg condition is not exactly satisfied.

Diffraction Analysis of Crystal Structure 95

Figure 2.35 Single crystal electron diffraction from more than one Laue zone: (a) mechanism of formation; (b) a diffraction pattern from a [100] oriented aluminium single crystal film.

will include all those reciprocal lattice points which surround the 000 spot (which corresponds to the directly transmitted beam), in so far as the (slight) curvature of the reflecting sphere permits them to fulfill the Bragg condition, but the pattern may also include additional points which lie in a layer of the reciprocal lattice above that containing the origin at 000 (Figure 2.35). The innermost set of diffracting reciprocal lattice points is referred to as the zero-order Laue zone, while the subsequent, outer rings of diffracting spots are termed higher order zones. Adequate calibration of the electron microscope enables distances in the diffractogram (the electron diffraction pattern) to be accurately interpreted as distances (angles) in reciprocal space, although there are several sources of calibration error. These include not only the physical limitations of the technique, especially ambiguity associated with the finite diameter of the reciprocal lattice points, but also experimental limitations associated with electromagnetic lens aberrations in the microscope, some inevitable curvature of the thin-film specimen, and the response of the recording medium. In general, it is not possible to specify lattice spacings derived from electron diffraction measurements to much better than 2 % of the lattice parameter. This is at least two orders of magnitude worse than can

96

Microstructural Characterization of Materials

be achieved by X-ray diffraction. While this is true for standard electron diffraction techniques, usually termed selected area diffraction, an alternative technique, convergent beam electron diffraction, can, for some crystals, be very much more accurate and may even be used to determine localized lattice strains, as well as to solve for the crystal structure. Convergent beam electron diffraction is, however, well beyond the scope of this book and the interested reader should consult the texts by Williams and Carter, or Spence and Zuo, that are listed in the Bibliography for this chapter. The electron beam can also be focused electromagnetically to a fine probe, or small apertures can be used to limit the diameter of a parallel electron beam, enabling selected area diffraction from very small specimen areas, of the order of 20 nm. This volume is a minute fraction of that which can be usefully sampled by X-rays, and can yield phase information on individual crystallites and precipitates in a polycrystalline material. If the area illuminated by the electron beam includes a large number of crystallites, then a powder pattern is generated, analogous to an X-ray powder pattern. In the electron diffraction case the fluorescent screen or recording medium is positioned normal to the incident beam, and records successive rings of reflections from each family of reflecting planes (Figure 2.36). The radius R of a specific ring on the powder pattern is related to the dspacing of the reflection and the wavelength of the electron beam l by the relation: d¼

2lL 2R

ð2:37Þ

where L is the effective camera length of the electron microscope when used as a diffraction camera. Equation (2.37) has been written with a factor of 2 in both the numerator and the denominator, since it is good practice to measure the distances 2R between two diffraction spots hkl and hkl in an electron diffraction pattern, in order to avoid errors associated with determining the position of the directly transmitted 000 beam. The parameter L can be varied in most microscopes, in order to select a value suitable for the lattice parameters of the phases being studied, and good calibration of the microscope should give the term lL, the camera constant of the microscope, with an accuracy of about 1%. 2.5.3

Kikuchi Patterns and Their Interpretation

For moderately thick transmission electron microscope specimens, a proportion of the incident electrons will undergo inelastic scattering. These electrons, having lost some energy, are deflected out of the path of the incident beam to form a diffuse halo around the central spot, but before exiting the specimen the same electrons may also be elastically scattered from the crystal lattice planes. If the specimen is a sufficiently perfect single crystal, this secondary elastic scattering will lead to a characteristic Kikuchi line pattern, which is usually superimposed on the single crystal spot pattern associated with Bragg diffraction of the primary incident beam. The Kikuchi line pattern arises because the diffuse angular distribution of inelastically scattered electrons falls off rapidly with angle, typically obeying an I ¼ I0 cos2 a law, where a is now the inelastic scattering angle. The crystal lattice planes will elastically scatter those diffuse electrons which are incident at the exact Bragg

Diffraction Analysis of Crystal Structure 97

Figure 2.36 Powder patterns in electron diffraction: (a) mechanism of generation; (b) a powder ring pattern from tempered carbon steel (indices in red are from a-Fe and indices in black are from Fe3C).

98

Microstructural Characterization of Materials

angle. Since very little energy is lost in the initial inelastic scattering event, the Bragg angle is effectively unchanged from that for the primary beam. However, more electrons will be elastically scattered away from the distribution of diffusely scattered electrons in the region closer to the incident beam, leading to a dark line in the diffuse scattering pattern close to the centre of the observed diffraction pattern and simultaneously generating a parallel white line at a fixed distance from the dark line, which is determined by the interplanar spacing of the diffracting planes (Figure 2.37). These pairs of dark and light Kikuchi lines are actually sections of hyperbolae, but since the g vectors are so much smaller in magnitude than the electron wave vectors k, they appear on the diffraction pattern as straight lines. From their geometry, the spacing of a dark/light pair of Kikuchi lines projected on the plane of observation is proportional to the value of 2y. Since the Bragg angles in electron diffraction are very small, the line bisecting each pair of Kikuchi lines is an accurate trace of the diffracting planes projected onto the plane of observation. Any displacement of this trace from the centre of the pattern is therefore an accurate measure of the angle which the reflecting lattice planes make with the primary incident beam. The Kikuchi pattern offers a means of calibrating crystal misorientations in the electron microscope extremely accurately. Because of the comparatively large size of the reciprocal lattice points in electron diffraction, spot patterns can only be used to determine crystal orientation to within a few degrees. Note that the Kikuchi pattern is formed by diffraction of the diffusely scattered electrons within the comparatively thick slice of the sample, while the spot pattern is formed by diffraction of the very much more intense beam of parallel primary electrons from those planes whose reciprocal lattice points intersect the reflecting sphere on the zero order or higher order Laue zones. If the dark Kikuchi line passes through the centre 000 spot of the primary beam, then the Bragg condition for that reflecting plane is accurately fulfilled, and the bright Kikuchi line must pass through the centre of the diffraction spot corresponding to that plane. The perpendicular distance from any diffracting spot to the corresponding dark Kikuchi line is therefore an accurate measure of the deviation of the primary beam from the exact Bragg condition for that spot. When the Kikuchi pattern is symmetrically aligned with respect to the incident beam (Figure 2.38), then the incident beam is accurately parallel to a symmetry zone of the crystal, and this zone can usually be readily identified from the Kikuchi pattern. A series of Kikuchi patterns, taken by tilting the specimen about two axes at right angles in the plane of the specimen, can be used to generate a Kikuchi map (Figure 2.39). Since the Kikuchi map accurately reflects the crystal symmetry, it can be used to identify the orientations of any specific grain in the electron microscope, almost by inspection. Kikuchi patterns are often used to align a crystal exactly on a zone axis, or to shift a crystal off a zone axis by any given angle. With the help of a Kikuchi map, you can tilt the crystal in a controlled way from one zoneaxis to another.

Summary The regular arrays of atoms in a crystal scatter short-wavelength radiation elastically (either X-rays, or electrons, or neutrons), at well-defined angles to the incident beam. The

Diffraction Analysis of Crystal Structure 99

Angular distribution of diffuse scattered intensity

Bragg diffracting plane

θ θ

θ

−θ

Intensity is Bragg scattered to form an excess line at an angle- θ from the projection of the diffracting plane and leave a deficit line at θ

Figure 2.37

Mechanism of formation of Kikuchi line diffraction patterns (see text).

100

Microstructural Characterization of Materials

(242)

(242) (224)

224

202 220

022 (202)

242

(202)

Figure 2.38 Kikuchi diffraction pattern from a [111] oriented aluminium crystal superimposed on the single crystal spot pattern.

scattering angle and the scattered intensity are functions of the radiation, the wavelength and the crystal structure, and this process is termed diffraction. A diffraction pattern is the angular distribution of the scattered intensity in space, and is recorded by collecting the scattered radiation, for example on a photographic emulsion or charge collection device. A diffraction spectrum is the intensity of the diffracted radiation collected as a function of the scattering angle. Laue showed that the allowed angles of scattering for a single crystal were a simple function of the lattice parameters of the crystal lattice unit cell, and Bragg simplified the scattering relations to yield Bragg’s law, l ¼ 2d siny, relating the angle of scattering 2y to the direction of the incident beam, the spacing d between the planes of atoms in the crystal lattice and the wave length l. of the incident radiation The allowed angles of scattering (the Bragg diffraction angles) always correspond to integer values of the Miller indices, but not all possible combinations of the Miller indices give rise to diffraction peaks, and some reflections (termed forbidden reflections) are disallowed, for example, if the Bravais lattice contains face-centred or body-centred lattice points. The intensities of the diffraction peaks are determined by the atomic number and position of the atoms associated with each lattice point, that is the space group, the Wyckoff positions of the atoms and the atomic species. Thus some ‘allowed’ peaks may be enhanced, if the atoms corresponding to each lattice point are scattering in-phase, while others may be reduced, or even absent if the scattering is out of phase and the interference is destructive.

Diffraction Analysis of Crystal Structure 101

Figure 2.39 identified.

Kikuchi map of a cubic silicon crystal with the principle reflecting planes

A convenient representation of the angular positions of the allowed diffraction peaks can be given in reciprocal space. The diffracting planes are then represented by points (reciprocal lattice points) whose positions are determined by the reciprocal of the interplanar spacing of the diffracting planes and the direction vector normal to these planes. In reciprocal space the Bragg equation defines a sphere (the reflecting sphere) and any reciprocal lattice point which can be made to intersect the surface of this sphere will give rise to a diffracted beam. Reducing the wavelength of the incident radiation increases the radius of the reflecting sphere, and will allow more reciprocal lattice points that are further from the origin in reciprocal space (corresponding to smaller interplanar spacings), to diffract when the Bragg condition is fulfilled. Rotating the crystal or reducing the grain size in a polycrystal increases the probability that a reciprocal lattice point will intersect the sphere and give rise to diffraction. The volume of the specimen sampled by the incident beam depends on the radiation used. In neutron diffraction, elastic scattering generally occurs over distances of the order of centimetres before energy losses due to inelastic scattering events become significant. However, even very high energy (MeV) electrons will be inelastically scattered if the sample thickness exceeds 1 or 2 mm. X-rays are an intermediate case. Most engineering materials irradiated with X-rays whose wavelengths are of atomic dimensions will generate an elastically scattered signal from a region 20–100 mm in depth. An X-ray diffractometer consists of a source of X-rays, a sample goniometer (which positions the sample accurately in space), a detector and a data recording system. The detector can be rotated about the sample to select the different diffraction angles. For

102

Microstructural Characterization of Materials

some applications, a photographic emulsion or a position-sensitive detector array may replace the single detector. White X-radiation may also be used, for example to record a diffraction pattern from a single crystal in a Laue camera. Many X-ray diffraction studies are based on the irradiation of a polycrystalline sample which diffracts monochromatic radiation to give a powder diffraction spectrum. In diffraction from a powder specimen, fine grains of any crystalline phase will diffract at the different Bragg angles in a sequence of diffraction cones which are intercepted in turn by the rotating detector. If the crystal grains are not randomly oriented in space, but possess some preferred orientation (crystalline texture), then the diffraction pattern will show intensity anomalies which can be analysed in a texture goniometer, to derive a pole figure that plots the distribution of the intensity diffracted by a particular family of crystal planes with respect to the sample coordinates, or a crystallite orientation distribution function, that plots the probability of finding a crystal having a particular orientation as a function of the Euler angles which describe the grain orientation with respect to the sample coordinates. In many cases, an accurate determination of the diffraction angles recorded in the diffraction pattern is sufficient to deduce the phases which are present and the orientation distribution of the individual crystals. More information can be derived from a measurement of the relative intensities of the diffraction peaks. Each unit cell in a crystal scatters a proportion of the incident beam into each diffraction peak associated with a specific hkl reflection. The structure factor gives the relative scattering power of the different hkl planes in the crystal, and can be calculated from a suitable model of the crystal structure to predict the relative peak intensities diffracted from a random polycrystal. The structure factor is only one of the parameters that determine the relative diffracted intensities, and diffraction analysis must also take into account several other effects: the Lorentz polarization factor, the multiplicity of the reflecting planes, the specimen geometry, the angular dependence of X-ray absorption losses and, at elevated temperatures, the effect of thermal vibration on the diffracted intensities. Electron diffraction differs from X-ray diffraction in many significant respects. First, the electron wavelengths that are of practical importance in an electron microscope are very small when compared with the interplanar spacings in crystals. Also, in electron diffraction the effects of inelastic scattering are pronounced and can lead to the formation of Kikuchi patterns. Useful transmission electron microscope sample thicknesses are always submicrometre. The short wavelengths in the high energy electron beam increases the diameter of the reflection sphere, which becomes very large compared with the spacing of the reciprocal lattice points, while the small volume of the sample region responsible for electron diffraction significantly relaxes the conditions for Bragg diffraction. This latter effect broadens the diffraction peak width and, equivalently, smears the reciprocal lattice points over a region of reciprocal space that is no longer small in comparison with the spacing of the reciprocal lattice points. Despite the comparative diffuseness of the reflecting sphere conditions for electron diffraction, it is still possible to identify accurately the orientation of a sample and to align any given crystal in the electron microscope column by using a Kikuchi line pattern. These patterns are formed by diffraction from the halo of electrons that forms around the primary electron beam in a thick sample as a result of weak inelastic scattering.

Diffraction Analysis of Crystal Structure 103

Bibliography 1. 2. 3. 4.

C. Barrett, and T.B. Massalski, (1980) Structure of Metals, Pergamon Press, Oxford. B.D. Cullity, (1956) Elements of X-Ray Diffraction, Addison-Wesley, London. J.B. Cohen, (1966) Diffraction Methods in Materials Science, Macmillan, New York. I.D. Noyan, and J.B. Cohen, (1987) Residual Stress: Measurement by Diffraction and Interpretation, Springer-Verlag, London. 5. D.B. Willions, and C.B. Carter, (1996) Transmission Electron Microscopy: A Textbook for materials Science, Plenum Press, London. 6. J.C.H. Spence, and J.M. Zuo, (1992) Electron Microdiffraction, Plenum Press, London.

Worked Examples Let us now use the techniques we have discussed in this chapter for the characterization of our selected materials. We start with a simple example: An automated Bragg–Brentano diffractometer is used to verify the crystal structure of a metal powder and measure its lattice parameters. Armed with a good powder diffraction spectrum from a sample which has been accurately mounted in the diffractometer, we use a literature database, the Joint Committee of Powder Diffraction Standards (JCPDS, now called the International Centre for Diffraction Data). JCPDS is a database of experimentally observed and calculated diffraction spectra, and lists both d-spacings and relative intensities. These diffraction data can be compared with our measured spectrum in order to identify the phases present in our sample. The JCPDS data come in two formats: 1. Tabulated cards for the different spectra, which can be accessed either from the common names or chemistry of the compounds, or by the d-spacings of the strongest observed reflections. 2. A computerized database which can be accessed using a computer program which automatically compares the major d-spacings derived from a spectrum with those listed in the database. While Tabulated cards for the different spectra still exist in many university libraries, most laboratories now use the computerized database. The X-ray diffraction spectrum of the powder we wish to identify is shown in Figure 2.40. We first generate a table of the d-spacings calculated from the prominent reflections by using Bragg’s law and the known wavelength for CuKa radiation (l ¼ 0.1540598 nm). We input these d-spacings into the computerized JCPDS database, and the output identifies for us the possible phases which best match the experimentally observed d-spacings and their relative intensities. Now we extract the data for each of the selected options from the JCPDS database to compare the standard with the measured values of d-spacing and intensity (since this is a powder sample, texture should not be a problem). The ‘unknown’ powder is nickel (Ni), whose JCPDS card is shown in Figure 2.41. Of course our sample could have contained several different phases, but in such cases additional information is usually available, either as prior knowledge of the phases that might exist, or from the chemical components expected to be present in the material.

104

Microstructural Characterization of Materials

Figure 2.40

X-ray powder diffraction pattern from an ‘unknown’ sample.

From the same powder diffraction spectrum of Ni powder, we can also determine the exact lattice parameter. Any variations in lattice parameter would indicate either the presence of residual stress (very unlikely in a powder specimen), or a nickel-based alloy that contains one or more alloy additions in solid solution. By careful calibration of the spectrometer, and assuming a linear dependence of the lattice parameter on composition

4-0850 MINOR CORRECTION d 4-0850

2.03

1.76

1.25

2.034

Ni

I/IS 4-0850

100

42

21

100

Nickel

Rad. CuKα λ 1.5405 Filter Ni Cut off Coll. Dia. I/It G.C.Diffractometer d corr.abs? Ref. Swanson and Taige,JC FEL. Reports, NBS Sys. CUBIC a0 3.5238 b0 α β Ref. IBID. ξα 2V Ref.

c0 γ

nαβ D,8.907 mp

dA 2.034 1.762 1.246 1951 1.0624 1.0172 5 S.G.O M - FM3M 0.8810 A C 0.8084 Z 4 0.7880

ξγ Color

I/I1

hkl

100 42 21 20 7 4 14 15

111 200 220 311 222 400 331 420

dA

I/I1

hkl

Sign

SPECTROGRAPHIC ANALYSIS SHOWS <0.01% EACH OF Mg, Si, AND Ca. AT 26C TO REPLACE 1-1258, 1-1260, 1-1266, 1-1272, 3-1043, 3-1051

Figure 2.41 The JCPDS card for nickel. Reproduced by permission of the International Centre for Diffraction Data.

Diffraction Analysis of Crystal Structure 105

Lattice parameter (nm)

0.355

0.354

Extrapolated lattice parameter: a=0.3522 (6)nm

0.353

0.352

0.351

0.350 -1

-0.5

0

0.5

1

1.5

2

cos(2θ)/sin(θ)

Figure 2.42 The lattice parameter of nickel determined from d-spacings taken from Figure 2.40, showing the method of extrapolating the data to 2y ¼ 180  to minimize errors.

(Vegard’s law), we could also determine the concentration of the alloy from an exact determination of the lattice parameter. Figure 2.42 shows the results for our sample, in which the apparent lattice parameter for nickel, determined from the individual reflections and using the known relationship between hkl and the lattice parameter for a cubic structure, has been plotted as a function of cos 2y/ siny. The systematic errors are quite small, and the best value for the measured lattice parameter is obtained by extrapolating to y ¼ p/2: a ¼ 0.3522(6) nm. This is close to the value listed both by the JCPDS (a ¼ 0.35238 nm) and in Pearson’s Handbook of Crystallographic Data for Intermetallic Phases (a ¼ 0.35232 nm). To confirm that a solid solution is present we would need to prepare a calibration curve for different alloy concentrations and use analytical techniques to determine the chemistry of the samples. The same approach can be used to determine the lattice parameters of a polycrystalline sample of a-Al2O3 (Figure 2.43). For alumina the situation is more complicated, since we have to refine two lattice parameters for the hexagonal unit cell that commonly defines the structure. This can be done using a simple computer program. We could also determine a correction factor to account for any systematic errors in our experimental results by using a standard sample, and then adjusting the measured lattice parameters for the phases of interest. Figure 2.44 shows a diffraction spectrum from a two-phase mixture of a-Fe and Fe3C in 1040 steel. The very weak reflections from the carbide reflect the low volume fraction of carbide in the steel. Some of the carbide reflections also overlap with those of iron and this further complicates the analysis of the spectrum. Careful inspection of the diffraction spectrum, and a comparison with simulated diffraction patterns, ensures that we identify each of the reflections correctly. Figure 2.45 is a diffraction spectrum from a completely different type of sample: a thin polycrystalline film of aluminium deposited on a thin film of TiN formed on an even thinner film of titanium. These films were deposited sequentially on a single crystal, silicon substrate. In order to detect such thin films in a Bragg–Brentano diffractometer, long counting times are required for each value of 2y, of the order of 20 s in the case of the pattern

106

Microstructural Characterization of Materials

Figure 2.43

X-ray powder diffraction pattern from a-alumina

shown in Figure 2.45. Specialized thin film diffractometers significantly reduce the counting time required for such specimens, but here we only consider the Bragg–Brentano diffractometer geometry. The reflections in Figure. 2.45 have all been indexed. The high intensity peak from silicon occurs because the silicon single crystal has been oriented in the diffractometer to diffract from the {400} planes. Even a slight misalignment of the silicon crystal would remove this reflection from the spectrum. Consider only the reflections due to the deposited thin films: just a few reflections are detectable, reflections that, according to the JCPDS, are not necessarily from the strongest reflecting planes. It follows that the deposited films must have a preferred orientation with respect to the plane of the silicon substrate crystal. Qualitatively, the titanium film has a

2000

α-Fe

Intensity (a.u.)

(011)

1500

1000

Fe3C (031)

500

α-Fe

α-Fe

0 30

40

50

60

α-Fe

(121)

(002)

(022)

70

80

90

100

2θ Figure 2.44

X-ray powder diffraction pattern from 1040 steel.

Diffraction Analysis of Crystal Structure 107

Figure 2.45 X-ray powder diffraction pattern from a thin polycrystalline film of Al/Ti/TiN deposited on a single crystal of Si: (a) has an expanded angular scale to show the details of the spectrum around the {400} Si substrate peak; (b) has an expanded intensity scale that shows the details of the reflections from the multilayer coating.

texture in which the [0001] direction in the titanium lattice is normal to the silicon substrate surface, while the TiN and aluminium films both have a texture in which a [111] direction is normal to this surface. We next consider a calibration curve prepared for quantitative phase analysis of a sample containing a phase mixture of alumina and nickel (Figure 2.46). The calibration curve was prepared by mixing known amounts of the two phases in a powder form. The strong deviation from linearity is due to differential X-ray absorption and confirms the need for such a calibration curve.

108

Microstructural Characterization of Materials

Figure 2.46 A calibration curve for quantitative analysis of a two phase mixture of nickel and alumina, showing the ratio of the integrated peak intensities summed for nickel and alumina, as a function of the known nickel content

Now let us move on to electron diffraction. As stated earlier, selected area electron diffraction does not have the precision which is available in X-ray diffraction, but we can easily obtain diffraction patterns from single grains in a polycrystalline sample, or from selected regions within any single grain. In Chapter 4, we will use selected area diffraction to correlate the crystallographic orientation of a grain with the contrast due to lattice defects observed in an electron micrograph. We need to solve a selected area diffraction pattern in order to identify the reflections responsible for contrast in the image, as well as to determine the zone axis parallel to the incident electron beam and the principle directions in the plane of the thin film sample. Our first example is a selected area electron diffraction pattern from a randomly oriented aluminium polycrystal (Figure 2.47). The ring pattern results from the intersection of the diffraction cones from the reciprocal lattices summed over a large number of fine, randomly oriented grains in the polycrystalline sample with the reflection sphere. The d-spacing for each ring of this diffraction pattern is determined by measuring the diameter of the ring and inserting the known value of lL (the microscope camera constant) in Equation (2.37). For Figure 2.47, lL was determined and the corresponding d-spacing of each ring is given. We can either use the information for aluminium from the JCPDS file to insert the hkl values responsible for each ring, or we can calculate a list of the d-spacings for every possible hkl in aluminium by using the relationship between the d-spacing and the lattice parameter for a cubic crystal. This calculation will not tell us if the reflection has a non-zero structure factor, but calculation of the structure factor for each reflection is also quite simple, especially if we have an appropriate computer program. The first step in solving any selected area diffraction pattern is to calibrate the values of lL for the different camera length settings of the microscope. Although most transmission electron microscope monitors display a value of the camera length L, this value is only

Diffraction Analysis of Crystal Structure 109

hkl

d (nm)

111

0.233

002

0.202

220

0.143

311

0.122

222

0.116

004

0.101

313

0.092

042

0.090

422

0.082

0.08 0.09 0.093 0.01 0.11 0.12 0.14 0.20 0.23

Figure 2.47 Selected area electron diffraction pattern from a polycrystalline aluminium specimen. Since the average aluminium grain size is much smaller than the selected area, a ‘ring’ pattern is formed. The measured d-spacings of the rings are indicated, and a table of dspacings for different planes in aluminium is given.

approximate, and calibration is always necessary using a standard specimen of known lattice parameter. We use aluminium (which is actually not a very good choice, since the lattice parameter of aluminium is sensitive to dissolved impurities). A single crystal region of an aluminium foil is oriented perpendicular to a low-index zone axis and a series of diffraction patterns is recorded for various values of the camera length L at an accelerating voltage of 200 kV (Figure 2.48). Can we index an electron diffraction pattern without knowing either lL or the zone axis of the crystal? Actually, for the case of aluminium there is a rather simple solution to this problem. Aluminium has a FCC structure, for which: 1 h2 þ k 2 þ l 2 ¼ 2 a2 d

ð2:38Þ

110

Microstructural Characterization of Materials

R1

R2

L = 100 cm

L = 80 cm

L = 60 cm

L = 40 cm

L = 20 cm

Figure 2.48 Selected area electron diffraction patterns of a single crystal of aluminium, recorded at different nominal camera lengths (L).

Putting (h2 þ k2 þ l2) ¼ N2, and using the ratio of the d-spacings for any two reflections we can write: d22 N12 ¼ d12 N22

ð2:39Þ

Diffraction Analysis of Crystal Structure 111

Figure 2.49 Calibration curve for the true value of lL as a function of the nominal camera length indicated on the microscope monitor.

To relate this equation between d and N to the values of R, the distance from the centre spot of the pattern to the reflections of interest, we substitute d ¼ lL/R into Equation (2.39): d22 N12 R21 ¼ ¼ d12 N22 R22

ð2:40Þ

This would be difficult to solve without a computer, but we can use the data from Table 2.1. We measure the distance from the central spot to each of two diffraction spots of interest (Figure 2.48) and find that R21 =R22 ¼ 0:5. Simple examination of Table 2.1 immediately identifies the reflecting planes as (200) and (220). Since the lattice parameter for pure Al is known (a ¼ 0.405 nm), the relevant d-spacings can be calculated: d200 ¼ 0.202 nm and d220 ¼ 0.143 nm, and we can now return to our original relation between d and l, d ¼ 2lL/ 2R, to calculate lL. The same procedure is possible for any diffraction pattern taken from aluminium at any arbitrary camera length, giving an accurate calibration of the camera constant lL for the microscope (Figure 2.49). In principle, standard statistical methods should be used to calculate the errors in lL, so its better to index the complete diffraction pattern. We now know that point 1 in Figure 2.48 corresponds to a 200 reflection, while point 2 corresponds to a 220 reflection, and the angle subtended by these two vectors at the origin (000), is measured to be 45 . We now take the vector cross product1 between the two directions [200] and [220], noting that the direction vectors are always perpendicular to the corresponding planes for the cubic systems. This gives the zone axis of the diffraction pattern, [001]. We check this by examining a stereographic projection for the cubic crystal structure, and note that the angle between (200) and (220) should be 45 , with both planes on a great circle whose zone axis is [001] (Figure 2.50). Now continue to move along this same great circle defined by [001] (the perimeter of the stereographic projection), taking angles between plane normals from the It is an accepted convention that the zone axis points up the microscope column, from the specimen to the electron source, that is, normal to the emulsion side of a negative recording film. It is important to follow this convention when relating crystallographic directions indexed from a diffraction pattern to specific features in electron micrographs (see Chapter 4).

1

112

Microstructural Characterization of Materials (010)

(110)

(110)

(011) (111)

(111)

(100)

(101)

(111)

(001)

(101)

(100)

(111) (011)

(110)

(110)

(010)

Figure 2.50 Stereographic projection for a cubic crystal in the cube orientation.

stereogram and correlating them to the measured angles between reflections on the diffraction pattern. Of course lL and 2R to should also be used to measure the dspacing for each reflection, and confirm that the indices we assign to each reflection are indeed correct. The result is a fully indexed selected area diffraction pattern (Figure 2.51). For the FCC structure this procedure is very straightforward, but things get more complicated for noncubic structures, for example the diffraction pattern shown in Figure 2.52 from alumina. Remember that directions are not usually perpendicular to planes in noncubic crystals, so to determine a zone axis (that is, a direction) from a set of planes, we need a stereographic projection which includes both planes and directions (find a computer program to generate your own stereographic projections). The equations listed in Appendix I can be used to determine a zone from any two indexed planes lying on a given zone, as well as the angle between these two planes, but the answer should be checked against a computer simulation. And, of course, you still have to solve your own patterns! As you may have noticed, the indices of all the diffraction spots in a single crystal pattern such as Figure 2.51 are related by a simple rule of vector addition or subtraction. This rule is valid for electron diffraction patterns corresponding to any lattice symmetry, since the lattice planes are directions in reciprocal space, and the diffraction pattern is just a twodimensional section through reciprocal space. This vector addition rule should always be used to confirm any solution to a single crystal spot pattern.

Diffraction Analysis of Crystal Structure 113

040

220

400

020

200

220

220

200

020

400

220

040 Figure 2.51 A fully indexed selected area diffraction pattern from aluminium for the same zone axis as the patterns shown in Figure 2.48.

1210 1120 3030

Figure 2.52 Partially indexed selected area diffraction pattern from a-alumina.

114

Microstructural Characterization of Materials

Finally, many electron microscopists use JCPDS data to help them solve their electron diffraction patterns. This can be misleading. Although the structure factor calculation is the same for both X-ray diffraction and electron diffraction, the atomic scattering factors for X-rays and electrons are very different, while the other coefficients in the calculation of total intensity also differ. Furthermore, the JCPDS always refers to a randomly oriented powder specimen, in which weakly scattering planes may remain undetected. Such a plane may actually diffract strongly in electron diffraction when its reciprocal lattice vector intersects the reflecting sphere.

Problems 2.1. The minimum lattice spacing which can be detected by diffraction of an incident beam is just half the wavelength of the incident radiation. Why? 2.2. When a first-order reflection is forbidden (for example, 110 in the FCC lattice), the second-order reflection (for example, 220 in FCC) is generally allowed. Why? 2.3. In a primitive cubic (PC) lattice the reflections 221 and 300 diffract at the same Bragg angle. Find another pair of reflections in this lattice that also diffract at the same Bragg angle. 2.4. What reciprocal lattice vector separates the lattice points 110 and
111 in reciprocal space? Write down the Miller indices of two other reflections that lie on the same zone. 2.5. In general, reciprocal lattice vectors are parallel to lattice directions having the same indices only in a cubic crystal. Nevertheless, this equivalence is found for some zones of high symmetry in other noncubic Bravais lattices. Give two examples. 2.6. Name three factors which may relax the exact diffraction condition, so that some diffracted intensity is measured for crystal orientations which deviate slightly from the Bragg condition. 2.7. Distinguish between white and characteristic X-rays and give one application for each type of radiation in X-ray diffraction. 2.8. Define the term mass absorption coefficient. Using literature data, estimate the thickness of an iron foil that will ensure 90 % transmission for Cu Ka radiation and Fe Ka radiation. How do you account for the large difference in the two calculated thicknesses, given the small difference in atomic number and the fact that the two wavelengths are quite close? 2.9. Estimate the minimum level of residual macrostress detectable by an X-ray line shift in steel (elastic modulus 220 GPa). Justify any assumptions you make. 2.10. Diamond has an FCC structure, but with additional forbidden reflections. Determine the first three additional forbidden reflections and explain their origin. 2.11. Should the measurement of lattice spacing be more accurate for thin films or for bulk specimens? (Hint: the answer is not quite as straightforward as it may appear.) Based on your answer, suggest a suitable specimen holder for the accurate measurement of lattice spacings in a powder.

Diffraction Analysis of Crystal Structure 115

2.12. What, if any, differences would you expect in the diffracted intensity from the same phase in a thin film as opposed to a bulk specimen (assume you are using a Bragg– Brentano diffractometer). 2.13. Index the diffraction pattern from Ni shown in Figure 2.40, using the JCPDS data from Figure 2.41. 2.14. Index the diffraction pattern from alumina shown in Figure 2.43. To help you index the pattern, write a computer program to calculate d-spacings for the sequence of planes (hkil). 2.15. A fully dense sample of PbTe (cubic, r ¼ 8.253 g cm3) is characterized by X-ray diffraction using Cu Ka in a Bragg–Brentano diffractometer. Calculate the mass absorption coefficient for PbTe (lCu Ka ¼ 0.1540598 nm). Calculate the depth of Xray penetration as a function of 2y. 2.16. For Mg (hexagonal): (a) Calculate the general structure factor. (b) Calculate the relative integrated intensities for the six strongest reflections listed in the JCPDS file. (c) Compare your calculated values with the values listed in the JCPDS file and explain any differences. 2.17. A cast sample consisting of one phase is characterized by X-ray diffraction in Bragg– Brentano geometry. The results are listed together with results from a powder sample of the same material in Table 2.2. Is there preferred orientation in the cast sample? 2.18. A mixture of Si and Mg powders is characterized by X-ray diffraction (l ¼ 0.1540598 nm). The results are summarized in Table 2.3. (a) Solve the diffraction pattern and identify each reflection. (b) Determine the relative amounts of each phase [(m/r)Si ¼ 60.3 cm2 g1; (m/r)Mg ¼ 40.6 cm2 g1]. 2.19. Calculate the energy of the X-ray photons for the following characteristic X-ray emissions: (a) Cr (l ¼ 0.2291 nm); (b) Co (l ¼ 0.1790 nm); (c) W (l ¼ 0.0209 nm). Table 2.2 X–ray diffraction data from a cast sample and a reference powder of the same material. Phase

h

k

l

Reference powder I (counts)

Cast sample I (counts)

a a a a a

1 2 1 0 3

1 2 3 4 1

1 0 1 0 1

2017 2339 6368 1255 2022

968 1736 4536 2380 880

116

Microstructural Characterization of Materials

Table 2.3 X–ray diffraction data from a mixture of Si and Mg powders. Phase

h

k

l

d-spacing (A˚)

2y 28.470 32.210 34.426 36.648 47.350 47.859 56.180 57.432 63.126 67.394 68.706 69.204 70.086 72.576 76.461 77.921

Unmixed powders IP (counts) 214863 2827 3140 11869 144157 1815 85800 2041 2244 301 2262 22982 1589 315 34785 378

Mixed powders I (counts) Quantity (%) 166747 633 703 2658 111875 406 66586 457 503 67 507 17835 356 71 26995 85

2.20. What is the minimum d-spacing that can be characterized using these wavelengths? Why? 2.21. Given a single crystal of Si (diamond structure with a ¼ 0.54329 nm): (a) At which 2y will (004) reflect using Cu Ka (l ¼ 0.1540598 nm)? (b) About which tilt axis and at what tilt angle must the sample be inclined to reflect from (111)? (c) At which value of 2y will the {111} planes reflect? 2.22. For GaAs and l ¼ 0.1540598 nm: (a) Calculate the positions of the atoms in the unit cell. (b) Under which conditions is the structure factor zero? (c) Are there reflections that would not appear if all the lattice sites were occupied by the same type of atom? (d) Calculate the structure factor for the first two reflections. (e) Compare these structure factors with those of the first two reflections for AlAs. (f) Is there a change in structure factor if the sites of the two types of atom in the lattice are exchanged? 2.23. A single crystal of silicon (diamond structure) is characterized by X-ray diffraction (l ¼ 0.1540598 nm) in a rotating spectrometer. (a) About which zone axis was the crystal rotated if reflections from both (004) and (111) were detected? (b) What was the rotation angle needed to acquire a reflection from (111) after first acquiring a reflection from (004)? (c) Give two additional allowed reflections that would be expected to appear in this experiment.

Diffraction Analysis of Crystal Structure 117

2.24. Cu3Au has a cubic structure. When fully ordered, the atoms are located in the following sites: Au at (0,0,0) and Cu at (1/2, 1/2, 0); (1/2, 0, 1/2); (0, 1/2, 1/2). Under certain conditions the structure becomes disordered and occupancy of these sites becomes random. Assuming the occupancy factor is given by the atomic concentration of the compound: (a) Describe the structure factor for the ordered and disordered states. (b) For which reflections will the structure factor remain unchanged when going through the order–disorder transition? 2.25. Cu has an FCC structure. Using the data from JCPDS: (a) Which are the three strongest Bragg reflections when using Cu Ka (l ¼ 0.1540598 nm)?   1 þ cos2 2
PjF 2 j, calculate the three strongest Bragg reflections and (b) Using I sin2
cos
 compare with the data from JCPDS.    1 þ cos2 ð2
Þ·cos2 2a PF 2 Að
Þ and (c) Repeat the above calculation using I a 2 2 sin
cos
ð1 þ cos 2aÞ assuming a graphite monochromator with a ¼ 26.4 . 2.26. Nickel and copper both have FCC structures. According to the equilibrium phase diagram, nickel and copper form a solid solution over the entire concentration range (that is, they are completely miscible), and the lattice parameter of the solid solution changes with concentration according to Figure 2.53. An X-ray diffraction pattern from an unknown Cu-Ni alloy, acquired using Zn Ka, is presented in Figure 2.54. (a) Solve the diffraction pattern given in Figure 2.54.

Figure 2.53 Lattice parameter as a function of Cu content in a Ni–Cu alloy.

118

Microstructural Characterization of Materials

Figure 2.54

X-ray diffraction pattern from an unknown Ni–Cu alloy.

(b) Determine the lattice parameter of the alloy by using the relation between the lattice parameter and (cos2y)/(siny). (c) Estimate the concentration of the alloy. (d) Calculate the mass absorption coefficient of the alloy for Zn Ka. (e) Determine the maximum penetration depth for each reflection in the diffraction pattern of Figure 2.54, assuming that the background level is 5%. Use the following data: l(Zn Ka) ¼ 0.1436 nm ANi ¼ 58.7 g mol1; ACu ¼ 63.5 g mol1 ralloy ¼ 8.925 g cm3 Mass absorption coefficients of Ni and Cu: (m/r)Ni ¼ 325 cm2 g1; (m/r)Cu ¼ 42 cm2 g1. 2.27. The X-ray diffraction pattern (acquired with l ¼ 0.1540598 nm) from a pure polycrystalline cubic material A is given in Figure 2.55. (a) Is the Bravais lattice of material A FCC or BCC? (b) Solve the diffraction pattern and index the reflections. (c) Determine the lattice parameter. (d) If an alloy of the same material (a solid solution of A containing B in Figure 2.56) is solidified by cooling from T1 to T2 fast enough to prevent diffusion, how would this affect the peak shapes of the reflections in X-ray diffraction? 2.28. A sample of polycrystalline Al (FCC) is coated with a 15 mm thick layer of Cu and characterized by X-ray diffraction using l ¼ 0.1540598 nm. Given that the lattice parameter of Al is 0.405 nm, the mass absorption coefficient for Cu is 52.7 cm2 g1,

Diffraction Analysis of Crystal Structure 119

Figure 2.55

X-ray diffraction spectrum from pure A.

and the background level in the X-ray diffraction pattern is 5 % of the maximum incident intensity, sketch an X-ray diffraction pattern for the sample. 2.29. Figure 2.57 is a selected area electron diffraction pattern of a-Fe. Find lL for the diffraction pattern, index the pattern and determine the zone axis. Mark both the zone axis and the great circle containing the normals to the diffracting planes on a standard stereographic projection for a cubic crystal. Is the zone axis of this pattern normal to this great circle? 2.30. Figure 2.58 is a selected area electron diffraction pattern from a polycrystalline region, taken from a cast metal block. There is concern that silicon may have been

T1

L

L+α

L+ β

T2 α

100% A

β

100% B

Figure 2.56 Schematic binary phase diagram of A and B.

120

Microstructural Characterization of Materials

Figure 2.57

Figure 2.58

Selected area diffraction pattern from a crystal of a-Fe.

Selected area diffraction pattern from a polycrystalline region of a cast steel block.

Diffraction Analysis of Crystal Structure 121

Figure 2.59 and Fe.

A selected area diffraction pattern from a polycrystalline thin film composed of Al

introduced into the casting as an impurity. Solve the pattern and index the rings. Is there any evidence for the presence of crystalline silicon? 2.31. Given a FCC crystal, calculate for the listed pairs of planes: (a) the zone axis, (b) an additional allowed reflecting plane that belongs to each zone axis.

Figure 2.60

Selected area diffraction pattern from a sapphire single crystal.

122

Microstructural Characterization of Materials

(c) the angle between the normals to all the reflecting planes for each zone. (110) : ð1
21Þ; ð2
20Þ : ð1
11Þ; (111) : ð
1
11Þ; (122) : (210). 2.32. For a BCC crystal, determine the normal to the planes defined by the following pairs of directions: (a) ½1
32½
201; (b) ½102½22
1; (c) ½
121½
1
11. Which of these planes are allowed reflections for the BCC lattice? 2.33. A thin, fine-grained, two-phase polycrystalline film composed of Al (FCC, a 0.405 nm) and Fe (BCC, a ¼ 0.2867 nm) is characterized by electron diffraction in transmission electron microscopy. A ring pattern acquired from the sample is presented in Figure 2.59. Determine which rings belongs to Al and which to Fe, and index the diffraction rings. 2.34. A selected area diffraction pattern of a sapphire single crystal is given in Figure 2.60. Solve the pattern, and determine the camera constant for this printed pattern. 2.35. A selected area diffraction pattern from an annealing twin in copper is shown in Figure 2.61. Solve the pattern, determine the camera constant, and define the twinning relationship

d=0.2087 nm

Figure 2.61

Selected area diffraction pattern from a twin in a copper single crystal.

3 Optical Microscopy The optical microscope is the primary tool for the morphological characterization of microstructure in science, engineering and medicine. In the medical sciences, thin slices of biological tissue and other preparations are prepared for transmission optical microscopy, with or without staining, and frequent use is made of additional image contrast available by the use of fluorescent dyes, dark-field optical microscopy, differential interference or phase contrast. The geologist also works primarily in transmission, polishing his mineralogical specimens down to a thickness of less than 50 mm and mounting them on transparent glass slides. For the geologist the anisotropy of the sample viewed in polarized light is the most frequent source of contrast, and provides information not only on the morphological characteristics but also on the optical properties and spatial orientation of any crystalline phases which are present in the sample. Metallurgical samples for metallographic examination were originally prepared by Henry Sorby (1864) as thin slices, using the same methodology developed earlier for mineralogical specimens, and his specimens have survived intact and are still available for examination (Figure 3.1). However, the presence of the conduction electrons renders metals opaque to visible light and all metallurgical samples must be examined in reflection. It follows that only the surface of the sample can be imaged, and that it is the surface topology and surface optical properties that are responsible for the contrast seen in specimens of metals and their alloys examined in the optical microscope. In reflection microscopy the contrast may be either topological, or due to differential absorption of the incident light, or the result of optical effects associated with reflection and optical interference. Polymers and plastics can be imaged in either reflection or transmission, but the amorphous, glassy phases present give poor contrast. However, crystalline polymer phases, frequently formed by slow cooling from a viscous liquid state, are often studied in transmission by casting thin films of the molten polymer onto a glass slide. In polarized light these polymer crystals show contrast that is characteristic of the orientation of the optically anisotropic crystal lattice of the polymer with respect to the polarization vector of the incident beam (Figure 3.2). Filled plastics and polymer matrix composites can also be Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

124

Microstructural Characterization of Materials

Figure 3.1 A ‘Widmansta¨tten’ microstructure in steel. Specimen prepared by Henry Sorby (ca. 1864). Reproduced with permission from Smith, A History of Metallography, p. 166. Published by the University of Chicago Press.

Figure 3.2 ‘Spherulites’ in polyethylene. Arrays of crystallites growing from a common nucleus are readily observed in polarized light. Reproduced by permission of John Wiley & Sons, Inc.

Optical Microscopy 125

examined in reflection, although the extreme differences in mechanical response between a low elastic modulus polymer and a high modulus filler or reinforcement makes specimen preparation problematic. Furthermore, the ready availability of scanning electron microscopes has reduced the motivation to study the morphology of these materials in the optical microscope. Nevertheless, contrast in the scanning electron microscope is insensitive to material anisotropy, while this is the principle source of contrast in polarized light microscopy. For example, elastomeric (rubbery) polymers exhibit molecular alignment at high elastic strains that gives rise to optical anisotropy which is readily observable in polarized light. Ceramics and semiconductors are usually examined by reflection microscopy, despite their obvious similarity to mineralogical samples, even though in some cases it may actually be easier (and more informative) to prepare a thin slice for transmission examination. Poor reflectivity, coupled in some cases with strong absorption of the incident light, makes for poor optical contrast in many ceramic samples when viewed by reflection, while their resistance to chemical attack often makes it difficult to find a suitable etchant to reveal the microstructure of a polished ceramic sample. In many cases, the presence of very small quantities of impurities or dopants alters the response of the sample to surface preparation, often through strong segregation of the dopant to grain boundaries and interfaces. In this chapter we will emphasize the contrast mechanisms which are typical of microstructural morphologies observed by optical microscopy, explaining, at an elementary level, the interaction between a specimen sample and an incident beam of visible light. The emphasis will be on reflection microscopy, that is, the metallurgical microscope, although much of the discussion is equally applicable for transmission samples.

3.1

Geometrical Optics

Rapid developments in the physical sciences over the past half-century and the technological revolution associated with semiconductors, microelectronics and communications, have left little place in the modern school science syllabus for the mundane topic of geometrical optics. It follows that the university science student’s understanding of image formation in either the telescope or the optical microscope can no longer be taken for granted. Nonetheless, in the context of the present discussion, some appreciation of what is happening inside an optical microscope is desirable. 3.1.1

Optical Image Formation

The lens of an optical magnifying glass forms an image of an object because the refractive index of glass is much greater than that of the atmosphere, and reduces the wavelength of the light passing through the glass. A parallel beam of light incident at an angle on a polished block of glass is deflected, and the ratio of the angle of incidence on the surface to the angle of transmission through the glass is determined by the refractive index of the glass (Figure 3.3).

126

Microstructural Characterization of Materials

Figure 3.3 A beam of parallel light (a planar wave front) is deflected on entering a block of glass because of the change in wavelength associated with the refractive index m of the glass.

In the case of a convex glass lens (a lens having positive curvature), the spherical curvature of the front and back surfaces of the lens results in the angle of deflection of a parallel beam of light varying with the distance of the beam from the axis of the lens, and bringing the parallel light beam to a point focus at a distance f that, for a given wavelength, is a characteristic of the lens, and is termed its focal length (Figure 3.4). If the lens curvature is negative, then the lens is concave and a parallel beam incident on the lens will be made to diverge. The beam of light will now appear to originate at a point in front of the lens: an imaginary focus corresponding to a negative focal length -f. There is no reason why the front and back surfaces of the lens should have the same curvature, nor even why one surface should not have a curvature of opposite sign to the

Optical Microscopy 127

Figure 3.4 A parallel beam transmitted through a convex lens along its axis converges to a focus at a fixed distance from the plane of the lens, the focal length.

other. It is the net curvature of the two surfaces taken together which determine whether the lens is convex or concave. Similarly, there is no reason why the refractive index should be the same for all the lenses in an optical system, and different grades of optical glass possess different refractive indices. The ‘lenses’ used in the optical microscope are always assemblies of convex and concave lens components with refractive indices selected to optimize the performance of the lens assembly. Depending on their position in the microscope, these lens assemblies are referred to as the objective lens, the intermediate or tube lens and the eyepiece. It is also quite common for the medium between the sample and the near side of an objective lens to be a liquid, rather than air. An immersion lens is one that is designed to be used with such an inert, high refractive index liquid between the sample and the objective lens. The assumption that a parallel beam of light will be brought to a point focus at a distance determined by the focal length of the lens is only a first approximation. Since the refractive index of glass varies with the wavelength of light, the focal length is only constant for monochromatic light (light of fixed wavelength). Visible (white) light is not brought to a single focus, but is rather dispersed, the shorter wavelengths (blue light) being brought to a focus at a greater distance from the lens than the longer wavelengths [red light,

Figure 3.5 (a) The focal length of a thin lens depends on the wavelength of the radiation, giving rise to chromatic aberration. (b) When the diameter of the lens is no longer small compared with the focal length, the outermost regions of the lens have a reduced focal length and result in spherical aberration.

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Figure 3.5(a)]. It follows that a parallel beam of white light is not brought to a sharp focus by a glass lens, but rather to a region of finite size (the ‘disc of least confusion’), a condition referred to as chromatic aberration. For thicker, larger diameter, lenses, even monochromatic light is not fully focused, and the outermost regions of the lens (corresponding to the largest angles of deflection of the incident light) have a shorter focal length than the region of the lens near its axis [Figure 3.5(b)]. This again results in a finite size for the optimally focused beam, another ‘disc of least confusion’, rather than a point focus (a condition termed spherical aberration). The lens systems used for the lens assemblies in an optical microscope, especially the objective lenses, correct for these aberrations very successfully, but it is important to check the manufacturer’s recommendations to ensure that the lens combinations (for example, an eyepiece and objective) are fully compatible and are appropriate for the type of specimen to be examined. Objective lenses for biological samples usually compensate for the thickness of a glass cover slip (often 0.1 mm in thickness) that protects the sample from the environment. As might be expected, there is a direct relationship between the cost of an objective lens and its optical performance, and several technical terms are used to describe this performance. The most common objectives are achromats, which are fully corrected for chromatic aberration at two wavelengths of light (blue and red), as well as for spherical aberration at an intermediate wavelength (green light). They achieve their best performance when used with monochromatic green light. Plan achromats are more fully corrected to ensure that not only the central field of view is in focus but also the periphery. They are designed for image recording, rather than simple viewing of the sample. The most expensive objectives are plan apochromats, that are fully corrected for the full visible range (red, green and blue light) and give an in-focus image over the full field of view. Specialized objectives are available for a wide range of viewing conditions, such as phase contrast, dark-field microscopy, differential interference and long-working distance configurations (see below). The human eye (Figure 3.6) forms an image of the visible world on a light-sensitive membrane, the retina, by focusing light transmitted through the lens of the eye. Unlike glass lenses, the curvature of the lens in the eye can be adjusted by a system of muscles in order to ensure that objects at different distances can be brought into sharp focus on the retina. Unfortunately, this control of the focus deteriorates with age, while for many of us the control is imperfect, even in childhood. Prescription spectacles usually do an excellent job of correcting the focus of our eyes. To prevent too much light entering the eye, the iris acts as a variable aperture, reducing the effective diameter of the lens in bright light. The light-sensitive retina is covered in a dense array of optical receptors, the rods and cones, which respond to the incident optical signal with remarkable sensitivity. In many animal species, including, of course, man, the retina responds not only to the intensity of the incident light, but also to the wavelength, resulting in colour vision. However, some 20% of the human race lack perfect colour vision, while the response of the eye to colour gradually fades as the intensity is reduced, so that as dusk falls the world becomes grey. The image formed by a simple lens can be analysed using a ray diagram in a thin lens approximation (Figure 3.7). A ray of light parallel to the axis and coming from a point in the object plane off the lens axis and at a distance -u in front of the lens will be deflected by the

Optical Microscopy 129

Figure 3.6 In the human eye the lens focuses an image onto the retina, while the iris acts as a variable aperture to limit the amount of light admitted. The space between the lens and the retina is filled with liquid, so that this is an immersion lens system.

lens to pass through the focal point at a distance f behind the lens. On the other hand, a ray travelling from the same point in the object plane but passing through the conjugate focal point -f in front of the lens, will be deflected parallel to the axis after passing through the lens. A third ray from the same point in the object plane, but now travelling through the

f -u

v

-f

Figure 3.7 A ray diagram relates the distance of the lens from the object u to both the focal length of the lens f and the position of the image plane v, and determines the magnification in the image, M ¼ v/u.

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centre of the lens will be undeflected. As we see from Figure 3.7, all three rays meet at a point in the image plane at a distance v behind the lens, to form the image. It is an elementary exercise in trigonometry to show that the magnification of the image created by the lens is given by M ¼ v/u, while the object and image distances are related by 1/u þ 1/v ¼ 1/f. It follows that the magnification achieved by a magnifying glass is controlled by its focal length, f and the distance between the magnifying glass and the object, u. 3.1.2

Resolution in the Optical Microscope

The wavelength of the electromagnetic radiation transmitted through the earth’s atmosphere from the sun varies from the infrared (at the long wavelength end of the spectrum) to the ultraviolet (at the short wavelength end), but the peak intensity for solar radiation reaching the earth is in the green region of visible light, very close to the peak sensitivity of the eye (approximately 0.56 mm). The temperature of the sun is about 5500 K, appreciably higher than that of the tungsten-halide light sources usually used for microscopy (about 3200 K). ‘Daylight’ filters have been developed that modify the halide spectrum to simulate sunlight and other filters are also used, especially ‘green’ filters for approximately monochromatic observation and grey-scale recording. 3.1.2.1 Point Source Abbe Image. The resolution of an optical lens is defined in terms of the spatial distribution of the intensity coming from a point source of light situated at infinity, when the point source is imaged in the focal plane of the lens. The calculated intensity distribution assumes a parallel beam of light travelling along the axis of a thin lens and brought to a focus at the focal distance, as shown in Figure 3.8, and for the cylindrically symmetric case, the ratio of the peak intensities for the primary and secondary peaks in this image intensity distribution is approximately 9:1. The width of the primary peak for this case is given by the Abbe equation: d ¼ 0:61

α

l msina

ð3:1Þ

Intensity

Figure 3.8 The Abbe equation gives the width of the first intensity peak for the image of a point object at infinity in terms of the angular aperture of the lens a and the wavelength of the radiation l.

Optical Microscopy 131

where l is the wavelength of the radiation, a is the aperture (half-angle) of the lens, determined by the ratio of the lens radius to its focal length, and m is the refractive index of the medium between the lens and the focal point (m
1 for air). 3.1.2.2 Imaging a Diffraction Grating. A diffraction grating consists of an array of closely spaced, parallel lines. When illuminated by a normally incident, parallel beam of light, a cylindrical wavefront is generated by scattering of the incident light from each of the lines in the grating. These wavefronts interfere to generate both zero-order and diffracted transmitted beams (Figure 3.9). When the spacing of the grating is large compared with the wavelength, the angle of diffraction for the nth-order beam is given by siny ¼ nl/d. It follows that a lens can only be used to image a diffraction grating if the angular aperture of the lens a is large enough to accept both the zero-order and the first-order beams, that is sina  siny ¼ l/d. We can compare this condition both with the Bragg equation for diffraction from a threedimensional crystal lattice, l ¼ 2dsiny, and with the Abbe relationship above, which is related to the Raleigh resolution criterion (see below), d ¼ 0.61l/msina. In effect, all three equations define the limiting conditions for transmitting information about an object when using electromagnetic radiation as the information carrier. The key parameter in all three cases is the ratio l/d, and all three criteria state that the resolution limit on the microstructural information available is directly proportional to the wavelength of the radiation used. 3.1.2.3 Resolution and Numerical Aperture. Raleigh defined optical resolution in terms of the ability of a lens to distinguish between two point sources at infinity when they are viewed in the image plane. His criterion for resolution was that the angular separation of two

Diffraction grating

Zero order diffracted wavefront

Incident wavefront

First order diffracted wavefront Scattered waves Figure 3.9 The diffraction pattern from a grating generates a series of diffracted beams. To image the grating at least the zero- and first-order beams must be admitted to the aperture of the lens.

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sources of equal intensity should ensure that the maximum of the primary image peak from one source should fall on the first minimum of the image from the second source. (Think of using an astronomical telescope to resolve a pair of identical stars with a small angular separation.) The Raleigh condition implies that the combined image of the two sources will show a small (13 %, actually), but detectable intensity minimum at the centre (Figure 3.10). It follows that the Raleigh criterion corresponds exactly to the width of the primary intensity peak given by the Abbe equation, d ¼ 0.61l/msina. We should note that the Raleigh resolution is defined in the focal plane of the lens for an image of a point source at infinity. In the optical microscope, the objective lens magnifies the object, placing the image far from the lens while the object itself lies close to the focal plane. For optical microscope objectives, it follows that the Abbe equation gives the minimum separation that can be distinguished in the focal plane of the object. Try not to get confused! It is important to recognize the fundamental significance of the Abbe relation: for any imaging system based only on wave optics, no image detail can be transmitted which is much below the wavelength used to transmit the information. If we wish to maximize the information in the image we should collect as much as possible of the optical signal generated by the object – that is, we should maximize the aperture of the objective lens. Actually, it is possible to make use of the particle properties of phonons in order to achieve a resolution that is below the wave optics limit. This is the basis of near-field microscopy, in which a sub-micrometre light pipe is scanned across the specimen surface. The resolution is then limited by the diameter of the light pipe and its distance from the surface and can be appreciably better than that set by the Abbe relation, but this technique goes well beyond the present text. Most recently, it has proved possible to beat the

d=

0.61λ µsinα

Figure 3.10 The Raleigh resolution criterion requires that two point sources at infinity have an angular separation sufficient to place the maximum intensity of the primary image peak of one source at the position of the first minimum of the second.

Optical Microscopy 133

Figure 3.11 Large objects of diameter d are blurred by the diffraction limit d derived from the Abbe relationship, but objects smaller than the Abbe width are still detectable in the microscope, although the intensity is reduced, while their apparent width remains that given by the Abbe equation.

diffraction limit on resolution by partially quenching the fluorescent emission from histological specimens stained with a fluorescent dye. The fluorescent emission is stimulated by pulsed laser irradiation, but a second, picosecond, laser pulse timed immediately after the first pulse and of appropriate energy, partially quenches the peripheral fluorescent signal, so that only light emitted from the central region is recorded. Resolutions of 30–40 nm have been claimed. Again, this experimental technique should mainly be taken to demonstrate that it is possible to beat the diffraction limit in some situations. As noted previously, objective lens aberrations can be corrected and should not limit the performance of the optical microscope. The parameter msina is termed the numerical aperture (NA) of the lens and is an important characteristic of any objective lens system. The maximum values of NA are of the order of 1.3 for an immersion lens system and 0.95 for lenses operating in air, and the value of NA is usually marked on the side of the objective lens by the manufacturer. It is important to distinguish between the resolution limit of the objective lens and the detection limit. As the signal intensity generated by a point source decreases it will become increasingly difficult to detect against the background noise of the system. In the reflection microscope, the smallest objects will scatter light outside the lens aperture, resulting in an intensity deficit in the image field. As the object becomes very small the intensity of the signal from the object will decrease while the apparent size of the object (the Abbe width) will remain unchanged. At some limiting size, when the signal approaches the background noise limit of the detection system, the object will cease to be detectable. The detection limit is well below the resolution limit dictated by the wavelength of the light and the NA of the lens. The situation is illustrated schematically in Figure 3.11. One reason for using darkfield illumination is that it is easier to detect small features that scatter light into the imaging field, rather than relying on the small amount of light which is scattered out of the objective aperture in a bright-field image. 3.1.3

Depth of Field and Depth of Focus

The resolution available for an object whose image is in focus in the image plane is finite and limited by the NA of the objective lens, so it follows that the object need not be at the exact object distance from the lens u, but may be displaced from this plane without sacrificing

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Microstructural Characterization of Materials

Figure 3.12 Since the resolution is finite, the object need not be in the exact object plane in order to remain in focus: there is an allowed depth of field d. Similarly, the image may be observed without loss of resolution if the image plane is slightly displaced: there is an allowed depth of focus D.

resolution (Figure 3.12). The distance over which the object remains in focus is defined as the depth of field: d ¼ dtana

ð3:2Þ

where a is the half the angle subtended by the objective aperture at the focal point. Similarly, the image will remain in focus if it is displaced from its geometrically defined position at a distance v from the lens. The distance over which the image remains in focus is termed the depth of focus: D ¼ M2d

ð3:3Þ

where M is the magnification. Both these expressions are approximate and assume that the objective can be treated as a ‘thin lens’, which is not the case in commercial microscopes. Since the resolution is given by d ¼ 0.61l/msina ¼ 0.61l/NA, it follows that the depth of field decreases as the NA increases. For the highest image resolution, the specimen should be positioned to an accuracy of better than 0.5 mm, and this is an essential requirement when specifying the mechanical stability of the specimen stage. The depth of focus is considerably less critical. Bearing in mind that a magnification greater than ·100 may be necessary if all the resolved detail visible in the optical microscope is to be recorded, then displacements in the image plane of the order of a millimetre are acceptable.

3.2

Construction of the Microscope

A simplified design for a reflection optical microscope is shown in Figure 3.13. The microscope is an assembly of three separate systems. The illuminating system which provides the source of light illuminating the sample, the specimen stage that holds the sample in position and controls the x, y and z coordinates of the area under observation, and the imaging system, which transfers a magnified and undistorted image to the plane of observation and to the recording medium. We will discuss each of these in turn. 3.2.1

Light Sources and Condenser Systems

There are two conflicting requirements for the light source. On the one hand, the area of the specimen being examined beneath the objective lens needs to be uniformly flooded with

Optical Microscopy 135

Eyepiece First Image Plane

Condenser Lens

Lamp

Half-Silvered Mirror Virtual Image Aperture

Objective Back Focal Plane

Condenser Aperture

Objective Specimen 2α Figure 3.13 The principle components of the reflection optical microscope and their geometrical relationship to one another.

light in order to ensure that all the microstructural features experience the same illuminating conditions, but on the other hand the incident light needs to be focused onto the specimen to ensure that the reflected intensity is always sufficient for comfortable viewing and recording. The source of light should be as bright as possible. Fifty years ago this was achieved by striking a carbon arc, which gave an excellent, though somewhat unstable, source of white light. Alternatively, a mercury arc lamp generated an intense monochromatic emission line in the green (l ¼ 0.546 mm), which corresponded well to the peak sensitivity of the human eye. Today, while some small instruments still use a conventional light bulb, high performance optical microscopes are now equipped with a tungsten-halide discharge tube that provides a stable and intense source of white light corresponding to a temperature of about 3200 K (compare the temperature of the sun, about 5500 K). Filters can then be used to select a narrow band of wavelengths, usually in the green, for monochromatic viewing, or to simulate sunlight more closely (‘daylight’ filters). In addition to the source itself, there are other important components in the illuminating system (Figure 3.13). The condenser lens assembly focuses an image of the source close to the back focal plane of the objective lens so that the surface of the specimen is uniformly illuminated by a near-parallel beam of light. The condenser aperture limits the amount of light from the source which is admitted into the microscope by reducing the effective source size. Contrast in the image can often be improved by using a small condenser aperture,

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Microstructural Characterization of Materials

although at the cost of reducing the image intensity and, if the aperture is too small, introducing image artifacts which are associated with the Abbe diffraction pattern of a ‘point’ source. A second aperture, the objective or virtual image aperture, is placed in the virtual image plane of the sample (Figure 3.13), so that only light illuminating the area under observation is admitted to the microscope. This ensures that light is not internally reflected within the microscope, leading to unwanted background intensity. The size of the virtual image aperture should be adjusted to the field of view of the microscope at the magnification used. Both the condenser and the virtual image apertures are continuously variable irises which can be adjusted to the required size. Many reflection microscopes also permit the illuminating system to be repositioned so that optically transparent specimens can be viewed in transmission. This is important not only for the thin tissue samples of biology and medicine, but also for mineralogical samples, partially crystalline polymers and thin-film semiconductor materials. It is also extremely useful when monitoring the quality of thin-film samples prepared for transmission electron microscopy. Alternatively, the virtual image aperture may be provided with a central stop that allows an annulus of light to illuminate the area under observation from the periphery of a special objective lens assembly. Such dark-field illumination (see below) may greatly enhance the contrast. 3.2.2

The Specimen Stage

The primary requirement for the specimen stage is mechanical stability and, given the expected
0.3 mm resolution for a good optical microscope, it is clearly essential that the positioning of the specimen be accurate to better than this limit. The accurate positioning of the specimen in the x–y plane is only one aspect of the stability required. The image is brought into focus by adjusting the vertical location of the specimen and the accuracy of this z-adjustment must be within the depth of field for the largest NA objective lens, typically also
0.3 mm. The necessary mechanical precision is commonly achieved by coarse and fine micrometre screws for all three (x, y and z) coordinates, and both the time-dependent drift of the stage and the mechanical ‘slack’ in the system need to be minimized. (The ‘slack’ is the difference in the micrometre reading when the same feature is brought into position from opposite directions.) In general, it is the z-adjustment that presents the most problems, since the necessary stage rigidity implies a fairly massive and hence heavy construction. Two possibilities exist, depending on whether the specimen is to be placed beneath or above the objective lens. In the former and more usual case (Figure 3.13), the plane of the prepared sample surface must be positioned accurately normal to the microscope axis. This is commonly achieved by supporting the specimen from below on soft plasticine and applying light pressure with a suitable jig (Figure 3.14). 3.2.3

Selection of Objective Lenses

Avery wide range of objective lenses are available, depending on the nature of the specimen and the desired imaging mode. The performance of the objective lens is primarily

Optical Microscopy 137

Plunger Glass Support Slide

Specimen Plasticine Base

Figure 3.14 If the specimen is to be placed beneath the objective lens, then it must be mounted with the plane of the specimen surface accurately normal to the microscope axis.

dependent on its NA, and this is almost universally to be found inscribed on the side of the objective lens assembly, together with the magnifying power for that lens. Most objective lenses are achromatic, that is they are not limited to monochromatic light, but are nevertheless recommended for viewing high resolution monochromatic images in the green. While achromatic lenses are corrected for both spherical aberration in the green and chromatic aberration at two wavelengths (red and blue), they only yield a focused image in the central region of the field of view. Plan achromat objectives ensure that the periphery of the field of view is also in focus and they are therefore more suitable for image recording. Apochromatic objectives are free of chromatic aberration for three wavelengths (red, green and blue), and, correspondingly, plan apochromats are designed for recording image detail in full colour. Histological examination of soft tissues, which accounts for the major proportion of the work of the optical microscope in the life sciences, requires that the specimen be protected from the environment by mounting a thin tissue slice on a glass slide and then protecting it with a thin cover slip. Similar techniques are used for many polymer specimens, particularly those that are partially crystalline. These materials can be cast onto the slide and the specimen thickness controlled by spinning or by applying uniform pressure to a cover slip. Objective lenses designed for use with such specimens are corrected for the refractive index and thickness (often 0.1 mm) of the optically flat cover slip. Not only the resolution, but also the brightness (the intensity per unit area of the image) depends on the NA of the objective lens. For any given conditions of specimen illumination, the brightness of the image decreases as the square of the magnification. However, larger NA lenses increase the cone acceptance angle for the lens, so that more light is collected. The NA may vary by an order of magnitude in going from a low-power (low magnification) lens to a high-power (high magnification) immersion objective. A similar order of magnitude increase in the magnification will then be required in order to observe all the image detail. It follows that there is still an overall reduction in brightness by a factor of 102/10 ¼ 10.

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Microstructural Characterization of Materials

Standard 4 mm Objective Normal Working 3 mm Distance

First Image Plane (unit magnification)

Half-Silvered Plate 12.8 mm Specimen

Figure 3.15 A long working distance attachment for studying specimens at high temperatures or in a hostile environment.

The working distance of the objective lens from the specimen surface also decreases dramatically as the NA of the objective lens assembly is increased, down to of the order of 0.1 mm for the highest-powered lenses. It is only too easy to damage a lens by driving the specimen through focus and into the glass lens, and good lenses are expensive to replace. Special long working distance lenses are available that allow high magnification observation without having the sample in close proximity to the objective lens. One inexpensive design creates an intermediate image at unit magnification by reflection (Figure 3.15) and permits a specimen to be imaged while in a hostile environment, for example in a corrosive medium or at an elevated or cryogenic temperature. Despite the attraction of in situ experiments, little optical microscopy has been done under such dynamic conditions. There are difficulties: the dimensional stability of the specimen and its support structure is one problem; another is to ensure that the optical path between the specimen and the objective lens assembly is not obscured by a condensate or by chemical attack. Cryomicroscopy is subject to the formation of ice crystals, while a high temperature stage may form opaque deposits that derive from the heating elements, the specimen or the supporting structure. To be successful, an in situ stage must combine a rapid response time with experimental stability. In the case of a heating stage, a compromise is required between the large heat capacity needed to ensure thermal stability, and the small heat capacity necessary to allow a rapid experimental response. Many other specialized objective lens and stage assemblies are available. One of the most useful, for both reflection and transmission work, is the dark-field objective which illuminates the specimen with a cone of light surrounding the lens aperture. The light scattered by the specimen into the lens aperture is then used to form a dark-field image in which the intensity is the inverse of that observed in normal illumination (Figure 3.16).

Optical Microscopy 139

Specimen

Specimen

(a)

(b)

Figure 3.16 A dark-field image (a) is formed by collecting the diffusely scattered light into the objective lens aperture, and is the inverse of the normal, bright-field image (b), where the scattered light falls outside the objective aperture and is lost to the objective lens.

3.2.4

Image Observation and Recording

The image magnification provided by the objective lens assembly is limited, and insufficient if the image is to be fully resolvable by the human eye. There are three options available. The first is to insert an eyepiece and an additional intermediate or tube lens, in order to view the image directly at a working magnification that is comfortable for the observer. Most microscopes now available are designed with a tube lens that allows the sample to be placed in the focal plane of the objective, so that the light returning to the microscope through the objective is essentially parallel, and only brought to an intermediate focus by the tube lens. This allows for a wide range of optical accessories to be inserted between the objective and the tube lens. The second option is to use the additional lenses to focus the image onto a light-sensitive, photographic emulsion or charge-coupled device (CCD), usually for subsequent enlargement. For the third option it is possible to scan the image in a television raster and display it on a monitor. For the recording of dynamic events in the microscope this may in fact be the preferred technology. In recent years the improved availability of high quality, CCD cameras has made it possible to record a digital image from an objective lens without any additional lenses. The consumer market for digital cameras has allowed high quality CCD technology to all but replace photographic recording, while conventional television camera technology is now seldom used, even for teaching purposes. Nevertheless, professional photographers still make use of photographic emulsions, since they usually require the highest performance CCD systems, which are still extremely expensive. More recently, there have been significant advances in a new technology, complementary metal oxide semiconductor (CMOS), which essentially places the camera on a single chip. For colour recording in both CCD and CMOS devices individual pixels (picture elements) can be filtered by red, green and blue dyed photodiodes, but while both CCD and CMOS cameras offer pixel sizes of 6 mm or even less, the CMOS cameras are limited to of the order of a million pixels per frame, an order of magnitude less than the CCD cameras. To make the most of high resolution digital colour recording it may be necessary to invest in planapochromatic objective lenses (Section 3.2.3). 3.2.4.1 Monocular and Binocular Viewing. Visual observation is most commonly performed with a monocular eyepiece, which enlarges the primary image by a factor of ·3 to ·15. A typical 0.95NA (nonimmersion) objective may have a primary magnification of ·40 and a resolution of 0.4 mm, so that, to ensure that all resolved features are readily visible

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Microstructural Characterization of Materials

to the eye (0.2 mm), some further magnification is required, simply calculated as: (0.2 · 103)/(0.4 · 40) ¼ ·12.5. Most good microscopes have an additional intermediate or tube lens (·4, for example), so that a ·3 or ·5 eyepiece should then be sufficient to resolve all image detail. Even without an intermediate lens there is no real reason to use a ·15 eyepiece, since superfluous additional magnification reduces the field of view, enlarging the resolved features to the point where they appear blurred to the eye of the observer. Some microscopes are equipped with a beam splitter and a binocular viewer. For those who have difficulty viewing comfortably through one eye this is undoubtedly a convenience, but the microscopist should be aware that there are some disadvantages. In particular, it is unusual for the focal plane of both eyes to be identical, so that one eyepiece of the pair needs to be independently focused. The user first focuses a feature of interest in the plane of the specimen, using just one eye and a fixed-focus eyepiece. He then adjusts the variable focus of the second eyepiece (without touching the specimen stage controls) until the images seen by both eyes merge into a single, simultaneously focused, image. This procedure of adjusting the binocular settings is completed by adjusting the separation of the two eyepieces to match the separation of the observer’s eyes. It is important to note that a binocular eyepiece does not provide stereoscopic (three-dimensional) viewing of the sample, which would require two independent objective lenses focused on the same field of view. Stereobinoculars (or stereomicroscopes), with twin objectives, are available, but with magnifications limited to about ·50. This limit is dictated by the geometrical problems associated with the positioning of the twin objective lenses close to the specimen surface. Stereobinoculars are important tools for inspection in the electronics industry. 3.2.4.2 Photographic Recording. A photographic emulsion and the human eye react very differently to light. The emulsions have their maximum sensitivity in the ultraviolet (about 0.35 mm) and both black and white, and colour films rely on dyes to extend the photosensitivity of silver halide emulsions beyond the green (Figure 3.17). Orthochromatic

10 Orthochromatic

100

Sensitivity

1 Ultraviolet 10 –1

Visible Range

Human Eye

80 60 40 20

10 –2 0.3

Relative Visibility

120

0.4

0.5 0.6 Wavelength (µm)

0.7

Figure 3.17 The eye is most sensitive to green light, whereas photographic emulsions have sensitivities which decrease steadily with increasing wavelength.

Optical Microscopy 141

emulsions are not sensitive to red light, which is a convenience in dark-room processing, and are a common choice for photographic recording of monochromatic microscope images in green light. Panchromatic film is a common choice for black and white photographic recording in daylight, but here too the sensitivity falls steadily with increasing wavelength in the visible range. In classical black and white photography the recording medium always yields a negative in which the clear areas, corresponding to zero excitation (no silver precipitation) and the opaque (black) areas correspond to maximum light excitation. This does not have to be the case, and Polaroid cameras commonly produce a positive grey scale image. Colour recording films may also be negative, and are then used for colour printing, or positive, for use as slides or transparencies. The convention is to use the suffix chrome for positive transparencies (Kodachrome, Ektachrome, etc.), but to use the suffix colour for negative film (Fujicolor, Agfacolor, etc.). The speed of an emulsion is its response to a fixed radiation dose at a standard wavelength and depends on three factors: the exposure time, the ‘grain’ of the emulsion and the development process. A photosensitive silver halide grain will react to subsequent development only if it can absorb a pair of photons. The time interval between the arrival of the two photons is important, since the grain may decay from its initial excited state in the interim. As the incident intensity decreases, the interval between photon excitations of the same grain increases, and the response of the emulsion is reduced, a phenomenon termed reciprocity failure. Larger halide grain sizes increase the photon collision cross-section and improve the photosensitivity (the speed of the emulsion). The price paid is a ‘grainier’ image with poorer inherent resolution. It follows that some compromise is usually required between fine-grained, slow-speed emulsions and coarse-grained, high-speed emulsions. During development, the grain of silver, which is nucleated at an activated halide crystal, grows into a cluster of silver grains which encompasses a much larger volume than that associated with the original halide crystal, so that the resolution in the final recorded image is affected by the growth of the silver grains during the development process. An emulsion designed for photomicroscopy, and developed according to the recommendations of the manufacturer, should have a resolution of the order of 10–20 mm and be capable of enlargement by a factor of ·10. It should therefore be possible to photograph a high resolution image without any loss of information at an appreciably lower magnification than that required to view the fully resolved microstructure. It follows that a low magnification, high resolution recorded image contains far more information than is available in the field of view required for observation at full resolution. The contrast attainable in a given emulsion is defined in terms of the dose dependence of the blackening in the developed emulsion. The dose is the amount of light per unit area E multiplied by the time of exposure t, while the blackening D is the logarithm of the ratio of the intensity incident on the emulsion I0 to the intensity of light transmitted through the exposed and developed emulsion I:
 I0 ð3:4Þ D ¼ log I The contrast g is defined as the maximum slope of the curve of D plotted against log Et. Emulsions with a high g lose image detail because they tend to register as black or white with few intermediate grey levels, while low g emulsions lack contrast because the grey levels are too close together. A major disadvantage of photographic recording is the nonlinearity of

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Microstructural Characterization of Materials 3.5

Overexposure

3.0

Ra ng e ea r

2.0 Li n

Density

2.5

1.5 1.0

Underexposure

0.5 0

–3.0

–2.0

–1.0

0

1.0

Log Exposure (lux s) Figure 3.18 The blackening of an emulsion is only linear over a selected range of exposure dose that depends on the speed and contrast of the emulsion. At very low doses reciprocity failure reduces the blackening, while at very high doses the blackening saturates.

the response of the emulsion and the difficulty of controlling the many parameters involved in exposing, developing, enlarging and printing the emulsion. It is very difficult to make quantitative measurements of image intensity or contrast based on photographic recording and digital recording, using a CCD camera, is therefore preferable. The range of information which can be recorded, either by a photographic emulsion or by a CCD camera, is never unlimited. This is illustrated in Figure 3.18. At very low values of the dose Et background noise will start to become a problem, while at very high doses the response of the recording media will saturate. High resolution, negative, black and white film is quite capable of responding to four orders of magnitude of the dose, but positive prints are limited to about two orders of magnitude. CCD cameras usually come somewhere in between (see below), but have the distinct advantage that there response is linear over this range. The response of photographic emulsions to high energy electrons is also linear in dose, since the halide grains are excited by a single electron impact (compare the excitation process for visible light, which requires two photons to strike a halide grain within a critical time interval). Similar considerations apply to CMOS cameras. 3.2.4.3 Television Cameras and Digital Recording. Television cameras and monitors have been attached to optical microscopes for some considerable time, primarily to allow presentation of microscope observations in ‘real time’ to large groups, but also for recording dynamic events occurring under the microscope (for example corrosion studies, or the effects of heating the sample). However, the number of pixels scanned in a standard television display is well below the number of image elements that the eye is capable of resolving across a single field of view. Nevertheless, the time-base of a television raster does permit any signal corresponding to two spatial dimensions to be recorded as a time-dependent analogue signal that can either be processed and displayed on a monitor with the same time-base or converted to a digital signal for further processing and storage or display. We will discuss digital signal processing for applications in microscopy more fully below (Section 3.5).

Optical Microscopy 143

The CCD and (CMOS) cameras are free of many limitations associated with photographic recording. The CCD camera can ‘grab’ a two-dimensional image frame in a few seconds and record a digitized image of 107 or even more pixel points (pixels are defined in Section 3.5). Furthermore, the response of a CCD or CMOS camera is essentially linear over a wide range of exposure dose, so that there is a one-to-one correlation between the recorded ‘brightness’ of any pixel point location and the original intensity of the signal from the object. Moreover, the digitized image can be efficiently processed, using standard computer programs to enhance image contrast, remove background noise, or analysed to extract quantitative morphological image data, such as grain size. Individual image frames can also be combined to extract comparative information from the image data sets, for example in the study of dynamic changes taking place in the sample during viewing. Finally, the diameter of the CCD camera is typically 25 mm or less, so that an image can be recorded using an objective lens alone, without the need for any intermediate lenses. The past few years have seen major changes in the design of optical microscopes, with computer control replacing manual control of the focus, and the CCD camera and computer monitor replacing the photographic system.

3.3

Specimen Preparation

For many students, good specimen preparation is a major obstacle to successful optical microscopy. It is unfortunate that every material presents its own individual and unique problems of specimen preparation. For example, the elastic modulus and the hardness of the material usually determine the response of the sample to sectioning, grinding and polishing, while the chemical activity determines the response to electrolytic attack and chemical etching. In what follows we will generalize as far as possible, while recognizing that each metal alloy, every ceramic material and all plastic compositions are almost certain to respond differently. 3.3.1

Sampling and Sectioning

The problem with all microscopes is that they lose the ‘larger picture’ by focusing on the details. It is only too easy to lose track of the relation between a microscope image (recorded from a particular position and in a specific orientation) and the engineering component from which the image was taken. Engineering systems are assembled from components which frequently have complex geometries and come in a wide range of shapes and sizes. They are produced by a variety of processing routes, and are unlikely to be of uniform microstructure. They often have a ‘right way up’, and the materials from which they are made are often inhomogeneous and anisotropic. Both the chemical composition and the microstructural morphology may vary across a section, even if it is only the surface layers that are ‘different’. Preferred orientation may be restricted to the microstructural morphology (elongated inclusions, aligned fibres or flattened grains), or it may be associated with crystalline texture, in which certain directions in the crystals are preferentially aligned along specific directions in the component (for example, the axis of a copper wire or the rolling direction in a steel plate). Crystalline texture

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Microstructural Characterization of Materials

may exist in the absence of morphological texture, the grains appearing equiaxed even though they all share a common crystallographic axis. As a consequence, it may not be easy to decide how best to section a component for microscopic examination. Nevertheless, it is always helpful to define the principle axes of the component and to ensure that the plane of any section is aligned with at least one of these axes. In even the simplest cases it is usually desirable to examine two sections, perpendicular and parallel to a significant symmetry axis of the component. In the case of rolled sheet, sections taken perpendicular to all three principle directions are desirable: the rolling direction, the transverse direction and the through-thickness direction. In a large casting the microstructure will vary, both due to differences in the cooling rate and to the effects of segregation. Sections taken from the first portion of a casting to solidify may have very different microstructures from the portion which solidified last. If a section has been taken perpendicular to a principle direction of the component, then it is also important to identify one other principle direction lying in the plane of the section: the trace of a free surface, or a growth or rolling direction. It is only too easy to confuse structurally significant directions, either during mounting and preparation of the section or as the result of image inversion, either in the microscope or during processing. (Note that newspapers frequently publish inverted images showing right-handed individuals engaged in apparently left-handed activities.) 3.3.2

Mounting and Grinding

For convenience during surface preparation many samples need to be mounted for ease of handling. A polymer resin or moulding compound is the commonest form of sample holder, and can be die-cast or hot-pressed around the sample without distorting the sample or damaging the microstructure (Figure 3.19). Clearly, the sample may need to be sectioned to fit into the die cavity. Very small samples can be supported in any desired orientation using a coiled spring or other mechanical device. Once the sample is securely mounted, the surface section can be ground flat and polished. Rough grinding requires some care, since it is easy to remove too much material, to overheat the sample, or to introduce sub-surface mechanical or thermal damage. It is helpful to ensure that the surface section is cut planar, even before the sample is mounted. A wide range of grinding media is available, designated as either cemented, metal-bonded

Die Moulding compound Sample Base Figure 3.19 Samples are commonly cast within a moulding compound for ease of handling during surface preparation.

Optical Microscopy 145

or resin-bonded. The commonest grinding media are alumina (corundum), silicon carbide and diamond, and all three are available in a wide range of grit sizes. The grit size is defined by the sieve size which will just collect the grit, and the quoted sieve size refers to the number of apertures per inch, so that the grit size is an inverse function of the particle size. A #320 grit has been collected by a #320 sieve, having passed though the larger standard size, a #220 sieve. Beyond #600 grit sieving is no longer practical since the grit particles aggregate, but the same definitions are used and very fine, sub-micrometre grits are available. In general, a #80 grit is used for coarse grinding, corresponding to particles a few tenths of a millimetre in diameter. Grinding is actually a machining process in which the sharp edges of the grinding medium cut parallel to the surface of the sample. The rate of material removal depends on the number of particle contacts, the depth of cut (a function of the applied pressure and the grit size) and the shear velocity at the interface between the grinding medium and the work piece. Heat generated at the interface and the debris resulting from material removal are two major obstacles to effective grinding. The former increases the ductility of the work piece, and hence the work required to remove more material, while the latter clogs the cutting surfaces. These effects can be inhibited by flushing the surface with a suitable coolant to remove both heat and debris from the grinding zone. The cutting edges of grinding media particles are blunted during grinding. New cutting surfaces can be exposed through wear of the surrounding matrix, which releases the blunted grit particles as debris and exposes fresh cutting surfaces. This may occur naturally during grinding of the work piece, but is more often accomplished by dressing the grinding wheel – using an alumina ‘dressing stone’ to remove the debris and expose new cutting surfaces. The choice of matrix is important. Metal-bonded grits are most frequently used in cutting discs for sectioning hard and brittle samples. They are also used for many coarse-grinding operations for which diamond is the preferred medium. Resin-bonded discs give a much coarser cut in sectioning operations, but are generally preferred for grinding, since they are less liable to lose cutting efficiency due to the build up of debris. The grinding direction may be important. For example, it is usually undesirable to grind a region near a free surface perpendicular to that surface, since the cutting particles are almost certain to introduce extensive sub-surface damage as they bite into the edge of the sample. Cutting in the reverse direction, so that the grit particles are exiting from the free surface during grinding, will result in much less damage. The extent of the sub-surface damage depends on the elastic rigidity and hardness of the material, so that soft metals are extremely difficult to grind. However, brittle materials are prone to sub-surface cracking. It is important to recognize that sub-surface damage is always introduced during grinding and to ensure that subsequent polishing fully removes this damaged layer. 3.3.3

Polishing and Etching Methods

The primary aim of polishing is to prepare a surface which is both flat and devoid of topographical features unrelated to the bulk microstructure of the sample. Each polishing stage is designed to remove a layer of damaged material resulting from the previous stage of surface preparation. There are three accepted methods of polishing a sample: mechanical, chemical and electrochemical. Of these three, mechanical polishing is by far the most important.

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Microstructural Characterization of Materials

In mechanical polishing the mechanical damage of the earlier stages of preparation are removed by resorting to finer and finer grit sizes. The number of polishing steps necessary to reduce topographical roughness to below the wavelength of light and, as far as possible, eliminate sub-surface mechanical damage may be as few as three (for hard materials) or as many as ten (for very soft samples). The carrier for the polishing grit may be a paper backing (for a silicon carbide grit, down to perhaps a #600 grit SiC paper), to be followed by a cloth polishing wheel (down to perhaps a 1/4 mm diamond grit). In a chemical polishing solution the products of chemical attack form a viscous barrier film at the surface of the sample which inhibits further attack (Figure 3.20). Regions of negative curvature (surface grooves and pits) develop thicker layers of the viscous, semiprotective barrier film, while ridges and protrusions, with positive curvature are covered by much thinner layers and are attacked faster. As a result the sample is rapidly smoothed to develop a topographically flat surface. Electrolytic polishing is somewhat similar, but the sample must be an electrical conductor, and is almost always a metal. Positively charged cations are dissolved in the electrolyte at the sample surface and form a viscous anodic film of high electrical resistance. For successful electrolytic polishing most of the voltage drop across the electrolytic cell should be across this anodic film, so that, as in chemical polishing, the rate of attack is again controlled by film thickness. External adjustment of the voltage across the electropolishing cell ensures better control for electrolytic polishing when compared with chemical polishing, and soft materials such as lead alloys, which are very difficult to prepare by mechanical polishing, can be successfully electropolished. Nevertheless, the development of increasingly sophisticated mechanical polishing methods, suitable for even the softest engineering materials, has decreased the importance of chemical and electrochemical methods of surface preparation. Etching of the sample refers to the selective removal of material from the surface in order to develop surface features which are related to the microstructure of the bulk material. If the different phases present differentially reflect and absorb incident light, then etching may be unnecessary. Most nonmetallic inclusions in engineering alloys are visible without etching, since the metallic matrix reflects most of the incident light while the inclusion often absorbs it and appears darker. Optically anisotropic samples observed in polarized light also

Figure 3.20 Both chemical and electrolytic polishing rely on a viscous liquid layer, to enhance the attack of protruding regions and inhibit attack at grooves and recesses, ultimately forming a mirror-like, polished surface.

Optical Microscopy 147

show contrast without etching, associated with differences in crystal orientation (Section 3.4.3). Etching may also develop the surface topography, for example by grooving grain boundaries or giving rise to differences in the height of neighbouring grain surfaces. Etching may also form thin surface films whose thickness reflects the underlying phase and grain structure. Such films may either absorb light or give rise to interference effects that depend sensitively on the film thickness. Most etching methods involve some form of chemical attack, which is more pronounced in those regions of the surface that have a higher energy (grain boundaries, for example). Thermal etching is an exception. In thermal etching heating of the sample allows short-range surface diffusion to occur, reducing the energy of the polished surface near a boundary and affecting the local topology. At grain boundaries, thermal grooves are formed. In many materials the surface energy of a single crystal is highly anisotropic, so that thermal etching reduces the total energy by forming surface facets on some suitably oriented grains. The commonest etching procedures make use of chemically active solutions to chemically etch the surface and develop a topology which is visible in the microscope. The solvents are commonly one or other of the alcohols, but molten salt baths may also be used for less reactive samples, such as ceramics. In most cases the sample is immersed in the solution at a carefully controlled temperature for a given time, then rinsed thoroughly and dried (typically, using alcohol). In some cases electrolytic etching is used to promote the localized attack, for example, on stainless steels. Chemical staining is sometimes used to form a surface film whose thickness depends on the surface features of the microstructure. A steel sample will develop such a film when oxidized in air at moderate temperatures, the different grains appearing in a rainbow of interference colours, controlled by the oxidation time and temperature that determine the thickness of the coherent oxide film formed on each grain (Section 3.4.5.4). 3.3.3.1 Steels and Non-Ferrous Alloys. Engineering alloys are perhaps the most common structural materials to be prepared for microscope investigation by mechanical polishing. The ferrous alloys cover a wide range of hardness, from brittle martensites to low yield-strength, transformer steels. There are few polishing problems which cannot be solved using simple rules of thumb. The most common forms of polishing defect are surface relief (differential polishing), the rounding of edges, and scratches and plastic deformation. Assuming that the polishing media are free of contamination (primarily debris from the earlier stages of surface preparation), then scratches and plastic deformation can be prevented by selecting a more compliant support for the polishing media, in order to reduce the forces applied to the individual particles and increase the number of particle contacts per unit area of the sample surface. Conversely, surface relief and edge rounding can be inhibited by selecting a less compliant support. 3.3.3.2 Pure Metals and Soft Alloys. The softest materials are the hardest to polish mechanically, because they are so susceptible to mechanical damage. If excessive force is applied to the polishing media, then wear debris will be embedded in the soft surface of the sample and dragged in the direction of shear. Even if no obvious signs of damage are visible on the polished surface, subsequent etching (see below) is liable to reveal sub-surface traces of plastic deformation. Successful preparation of these materials is accomplished by

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Microstructural Characterization of Materials

polishing at low shear velocity and applied pressure, so as to inhibit polishing debris from adhering to the sample and smearing across the surface. 3.3.3.3 Semiconductors, Ceramics and Intermetallics. Brittle materials are usually easier to polish mechanically than soft materials. In particular, there are unlikely to be problems associated with polishing debris adhering to the surface. However, microcracking and grain pull-out in these materials is very possible, associated especially with adhesive failure at interfaces and boundaries between regions of different compliance. The selection of the polishing medium is important, and its hardness should exceed the hardness of the sample. Neither alumina nor silicon carbide can be successfully polished with SiC grit, and diamond is the medium of preference. The same is true, to a lesser extent, for silicon nitride samples. Cubic boron nitride (CBN) has some advantages as a grinding medium. The hardness exceeds that of SiC but the oxidation resistance is better than that of diamond. An important point to realize is that even the most brittle of materials can deform plastically in the high pressure zone beneath a point of contact with a grit particle. Plastic flow in this region will generate internal stresses which, when the contact stresses are removed, tend to cause cracking and chipping around the original contact, so that damage to the surface and near-surface region develops adjacent to the original contact area. Again, the solution is to limit the applied pressure and select a larger compliance for the carrier of the grinding media. 3.3.3.4 Composite Materials. Some of the most difficult materials to prepare for optical microscopy are engineering composites in which a soft, compliant matrix is reinforced with a stiff but brittle fibre. While such materials have been successfully prepared in crosssection, fewer optical micrographs have been published showing the distribution of the reinforcement parallel to the fibres. Attempts to prepare such samples often result in loss of fibre adhesion, fracture of loose fibres, and damage to the soft, supporting matrix by fragments of the hard reinforcement. In addition to an awareness of the problems and artifacts (features which are associated with preparation defects) involved in specimen preparation for the optical microscope, it is important to recognize that some samples may just be unsuitable for optical microscopy.

3.4

Image Contrast

Image contrast in the optical microscope may be developed by several alternative routes, most of which require careful surface preparation. A brief description of the various etching procedures commonly used to develop contrast in alloy and ceramic samples has been given previously (Section 3.3.2). A quantitative definition of contrast is best given in terms of the intensity difference between neighbouring resolved image features, C ¼ ln(I1/I2). For small intensity differences this reduces to: DC ¼ DI=I

ð3:5Þ

and a comparison with the Raleigh criterion for resolution (Section 3.1.2.3), suggests that DC should be at least 0.14 if features separated by a distance equal to the resolution are to be visible.

Optical Microscopy 149

At larger separations very much smaller contrast variations are distinguishable, and many computerized image processing systems operate with 256 grey levels, corresponding to intervals of DC ¼ 0.004. However, the largest NA objectives accept light scattered from the surface at much higher angles than objectives of lower NA, so that the contrast obtained in the image from any given feature is reduced at high NA. It follows that higher magnification images tend to show lower contrast. 3.4.1

Reflection and Absorption of Light

Electromagnetic radiation incident on a polished solid surface may be reflected, transmitted or absorbed. Specular (mirror-like) surfaces, highly reflecting to visible light, are characteristic of the presence of free conduction electrons in the sample material, and hence of metallic materials. However, most metals absorb a significant proportion of the incident light. Thus copper and gold absorb in the blue, so that the reflected light appears reddish or yellow. However silver and aluminium reflect over 90% of normally incident visible light, and both these metals are used for mirror surfaces. The high reflectivity of polished aluminium is not affected by the presence of the thin, amorphous oxide protective film formed on the surface in air, since the thickness of this film is well below the wavelength of visible light. Indeed most samples, and certainly all metals and alloys (with the partial exception of gold), are normally covered by some kind of surface film, either as a result of surface preparation (polishing and etching), or due to reactions in the atmosphere. As long as these films are uniform, coherent and of a thickness less than the wavelength of the incident light, they do not interfere with the reflectivity. The relation between the fraction of the incident light which is reflected and that transmitted or absorbed depends on the angle of incidence of the light. The refractive index of the solid, the ratio of the wavelength in free space to that in the solid, determines the critical angle (the Brewster angle) beyond which no light can be transmitted, and in darkfield illumination this critical angle may be exceeded, increasing the light signal scattered into the objective lens (Figure 3.21). The fraction of the incident light reflected from the surface is sensibly independent of the sample thickness for all thicknesses exceeding the wavelength, and depends only on the material and the angle of incidence, but the fraction that is transmitted depends sensitively on the thickness, and decreases exponentially as the thickness increases, in a manner exactly analogous to the case of X-ray absorption (Section 2.3.1). Mineralogical samples are commonly prepared as thin sections and examined in polarized light (Section 3.4.3). It is important that they should be sufficiently thin, primarily to allow adequate transmission, but also to ensure that resolved features in the object do not overlap in the projected image of the slice. In general samples of thickness 50 mm or less are suitable. Ceramic and polymer samples often transmit or absorb an appreciable fraction of the incident light, giving poor contrast, both because of the weak reflected signal and also due to light that is scattered back into the objective from sub-surface features which are below the focal plane in the image. An evaporated or sputtered coating of aluminium high lights the surface topography of such samples, but at the cost of losing information associated with variations in reflectivity and absorption. Some of these materials (especially crystalline polymers and glass-ceramic materials) can be studied in transmission, using polarized light. Convincing separation of the topological features from effects uniquely associated with the

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Microstructural Characterization of Materials

Incident Intensity

Reflected Light

At the critical angle of incidence no light is transmitted

Light which is not transmitted or reflected is absorbed

αc

Transmitted Light Figure 3.21 The relation between the intensity reflected, transmitted and absorbed, and the ‘critical’ Brewster angle for a specularly reflecting surface.

bulk microstructure is best accomplished by imaging the same area both before and after coating with the aluminium reflecting film. 3.4.2

Bright-Field and Dark-Field Image Contrast

In normal, bright-field illumination only a proportion of the incident light is reflected or scattered back into the objective lens. There are two limiting contrast conditions in bright field illumination. The first is that discussed previously and is often termed Kohler illumination. The light source is focused at the back focal plane of the objective lens, so that the light from a point source is incident normally on the sample surface while that from an extended source is incident over a range of angles determined by the size of the source image in the back focal plane of the objective [Figure 3.22(a)]. If, however, the light

(a)

(b)

Extended Source Image

Sample

Extended Source Imaged in Sample Plane

Figure 3.22 Two limiting conditions for bright-field illumination: (a) the source is imaged in the back focal plane of the objective; (b) the source is imaged in the focal plane of the objective.

Optical Microscopy 151

Sample

Figure 3.23 In bright-field illumination topographical microstructural features are revealed by scattering of light outside the objective lens aperture.

source is imaged in the plane of the sample, then the distribution of the intensity over the sample surface will reflect that in the source. If the sample is a specular reflector (a mirror) then an image of the source will be visible in the microscope [Figure 3.22(b)]. For high quality light sources, which emit uniformly, there may be some advantage in increasing the incident intensity by focusing the condenser system so that the source and sample planes coincide, but with most light sources, especially at low magnifications, uniform illumination of the sample is best achieved by focusing the light source at the back focal plane of the objective. Topographical features which scatter some of the incident light outside the objective lens will appear dark in the image. This is true of both steps and grain boundary grooves (Figure 3.23). The features themselves may have dimensions considerably less than the limiting resolution of the objective, but two features will only be observed in the image if they are separated by a distance greater than the Raleigh resolution and give rise to sufficient contrast. In most cases, the size of a topological feature at the surface reflects the surface preparation as much as it does the bulk microstructure. An obvious example is the ‘grooving’ at grain and phase boundaries that is associated with chemical or thermal etching. It follows that considerable care may need to be exercised when measuring a microstructural parameter, such as particle size or porosity. We will return to this in Chapter 9. Since contrast is mainly determined by comparing the intensity of the signal from some feature with that of the background, features which appear faint in the normal, bright-field image can often be enhanced by using a dark-field objective (Figure 3.16). In some cases topographical information can be enhanced by deflecting the condenser system, so that the specimen is illuminated from one side only. Such oblique illumination is sometimes available as a standard attachment. The apparent shadowing of topological features in oblique illumination helps to bring out the three-dimensional nature of the surface, but usually with some loss of resolution. Such images may also be somewhat misleading, since the apparent identification of ‘hills’ and ‘valleys’ depends on the direction from which the light comes. In the ‘real’ world light comes from above, and arranging the illumination in the microscope so that the sample is illuminated from ‘below’ appears to turn the ‘valleys’ into ‘hills’! The contrast obtained from a single feature by using bright-field, dark-field and oblique illumination are compared schematically in Figure 3.24.

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Microstructural Characterization of Materials

I

I

Bright Field

I

Dark Field

Oblique

Figure 3.24 A schematic comparison of contrast in bright-field, dark-field and oblique illumination.

3.4.3

Confocal Microscopy

Confocal microscopy, which has been developed primarily for the biological and health sciences, should also be useful in materials science, especially in the study of transparent glasses and polymers. In confocal microscopy a parallel beam of light is brought to a sharp focus at a designated location in the sample. The light source is usually a laser, although high intensity white light from a xenon arc and monochromatic (green) light from a mercury arc have also been successfully used. The point probe is scanned across the sample in an x–y plane television raster and generates a signal from a thin slice of the sample located at a specific depth beneath the surface. The resulting image is termed an optical section. A series of such optical sections taken at different depths is remarkably free of background noise. By focusing the point source using a cone of light, as in dark-field microscopy, and selecting a suitably excited fluorescent signal, it has proved possible to obtain very high resolution three-dimensional images of specific sites in a biological tissue sample which correspond to different fluorescent ‘labels’. The use of this technique is beyond the scope of the present text. 3.4.4

Interference Contrast and Interference Microscopy

In interference microscopy the light reflected from the sample interferes with light reflected from an optically flat standard reference surface. In order to achieve this, the two beams must be coherent, that is, they must have a fixed phase relation and this is achieved by ensuring that both beams originate from the same source, using a beam splitter to first separate and then recombine the two signal amplitudes from the sample and reference surfaces. 3.4.4.1 Two-beam Interference. The simplest arrangement for achieving interference contrast is to coat an optical quality glass coverslip with a thin layer of silver or aluminium,

Optical Microscopy 153

Figure 3.25 Condition for two-beam interference using a half-silvered reference surface.

such that rather more than half of a monochromatic incident light beam will be transmitted by the cover-slip (see below). The light reflected from the thin metal film constitutes the reference beam, while that transmitted through the reference coating and then reflected from the sample surface is the interfering beam (Figure 3.25). Multiple reflections are ignored for the time being, but we should note that these will affect both image resolution and contrast. It is also assumed that there is no absorption of the light, so that the incident light is either reflected or transmitted, but not absorbed. If the reflection coefficient of the metal film is R, then, in the absence of absorption, the transmission coefficient will be (1  R). Assuming that the reflection coefficient of the sample surface is 1, then no light is absorbed by the sample and the intensity reflected back from the sample will be (1  R). The reflected beam from the sample will then be transmitted back through the metal film, with a transmission coefficient (1  R), and the remaining intensity will be re-reflected. Hence the intensity of the second beam from the sample that is transmitted back through the metal film will now be (1  R)2, as compared with the intensity of the reference beam, R reflected by the metal film. For the two beams to interfere strongly they should be of comparable intensity, R ¼ (1  R)2. Solving this pffiffiffi equation, the required value of R is (3  5)/2, or approximately 0.38. The condition for destructive interference is that the two beams differ in path by (2n þ 1)l/2, so that the phase difference between the beams is equal to p. But the path difference is just twice the separation h of the partially reflecting reference surface from the sample, that is 2h ¼ (2n þ 1)l/2, and the first destructive peak occurs when the two surfaces are separated by l/4. Successive interference fringes correspond to height contours separated by Dh ¼ l/2. Assuming that shifts in an interference fringe are detectable to of the order of 10 % of the fringe separation and that the wavelength of the incident light is in the green, then simple two-beam interferometry should be capable of detecting topological height differences to an accuracy of 20 nm, roughly the same sensitivity as that for phase contrast (Section 3.5.4). An example of two-beam interference fringes from a system of grooved grain boundaries is shown in Figure 3.26. The reference surface is at a slight angle

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Microstructural Characterization of Materials

Figure 3.26

Two-beam interference pattern due to grain boundary grooving.

to the sample surface, and the separation of the fringes in the image corresponds to differences in separation between the reference and sample surfaces of l/2. Interference contrast and sensitivity can be improved by placing a drop of immersion oil between the coverslip and the sample, reducing the effective wavelength by a factor m, the refractive index of the oil. The objective lens NA for good two-beam interference is limited, since for an NA greater than about 0.3 the difference in pathlength for light passing through the periphery of the lens and along the axis is sufficient to destroy the coherency of the beam. The condenser system should be focused on the back focal plane of the objective (Kohler illumination) to ensure that the incident light is normal to the sample surface. It may also be useful to view the specimen in white light, when the orders of the interference fringes correspond to Newton’s colours. These colours correspond to the subtraction of a single wavelength from the white spectrum, generating the complementary colour. For very small separations, the shortest wavelengths interfere in the blue and the corresponding Newton’s colour is yellow. As the separation increases the wavelength that interferes moves towards the green and the complementary Newton’s colour is magenta. Finally, the interfering wavelength moves into the red, and the complementary interference colour is cyan. For still larger separations, the interference conditions move to higher orders of (2n þ 1)l/2 and the colour sequence is repeated, but with increasingly dull contrast as several different wavelengths start to contribute to the interference. A region of the specimen that is in contact with the reference plate then appears white. It is a sobering experience to see just how few points of contact there are between the reference and sample surfaces, even though the reference surface is optically flat and the sample has been well-polished. 3.4.4.2 Systems For Interference Microscopy. The Mercedes (or Rolls Royce) of microinterferometers is that designed by Linnik (Figure 3.27), but it is seldom used. Two identical objective lenses ensure that no path differences are introduced by the optical system. The position of the reference surface can be adjusted along the optic axis and the reference

Optical Microscopy 155

Eyepiece

Beam Splitter

Source

Reference Surface

Matched Objective

Objective Specimen Figure 3.27 Optical system for a Linnik microinterferometer.

surface can be tilted accurately about two axes at right angles and in the plane of the surface, in order to adjust both the spatial separation and the orientation of the interference fringes. The position of the reference surface is adjusted in white light so that, at a small angle of tilt, Newton’s interference colours are observed either side of the white contour marking the line of coincidence of the reference image with the image of the sample. Far simpler systems for interference microscopy are commercially available as standard attachments, but most have a rather limited life, since they rely on half-silvered reflecting surfaces that are easily damaged if brought into contact with a sample. The disadvantage of a metal-coated coverslip (which is cheap and can be regarded as a consumable) is the inability to adjust the separation between the reference surface and the sample or the distance between the interference fringes, but for many applications this is not critical. 3.4.4.3 Multi-beam Interference Methods. As noted previously, in simple two-beam interference a proportion of the light is multiply reflected. With the previous assumption of no absorption loss and a reflection coefficient of 0.38 for the half-silvered coverslip, these multiply reflected intensity losses amount to 24 % of the incident light and result in unwanted background. By increasing the reflection coefficient of the reference surface the proportion of light which is multiply reflected can be increased, until the value is close to 1. With the geometry shown in Figure 3.28, the dependence of the total reflected intensity I is

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Microstructural Characterization of Materials

RT 2

R

θ

R 3T

RT

R 3T 2

R 5T

R 5T 2

Thin film (reference surface)

t T

R 2T

R 4T Specimen

Figure 3.28 Geometry for multiple beam interference.

given by:

3 #2 T 1 4 5 I¼ 4R sin2 ðd=2Þ ð1RÞ2 1þ ð1R Þ2 "

2

ð3:6Þ

where Tand R are the transmission and reflection coefficients, and the reflection coefficient of the sample is now assumed equal to that of the reference surface. The parameter d is given by: d ð2phcosyÞ ¼ ð3:7Þ 2 l where h is the separation of the reference and sample surfaces. If, once again, we assume that there is no absorption in the reference film, so that T þ R ¼ 1, and the intensity collected falls to zero when 2hcosy ¼ nl. The interference fringes that are formed are now localized at the reference surface, and the number of beams which contribute is determined by the angular tilt of the reference surface with respect to the optic axis and the separation between the reference surface and the sample. The width of the black interference fringes depends on the number of beams taking part and this width can be very narrow when compared with the cos2 intensity dependence obtained in the two-beam case. For multiple beam interferometry to be effective, the incident beam must be parallel and the surfaces separated by no more than a few wavelengths. The best patterns have been produced by spin-coating the sample with a thin plastic film and evaporating silver onto the atomically smooth surface of the plastic. Under these conditions the very sharp interference fringes can reveal topological changes in the surface at the nanometre level, and it is quite possible to image growth steps on crystals that are only a few atoms in height (Figure 3.29). Of course, there is a penalty to be paid: since the incident and reflected light must be parallel, and multiple beam interferometry is only possible with a low NA objective, and the lateral resolution is typically no better than 1 or 2 mm. 3.4.4.4 Surface Topology and Interference Fringes. A few more words are in order concerning the information which can be derived using interference microscopy. The small depth of field of the optical microscope means that the region of the object in focus is a thin slice of thickness of the order of 1 mm or less. The image is then a planar projection of this slice of material, and contains information on both the topology and the physical properties

Optical Microscopy 157

Figure 3.29

Multiple beam interference fringes from a polished quartz specimen.

of the surface. The lateral resolution is limited by the wavelength of the light and the NA of the objective lens, but also by the method of surface preparation and the means used to record the image. Interference micrographs contain image contrast based on variations in the phase of the light reflected from the surface. These phase variations may be associated with three quite distinct optical effects. 1. They may be associated with surface anisotropy and best revealed by polarized light (Section 3.4.4). The phase shift arises from differences in the wavelength of light polarized parallel to the two optic axes of the anisotropic sample surface. 2. If they are due to small topological features or spatial inhomogeneities in the optical properties (variations in refractive index), then they may be detected by phase contrast microscopy or differential interference (Nomarski) contrast (Section 3.4.5), which identifies the sign of the phase shift and its approximate magnitude. 3. Finally, they may be due to variations in surface topology, and then interference microscopy is often the most appropriate tool, quantitatively monitoring the separation between an optically flat reference surface and the sample surface. The vertical resolution for topological features in interference microscopy is much better than the usual lateral resolution of the optical microscope, of the order of 20 nm for twobeam interference (or phase contrast) and as little as 2 nm for multiple beam interference. This vertical resolution is only available at the expense of lateral resolution, since the requirements for good interference images limit the NA of the objective lens to of the order of 0.3. In spite of this, quantitative measurements using interference microscopy have proved extremely valuable, and applications range from measurements of grain boundary energy using thermal grooving at grain boundaries, to the height of facets on the surfaces of a growing crystal, and the size of the slip steps in ductile materials. 3.4.5

Optical Anisotropy and Polarized Light

Many samples are optically anisotropic, that is the refractive index, and hence the wavelength of light in the material, are a function of the direction of propagation. Crystals of cubic symmetry are optically isotropic, while those with tetragonal, rhombohedral or hexagonal symmetry are characterized by two refractive indices, parallel and perpendicular to the primary symmetry axis. Crystals of even lower symmetry are characterized by three refractive indices. If a beam of light is incident on an optically anisotropic, transparent

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Microstructural Characterization of Materials

Crystal

Isotropic

Anisotropic

Figure 3.30 A monochromatic beam of light incident on a transparent crystal will propagate as two beams with different refractive indices (corresponding to two different wavelengths) when the crystal is optically anisotropic.

crystal, the beam will be transmitted through the crystal as two beams whose electromagnetic vectors are aligned parallel to the principle optical axes of the crystal in the plane perpendicular to the incident beam (Figure 3.30). The two beams correspond to different wavelengths for the light passing through the crystal. On exiting from an anisotropic crystal the two polarized beams will recombine, but any object viewed through a slice of the crystal will appear doubled, due to the different deflections of the two beams travelling through the slice. A beam of light is said to be polarized when the electromagnetic wave vectors are not randomly oriented perpendicular to the direction of propagation, but are instead aligned in a specific direction. Sunlight reflected at a shallow angle from a car roof is partially polarized, with the direction of polarization in the plane of the roof. ‘Polaroid’ sunglasses only transmit light which is vertically polarized, so the glare from the (approximately horizontal) car roof is cut out by the sunglasses. If the sunglasses are removed and rotated 90 , the glare will be transmitted, since the plane of polarization of the lens is now parallel to that of the reflected glare from the car roof. 3.4.5.1 Polarization of Light and Its Analysis. In the polarizing microscope the incident illumination is plane-polarized by inserting a ‘polarizer’ into the path of the condenser assembly. To avoid any spurious changes in polarization of light during transmission through the optical system the plane of polarization is chosen to be perpendicular to the plane of assembly of the microscope components, A second polarizing element, termed the ‘analyser’, is placed in the path of the imaging lenses and can be rotated about the optic axis of the microscope so that the plane of polarization of the analyser may be at any angle between 0 and 90 to that of the polarizer. When the angle between the planes of polarization of the polarizer and analyser is set at 90 the sample is said to be viewed through crossed polars, and no light reflected from an isotropic sample can be transmitted through the analyser to the final image. When a beam of plane-polarized light is reflected from an optically anisotropic surface, the components of the electromagnetic wave vector reflected from the surface are themselves resolved into two components, parallel to the principle optic axes of the surface

Optical Microscopy 159

(a)

(b)

Plane of polarizer

Resolved components

Rotating wave vector

Principle axes of surface anisotropy

Incident wave vector Figure 3.31 (a) Plane-polarized light is resolved into two components when it interacts with an anisotropic sample. (b) The out-of-phase component then recombines to form an elliptically polarized wave.

(Figure 3.31). Due to the optical anisotropy, these two components are now no longer exactly in phase, and the combined reflected beam therefore has a wave vector whose tip will rotate as the beam propagates, varying in amplitude as it does so. The tip of the amplitude vector viewed in the plane normal to the direction of propagation of the beam then describes an ellipse, and the beam is said to be elliptically polarized. If an analyser with a plane of polarization at 90 to that of the polarizer intercepts this reflected beam, then only that component of the amplitude parallel to the plane of polarization of the analyser will be transmitted. This situation is summarized in Figure 3.32. From Figure 3.32 it is clear that the maximum amplitude reflected from the sample will be transmitted through the analyser when the principal axes of the sample are set at 45 to the crossed polars. Rotating the analyzer will result in more light being accepted, and when the two polars are set parallel, then all the light reflected will be admitted to the final image. To maximize the contrast in polarized light the condenser aperture should be stopped down and the source focused on the back focal plane of the objective (Kohler illumination), so that nearly all the incident light is normal to the sample surface. The NA of the objective lens has a pronounced effect on the contrast, and only at low values of NAwill the path of the reflected light be sensibly normal to the specimen surface. At high values of the NA the light will be collected from the specimen surface over a wide range of angles, and the effects of optical anisotropy are then considerably reduced. 3.4.5.2 The 45 Optical Wedge. If a thin slice of a transparent, optically active, birefringent crystal, such as quartz, is inserted into the optical path between the polarizer and the analyser, and the optical axes of the quartz slice are set at 45 to those of the crossed polars, then the amplitude of the two components of the beam travelling through the quartz crystal will be identical while the phase difference between the two beams at the exit surface will depend on the thickness of the crystal. If the crystal is cut as a wedge and a white light source is used, then the two beams exiting the wedge will generate bands of interference whenever the phase difference Df is equal to p, or more generally, whenever Df ¼ (2n þ 1)p. The path difference between the two waves travelling through the quartz crystal

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Microstructural Characterization of Materials

Plane of polarizer

Incident amplitude

Resolved components reflected from the sample

Plane of analyzer

Amplitude accepted by analyzer

Figure 3.32 A plane-polarized beam of light incident on an optically anisotropic surface will be reflected as two beams with wave vectors parallel to the principal optic axes of the surface. These combine to form an elliptically polarized reflected beam, which is resolved by the analyser, whose plane of polarization is set at 90 to that of the polariser.

Dd is related to the difference in the refractive index in the two principal directions and the wavelength of the incident light. If the refractive index Z is given by Z ¼ l0/l ¼ c/v, where l0 is the wavelength in space, l is that in the crystal, c is the velocity of light in space and v the velocity of light in the crystal, it follows that: Dd ¼ tðZ1 Z2 Þ ¼

l0 D
2p

ð3:8Þ

where t is the thickness of the crystal. For each thickness of crystal in the wedge there will be a specific wavelength in the incident white light that will interfere destructively and be removed from the transmitted beam, as discussed previously (Section 3.13). The shortest wavelengths of the visible spectrum are removed first (violet), leaving the transmitted band of light pale yellow, this is followed by ‘blue’ interference, leading to a magenta band of transmitted light, then green followed by red (transmitting cyan). For thick crystals Dd ¼ (2n þ 1)l0/2, where n is an integer, and the same order of ‘interference’ colour bands will be repeated, although with decreasing brilliance for each successive order, since for the thicker regions the interference condition at each thickness can be fulfilled for more than one wavelength in the visible range.

Optical Microscopy 161 3300

Crossed Polars

Parallel Polars

6th order

3000 Verypale blue

Verypale yellow

Optical path difference (nm)

2700

5th order

2400 Pale blue grey

Pale yellow-green

2100

4th order

1800 Grey blue

Greenish-yellow

1500

3rd order

1200 Violet-red

Yellow-green

900

2nd order

600 Violet

300

Greenish-yel

1st order

0

Figure 3.33

Newton’s thickness interference colours in crossed and parallel polars.

This sequence of interference colours was first noticed by Newton, and is referred to as Newton’s colours. Using parallel, rather than crossed, polars the colours are inverted (Figure 3.33). 3.4.5.3 White Light and The Sensitive Tint Plate. Instead of a quartz wedge it is possible to insert a thin, transparent slice of quartz of uniform thickness, selected to introduce a phase difference equivalent to destructive interference of green light, tinting the light accepted into the imaging system by the analyser magenta. If the sample is optically isotropic, reflection from the surface introduces no further phase shift and, when viewed under white light using this sensitive tint plate between crossed polars, the microstructure appears coloured mauve or magenta. However, if the surface is anisotropic, and so introduces an additional phase shift into the reflected beam, then this phase shift will result in a colour shift along the sequence of Newton’s colours. A positive phase shift will increase the wavelength for destructive interference, and that area of the specimen surface will now appear more blue or cyan. Any negative phase shift will result in destructive interference at a shorter wavelength and a colour shift in the image of the sample towards the yellow. Rotating the sample by 90 will reverse the sign of these phase shifts introduced by optically anisotropic features present in the microstructure. The sensitive tint plate is a very powerful tool for exploring the optical anisotropy of a sample viewed in reflection, and is also very important when studying anisotropic

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Microstructural Characterization of Materials

Figure 3.34 Crystalline polymers viewed in transmission by polarized light reveal a wealth of detail associated with the crystallization process. The arms of the black crosses are parallel to the planes of the polarizer and analyser in the microscope.

crystalline polymers in transmission. In Figure 3.34 nodules of a crystalline polymer contain well-ordered microcrystals. The contrast from each nodule viewed in polarized monochromatic light includes a black, ‘Maltese cross’ that is aligned along the axes of the polarizer and analyser with dimensions that reflect the NA of the objective lens. Between the arms of the crosses the alignment of the microcrystals around the nucleation centre for each nodule is clearly visible. The high optical anisotropy ensures excellent contrast, even when working at high resolution and with large NA objectives. 3.4.5.4 Reflection of Polarized Light. A few further words are necessary concerning the interaction of polarized light with a reflecting surface. As noted earlier (Section 3.4.4), unpolarized light is partially polarized when it is reflected at an angle, with the direction of polarization perpendicular to the plane containing the incident and reflected beams. Furthermore, most surfaces prepared for microscopic examination are covered by a surface film. It follows that linearly polarized light incident on the surface is likely to undergo some phase change and, to some extent, become elliptically polarized. The cause may be topographic, for example, fine surface facets or aligned grooves which impart optical anisotropy to an otherwise isotropic surface, or it may be associated with the conditions of illumination (a primary beam incident at a glancing angle, as in dark-field illumination) or it may result from the conditions of collection of the reflected light (a large NA objective accepting a wide cone angle). In many cases complete extinction is not obtained in crossed polars, and extinction does not necessarily mean that the sample itself is anisotropic. Alternative explanations could be loss of linear polarization resulting from an oblique angle of incidence or the existence of ‘anisotropy’ due to surface topology. Changing the plane of focus of the source by adjusting the condenser lens setting should give an indication of whether or not the illuminating conditions are responsible for any apparent anisotropy.

Optical Microscopy 163

3.4.6

Phase Contrast Microscopy

In phase contras microscopy, intensity variations in the image are due to interference between a specularly reflected beam and an elastically scattered beam from the same sample surface. Phase contrast is important in the histological examination of organic tissues viewed in transmission, when small differences in the scattering from features giving weak contrast need to be amplified, but the technique is also useful for revealing fine topological features in reflection microscopy. If light is reflected from two neighbouring regions which differ slightly in height by a small amount h, that is much less than the wavelength, then the reflected beams will be out of phase by a small phase angle:
 h ð3:9Þ df ¼ 2p l If h is sufficiently small, the phase shift between the two specularly reflected wave vectors is equivalent to a small elastically scattered amplitude whose wave vector is sensibly out of phase with the specularly reflected beams by p/2 (Figure 3.35). The problem is to convert this very weak scattered signal into an observable intensity difference. This requires both that the scattered amplitude be comparable with the amplitude of the specularly reflected signal and that the phase difference be shifted by an additional p/2, so that constructive or destructive interference may take place. One solution is illustrated in Figure 3.36. In this figure the source of light is an annular condenser aperture which ensures that the path length to the specimen is sensibly identical for all rays. This annular aperture is focused on the back focal plane of the objective. An intermediate, auxiliary lens is used to bring the image of the source, reflected by the sample

k1

k2

Beam 1

Beam 2

h h<<λ δφ

k2 k1

|k| = 1/λ

Diffracted amplitude

Figure 3.35 Phase contrast is based on the light scattered from small features associated with a difference in height h. The scattered amplitude is approximately p/2 out-of-phase with the specularly reflected beams.

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Microstructural Characterization of Materials

Second Image of Aperture

Eyepiece Phase Plate Auxiliary Lens Annular Aperture Lamp

Half-Silvered Mirror Condenser Lens First Image of Aperture Objective Lens Specimen

Figure 3.36 An optical system for phase contrast reflection microscopy.

surface, into focus on a grooved phase plate which is placed behind the final imaging lens. The diameter and width of the groove in the phase plate is chosen to exactly match the annular source aperture, so that specularly reflected light only passes through the groove, while the scattered light is transmitted through the ungrooved portion. The depth of the groove is selected to introduce the additional required phase shift of p/2, so that the scattered signal now either reinforces (þp) or interferes (p) with the specularly reflected signal. That is ‘bumps’ and ‘hollows’ in the surface topology will appear either brighter or darker than the background in the image. To ensure that the amplitude of the specularly reflected signal is comparable with that of the scattered signal, the grooved region must be coated with an absorbing layer that reduces the directly transmitted amplitude by approximately 90%, ensuring that the interference contrast will be clearly visible. A series of phase plates may be provided, especially for use with transmission samples, having phase shifts of (2n þ 1)p/2. These phase plates are particularly useful in studying biological cell and tissue samples in which small differences in refractive index can be characterized by using the phase contrast microscope. Bearing in mind that the height (or phase) differences must be small, and assuming that a reasonable value of h/l to achieve this is 0.1, it follows that the phase contrast microscope will have no difficulty in picking up variations in surface topology of the order of 10 % of the wavelength, or of the order of 50 nm. In practice, surface steps of the order of 20 nm are readily detectable. 3.4.6.1 Normarski or Differential Interference Contrast. Nomarski or differential interference contrast is based on a rather different concept that provides a simple alternative to reflection phase contrast microscopy. A double quartz wedge is inserted into the optical system after the polarizer in the 45 position and illuminates the sample with two slightly

Optical Microscopy 165

displaced beams that are separated in the object plane by a small lateral shift. After reflection from the sample the light collected by the objective is brought back into registry by a second double prism and the image is viewed through the analyser. Since the path length of the twin beams is identical, any residual difference in phase must be due to microstructural features that are on the scale of the separation of the twin images. These are usually height changes, for example, those associated with slip steps or boundary grooves. The contrast obtained in Nomarski interference reflects these small topological differences, and the interpretation is very similar to phase contrast.

3.5

Working with Digital Images

In this section we introduce some of the terminology associated with digital imaging, and compare analogue with digital data. Much of the discussion is quite general, and applies equally to data focused by a system of lenses in an optical or electron microscope, as well as to images derived from raster scans, as in a television camera or scanning electron microscope. 3.5.1

Data Collection and The Optical System

In a simple optical or transmission electron microscope the image intensity is a continuous function of the x–y coordinates in the image plane, I ¼ f(x,y). The number of photons or electrons contributing to the signal is usually large enough to be able to ignore the statistics of image formation, although for damage-sensitive materials this is not necessarily the case, while for some weak signals the signal-to-noise ratio may also be a problem. The signal is dependent on the probe energy (the wavelength of the light used or the accelerating voltage for the electrons), so that we should really write: Il ¼ fl(x, y). For analogue optical images obtained in monochromatic light and transmission electron microscope images this equation is adequate, however recording optical images in colour relies on filtering the intensity through colour filters. These filters may be red, green and blue (RGB) for positive images, or yellow, magenta and cyan (YMC) for negative images. That is, the analogue intensity is dependent on three colour parameters in addition to the two spatial parameters. For images recorded in monochromatic light, only one wavelength is involved, but this wavelength still needs to be specified. 3.5.2

Data Processing and Analysis

We first consider the format of the image that we wish to process and analyse. The variation in intensity with position in the image plane of an optical system I ¼ f(x, y) represents information from the object (the specimen) which has been transmitted by radiation that has passed through the optical system. The image intensity is a continuous function in x and y. When recording in an analogue data format, such as a photographic negative, this continuous function is modified by the response of the recording medium, but it is still a continuous function. In a digital, rather than analogue, data format, the intensity is no longer a continuous function in the x–y image plane, but rather a discrete intensity function. That is, the image

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Microstructural Characterization of Materials

stored in a digital format exists as discrete picture elements, or pixels, in which each pixel stores information on the intensity of the image at a specific x,y location in the image. The location x,y is the position of a pixel, whose dimensions are determined by the total area of the recorded image and the total number of pixels. It is important to question the equivalence between the area sampled by a specific pixel and the site on the sample, propagated through the optical system that generates the signal. There are two factors to be considered. The first is the resolution of the microscope and the second is the signal collection efficiency. To be sure that the full resolution of the microscope is available in a digital data collection system, the pixel size needs to be smaller than the resolution projected onto the image plane, namely Dx,Dy < Md, where M is the magnification. In practice a 3 · 3 array of pixels is usually selected to represent a feature that is just resolvable, and this ensures that no resolution is lost in the digitized image. Images are often recorded in a rectangular format (usually for aesthetic reasons), so the number of pixels along the x and y axes will not be the same. The recommended aspect ratio of width to height for television and video equipment is 4:3, but CCTV cameras are usually 1:1, while a widescreen format (high definition television, HDTV) is 16:9. It follows that an image recorded for one format is often distorted when transferred to another format. In scanning electron microscopes the magnification on the x and y axes is seldom identical. The magnification may also vary from the centre to the periphery of the field of view. A square grid will then appear either ‘barrelled’ or as a ‘pin-cushion’, depending on whether the magnification is larger or smaller near the centre of the scanned area. Tilting the sample, which can usually be done without losing the focus in SEM, always results in some foreshortening perpendicular to the tilt axis. It follows that scanned images may have a different magnification in the x and y directions that requires either calibration or correction for quantitative work. The eye, as we noted earlier, only exercises its full resolution (
0.2 mm) over a limited area of the retina, but the eye typically scans over distances of the order of 20 cm at the near point. It follows that the minimum number of pixels to be recorded for a digital image is 106. In dynamic imaging, the eye is very good at following changes in the image and rather fewer pixels can be used for each frame, but for colour images the digital system relies on dyes to select the RGB components in neighbouring pixels. Typically, this is done in a 2 · 2 group of four pixels, in which two diagonal components are red and blue (R and B), while the other two are both green (G). 3.5.3

Data Storage and Presentation

So far we have only considered digitizing the signal with respect to its location in the image, not the digitizing of the intensity recorded from a given pixel, that is, from a given location. We need to be quite clear about the terminology we are using. The intensity is the signal acquired from a given pixel at a specific location, x,y. The brightness, however, is expressed as intensity per unit area, and is therefore the intensity summed over the total number of pixels that define an area of interest and divided by the area covered by these pixels. To identify differences in contrast, we therefore need to ensure that the pixels sample the intensity at spatial intervals which are appreciably smaller than the distances that separate the variations in intensity that determine contrast. In other words, a sufficiently high density of pixels will preserve the spatial frequencies in the original (analogue) image that contain

Optical Microscopy 167

this information. The Nyquist criterion requires that the pixel sampling interval corresponds to twice the highest spatial frequency we wish to record, while Shannon’s sampling theory states that the sampling interval should be no more than one-half the size of the resolution limit of the microscope optical system. For all practical purposes these two criteria are equivalent, but the reader should also remember that there will be a noise limit that restricts the significance of the differences in contrast (see below). 3.5.3.1 Data Storage Systems. Two quite separate factors will limit the useful range of intensity that can be detected and recorded by any system. The first is the lower limit set, either by background noise or by the failure of the system to retain a signal at low doses, while the second is the upper limit set by the saturation of the detection or recording system. In photography, reciprocity failure at low doses and precipitation overlap at high doses are the limiting conditions. In addition, the emulsion excitation process and the definition of contrast dictate that the response curve for an emulsion is best plotted on a curve of density (the logarithm of the ‘blackening’) against the logarithm of the dose (the intensity of the signal multiplied by the exposure time). The useful range is then the linear portion of this curve (Figure 3.18), and the slope of this region determines the contrast response, or g of the film. The analogous curve for digital recording technologies is very similar, but in this case the low dose limit is determined by signal noise, while the high dose limit is due to saturation and ‘blooming’ of the detector, that is, excitation of neighbouring pixels by charge overflow. Two systems are in common use. CCD detectors collect an electrical charge excited by the image signal formed in the plane of the detector. When exposure is complete, the accumulated charge is read off from each pixel in sequence. Pixel sizes may be anything from a few micrometres up to tens of micrometres and the frame size (the total number of pixels in the array) is also variable. However, a typical frame size would be some 20 mm across and may contain 107 pixels or even more. The second digital image system is the CMOS detector. In this system the pixel signals are read out line by line, rather than as one complete frame of data. The CMOS system is essentially a ‘camera’ on a single silicon chip, with the read-out electronics surrounding the rectangular pixel array. Both CCD and CMOS detectors are used for optical image recording and data collection. At the time of writing, the CMOS technology is inherently faster, and is slowly replacing CCD technology in optical image recording, even though the total number of pixels in a large array is still less by almost an order of magnitude. Certainly for dynamic, real-time recording the CMOS technology is preferred. In electron microscopy the CCD technology is almost exclusively used, probably because of radiation damage susceptibility under the high energy electron beam. As noted above, the use of dye-filter arrays over the detectors allows both systems to be used for colour photography. 3.5.4

Dynamic Range and Digital Storage

The eye is capable of distinguishing about 20 intensity levels in a grey scale image, as well as an extraordinarily wide range of colour tints and shades (many thousands). However, no two individuals have precisely the same colour response. Moreover, what we see in nature is often impossible to reproduce in any recorded image (think of the colours of a peacock’s tail, a butterfly’s wing or the iridescent carapace of a beetle). In recording a ‘true’ image,

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Microstructural Characterization of Materials

Figure 3.37

Bit depth and grey levels for binary images.

especially a colour image, we have an almost impossible task. What we see through the eyepiece of an optical microscope will always appear more vibrant than what we are able to record on a positive colour transparency or reproduce on a digital colour monitor. The positive colour print is an even poorer record than a transparency of what we saw through the eyepiece of the microscope. Moreover, the tonal values will in each case be different, and will vary, both from one recording system to another and with the settings employed by the operator. In both digital and analogue image colour observation and recording, the range of shades and tints is limited by the performance of the phosphors, dyes and pigments used. Let us start with the grey scale. A black and white image is referred to as a 1 bit binary image (2n, where n is the number of bits). Every pixel in the image is registered as either black or white, as in a newspaper photograph. If there are four levels of intensity recorded per pixel, then this is a 2 bit image, and 2n grey levels per pixel can be stored in n bits of image data. Most systems store many more than the 4 or 5 bits needed to record all the grey levels the eye can detect (Figure 3.37). Colour can add a great deal to the recorded image, and usually 8 bit data are stored, corresponding to 256 intensity levels in each of the three colours red, green and blue (RGB). When processing the image it is desirable to manipulate the intensity levels selected for an image in order to make full use of the range of levels that can be detected by the eye. This is true of both grey scale and colour digital images. Digital image data storage is available in a somewhat confusing variety of computer formats, for example those labelled GIF, TIFF (tagged image file format), BMP (Bitmap) or JPEG (Joint Photographic Experts Group) (Table 3.1). These all differ in their digital data storage algorithms, and correspond to compressed data files. Usually, microscopists will select the storage format which is convenient for the software they are using. However, all of the formats compress the data unless we intentionally disable the compression option during file storage. Compression is

Optical Microscopy 169 Table 3.1 Pixel dimensions of different image sizes, and their relative file size when stored in different compression modes. Pixel dimension 16 · 16 64 · 64 128 · 128 256 · 256 512 · 512

Grey scale (8 bit) 2k 6k 18 k 66 k 258 k

Bitmap (24 bit)

JPEG (24 bit)

TIFF (24 bit)

2k 13 k 49 k 193 k 769 k

2k 5k 12 k 22 k 52 k

2k 13 k 49 k 193 k 770 k

convenient when storing or transferring large files, but we should remember that there is no such thing as compression without loss of data (Figure 3.38). Finally, a few words about printing digital images are in order. Most printers are halftone, that is they print a dot-matrix array. The tones depend only on the density of dots per unit area of the printed page. Two types of half-tone printers are in common use, ink-jet printers in which a droplet stream is ejected by a piezoelectric actuator from an ink reservoir that contains either black or a coloured ink. The coloured dyes may be red, green and blue, or yellow, magenta and cyan. The ink droplets are then absorbed and dry on the paper with very little spreading. Laser printers use a totally different principle, creating a separate charge pattern on a photoelectric surface and transferring toner of the required colour onto this surface. Laser printers are faster, but more expensive, but the inks for ink-jet printers are more expensive. In both cases it is the cost of the dyes and toners that largely determines the economics of the process.

Figure 3.38 Compression algorithms reduce the file size, but may also introduce noise into the image.

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Microstructural Characterization of Materials

Dye sublimation printers sublimate solid dye from a source which is located very close to a special paper, followed by heating, which provides a continuous tone print. Dye sublimation printers are expensive and primarily used for commercial applications. Wax printers use a liquid source of coloured wax to provide a continuous tone print, and cost about the same as a good colour laser printer. As noted earlier, the printed image always compares poorly with what the eye observes directly in the microscope, or even what can be seen on a computer monitor, for which an array of coloured phosphor pixels is excited electronically. Nevertheless, the technology continues to improve, and the gap in image quality between the observed and the recorded image is shrinking. Photographic recording is no longer an essential component of microscopic investigation. Digital images are capable of recording all image detail, and digital data files can be processed either to optimize the printed presentation of image information or to quantitatively analyse the microstructural information.

3.6

Resolution, Contrast and Image Interpretation

Finally, we summarize some of the problems and pitfalls associated with the interpretation of microstructure based on optical images. It should be clear by now that the observed or recorded image of a selected sample only contains information on the microstructure of the material if the observer understands the sequence of processes used to obtain the image: 1. preparation of the sample; 2. imaging in the microscope; 3. observing and recording the image. The ultimate resolution of the features observed in a recorded image includes contributions from all three of the above processes. The width of grain boundary grooves resulting from etching may limit the resolution in fine-grained microstructures far more than the NA of the objective lens or the wavelength of the light used. However, so may the grain size of a photographic emulsion used to record the image and the photographic process used to develop the negative and print the final image. In digital imaging, the pixel size in relation to the optical resolution, the number of pixels available and the number of grey levels used to record image brightness will place limitations on subsequent data processing and image evaluation. Contrast in the optical microscope can be due to the surface topology, the optical properties of the sample (reflection coefficient and/or optical anisotropy), or the presence of a surface film. The contrast observed will depend on the wavelength and may be associated with variations in reflected amplitude, or result from phase shifts, as in the changes in colour observed with a sensitive tint plate. The enhancement of contrast, either by photographic or digital processing, may increase the g in the field of view, but reduce the number of grey levels and result in a loss of information. Image enhancement using phase contrast or differential, Normarski interference may result in some loss of lateral resolution. Contrast is generally better for images viewed with a lower NA objective, but only at the cost of reduced resolution.

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Finally, resolution by itself is useless without image contrast, but poor resolution, just for the sake of good contrast, is equally self-defeating. The eye is probably still the best judge of what constitutes a good compromise, both in image observation and for a recorded analogue or digitized image, but computer-assisted data processing can provide the microscopist with reliable and objective assistance.

Summary The optical microscope is the tool of preference for the microstructural characterization of engineering materials, both because of the wealth of information available in the magnified image and the ready availability of high quality microscopes, specimen preparation facilities and inexpensive methods of image recording and data processing. The visual impact of the magnified image is immediate, and its interpretation is in terms of spatial relationships which are already familiar to the casual observer of the macroscopic world. Geometrical optics determines the relationship between the object placed on the microscope stage and its magnified image. The ability to resolve detail in the image is limited primarily by the wavelength of the light used to form the image, together with the angle subtended by a point in the object plane at the objective aperture. The aperture of the eye and the range of wavelengths associated with visible light limit the resolution of the eye to approximately 0.2 mm. That is, the unaided eye can distinguish between two features at a comfortable reading distance (30 cm) if they are separated by 0.2 mm. The numerical aperture (NA) of an objective lens is the product msina, where m is the refractive index of the medium between the lens and the object and a is the half-angle subtended at the objective (the angular aperture of the lens). Values of NA vary from of the order of 0.15 for a low magnification objective used in air to about 1.3 for a high-power, oilimmersion objective, leading to a limiting (best possible) resolution of the order of half the wavelength of visible light, about 0.3 mm. A sharply focused image will only be obtained if the features to be imaged are all in the plane of focus, and high NA lenses require accurate focusing. It follows that specimens to be imaged at the best resolution must be accurately planar. This limited depth of field is the primary reason why samples of opaque materials must be polished optically flat, while those of transparent materials must be prepared as thin, parallel-sided sections. The components of the optical microscope include the light source and condenser system, the specimen stage, the imaging optics and the image recording system. Each of these components has its own engineering requirements: the intensity and uniformity of the light source; the mechanical stability and positional accuracy of the specimen stage; the optical precision and alignment of the imaging system, and the sensitivity and reproducibility of the image data-recording system. While in the past photographic recording was the primary option, the rapid development of charge-coupled device (CCD) and complementary metal oxide semiconductor (CMOS) digital image data systems has led to the development of a new range of microscopes in which digital recording and computeraided image enhancement have, to a large extent, made the photographic darkroom obsolete.

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A major consideration in the application of the optical microscope is the selection and preparation of a suitable sample. In the first place, the sample must be representative of the features which are to be observed. That is, the sampling procedure must take account of both inhomogeneity (spatial variations in the features and their distribution) and morphological anisotropy (orientational variations, as in fibrous or lamellar structures). Secondly, preparation of the sample surface must reveal the intersection of bulk features with the plane of the section without introducing artifacts (such as scratches or stains). In most cases, surface preparation is a two-stage process. The first step is to prepare a flat, polished, mirrorlike surface, while the second is to develop contrast by the use of suitable chemical etchants, solvents or differential staining agents. Image contrast is a sensitive function of the mode of operation of the microscope, but most engineering materials are examined by reflection microscopy. The contrast then reveals local differences in the absorption and scattering of the incident light. Chemical etchants commonly develop topographic features which scatter the incident beam outside the objective aperture, but they may also differentially stain the surface, so that some features absorb more light than others. In many cases, the different phases and impurities present (‘inclusions’) will also give contrast that is associated with differences in the reflectivity of the different phases for the incident light, quite independent of the action of an etchant. The optical imaging conditions can also be controlled in order to enhance the image contrast. A ‘dark-field’ image is formed by collecting the scattered light from the object, rather than the specularly reflected light used to form a ‘bright-field’ image. If the specimen is illuminated with plane-polarized light, then the changes in polarization that accompany the interaction of the light with the specimen can be analyzed using ‘crossed polars’ that convert any rotation of the plane of polarization into variations in image intensity. An optical wedge or sensitive tint plate can be used to introduce a controlled phase shift and further enhance the image contrast in polarized light, often yielding quantitative information on the optical properties of the sample and its constituents. In phase contrast reflection microscopy small (>20 nm) topological differences at the sample surface can be converted into variations in image intensity, while in transmissionmicroscopy differences in refractive index can be similarly imaged. By combining the light reflected from the sample surface with that reflected from an optically flat reference surface it is possible to obtain two-beam optical interference, in which the interference fringes again reflect the topology of the sample surface. Multiple beam interference allows very small differences in the height of surface features to be detected with a sensitivity of a few nanometre, and this vertical ‘resolution’ is several orders of magnitude better than the lateral image resolution, which is limited by the NA of the objective lens. The quality of digital and analogue recorded images depends on three independent factors: the preparation of the sample, the optical imaging in the microscope; and the system used to record and process the final image data. The observed image in the microscope, the image recorded on a transparency, the digitized image viewed on a computer screen and the printed image reproduced in a published text will all differ in quality, reflecting the different technologies being used. The eye is the best judge of that combination of resolution and contrast which yields the most information, but digital image data processing software now provides the microscopist with some excellent tools for making that judgement and ensuring that results will be recorded and presented with minimal loss of image quality.

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Bibliography 1. Metals Handbook, Volume 7: Atlas of Microstructures of Industrial Alloys, American Society for Metals, Metals Park, OH, 1972. 2. J.L. McCall and P.M. French (eds), Metallography in Failure Analysis, Plenum Press, London, 1978. 3. J.B. Wachtman, Characterization of Materials, Butterworth-Heinemann, London, 1993. 4. J. Russ, The Image Processing Handbook, 4th edn, CRC Press, Boca Raton, FL, 2002.

Worked Examples We begin by considering the influence of the polishing process on the surface finish of 1040 steel. Small samples are cut from a large block and embedded in a thermoplastic mount. Granules of the thermoplastic are poured around a specimen that has been mounted in a metal die. The granules are then compacted at moderate pressure and temperature. Alternatively, and to avoid any damage to the specimen, a thermosetting resin can be cast into the mould containing the specimen and then cured at room temperature. Figure 3.39 compares optical micrographs of a 1040 steel surface at different stages of polishing. A ‘good’ polish results in a planar surface with no scratches visible under the

Figure 3.39 Optical micrographs of 1040 steel after polishing with a sequence of diamond grits: (a) Rough-grinding to achieve a planar surface; (b) after polishing with 6 mm diamond grit; (c) after polishing with 1 mm diamond grit; (d) after polishing with 1/4 mm diamond grit.

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Figure 3.40 The same 1040 steel from Figure 3.39 after etching for different lengths of time in a very dilute nitric acid: (a) under – etched; (b) a good etch; (c) over etched.

Optical Microscopy 175

optical microscope. A final polish with 1/4 mm diamond particles removes most surface artifacts, and provides a specimen that is ready for etching [Figure 3.39(d)]. The choice of etchant must be consistent with the information sought from the sample. A good compendium of etchants for metals and alloys is the Metals Handbook that describes the experimental sample preparation procedures and gives examples. For our 1040 steel we wish to determine the average grain size and shape, so we chose a very dilute solution of nitric acid in ethanol (nital). Figure 3.40 shows optical micrographs of the polished steel after etching in nital for increasing times. The two main variables for a given etchant are time and temperature, and the higher the temperature the faster the etching rate. Too light an etch gives poor contrast and fails to reveal the microstructure adequately, while overetching results in pitting of the sample surface and loss of resolution. Alumina is an inert oxide, and the polished surface of an alumina sample is difficult to prepare for optical microscopy by chemical etching, although some chemical etchants that are based on molten salts or hot, concentrated acids do exist. Instead, microstructural contrast in alumina and many other ceramics is commonly developed by thermal etching. In this process a polished specimen is heated to a temperature at which surface diffusion can occur. For example, the intersection of a grain boundary with a polished surface is thermodynamically unstable, since the surface tension forces are not in equilibrium [Figure 3.41(a)]. If surface diffusion takes place, then groove formation along the line of intersection of a grain boundary with the free surface can reduce the total surface energy [Figure 3.41(b)]. When viewed under the microscope, the ‘thermally etched’ grain boundary grooves will scatter light outside the objective lens aperture, and provide the contrast needed to identify the grain boundaries.

(a)

γA

γB

GrainA

Grain B γGB

non-equilibrium

γA

θ

φ γB

(b)

GrainA

γGB

Grain B

equilibrium Figure 3.41 A grain boundary intersecting a polished surface is not in equilibrium (a). At elevated temperatures surface diffusion forms a grain boundary groove to balance the surface tension forces (b).

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Figure 3.42 Polished alumina thermally etched at 1200  C for 30 min shows poor contrast and fails to reveal all the boundaries (a). Thermal etching for 2 h at the same temperature clearly reveals all the grain boundaries but with some loss of resolution for the finest grains (b).

Mass transfer by surface diffusion is a function of both temperature and time, and insufficient thermal etching will result in poor contrast for the alumina grain boundaries [Figure 3.42(a)]. Increased thermal etching increases the depth and width of the grooves, improving the grain boundary contrast, but with some loss of resolution [Figure 3.42(b)]. The same diffusion processes promote grain boundary migration and grain growth, changing the bulk microstructure, while exaggerated ‘over-etching’ will result in ‘wide’ grain boundaries and a serious loss of resolution.

Problems 3.1. What optical properties of an engineering material are important in determining the preparation of a sample for optical microscopy? 3.2. Are short-sighted individuals blessed with ‘better’ resolution? Discuss! 3.3. The Raleigh resolution criterion assumes two point sources of light and implies unlimited contrast in the image. What factors are likely to prevent the attainment of the Raleigh resolution? 3.4. The better resolution obtained with a high numerical aperture objective is accompanied by reduced contrast and depth of field. Why? 3.5. Mechanical stability is a necessity for the specimen stage of an optical microscope. Compare (quantitatively) the mechanical stability required parallel to the optic axis of the microscope with that required in the plane of focus. 3.6. Photographic recording of the optical image is being replaced by digital recording. Why? 3.7. What sections from the following components would you select for examination in an optical microscope? a. A multilayer capacitor, b. Steel wire, c. A ‘nylon’ plastic sheet, d. A sea shell

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3.8. Chemical etching is often used to develop image contrast on a polished sample surface. Give three examples of contrast developed by chemical etching. 3.9. Give one example of a microstructure where you consider that dark-field illumination would give more information than bright field illumination, and justify your opinion. 3.10. What is the orientation of the principal axes of an optically anisotropic sample with respect to the axis of polarization of the incident light when the observed intensity is a maximum using crossed polars? Explain! 3.11. What is a sensitive tint plate and when would you consider it useful? 3.12. Two-beam interference images have been used to analyze the shape of grain boundary grooves formed by thermal etching. Derive an expression to show how the spacing of the fringes and the angle they make with the boundary will affect the vertical resolution.

4 Transmission Electron Microscopy The electron microscope extends the resolution available for morphological studies from that dictated by the wavelength of visible light to dimensions which are well into the range required to image the lattice planes in a crystal structure, that is from of the order of 0.3 mm to of the order of 0.1 nm. The first attempts to focus a beam of electrons using electrostatic and electromagnetic lenses were made in the 1920s, and the first electron microscopes appeared in the 1930s, pioneered primarily by Ruska, working in Berlin. These were transmission electron microscopes, intended for samples of powders, thin films and sections prepared from bulk materials. Reflection electron microscopes, capable of imaging the surfaces of solid samples at glancing incidence, made their appearance after the Second World War but these were soon superseded by the first scanning electron microscopes (Chapter 5), and these were almost immediately combined with the microanalytical facilities available in the microprobe (Chapter 6). Sub-micrometre resolution was demonstrated on the earliest transmission electron microscopes that had been manufactured in Europe, and later in Japan and the USA. The early developments in electron microscopy are an international success story: in the immediate post-war period commercial transmission instruments were manufactured in Germany, Holland, Japan, the UK and the USA. The first scanning instruments were made in the UK, while the first microprobe was a French development. As we will discuss in ˚ ngstrom, resolution is currently available Section 4.1.2, sub-nanometre and even sub-A from advanced transmission electron microscopes. However, perhaps the most important characteristic of the transmission electron microscope is that it combines information from objects in real space at excellent resolution, with information from the same object obtained in reciprocal space, that is, electron diffraction patterns can be recorded (Chapter 2). Together with microanalytical techniques that can be integrated into the same instrument (Chapter 6), this makes the transmission electron microscope one of the most versatile and powerful tools available for microstructural characterization. In this chapter we will outline the basic principles involved in focusing an image with a high energy beam of electrons before discussing the factors that limit resolution in electron Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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microscopy. We will then compare the requirements for transmission electron microscopy (TEM) before discussing in detail the specimen preparation procedures, and the origin and interpretation of contrast in the transmission electron microscope image. In Chapter 5 we will extend this discussion to scanning electron microscopy (SEM). The transmission electron microscope is in many ways analogous to a transmission optical microscope, but the microscope is usually ‘upside down’, in the sense that the source of the electron beam is at the top of the microscope while the recording system is at the bottom (Figure 4.1). An electron gun replaces the optical light source and is maintained at a high voltage with respect to earth (typically 100–400 kV). A number of different electron sources have been developed, but the basic design of these different electron sources is similar (Figure 4.2). In a thermionic source, electrons are extracted from a heated filament at a low bias voltage that is applied between the source and a cylindrical polished cap (the Wehnelt cylinder). This beam of thermionic electrons is brought to a focus by the electrostatic field and accelerated by an anode held at earth potential beneath the Wehnelt cylinder. The beam that enters the microscope column is characterized by the effective source size d, the divergence angle of the beam a0, the energy of the electrons E0 and the energy spread of the electron beam DE. In general, the temperature of the source limits the energy spread of the electron beam DE to approximately kT. A smaller source size d improves the beam coherence, and hence the contrast that can be obtained from phase shifts due to interactions of the beam when traversing a thin specimen (Section 4.3.3).

Figure 4.1 As in the transmission optical microscope, the transmission electron microscope includes a source, a condenser system, a specimen stage, an objective lens and an imaging system, as well as a method for observing and recording the image.

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Figure 4.2 Schematic drawing of (a) a tungsten filament thermionic electron source and (b) a LaB6 tip for enhanced thermionic emission. A sharp tungsten tip (c) is for a common field emission gun source. In thermionic sources the filament or tip is heated to eject electrons, which are then focused with an electrostatic lens (the Wehnelt cylinder) (d). In field emission guns (e) the electrons are extracted by a high electric field applied to the sharp tip by a counter electrode aperture, and then focused by an anode to image the source.

Three electron sources are in common use: a heated tungsten filament is capable of generating electron beam current densities of the order of 5 · 104 A m
2, from an effective source size, defined by the first cross-over of the electron beam, that is some 50 mm across. Thermionic emission temperatures are high, resulting in an appreciable energy spread of the order of 3 eV, and the coherency of the beam is also limited. A lanthanum hexaboride (LaB6) crystal can generate an appreciably higher beam current, about 1 · 106 A m
2, at much

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lower temperatures. Cerium hexaboride (CeB6), is now also in use. The energy spread of the beam is significantly reduced to about 1.5 eV, although the vacuum requirements are more stringent compared with a tungsten filament source. A ‘cold’ field emission gun, in which the electrons ‘tunnel’ out of a sharp tip under the influence of a high electric field, can generate current densities of the order of 1 · 1010 A m
2. The sharp tip of the tungsten needle that emits the electrons is no more than 1 mm in diameter, so the effective source size is less than 0.01 mm and therefore quite coherent. Moreover the temperature of this source is low, and an energy spread of 0.3 eV is typical. More often, a ‘hot’ field emission source replaces the ‘cold’ source. In this case a tungsten needle is heated to enhance emission by electron tunnelling, a process termed Schottky emission. ‘Hot’ Schottky sources often contain zirconium in order to reduce the work function. They have slightly greater energy spread and a larger effective source size than cold field emission sources, but they are more stable and reliable and have a longer useful life and less stringent vacuum requirements. Recently, the introduction of electron beam monochromators has further reduced the energy spread from an electron source to less than 0.15 eV, although at the expense of the achievable electron current density. When available, the reduction in energy spread provided by a monochromator can be very important, both for analytical analysis (Chapter 6) and to improve the information limit in the transmission electron microscope (Section 4.3.3). The high energy electrons from the gun are focused by an electromagnetic condenser lens system, whose focus is adjusted by controlling the lens currents (and not the lens position, as would be the case in the optical microscope). The specimen stage is mechanically complex. In addition to the x–y controls, the stage allows the specimen to be tilted about two axes at right angles in the plane of the specimen, and the tilt axes to be adjusted (this is termed a eucentric stage). Some z-adjustment along the optic axis is important. It is also often possible to rotate the specimen about the optic axis of the microscope. In TEM the standard specimen diameter is only 3 mm, while only samples less than about 0.1 mm are thin enough to allow most of the high energy electrons to pass without suffering serious energy loss. In SEM, by contrast (Chapter 5), the signal is collected from the specimen surface and the electron beam loses energy by inelastic scattering as the electrons penetrate beneath the sample surface. Focusing of the image in the transmission electron microscope is not obtained by adjusting the position of the specimen along the z axis, so as to alter its distance from the objective lens, but rather by changing the lens current in order to adjust the focal length of the electromagnetic lens in order to focus a first image from the elastically scattered electrons that have been transmitted through the thin film specimen. The final imaging system also employs electromagnetic lenses, and the final image is observed on a fluorescent screen that converts the high energy electron image into an image that is visible to the eye. Typical electron current densities at the screen are of the order of 10
10–10
11 A/ m
2, but they may be even lower when studying damage-sensitive materials or at high magnifications. Photographic emulsions have been used to record the final image but, as in optical microscopy, advances in the development of charge-coupled devices (CCDs) combined with computerized image processing (Section 3.5) have now made digital image recording the technology of choice. The high energy electron beam has a limited path length in air, so that the whole electron microscope column must be kept under vacuum. Specimen contamination under the beam, that is, the development of a carbonaceous layer on the specimen surface, is a serious

Transmission Electron Microscopy 183

Figure 4.3 In the scanning electron microscope a fine probe of electrons is focused onto the sample surface and then scanned across the surface in a television raster. A signal generated by the interaction of the probe with the sample is collected, amplified and displayed on a monitor with the same time base as the raster used to scan the sample.

problem in electron microscopy, and may restrict viewing time for any selected area of the sample and limit the achievable resolution. In general, the vacuum needs to be better than 10
6 Torr, while for the highest resolution a vacuum of 10
7 Torr is desirable. The sources of contamination include the sample, the components of the microscope and the pumping system itself, and they should be trapped, usually by cryogenic cooling of the specimen and its surroundings. For comparison, the scanning electron microscope (Figure 4.3) again has a source of high energy electrons and a condenser system, but now employs a probe lens to focus the electron beam into a fine probe that impinges on the specimen. The electromagnetic probe lens in the scanning electron microscope fulfills a similar function to the objective lens in the transmission electron microscope, since it determines the ultimate resolution attainable in the microscope. However, the probe lens is placed above the specimen, and plays no part in collecting the image signal from the specimen. Indeed, in SEM the elastically scattered electrons are of no especial importance in providing morphological information. Rather it is the inelastic scattering processes which occur when the electron probe interacts with the sample, that provide the microstructural information collected in this instrument. The electron energy of the beam in scanning electron microscopy is appreciably less than that used in transmission, usually of the order of 3–30 keV, although much lower energies, as low as 100 eV, may prove useful. The ‘image’ in SEM is obtained by scanning the focused electron probe across the sample surface in a television raster, and then collecting an image signal from the surface and

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displaying it, after suitable amplification and processing, on a monitor with the same time base as that used to scan the probe across the sample (Figure 4.3). Line scans of the sample are made along the x-axis, and at the end of each line scan the beam is switched back to the zero of x and the y coordinate is incremented by Dy. The signal collected I is thus a function of time I(t), where each value of t corresponds to specific x, y coordinates in the plane of the specimen surface. Once the signal from a complete set of line scans has been collected, corresponding to the range of y selected, the beam is switched back to the zero of the x, y coordinates, ready to collect data for another image frame. This is also the principle employed in the cathode ray tube, where the image is formed by scanning the screen with a modulated beam of electrons. There is no reason why the position of the x-scan should not also be incremented by Dx. In this case the x, y coordinates for each frame scan will constitute a set of pixels (picture elements) of size Dx, Dy, and the total time required to acquire an image frame will equal the total number of points in the frame multiplied by the dwell time at each pixel position. Of course, the signal intensity for each point still has to be digitized before full digital image processing and analysis can be done. This means that the intensity of the signal collected at each point has to be amplified and binned, typically in an 8 bit (256 grey level) register and the total number of pixels should correspond approximately to the number of points scanned in the image frame on the sample. The power of the scanning electron microscope derives from the wide range of signals that may result from the interaction of the electron probe with the sample surface. These include: characteristic X-rays, generated by excitation of inner shell electrons; cathodoluminescence, excitation in the range of visible light that is associated with valency electron excitation; specimen current passing through the sample due to the net absorption of electric charge; and backscattered electrons, that are elastically and inelastically scattered out of the surface from the probe beam. The most commonly used signal is that derived from low energy secondary electrons, which are ejected from the surface of the target by the inelastic interaction of the sample with the primary beam. The secondary electrons are emitted in large numbers from a region that is highly localized at the point of impact of the probe. They are therefore readily detected and are capable of forming an image whose potential resolution is limited primarily by the diameter of the focused probe at the sample surface. The fundamental difference in the operation of the transmission and the scanning electron microscopes can be summarized in terms of the two modes of data collection that are employed to form the image. In both optical microscopy and in transmission electron microscopy information is simultaneously collected over the full, magnified field of view and focused by suitable lenses to build up a magnified image as a function of the integrated data collection time. In the scanning electron microscope the information is collected sequentially, for each data point in turn, while the focused probe is scanned across the sample field of view. The rate of scan must be restricted in order to ensure that the specimen signal recorded for each image point is statistically adequate, and the total time required to form a scanning image is determined by the minimum scanning speed that will achieve this goal for each pixel multiplied by the number of image pixels that are to be collected. This distinction between an optical image, in which the image data are acquired for all image points simultaneously, and a scanning image, in which the image is developed sequentially (that is, one pixel at a time) cannot be overemphasized.

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4.1

Basic Principles

The design and construction of an electron microscope are well beyond the scope of this text, but it is still important to have some appreciation of the basic physical principles that determine the behaviour of electrons in a magnetic field and their interaction with matter. 4.1.1

Wave Properties of Electrons

The focusing of an electron beam is possible because of the dual, wave–particle character of electrons. This wave–particle duality is expressed in the de Broglie relationship for the wavelength of any particle: l¼

h mv

ð4:1Þ

where m is the mass of the particle, v is its velocity and h is Planck’s constant. Assuming that the accelerating voltage in the electron gun is V, then the electron energy is eV ¼

mv2 2

ð4:2Þ

where e is the charge on the electron. It follows that l ¼ h/(2 meV)1/2, or l ¼ (1.5/V)1/2 nm when V is in volts. This numerical value is rather approximate, since, at the accelerating voltages commonly used in the electron microscope, the rest mass of the electron m0 is appreciably less than the relativistic mass m, and a correction term should be included in the de Broglie equation: h l ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2m0 eV 12mþ0eV 2 c

ð4:3Þ

where c is the velocity of light. The relativistic correction amounts to about 5% at 100 kV, rising to 30 % at 1 MV. The electron wavelength at 100 kV is 0.00370 nm, nearly two orders of magnitude less than the interatomic spacings in the solid state. At 10 kV, typical of many applications of SEM, the wavelength is only 0.012 nm, still appreciably less than the interatomic distances in solids. 4.1.1.1 Electrostatic and Electromagnetic Focusing. Electrons are deflected by both electrostatic and magnetic fields, and can be brought to a focus by suitably engineering the electrostatic or magnetic field geometry. In the region of the electron gun the electron beam is influenced by the electrostatic field created by the anode and Wehnelt bias cylinder. These usually result in a first focus, ‘virtual’ electron source. With just one exception, to be noted below, all subsequent focusing in the electron microscope is electromagnetic and is achieved by electromagnetic lenses equipped with soft iron (essentially having zero magnetic hysteresis) pole pieces. Unlike optical lenses, made from glass, the focal length of an electromagnetic lens is variable and can be controlled by varying the lens current that flows in the coil surrounding the pole pieces. An electron travelling in a magnetic field is deflected in a direction at right angles to the plane that contains both the magnetic field vector and the original direction of travel of the

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H (a)

Object (b)

Rotated Image

Figure 4.4 An electron in a magnetic field is deflected at right angles to both the momentum and magnetic field vectors. (a) An off-axis electron follows a spiral path. (b) Electrons originating at a point off the axis are brought to a rotated focus.

electron (its momentum vector). In a uniform magnetic field an electron that is travelling offaxis will follow a helical path [Figure 4.4(a)]. To a first approximation, electrons of the same energy and travelling in a cone of directions originating from any one point within a uniform magnetic field will be brought together to a focus at a second point after spiralling along the axis of the field [Figure 4.4(b)]. The image of an object, formed by focusing electrons using electromagnetic lenses, differs in several important respects from that formed by focusing light using glass lenses. In the first place, although the image of an object in a plane perpendicular to the axis of an electromagnetic lens is also in a plane perpendicular to this same axis, it is rotated about this axis, so that focusing of the objective lens by adjusting the lens current is accompanied by rotation of the image about the optic axis. It follows that two images of the same object taken at different magnifications will also be rotated with respect to one another. This rotation can be compensated by reversing the magnetic field vector over a proportion of the optical path, and it is now common practice to design electromagnetic lenses to carry the current in the windings of the lens coil of the upper and lower halves of the lenses in opposite directions. With such a compensated lens the electrons are not confined to travel in a plane, so that the behaviour of the electron beam passing through the electromagnetic lens system is still quite different from the behaviour of a light beam in the optical microscope. In the light optical microscope there is an abrupt change in refractive index when the light is deflected as it enters a glass lens, but the refractive index is constant within the glass lens. With

Transmission Electron Microscopy 187

an electromagnetic lens the deflection of the electrons is continuous, and the magnetic field created by the lens pole pieces varies continuously throughout the optic path within the lens. Finally, the angle subtended by the path of an electron with respect to the optic axis is always very small (less than 1 ) so that the optic path length through the electromagnetic field of the lens is always very long when compared with the angular spread of the beam perpendicular to the optic axis. This means that the numerical aperture in the transmission electron microscope is always very small. This may be compared with the case of the optical microscope, for which the numerical aperture of the objective may correspond to an acceptance angle for scattered light of between 45 and 90 . The numerical aperture (note that this term is not used in electron microscopy) of an electromagnetic lens never exceeds 10
2. 4.1.1.2 Thick and Thin Electromagnetic Lenses. The physics of electromagnetic focusing means that the simple geometrical optics which we applied to the light optical microscope (Section 3.1) is a very inadequate approximation to the optics of image formation in the transmission electron microscope. In particular, the simple relationships between the focal length, the magnification, and the relative positions of the object and the image along the optic axis no longer hold, because they are based on the assumption that the lens is thin when compared with the total optical path between the object and the image. The thin lens approximation is also insufficient for high powered objectives in the light microscope, when it is replaced by much more complex, thick lens calculations. In electron microscopy all the electromagnetic lenses are ‘thick lenses’. Nevertheless, it is still common to illustrate the imaging modes in the electron microscope using ray diagrams which are presented in two dimensions, as though the electrons were not rotated, and the electron paths changed direction abruptly at the lens positions, as though the thin lens approximation were still valid. No quantitative calculations are possible in such a qualitative model. 4.1.1.3 Resolution and Focusing. Given that the maximum beam divergence in the electron microscope is less than 1 , the Raleigh criterion for the image of a point source can be reduced to: d ¼ 0:61l=msina  0:61l=a > 60l. Inserting the value for the wavelength at 100 kV, 0.0037 nm, the potential resolution of the transmission electron microscope should be of the order of 0.2 nm. However, at higher operating voltages significantly better resolution is possible (see below). As a gross approximation we can use the light-optical expression for depth of field, d  d/a, so that the thin film specimens used in transmission electron microscopy should be of the order of 20–200 nm in thickness if both top and bottom of the film are to be in focus simultaneously. Similarly, for depth of focus, D ¼ M2d, so that at a magnification M of 10 000 the expected depth of focus is of the order of metres! There is therefore no problem in focusing an image on a fluorescent screen and subsequently recording the same image on a photographic emulsion or CCD detector that is placed some distance beneath the focusing screen. 4.1.2

Resolution Limitations and Lens Aberrations

At this point we should consider the optical performance of the electron microscope in more detail and the reasons why the angular divergence of the electron beam has to be limited to such small values.

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4.1.2.1 Diffraction Limited Resolution. The diffraction limit on resolution is, as in light optical microscopy, that given by the Raleigh criterion, dd ¼ 0:61l=msina. In vacuum, m ¼ 1 and at small angles sin a ¼ a. Inserting the expression given previously for the wavelength of the  electron beam  in terms of the accelerating voltage: pffiffiffiffi dd ¼ 0:61l=a ¼ 0:75= a V ð1 þ 10
6 VÞ . It follows that, for a given divergence angle, it should be possible to improve the resolution by increasing the accelerating voltage. Experimental electron microscopes have been constructed with accelerating voltages up to 3 MV, but commercial instruments have been limited to about 1 MV. At these voltages many samples experience extensive radiation damage, which increases with prolonged exposure to the electron beam. Most high resolution TEM for imaging crystal lattices is performed at 300 or perhaps 400 kV, close to the threshold for the onset of radiation damage in most nonorganic engineering materials. At these voltages a point-to-point resolution in non-crystalline samples, of the order of 0.15 nm, is readily and routinely attainable. 4.1.2.2 Spherical Aberration. Analogous to light optics, an electron beam parallel to but at a distance from the optic axis of an electromagnetic lens will be brought to a focus by an electromagnetic lens at a point on the axis that depends on the distance of the beam from the axis [Figure 4.5(a)], while a beam further from the optic axis will be focused closer to the lens, so that the plane of ‘best’ focus will correspond to a disc of least confusion whose size will depend on the angular spread of the beam. This phenomenon is simply spherical aberration, as in the optical microscope, and the radius of the disc of least confusion constitutes an aberration-dependent limit on the resolution that is given approximately by: ds  C s a3

ð4:4Þ

r2

(a)

r1

Disc of Least Confusion (b)

V

V–∆V

Figure 4.5 Spherical (a) and chromatic aberration (b) prevent a parallel beam from being brought to a point focus. Instead a disc of least confusion is formed in the focal plane of the lens.

Transmission Electron Microscopy 189

d

s

Figure 4.6 The diffraction and the spherical aberration limits on resolution have an inverse dependence on the angular aperture of the objective lens, so that an optimum value of a exists.

where Cs is the spherical aberration coefficient of the electromagnetic lens. By comparison, the diffraction limit on resolution dd is inversely proportional to the angular aperture of the objective a, while the spherical aberration limit ds is proportional to the third power of the angular aperture (Figure 4.6). It follows that for any given lens of fixed spherical aberration coefficient there should be an optimum angular aperture at which dd equals ds, corresponding to Cs a3 ¼ 0:61l=a, or a4 ¼ 0:61l=Cs . The required angular aperture is therefore a sensitive function of both the accelerating voltage (the electron wavelength) and the spherical aberration coefficient of the lens. Typical values for the spherical aberration coefficient of an electromagnetic lens are somewhat less than 1 mm. Inserting a moderate value of 0.6 mm for Cs and the wavelength appropriate to 100 kV electrons, 0.0037 nm, we obtain a value of about 8 · 10
3 for this optimum value of a. Electromagnetic lens design has improved steadily over the past half-century, and it has now proved feasible to introduce multipole electrostatic spherical aberration correctors into the microscope column that are able to reduce the spherical aberration coefficient to an arbitrarily small value. These correctors are not yet generally available, but there is little doubt that they soon will be. Figure 4.7 shows the arrangement of twin hexapole correctors in the FEI Titan, a top-of-the-range, ultra-high resolution transmission electron microscope, in which a corrected Cs of zero has been achieved. 4.1.2.3 Chromatic Aberration. Chromatic aberration arises because higher energy electrons are less deflected by a magnetic field than those of lower energy, so that they are brought to a focus at a point on the optic axis that is further from the center of the lens, once again giving rise to a disc of least confusion, this time determined by the energy spread in the electron beam and the chromatic aberration coefficient of the electromagnetic objective lens [Figure 4.5(b)].

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+ -

-

+

+ -

Figure 4.7 Schematic of the influence of a double hexapole electrostatic aberration corrector on the shape of an electron beam. The corrector system results in ray paths through the axis of the column (red circles) that are compensated (white circles) for the residual spherical aberration of the objective lens.

There is more than one source of chromatic aberration, although that due to the kinetic spread in beam energy is generally the most important. If the electrons are thermally emitted (as is, to some extent, always the case, even for commercial field emission sources), then the relative energy spread will be given by DE/E0 ¼ kT/eV, where k is Boltzmann’s constant and e is the electronic charge. For T ¼ 2000 K, a reasonable temperature for a tungsten filament source, and 100 kV electrons, the energy spread DE/E0 is about 1.5 · 10
6. Electrons may also lose some energy due to inelastic scattering in the thin sample, adding to the chromatic aberration due to the thermal energy spread of the beam. This may affect an appreciable fraction of the incident electrons if the specimen is thick or of high atomic number, as in a heavy metal sample. Fluctuations in the electromagnetic objective lens current may also contribute to chromatic aberration, since they will change the focal length of the lens. The equation that relates the chromatic aberration limit on resolution to these variations in beam energy and lens current is: dc ¼ C c

DE a E0

ð4:5Þ

where Cc is the chromatic aberration coefficient of the lens, while DE includes instabilities in both the accelerating voltage and the objective lens current. As with spherical aberration,

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the resolution limit increases with a, but this time only linearly. Providing the voltage stability of the electron gun and the current stability of the electromagnetic lenses is adequate, chromatic aberration should be limited only by the temperature of the electron source, although it may also be affected by inelastic interactions in the specimen. 4.1.2.4 Lens Astigmatism. The axial symmetry of the electro-optical system is an extremely important factor limiting the performance of the electron microscope, and the exact alignment of the lens components within the microscope column is a critical factor in optimizing the performance of the instrument. The objective lens is the component that is most affected by misalignment. The axial symmetry of this lens is especially sensitive to minor disturbances associated with the geometry, size, position and dielectric properties of the sample, as well as to small amounts of carbonaceous contamination that may be deposited on the sample or the objective aperture. A basic feature of any loss of axial symmetry is a variation in the focal length as the electrons spiral about the optic axis. This results in two principal focal positions on this axis that give two line foci at right angles (Figure 4.8). This condition is termed astigmatism. Astigmatism cannot be prevented, both because of the inherent residual asymmetry in the lens and pole piece construction, and because of the extreme sensitivity of the astigmatism to minor misalignment, specimen asymmetry and contamination in the microscope. However, it can be corrected. Complete twofold astigmatism correction is achieved by introducing sets of correction coils whose variable magnetic fields are at right angles to both the optic axis and the magnetic field of the main lens coils. The correction coil currents can be periodically adjusted during operation of the microscope to balance exactly any changes in the magnetic asymmetry that is due either to the build-up of contamination or displacement of the sample during viewing. This on-line correction is especially important when working with thick samples of high dielectric constant, and for magnetic materials in particular. A number of geometrical arrangements for the astigmator assembly are possible, for example four pairs of coils forming an ‘octet’ or octopole correction system.

x

y

y focus

x focus

Figure 4.8 Astigmatism is the result of axial asymmetry and leads to variations in the focal length about the optic axis, resulting in two principle line foci at right angles along the axis.

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4.1.3 Comparative Performance of Transmission and Scanning Electron Microscopy We now summarize the principal differences between the features and the performance of the transmission and scanning electron microscopes. 4.1.3.1 The Optics of Image Formation. As noted previously, the primary image in TEM is obtained by focusing the objective lens. A series of additional imaging lenses then enlarge this image to the final magnification on a fluorescent screen, a photographic emulsion or a CCD recording array. In the scanning electron microscope the image is formed, point by point, by collecting a signal that is generated by the interaction of a focused electron beam probe as it is scanned across the surface of the sample in a television raster. 4.1.3.2 Depth of Field and Depth of Focus. As in the light optical microscope, the depth of field of the transmission electron microscope is limited by the NA of the objective lens and the resolution of the microscope, but since the angular aperture of electromagnetic lenses is so small, the depth of field in TEM exceeds the resolution by some two orders of magnitude. In SEM the electron probe is focused by a probe lens, whose operation is analogous to that of the objective lens in transmission microscopy. However, the inelastic scattering processes that occur during interaction of the probe with the specimen, together with the requirement for an adequate signal current, restrict the effective probe size, and hence the resolution, to the nanometre range. It follows that, with an angular aperture for the probe lens of the order of 10
3, the depth of field in the scanning electron microscope (d/a) is typically of the order of micrometres, considerably better than the depth of field that can be achieved in optical microscopy and at a much improved resolution. Both the light optical and the transmission electron microscopes generate a twodimensional image of a thin, planar section that has been prepared from the bulk material. By contrast, the image in the scanning electron microscope contains considerable in-focus information on the three-dimensional topography from the surface of any solid sample. Furthermore, since data collection in the scanning image is collected point-by-point and line-by-line, the question of depth of focus does not arise: there is now no focused image in the sense of classical optics. 4.1.3.3 Specimen Shape and Dimensions. The electro-optical image requirements in TEM usually place the specimen within the magnetic field of the objective lens (thus making the study of magnetic materials problematic). The space available for the specimen is therefore very restricted, so that in addition to the stringent limitations on sample thickness that are dictated by the onset of inelastic scattering in the electron beam, there are also limitations on the sample’s lateral dimensions. The standard external specimen diameter is 3 mm, but only the central 2 mm or so of the sample is actually available for examination. By contrast, samples for SEM sit well below the probe lens, and well outside the lens magnetic field. With a long working distance probe-setting, reasonable resolution is available even when the lens–sample separation is over 50 mm. In addition, there are no limitations on lateral dimensions, other than those imposed by the design of the sample chamber. Most samples have lateral dimensions similar to those used for optical microscopy

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(20–30 mm), but much larger assemblies have been inserted into the specimen chamber, and specimen chambers are available that will accept specimens 10 cm or more across. Complete sections of failed engineering components and complex solid-state devices are commonly inserted into the scanning electron microscope for detailed evaluation. 4.1.3.4 Vacuum Requirements. The vacuum requirements of all electron microscopes, transmission and scanning, are determined by three factors: 1. The need to avoid scattering of the high energy electrons by residual gas in the microscope column. 2. The necessity for thermal and chemical stability of the electron gun during microscope operation. 3. The need to minimize or eliminate beam-induced contamination of the sample during observation. The least stringent requirement is actually the first, since a vacuum of 10
5 Torr is quite sufficient to ensure that the cross-section for scattering of the high energy electrons by residual gas is negligible. The second factor is very much more important. A heated tungsten filament is steadily eroded, primarily by oxidation during operation at 10
5 Torr. Both alternative sources, either a low work-function lanthanum hexaboride (LaB6) crystal or field emission source (both of which are operated at much lower temperatures, and hence generate a beam with lower chromatic aberration), require a much better vacuum: typically of the order of 10
7 Torr for LaB6 and down to the 10
10 Torr range for field emission guns. However, the third factor listed above is equally important, since specimen contamination is most frequently the result of inelastic interaction between contaminant gases absorbed on the sample surface and the incident high energy electron beam. Hydrocarbons arriving at the sample are both polymerized and pyrolysed by the incident electrons to form an adherent, amorphous, carbonaceous layer on the sample surface. After extended electron irradiation of a specific area during observation, the layer of amorphous ‘carbon’ contamination may even obscure all morphological detail. Contamination can be significantly inhibited by cryogenic cooling of the specimen surroundings, in order to trap the condensable contaminant species, and this is the procedure generally adopted for TEM. However, the large specimens employed in the scanning electron microscope make a cryogenic trap much less effective, while the very high beam current concentrated in the focused electron probe exacerbates the rate of contamination. The only adequate solution is to ensure that the source of contamination is not the specimen, for example, by plasma etching of the sample in an argon and oxygen gas mixture to oxidize carbon deposits on the surface, and then work with the best possible chamber vacuum. 4.1.3.5 Voltage and Current Stability. While chromatic aberration ought not to be a problem, in either transmission or scanning microscopy, it is a mistake to assume that current and voltage instabilities only affect the performance through their influence on the objective or probe lens. In particular, the scanning electron microscope may be susceptible to image distortion that arises from electrical instability of the scanning systems. Several causes may be responsible, but it is the results that concern us: these include differences in the effective magnifications for the x- and y-scan directions, possible shear distortion of the image in the x-direction, drift of the image or a dependence of the magnification on distance from the optic axis: barrelling if the central region is enlarged, but a pin-cushion effect if it is

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the peripheral region that is enlarged. Many of these imaging defects are a result of the point-by-point data collection process and are a consequence of distortions in the scanning raster of the probe x–y coordinates with respect to that of the image. Electrostatic charging of a sample that is an electrical insulator can also be a major source of instability, but can usually be prevented by a conductive coating or by working with very low beam voltages and currents.

4.2

Specimen Preparation

It is not always easy to prepare good specimens for examination in the transmission electron microscope, but many techniques are now available for doing so. If a good specimen has been prepared, the information that can be obtained by TEM is unique. However, there is nothing more frustrating than attempting to extract useful information from a poorly prepared transmission specimen, and more than one graduate student has acquired a lifelong aversion to the transmission electron microscope solely for this reason. Successful transmission electron microscopy depends on three diverse skills: preparing a good specimen; acquiring good data; and possessing the understanding needed to interpret the data. In what follows we will try to provide a sound foundation for success in all three domains. Awide range of experimental methods and a good choice of commercial equipment exist to ‘ease the pain’ of specimen preparation for the transmission electron microscope. Providing adequate facilities are available, there is no real barrier to the preparation of good, thin-film specimens from any engineering material, and no excuse whatsoever for wasting time on the examination of a poor specimen in the microscope. Good samples for conventional TEM commonly need to be thinned uniformly to less than 100 nm, while those for lattice imaging in high resolution TEM or sub-micrometre microanalysis by electron energy loss spectroscopy (EELS) should be less than 20 nm thick. Preparing such samples reliably is not trivial, but, with the help of the tools that are now available, success should be well within reach of a careful and competent electron microscopist. In what follows we concentrate on the preparation of thin film sections from a bulk sample of an engineering material. We will summarize briefly the methods of sample preparation that have been developed for soft biological tissues, and those techniques that are available for the dispersion of particulate samples or the deposition of thin film samples from the gaseous or liquid phases. It should be emphasized that every material is a ‘special case’, and that every engineering component to be investigated has to be sectioned and a sample selected. The characterization of powder samples by transmission electron microscopy is an important task, with applications ranging from ceramics technology to cosmetic preparations. The trick is to prepare a sample that reflects the composition, particle shape and particle size distribution of the bulk powder without introducing artifacts associated with either particle agglomeration or fragmentation. This usually involves preparing a stable dispersion in a suitable liquid medium, often with the help of surface-active additives. A drop of the dispersion placed on a glassy carbon film mounted on a microscope grid is often used but this is seldom satisfactory, since the particles collect at the meniscus as the drop dries, leaving irregular aggregates on the grid that are difficult to interpret. Far better is to

Transmission Electron Microscopy 195

spray the dispersion over the carbon coated grid using a commercial nebulizer (atomizer). The spray droplets are only a few micrometres in diameter, and, for sufficiently dilute dispersions, each droplet will contain no more than one or two particles, so that there is no danger of aggregation during drying and the size distribution observed in the electron microscope will be that characteristic of the original dispersion. Soft tissues and many polymer samples can be sectioned using a microtome that generates a sequence of very thin slices from the stub of a suitably prepared sample, rather like sliced bread. By examining an ordered sequence of slices (serial sectioning) it is possible to build up a picture of the three-dimensional structure of soft biological tissues and cellular structures. Glass knives are still commonly used, with the advance of the specimen stub controlled by the thermal expansion of a mounting rod. More common today, better control of the slice thickness is achieved using a diamond knife and piezoelectric control for the advance of the specimen mount. Thin films deposited from the liquid or vapour phase can often be stripped from a suitable substrate and, if they are thin enough, examined directly by TEM. However, it is often the cross-section of a thin film microelectronic device that is of interest, and this requires that the sample be rigidly mounted and sectioned at a known location. We discuss the preparation of such samples below. 4.2.1

Mechanical Thinning

The usual starting point for the preparation of a thin film specimen for transmission electron microscopy is a sample taken from a bulk component. This sample is typically a 3 mm diameter disc several hundred micrometres in thickness. The disc may be punched out of a ductile metal sheet, trepanned from a brittle ceramic, cut from a bar, or machined from a larger section (Figure 4.9). In all cases it is necessary to select the axis of symmetry and the centre of the disc with respect to the coordinates of the bulk component, since these determine the location that is sampled and the direction of viewing in the microscope. At this stage it is essential to minimize mechanical damage to the material and preserve a flat and smooth surface. The next task is to reduce the thickness of the disc sample. This is accomplished by the same procedures of grinding and polishing as were described previously for the preparation of optical microscope specimens (Section 3.3). As in the previous discussion, the stiffness (elastic modulus), hardness and toughness of the material determine the optimum choice of grinding and polishing media, and ductile metals, brittle ceramics, reinforced composites and tough alloys will all respond quite differently. Four mechanical thinning treatments are possible (Figures 4.10 and 4.11): 1. A polished, parallel-sided disc is prepared and then thinned from one or both sides using a rigid jig to maintain a planar geometry. A crystalline wax is used to fix the specimen to a flat, polished support. The wax is easily melted on a hotplate, both when the sample is first attached and when it is turned over on the baseplate. As the thickness decreases, so should the grit size of the grinding and polishing media be reduced finishing the thinning process with sub-micrometre diamond grit. The sample thicknesses should then be 100 mm or less. At this stage, the greatest problem is often the relief of internal residual stresses in the material that may lead to curvature and buckling of the sample as soon as it is removed from the baseplate support (see below).

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Microstructural Characterization of Materials

Machine and slice

Grind and trepan Figure 4.9

Some methods used to section a disc from a bulk component.

2. Once the thickness of the sample is reduced to 100 mm or less and the surfaces have been polished to a micrographic finish (Section 3.3.2), the disc is secured on an optically flat baseplate and ‘dimpled’. In this process a fine grinding medium removes material from the central area of the disc sample as it is rotated in contact with a polishing wheel. This is, essentially, a lapping process in which particles of the grinding media are displaced in

Figure 4.10 Mechanical thinning of a disc may be achieved by several methods: (a) a simple, parallel-sided geometry; (b) dimpling to thin the centre while retaining a thicker peripheral support; (c) from a wedge in which the region of interest is at the thinned side of the wedge.

Transmission Electron Microscopy 197

Figure 4.11 Schematic showing a process to prepare a cross-section transmission electron microscope specimen from a thin film on a flat substrate. (a) Rectangular sections are cut from the wafer. (b) They are glued together to form a block greater than 3 mm in thickness, from which a 2.8 mm diameter rod is trepanned. (c) The rod is inserted and glued into a 3.0 mm outer diameter metal tube. (d) Thin sections are then cut from this assembly. (e) Finally, these sections are mechanically thinned (f–g).

the region of contact shear, continuously fracturing to expose new cutting edges. Several commercial dimplers are available, and the process has been successfully used to thin samples of hard and brittle materials to thicknesses of 20 mm or less in the central region of the ‘dimpled’ disc. ‘Dimpling is an excellent method for avoiding the complications

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Microstructural Characterization of Materials

of distortion by residual stress, since the thicker rim of the sample now acts as a ‘frame’ to constrain the thin central region. 3. For some samples it may be an advantage to prepare a ‘wedge’ rather than a ‘dimple’. This would be the case for an interface, where the microstructure of the interface is the subject of investigation, and the maximum thin area from the interface region is obtained when the interface is aligned parallel to the edge of the wedge. In the case of thin films and multilayer sandwiches of different phases (for example, semiconductor devices and composite materials) it is common practice to section the sandwich at right angles, and mount the plane of the sandwich perpendicular to the axis of the wedge. In this configuration, each layer is sectioned as a wedge and the morphology and interface microstructure of each layer can be studied in a single thin-film sample. The wedge is prepared using a rigid jig, for which the wedge angle is pre-selected and is typically less than 10 . 4. Cross-sections of microelectronic devices are more generally prepared by either of two techniques that have become standard for the industry (Figure 4.11). The second technique will only be discussed after we introduce focused ion beam systems in Chapter 5. In the first method the device to be sectioned is first diced into squares and the diced slices glued together, with the central slices face-to-face, in order to form a block more than 3 mm in thickness. A rod is then trepanned from this block with the axis of the rod in the plane of the face-to-face layers and parallel to the plane of the device. This rod is now glued into a 3 mm outer diameter metal tube (often made of copper or brass). This tube assembly can now be sliced and mechanically thinned to form a series of 100 mm thick sections through the device, each of which can be dimpled. There is little danger of failure at the device interfaces, since the disc is constrained by the metal ring around the periphery. These dimpled samples are then ion-milled (see below). 4.2.2

Electrochemical Thinning

No mechanical thinning process can avoid introducing some sub-surface mechanical damage, either through plastic shear or by microcracking. If the material is a metallic conductor, then it is frequently possible to thin the sample by chemical dissolution rather than mechanical abrasion. This is most commonly achieved electrochemically. The techniques that have been developed for electrochemical thinning are based on standard electropolishing solutions. However, the conditions required for electropolishing bulk components often differ markedly from those we are concerned with in the preparation of specimens for the transmission electron microscope, primarily because the electrically conducting area that is to be thinned is so small. Far higher current densities are used than would be possible in bulk electropolishing. The problem of dissipating the heat generated during chemical attack at high current densities can be solved by passing the current through a jet of the polishing solution that impinges on one or both sides of the disc sample (Figure 4.12). While thin films of all metals and alloys, as well as many other materials that possess the necessary electrical conductivity, have been successfully prepared by jet-polishing, each of these materials requires its own polishing conditions, especially the composition and temperature of the solution, and the current density. These ‘recipes’ are available in the literature or from the manufacturers of jet-polishing units.

Transmission Electron Microscopy 199

V

Pumped electrolyte jet

Pumped electrolyte jet Disc specimen

Figure 4.12 In jet polishing a current is passed through a stream of the polishing solution as it impinges on the disc sample. Thinning is accomplished by electropolishing at a high current density.

Electrochemical thinning of a thin-film specimen is complete as soon as the first hole appears near the centre of the disc sample. Initial hole formation is detected either by eye or, more usually, by using an optical laser signal passing through the hole to automatically switch off the current and the jet. Once the specimen has been rinsed and dried, it is ready for insertion in the microscope. The regions around the central hole in the sample should be transparent to the electron beam (typically 50–200 nm in thickness). If the areas available for investigation are found to be too small, then this is most commonly because the jetpolishing process was allowed to continue after formation of the first hole, leading to rapid attack at the edge of the hole and consequent rounding of the rim. Sometimes these areas are found to have a roughened, etched appearance, usually because the polishing solution is exhausted, contaminated or over-heated, or possibly because the current density is insufficient. 4.2.3

Ion Milling

The earliest successes in preparing thin-film specimens of ductile metals and alloys by ion milling were pioneered by Raymond Castaing in France, in the mid 1950s. Castaing employed a beam of energetic inert gas ions to sputter away the surface of thin aluminium alloy foil samples. This technique was initially superseded by chemical thinning methods, based on electrolytic polishing, but the development of increasingly sophisticated ion milling systems has led to a resurgence of ion milling, which is now the preferred method for removing the final surface layers from samples intended for TEM (Figure 4.13).

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Microstructural Characterization of Materials

Ion gun

Specimen

Ion gun

Figure 4.13 Precision ion milling permits thin film multilayer assemblies to be thinned perpendicular to the plane of the assembly. Two ion guns are normally used in order to sputter from both sides of the sample. During the process the sample is rotated about the axis perpendicular to the plane of the assembly.

There are several reasons for the current preference for ion milling: 1. Ion milling is a clean, low-pressure, gas-phase process, so that contamination of the surface is easier to control. 2. Electrochemical methods are largely restricted to metallic conductors, while ion milling is more generally applicable, for example to ceramics and semiconductors. 3. Although some sub-surface radiation damage is often introduced during ion milling, this can be minimized by suitable choice of the milling parameters, for example the ion energy and the angle of incidence of the ion beam on the sample, and there is no danger of mechanical damage. 4. Ion milling also removes surface contaminant films, such as the residual anodic oxide layers associated with electropolishing. In addition, no changes in surface composition are expected, since milling is normally performed at temperatures well below those at which diffusion occurs in the specimen. 5. Thin-film multilayers of different materials deposited on a substrate can seldom be chemically thinned in cross-section. 6. The sophisticated ion milling units now available are able to ensure that the thinned region of the sample selected for examination can be localized to better than 1 mm, and this is often a primary consideration in the study of thin-film microelectronic and optronic devices. Such accuracy in selecting the area of examination for thin-film electron microscopy is quite impossible using chemical methods. Of course, ion milling also has its problems. Sputtering is a momentum transfer process. The rate of sputtering is a maximum when the ion beam is normal to the sample surface and the atomic weight of the sputtering ions is close to that of the sample material. However, an ion beam incident at right-angles also maximizes both the sub-surface radiation damage and

Transmission Electron Microscopy 201

topographic surface irregularities associated with microstructural features. In addition, the need to avoid ion-induced chemical reactions usually restricts the choice of sputtering ion to the inert gases (typically argon). The sputtering rate can, in principle, be improved by increasing the incident ion energy, but only at the cost of occluding large numbers of the sputtering ions in the sub-surface region. These contribute to a radiation-damaged layer. In practice, the ion energy is therefore limited to a few kilovolts. At these energies, the depth of ion injection is sufficiently limited to allow most of the occluded ions to escape to the surface by diffusion, rather than nucleate sub-surface damage. In general, the angle of incidence for ion milling is restricted to no more than 15 , and, at an energy of the order of 5 kV, the rates of sputtering are then often no more than 50 mm h
1. It follows that the preferred specimen for subsequent transmission electron microscopy is one that has already been thinned mechanically (usually by dimpling) or electrochemically (usually by jet-polishing) to a thickness of the order of 20–50 mm before it is thinned in the ion miller. The disc specimen is rotated during ion milling, in order to ensure that thinning is as uniform as possible. The initial stages of milling are performed from both sides of the sample simultaneously, often at an angle of up to 18 to improve the thinning rate. The angle of incidence of the ions (and hence the rate of milling) is then reduced in the final stages of the thinning process to minimize surface roughness. The minimum angle that will result in a optically planar surface finish with large, uniformly thinned areas is dictated by the ion beam geometry. At glancing angles a high proportion of this beam could sputter material from the specimen mounting assembly, and hence contaminate the specimen. Primarily for this reason, the minimum sputtering angles in the final stages of ion milling are usually between 2 and 6 . In general, milling is judged to be complete when the first hole is formed in the sample. In a precision ion-milling system the area selected for thinning is monitored in situ using a light transmission detector. Milling is terminated as soon as the required region is perforated. Unfortunately, this may not work if one or other of the layers in the sample is transparent to light.

4.2.4

Sputter Coating and Carbon Coating

The electron beam carries a charge, so electrically insulating specimens will generally acquire some electrostatic charge during examination. In many instances charging is not a problem, since the small size of the sample and surface conductivity limit the charge. If necessary, the specimen can be coated with a thin electrically conducting layer. The preferred conducting material is carbon, since this element has a low atomic number (6) and deposits as a uniform, amorphous thin film. Any substructure due to the carbon coating is of very low contrast and on a nanometre scale. The carbon coating may be evaporated onto the surface by passing a high electric current through a point contact between two carbon rods, or sputter-coated by bombarding a carbon target with inert gas ions and depositing the sputtered material on the sample surface. ‘Difficult’ samples, such as ceramics, may require coating on both sides. The nanometrescale morphology of the thin (5–10 nm) coating is sometimes faintly visible in recorded microscope images.

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4.2.5

Microstructural Characterization of Materials

Replica Methods

Instead of preparing a thin slice from a component for direct examination in the transmission electron microscope, it is possible to take a replica from a surface. This may not be necessary if the surface can be examined at sufficient resolution in a scanning electron microscope, but this is not always possible and there are several reasons why a replica for insertion in the transmission electron microscope may be desirable. 1. Nondestructive examination may be necessary. For example, in failure analysis, and often for legal reasons, there is a reluctance to section the component and destroy evidence. Moreover, taking a replica from the surface can be done quite easily in the field, far from the laboratory, and without destroying the component. This is especially true of forensic investigations at crime scenes and crash sites, where the evidence must be preserved for the court. 2. Selecting one component of a complex sample for investigation. When evidence is sought for the presence of a specific phase on the surface, collection on a replica can preserve both the phase morphology and its distribution. Again, forensic examples are easy to imagine: gun shot residues recovered from skin, or paint pigment particles at the scene of a crash. In many failure investigations corrosion products can also be isolated conveniently on a replica. Since analysis of the chemical composition and phase content of a corrosion product will not be obscured by the composition and structure of the bulk material, there is good reason to identify these products by isolating them on a replica. 3. Extracting specific phases from a polyphase material. Suitable chemical etchants can sometimes be used to isolate a selected phase from the bulk material on a replica, while still preserving the original phase distribution in the surface section taken from the bulk. The chemistry, crystallography and morphology of the extracted phase can then be studied in the transmission electron microscope with no interference from the microstructural features of the remaining constituents of the bulk matrix. An excellent example is the study of carbide precipitation in steels, where an extraction replica can reveal the composition, crystal structure and morphology of the carbide phases that would normally be obscured by the strong diffraction contrast and X-ray excitation of the ferrous alloy matrix. 4. Correlation of microstructures using alternative imaging methods. Finally, it may be desirable to compare observations made on a replica taken from a solid surface with observations made on the same surface in scanning electron microscopy. Surface markings associated with mechanical fatigue are one example in which combining these two techniques may be an advantage. Dimpled ductile failures can be observed in the scanning electron microscope and the nonmetallic inclusions that are associated with the nucleation of the dimples can be extracted on a replica and then identified in the transmission electron microscope. The usual procedure (Figure 4.14) is to obtain a negative replica of the surface on a flexible, soluble plastic. The plastic may be cast in place and allowed to harden, or it may be a plastic sheet that has been softened with a suitable solvent and then pressed onto the surface before allowing the solvent to evaporate. In some cases it will first be necessary to remove loose contamination by cleaning the surface ultrasonically or by using an initial

Transmission Electron Microscopy 203

Negative Plastic Replica

Carbon Replica

Sample

Plastic Replica

Heavy Metal Shadow

Extracted Particle Figure 4.14 A negative plastic replica can be used to prepare a thin film carbon replica of the original surface that contains extracted particles. The replica can be shadowed to reflect the original particle morphology.

‘cleaning’ replica before making the final plastic replica. In other cases it may be precisely the ‘contamination’ that is the subject of interest, as in the case of gunshot residues. Once a plastic replica has hardened, it can be peeled away from the surface and then shadowed with a heavy metal, such as a gold–palladium alloy, to enhance the final contrast in the electron microscope. The shadowing metal is selected for minimum particle size and maximum scattering power, and the particle cluster size is typically about 3 nm. After shadowing, a carbon film 100–200 nm in thickness is deposited on the plastic replica and the plastic is then dissolved in a suitable organic solvent. The carbon film retains any particles removed by the plastic from the replicated surface, as well as the heavy metal shadow that reflects the original surface topology. The carbon replica is then rinsed and collected on a fine-mesh copper grid for examination in the transmission electron microscope. There may be no need for a ‘negative’ plastic replica. For example, an alloy sample that has been polished so that the particles of a second phase stand proud of the surface can be coated with a carbon film that is deposited directly onto the surface and will adhere strongly to the particles. Further etching of the matrix will release a carbon extraction replica on which are distributed, in their original configuration, the particles of the second phase.

4.3

The Origin of Contrast

The electron beam interacts with a thin-film specimen both elastically and inelastically, but it is the elastic interactions that dominate the contrast observed in the transmission electron microscope. On the other hand, it is the inelastic scattering events that contain information on the chemical composition of the sample, and we will return to this in Chapter 6 when we discuss EELS. Contrast arises from three quite distinct image-forming processes, and these are termed mass–thickness contrast, diffraction contrast and phase contrast, respectively. Figure 4.15 illustrates the electron scattering processes schematically. If the sample is amorphous, that

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Microstructural Characterization of Materials

Incident Beam

Specimen Envelope of Inelastic Scattering Intensity

Envelope of Elastic Scattering Intensity

Coherently Scattered Diffracted Beams

Figure 4.15 Most of the incident beam is elastically scattered by the sample, either randomly (as in a glassy or amorphous specimen) or coherently to discrete angles (as in a crystalline phase). The image may be formed from either the direct transmitted beam, by a diffracted beam, or by the interference of the diffracted beams, both with each other and with the direct transmitted beam (see text).

is it has a glassy microstructure with no long-range crystalline order, then the elastic scattering gives rise to an envelope of transmitted intensity that varies with the scattering angle according to an approximate cos2y law. The intensity scattered out of the direct beam from such a glassy specimen depends on the energy of the electron beam, the sample thickness and the sample density. The image contrast is then said to be due to variations in mass–thickness. If an aperture is placed in the electro-optical column at an image plane of the electron source that lies beneath the plane of the specimen, then the aperture will intercept most of the scattered electrons and the image will be dominated by those direct transmitted electrons which have not been scattered in passing through the thin-film sample. Mass–thickness contrast usually dominates the features seen in transmission electron microscopy of biological samples taken from soft tissues, as in histological studies. In a crystalline sample, the electrons are elastically scattered according to Bragg’s law and generate diffracted beams at discrete angles 2yhkl to the direct transmitted beam which correspond to those crystal planes whose Miller indices hkl satisfy the Bragg condition (or, more exactly, those crystal planes whose reciprocal lattice vectors touch the reflection sphere) (Section 2.5). An aperture can now be inserted into the optical column at an image plane of the electron source beneath the specimen, so that this aperture allows either the directly transmitted beam to pass into the imaging system, to form a bright-field image, or selects one of the diffracted beams to be accepted, forming a dark-field image. In both cases the image contrast is determined primarily by the presence of crystal lattice defects that affect the local diffracted intensity generated near the lattice defect. This imaging mode is termed diffraction contrast. Both mass–thickness and diffraction contrast create what are

Transmission Electron Microscopy 205

essentially magnified shadows of microstructural features. This is not too dissimilar from the elongated shadow of a tree on the grass that shows the leaves and branches in a twodimensional projection that depends on the angle of the sun. Finally, if the resolving power of the microscope is adequate, a larger diameter aperture can be inserted to admit several diffracted beams simultaneously into the imaging system, with or without the direct transmitted beam. These beams interfere in the image plane to yield a lattice image that reflects the periodicity in the crystal in the plane normal to the optic axis of the microscope, an effect termed phase contrast. Unlike mass–thickness and diffraction contrast, the phase contrast image makes use of the elastically scattered electrons from several different crystal lattice planes and is an image of the crystal structure. Since not all diffracted beams can be included, this lattice image is incomplete. 4.3.1

Mass–Thickness Contrast

The probability of an electron being elastically scattered out of the incident beam depends on the atomic scattering factor, which increases monotonically with the atomic number and the total number of atoms in the path of the beam, that is the total thickness of the film. Mass–thickness contrast thus reflects a combination of variations in specimen thickness and specimen density, and is therefore similar to the effects of mass absorption discussed previously for X-rays (Section 2.3.1). In the life sciences, mass–thickness contrast almost always dominates the image. The contrast of soft tissue, biological samples in the electron microscope is often enhanced by a heavy-metal, tissue-staining procedure. In the natural sciences and in engineering studies, mass–thickness contrast only predominates in noncrystalline materials, such as two-phase glasses, as well as in replica studies, where the contrast is often due to variations in the thickness of a metal shadow deposited on the replica, or to the presence of an extracted phase (Section 4.2.5). 4.3.2

Diffraction Contrast and Crystal Lattice Defects

In the case of a perfect crystal, the contrast in the microscope is associated with the amplitude scattered from the incident beam into a diffracted beam by diffracting planes which have a specific reciprocal lattice vector g. This amplitude can be calculated by summing the diffracted amplitudes from all the unit cells that lie along the path of the diffracted beam (Figure 4.13). The phase difference f in the amplitude scattered by a unit cell that is at a position r with respect to the origin in the column of material responsible for the contrast is given by: f ¼ 2pðg  rÞ

ð4:6Þ

while the amplitude that is scattered by this unit cell can be written: Aeif ¼ A exp½
2piðg  rÞ

ð4:7Þ

The unit cells in the column of crystal being considered are each separated by the lattice parameter a. For Bragg diffraction, that is scattering in phase, g  r ¼ n, where n is an integer. Since the electron wavelength is very much less than the lattice parameter, each unit cell in the crystal scatters independently of the others, so that the amplitude scattered by a single unit cell will be proportional to the vector sum of the atomic scattering

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Microstructural Characterization of Materials

factors and hence to the structure factor for the cell F for the specific diffracting planes (Section 2.4.2). If the amplitude scattered is small compared with the incident amplitude, we can ignore any reduction in the amplitude of the direct transmitted beam, so that each unit cell in the column scatters the same amplitude. It follows that, at the Bragg position, the total amplitude scattered by a column of n unit cells will be a linear function of n: X F n exp½
2piðg  rÞ ð4:8Þ An ¼ n

If the number of unit cells in a column is sufficiently large, we can replace the sum by an integral, which for convenience we now take over the thickness t of the thin-film sample measured from the mid-thickness, that is the integral is taken over the range t/2: F At ¼ a

Zt=2 exp½
2piðg  rÞdr

ð4:9Þ


t=2

If the angle of incidence of the incident beam deviates slightly from the Bragg condition, then the scattering vector g in reciprocal space must be replaced by g þ s, where s is the angular deviation from the Bragg position measured in reciprocal space. Similarly, if the structure of the ‘perfect’ crystal lattice is distorted, due to the presence of a lattice defect generating a displacement R, then the position of the scattering element at the position r in the column of crystal is shifted to r þ R. The phase angle for the amplitude scattered from a unit cell at the position r in a column of the crystal that is both misoriented and defective is therefore given by: f ¼ 2pðg þ sÞ  ðr þ RÞ

ð4:10Þ

Expanding the brackets we have: f ¼ 2pðg  r þ g  R þ s  r þ s  RÞ

ð4:11Þ

Of these terms g  r ¼ n is an integer and has no effect on the phase of the amplitude scattered. The term s  R is obtained by multiplying two small vectors together, and so can be neglected. The two remaining terms, g  R and s  r, are additive. They represent the phase shift in the amplitude scattered at the position r into the diffracted beam g due either to deviations from the exact Bragg condition s  r, or to distortions of the crystal lattice, that is, lattice strains associated with the presence of lattice defects g  R. It follows that in microstructural features imaged by diffraction contrast we are observing the summed contrast effects of both deviations in reciprocal space (changes in the specimen orientation with respect to the incident beam), as well as displacements in real space (displacements of the crystal lattice due to lattice strains). Complete interpretation of diffraction contrast requires that these two effects be separated and their origin identified. This requires that the operating reflections, the g vectors, be known and that the causes of the displacements in both real space R, and reciprocal space s, be identified. This is often a difficult process that necessitates knowledge of the thickness of the sample, together with a series of both bright-field and dark-field images taken using different g reflections. In practice, a complete analysis of diffraction contrast may not be

Transmission Electron Microscopy 207

necessary, and image analysis can be limited to a generic identification of the types of lattice defects present: dislocations, stacking faults and point defect clusters, rather than a complete quantitative analysis of a defect that includes, for example, the sign and magnitude of a dislocation Burgers vector or the value of a stacking fault vector. 4.3.3

Phase Contrast and Lattice Imaging

The transmission electron microscope is commonly operated either as a microscope (bright-field or dark-field images) or as a diffraction camera (Figure 4.16). When operated as a microscope, imaging either mass–thickness or diffraction contrast, an objective aperture, placed in the back focal plane of the objective lens, limits the angle of acceptance for the beam to an angle a, which is less than the Bragg angle y for any of the coherently scattered electron beams. The image is then a bright-field, shadow projection image of the electrons coming through the objective aperture. In this image the intensity variations in the image plane reflect the variations in the electron beam current passed down the microscope column after traversing the different regions in the field of view on the thin-film sample. When the incident beam is tilted by a Bragg angle, so that one or other of the coherently diffracted electron beams, rather than the direct transmitted beam, passes down the optic axis of the microscope, then this diffracted beam is accepted into the imaging system. The objective aperture now cuts out both the direct transmitted beam and all other diffracted beams, so that the bright field image is replaced by a dark field image. If the objective aperture is removed or replaced by an aperture, that is large enough to accept both the direct transmitted beam and one or more of the diffracted beams into the imaging system, that is a >2y, then the differences in the path length followed by the different beams will result in an interference pattern in the image plane (Figure 4.17). In order for this interference pattern to be observed and interpreted, several parameters characteristic of the microscope have to be known. These include the values of the chromatic and spherical aberration coefficients, the exact focusing plane of the image, and the coherence and energy spread in the incident electron beam which is determined by the electron source. That phase contrast in the transmission electron microscope corresponds to an interference pattern is extremely important. Imagine an object represented by a number of point sources and having a total electron wave distribution in the object plane defined by f (x,y). The corresponding function g(x,y) in the image plane to this object function represents the amplitude and phase of f(x,y) after travelling down the microscope column. Each point in the image plane will contain contributions from all the beams that have been transmitted by the objective aperture, so: Z ð4:12Þ gðrÞ ¼ f ðr 0 Þhðr
r 0 Þdr 0 ¼ f ðrÞ  hðr
r 0 Þ where the more convenient radial coordinates r have been used instead of Cartesian coordinates (x,y), and h(r) represents the contribution to the electron wave distribution function from each individual object point to any given point in the image. The function h(r) is termed a point-spread function or impulse-response function, and g(r) is the convolution of f(r) with h(r). Thus the electron wave function of the incident beam is modified by the electron density distribution in the specimen, but also convoluted with a function which

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Microstructural Characterization of Materials

Specimen

Objective Lens Objective Aperture

SAD Aperture

Intermediate Lens

Projector Lens

Screen Diffraction Pattern

Image

Figure 4.16 The transmission electron microscope can be used to image the specimen by focusing the final image in the plane of the fluorescent screen, or it can be used to image the diffraction pattern from the specimen. To an excellent approximation, the image of the specimen is observed when the imaging system is focused on the front focal plane of the objective (the position of the specimen), while the diffraction pattern is observed when the imaging system is focused on the back focal plane of the objective (The back focal plane of the objective corresponds to the first image plane for the electron diffraction pattern in the microscope). SAD, selected area diffraction.

Transmission Electron Microscopy 209

Figure 4.17 If the objective aperture accepts a Bragg-diffracted beam as well as the direct transmitted beam, a > 2y, then an interference pattern will be formed in the image plane as a result of the difference in path lengths of the two beams.

describes the response of the microscope column: the electron beam source, the electromagnetic lenses, the aperture sizes and the lens aberrations. It is now convenient to move to reciprocal space where we can represent g(r) by a Fourier transform for which: X GðuÞexpð2piu  rÞ ð4:13Þ gðrÞ ¼ u

where u is the reciprocal lattice vector. We define the Fourier transform of f(r) as F(u) and that of h(r) as H(u), and find them to be related [compare Equation (4.10)] by: GðuÞ ¼ HðuÞFðuÞ

ð4:14Þ

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Microstructural Characterization of Materials

where H(u) is termed the contrast transfer function (CTF) characteristic of the microscope. The CTF is dimensionless and is plotted as a function of r
1, that is, as a function of the spatial frequency reaching the image plane. The CTF describes the total influence of all the various microscope parameters on the phase shift for an electron wave propagating down the microscope column. There are three major contributions to the CTF: 1. An aperture function that defines the cut-off limit for spatial frequencies above a critical frequency determined by the aperture radius. 2. An envelope function that describes the damping of the spatial frequencies due either to chromatic aberration, or to instabilities in the objective lens current, or to the coherence limit of the electron source. 3. An aberration function that limits the spatial frequencies available for imaging and is usually dominated by the spherical aberration coefficient of the electromagnetic objective lens. The aberration function can be given as: BðuÞ ¼ exp½ixðuÞ

ð4:15Þ

where xðuÞ ¼ pDf lu2 þ

1 pC s l3 u4 2

ð4:16Þ

and Df is the underfocus of the objective lens, that is the distance between the object plane and the focal plane of the lens, l is the wavelength, and Cs is the spherical aberration coefficient of the objective lens. We now examine the influence of the above three contributions on the form of the CTF. Figure 4.18 shows the CTF for a 300 kV transmission electron microscope with a spherical aberration coefficient of 0.46 mm at an objective lens underfocus of
36 nm. The parameters used in calculating Figure 4.18 include the relative structure factors for the lattice spacings in aluminium. Figure 4.18(a) shows the coherent CTF and is defined only by Equation (4.16.) Figure 4.18(b) shows the spatial coherence envelope, defined by the aperture function, and Figure 4.18(c) shows the temporal coherence envelope that is defined by the damping of the spatial frequencies due to lens instabilities and the limited coherence of the electron source. Finally, Figure 4.18(a)–(c) is combined in Figure 4.18(d), which therefore corresponds to the CTF for this specific microscope column at a selected objective lens defocus. From Figure 4.18(d) we see that the CTF of this microscope can only transfer a limited range of reciprocal spacings, that is, spatial frequencies, to the image. These frequencies can be correlated with the lattice spacings in the crystal structure of the specimen that are responsible for the diffraction pattern observed in the microscope. In regions where the CTF is zero, no information can be transferred down the microscope column to the image. Thus, in the present example, only the {1 1 1}, {0 0 2}, and {2 0 2} planes of aluminium can be resolved in the image, always provided that the crystal is oriented in the correct zone axis! Since the CTF depends on the defocus of the objective lens, the CTF can be tuned to selected spatial frequencies by adjusting the objective lens current. This is illustrated in Figure 4.19, which uses a different presentation for full CTF. The y-axis is the value of objective lens focus, with negative values corresponding to an underfocus, and the x-axis is the spacing in reciprocal space, that is the spatial frequencies, and the intensity is plotted on

Transmission Electron Microscopy 211

Figure 4.18 A CTF for a high-resolution objective lens shown at Scherzer defocus and including the normalized structure factors for some crystallographic planes in aluminium. The functions were calculated using a 300 kV accelerating voltage, a focal spread of 10 nm, and a semi-convergence angle of 0.3 mrad. (a) The coherent CTF defined only by Equation 4.16. (b) The spatial coherence envelope, defined by the aperture function (c) The temporal coherence envelope defined by the damping of the spatial frequencies due to electromagnetic lens instabilities and the coherence of the electron source. Combining (a), (b), and (c) yields (d), the CTF of the microscope at Scherzer defocus.

Figure 4.19 Absolute values of the CTF plotted using the same values used in Figure 4.18, but here the relative values of the CTF correlates to the plotted brightness (white corresponds to CTF maxima, while black is a CTF of zero). The y-axis is the objective lens defocus. The selected value of objective lens defocus (the Scherzer focus) plotted in Figure 4.18 is indicated in Figure 4.19 as a horizontal red line.

212

Microstructural Characterization of Materials

a shaded, grey scale. The value of objective lens defocus used to calculate Figure 4.18 is indicated in Figure 4.19 as a red, horizontal line. In this case only the absolute values of the CTF are plotted, so a ‘white’ intensity corresponds to both positive values of the CTF, and then, after crossing the zero values (black), negative values. By controlling the objective lens defocus, that is the lens current, we can shift the CTF and enhance the phase contrast for specific, selected crystallographic planes. In order to get the best performance from the microscope, a calculation of the CTF and some careful thought about the information that is being sought should precede the microscope session! Since no information is transferred to the image when the CTF has a value of zero, the best CTF, that is the best objective lens performance, will have few oscillations about the zero values at high spatial frequencies. Under-focusing the objective lens moves the first CTF crossover (first value of zero) to larger values of nm
1 (that is, smaller d-spacings), and can partially compensate for the spherical aberration of the lens. The position of optimum compensation is termed the Scherzer defocus (Scherzer predicted this effect in 1949). The Scherzer defocus can be calculated from: pffiffiffiffiffiffiffiffiffiffiffiffi ð4:17Þ Df Sch ¼
1:2 ðC s lÞ The Scherzer defocus crossover determines the resolution limit of the microscope, usually termed the point resolution, and corresponds to the minimum defocus value at which all beams below the first CTF crossover have approximately constant phase. Below the Scherzer defocus, crystallographic information will still be available from spacings below the point resolution, but complete computer simulation of the image is necessary to interpret the information. Under such conditions the information transferred to the image by the microscope is severely damped, so that the contrast available from crystallographic planes with small d-spacings is limited. This sets a second resolution limit for the microscope, termed the information limit. The distinction between the point resolution and the information limit is the reason why lattice images can be readily obtained from crystal lattice planes that have d-spacings appreciably less than the point resolution of the microscope. Point resolution is an important criterion for microscopists working on noncrystalline biological tissue and cell samples, but for the materials community it is the information limit that is the more important criterion, since this defines the minimum interplanar spacings that can be resolved in a lattice image. The introduction of hexapole electrostatic stigmators for spherical aberration correction (Section 4.1.2.2) had a major impact on the electron microscope community. The correction system can reduce the value of the spherical aberration coefficient, so that the point resolution and information limit of the microscope coincide. Using a monochromatic electron source to reduce the influence of the chromatic aberration, resolutions of less than 0.07 nm are now available. In addition to this improved resolution limit, it is also possible to extract phase information from the lattice image, so that the information derived is no longer limited to a two-dimensional intensity distribution in the x–y plane. Initially this was achieved by analysis of a through-focus series of images, but, in principle, a single recorded image can be compared with a virtual image based on the known crystal structure to extract in-depth information. Lattice imaging has brought the electron microscopist

Transmission Electron Microscopy 213

a long way from the ‘shadow’ images based on mass–thickness and diffraction contrast. It is important to recognize that the computer interpretation of lattice image contrast is now an essential component of image analysis. The microscopist should never be misled into thinking that the periodicity and contrast observed in the lattice image represent directly the position and electron density of the atoms in the nanostructure being studied.

4.4

Kinematic Interpretation of Diffraction Contrast

The basic assumption of kinematic diffraction theory, that the amplitude diffracted out of the incident beam does not affect the intensity of this beam, is so patently false that it may seem odd to include this section in the text. However, the use of kinematic arguments, using amplitude–phase diagrams, drastically simplifies the discussion of diffraction contrast from lattice defects and allows the major qualitative features of defect contrast to be demonstrated, so that the loss of quantitative rigour should be forgiven. 4.4.1

Kinematic Theory of Electron Diffraction

The basic equation for the kinematic theory of electron diffraction has been given in Section 4.10, and the generalized form that we will discuss here is: Z þ t=2 exp½
2piðg þ sÞ  ðr þ RÞ ð4:18Þ At ¼ F=a
t=2

We justify the approximations of the kinematic theory by assuming that the intensity of the diffracted beam is much less than that of the incident beam, so that the intensity of the direct transmitted beam approximates that in the original incident beam. 4.4.2

The Amplitude–Phase Diagram

Returning to the ‘column’ approximation, which assumed that each unit cell in a column of crystal in the thin-film specimen scattered independently when viewed along the direction of the diffracted beam (Figure 4.20) and further assuming a perfect crystal free of lattice strains (R ¼ 0), then the phase mismatch between successive unit cells in the column Df is uniquely determined by the deviation in reciprocal space from the Bragg condition s, and is given by Df ¼ 2pa, where a is the thickness of a unit cell in the column. Replacing the increment Df by df and the sum by the integral, that is, taking df ¼ 2psdr, and ignoring the structure factor F, since this will be a constant for any given operating reflection, while noting that, for g  r ¼ 1, exp[2pi(g  r)] ¼ 1, then we obtain the total amplitude diffracted by the column as: Z þ t=2 sinðptsÞ exp½
2pis  rdr ¼ ð4:19Þ A¼ ps
t=2 which corresponds to a relative diffracted intensity: I=I0 ¼

sin2 ðptsÞ ðpsÞ2

ð4:20Þ

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Microstructural Characterization of Materials

Incident Beam

Unit Cell at r

Column of Crystal

Direct Transmitted Beam

Thin Film Specimen

Diffracted Beam

Figure 4.20 The unit cells in a column of crystal that corresponds to a Bragg scattering direction will each scatter a proportion of the incident beam into the diffracted beam.

Although this relation is based on an oversimplified model, it demonstrates the effects of both the specimen thickness t and the deviation from the Bragg condition s on the diffracted intensity. This equation is best plotted as an amplitude–phase diagram (Figure 4.21). The radius of the circle is equal to (2ps)
1 and decreases rapidly as the specimen is tilted out of the

Bottom of thin film

2πst 1 2πs

Resultant amplitude

t

Top of thin film Figure 4.21 In kinematic theory, the amplitude–phase diagram for the diffracted amplitude is a function of specimen thickness t and deviation from the Bragg condition s.

Transmission Electron Microscopy 215

Bragg condition (increasing s). As the thickness increases at a fixed value of s, the diffracted amplitude oscillates sinusoidally between two maxima that are proportional to (ps)
1, while the relative diffracted intensity oscillates between zero and a maximum value that is proportional to (ps)
2. It follows that a tapered thin-film sample, viewed at a fixed crystal orientation which is just off the Bragg condition for a particular reflection, will show a series of bright fringes, termed thickness fringes, that are parallel to the edge of the specimen. The fringes will get closer together and grow fainter as the sample is tilted further from the Bragg position. However, a bent crystalline film of uniform thickness will also show fringes, termed bend contours, whose intensity and separation decrease with increasing s, the first extinction corresponding to the condition st ¼ 1. Both thickness fringes and bend contours are prominent features of the contrast from thin crystalline films. Although dynamic diffraction and absorption effects (Section 4.5) markedly modify the quantitative analysis of these features, the kinematic theory is quite adequate for a qualitative understanding of this thickness- and orientation-dependent diffraction contrast.

4.4.3

Contrast From Lattice Defects

The presence of a lattice defect modifies the amplitude–phase diagram of Figure 4.21 by introducing a second term into the phase shift, so that the integral determining the amplitude becomes: Z þ t=2 exp½
2piðs  r þ g  RÞdr ð4:21Þ A¼
t=2

The additional phase shift due to the displacement field of the defect may either increase the curvature of the amplitude–phase diagram, so that the crystal lattice is tilted locally towards the Bragg condition, or decrease this curvature, so that the effect of the lattice defect is to tilt the lattice in this region further from the exact Bragg condition (Figure 4.22). In the first case, the radius of the amplitude–phase diagram is decreased and the diagram starts to collapse in the region of the lattice near the defect. In the second case the diagram is expanded in the region near the defect, opening up the amplitude–phase diagram. For the sake of convenience, the zero of coordinates in the unit cell column of diffracting crystal is now shifted to coincide with the position of maximum lattice displacement due to the defect, that is the ‘centre’ of the defect displacement field, rather than the mid-thickness of the thinfilm sample. It follows that the displacement field R due to the defect has a result equivalent to either amplifying or suppressing the effect of s, by increasing or reducing the effective deviation of the crystal from the Bragg condition. An important consequence of this conclusion is that the effect of the defect will be reversed if the sign of s is reversed. This will occur whenever a crystalline film that contains a defect is tilted in order to image the defect with the incident beam on opposite sides of the exact Bragg orientation. The position of the maximum value of R in the column is also important, and in general maximum diffraction contrast from defects is expected to occur when R and s are of opposite sign and when the position of maximum R is near the mid-point of the foil. In thicker regions the intensity is expected to oscillate as the defect position moves from the top to the bottom of the foil, reflecting the

216

Microstructural Characterization of Materials Top of Thin Film

Resultant Amplitude

Position of Defect

Resultant Amplitude

Position of Defect Bottom of Thin Film

Figure 4.22 The presence of a lattice defect introduces a displacement field that either collapses the amplitude–phase diagram (increases the curvature) or tends to open it out (decreases the curvature). (The origin of the amplitude–phase diagram has been moved to correspond to the position of maximum displacement R in the column of crystal considered)

oscillations in amplitude that are associated with the diameter of the amplitude–phase diagram for a perfect crystal. We have concentrated on the intensity in the kinematic diffracted beam, and hence on the dark field image, but in practice samples are usually first viewed in bright field before any defect analysis is attempted in dark field. In general the kinematic intensity observed in bright field is approximately the inverse of that observed in dark field. That this is not always the case is due to the non-kinematic, dynamic and absorption effects that become important for thicker films and will be discussed later (Section 4.5). Grain and phase boundaries constitute a special class of lattice defect whose contrast can be qualitatively explained in terms of thickness extinction fringes. The two crystals on either side of the boundary are unlikely to deviate from the nearest Bragg position by the same amount, so that the crystal with the minimum value of s will dominate the contrast and generate a series of thickness extinction fringes at an inclined boundary whose separation is dictated by the value of s and the tilt angle of the boundary with respect to the incident beam. The number of fringes will depend on the thickness of the thin-film crystal and the structure factor for the reflection g. Diffraction from the neighbouring crystal will interfere with this simple, single crystal thickness contrast. This can occur when the second crystal is also near a Bragg diffraction condition, so that the appropriate value of s for the second crystal approaches that for the first. 4.4.4

Stacking Faults and Anti-Phase Boundaries

Stacking faults and anti-phase boundaries (APBs) constitute a special case since the displacement vector does not vary continuously along the column of the diffracting crystal,

Transmission Electron Microscopy 217

Above the fault Fault Position

Below the fault 2πg. R Phase Shift

Figure 4.23 An amplitude–phase diagram for a column of crystal containing a stacking fault or APB at the origin of the coordinate system.

but rather changes discontinuously across the plane of the stacking fault or the APB. The amplitude–phase diagram for the crystal above the point of intersection of the diffracting column with the fault plane is therefore undisturbed, while that below the fault plane, now positioned for convenience at the origin of the diagram coordinates, is rotated by an angle equal to the phase angle associated with the fault vector, 2pg  R (Figure 4.23). For example, a stacking fault in the lattice of an FCC metal has a fault vector 1/6ah1 1 2i. This vector is exactly equivalent to a fault vector of 1/3ah1 1 1i in the FCC lattice, since it can be combined with an appropriate unit lattice vector of 1/2ah1 1 0i, as in the dislocation 1 1 0 ¼ a3 ½ 1 1 1. The phase shift 2pg  R, where R is now the fault reaction, a6 ½1 1 2 þ a2 ½ vector, will be zero if g is perpendicular to R. If |R| is 1/3ah1 1 1i and |g| is an allowed 1/ 2ah1 1 0i reflection, then the phase shift can take one of three values, 0 (when the two vectors are mutually perpendicular) or 120 . The sign of R is determined by the type of stacking fault, which may be either intrinsic or extrinsic that is the fault may be due to a missing plane of atoms (ABC|BCABC) or to an extra plane (ABCBABC) inserted into the stacking sequence. In the case of an APB in the crystal, the fault will only be visible if g  R is a noninteger. So it is not enough for the reciprocal lattice vector to have a resolved component parallel to the fault vector. The reciprocal lattice vector must be a partial lattice vector of the parent lattice, termed a superlattice vector. Hence the APB will only become visible if the diffracted beam is from a superlattice reflection. If the sample is thick enough, then the amplitude–phase diagram from a column of crystal containing a fault plane will exhibit thickness fringes as the fault position moves from the top to the bottom of the crystal, similar to those observed in a wedge of crystal or at a grain

218

Microstructural Characterization of Materials

boundary. However, the origin of these fringes is very different, since diffraction is now occurring in both the regions of the crystal, above and below the fault plane. 4.4.5

Edge and Screw Dislocations

Diffraction contrast from dislocations is dominated by the Burgers vector of the dislocation line b, since the strain field displacements associated with the dislocation are proportional to the value of b. However, the direction of the dislocation line l with respect to the Burgers vector is also important. Dislocations that lie parallel to the Burgers vector are termed screw dislocations, and, in an isotropic crystal, have a cylindrically symmetric strain field, with no strain component perpendicular to the Burgers vector. It follows that a reciprocal lattice vector perpendicular to a screw dislocation in an isotropic crystal should not result in any contrast. The condition g  R ¼ 0 for no diffraction contrast can therefore be replaced by the condition g  b ¼ 0 for no contrast from a screw dislocation. By observing the contrast due to the presence of the dislocation in different dark-field images as a function of the diffracting vector g in each case, and finding two values of g for which no contrast is observed, it is possible to identify the direction of the Burgers vector unambiguously, but deriving the magnitude of the Burgers vector from diffraction images requires a more complete analysis. By contrast, edge dislocations have a residual component of the strain field perpendicular to b that results from the dilatation (volume change) associated with the presence of an edge dislocation, so that some contrast is expected even if g  b ¼ 0. Even so the weakness of the contrast compared with that observed with other reflections, together with some knowledge of the Burgers vectors to be expected in the crystal lattice, usually permits some tentative conclusions to be drawn. The vector product b · l gives the normal to the allowed glide plane for the dislocation in the crystal. This plane is undefined for screw dislocations, which are therefore free to crossslip. In general, a preferred glide plane exists, even for screw dislocations, suggesting that screw dislocations are partially dissociated, even when the separation of the partial dislocations is small and no stacking fault contrast is observable. The sign of the strain-field associated with a dislocation is reversed when the column of diffracting crystal is moved across the projected position of the dislocation line in the image, so that the sign of the phase shift is also reversed. It follows that on one side of the dislocation the amplitude–phase diagram is expanded, enhancing contrast, while on the other side it collapses, reducing the observed contrast, and this depends on whether or not g  R has the same sign as s  r (Figure 4.22). Thus, when the projection of a dislocation line in the image crosses a bend contour, the position of maximum contrast changes sides, and the true dislocation position can be inferred from the mid-point between the two maxima (Figure 4.24). The apparent width of a dislocation depends on the numerical value of g  b and the value of s. Close to the Bragg position, the contrast will be a maximum and the width of the observed contrast maxima is typically of the order of 10 nm. This is a very poor resolution for transmission electron microscopy and constitutes a major barrier to the study of dislocation–dislocation interactions in sub-boundaries or during plastic shear. Much better resolution can be obtained in the dark-field image by moving away from the Bragg position, that is, by tilting the sample to large values of s. The contrast is then very weak, and may be

Transmission Electron Microscopy 219

–R +R Dislocation Line +s Bend Contour –s

Figure 4.24 The contrast expected from a dislocation line as it crosses a bend contour. When s and R are of the same sign the amplitude–phase diagram is expanded and strong contrast is expected. The region of maximum contrast lies to one side of the dislocation line and moves to the opposite side if the dislocation crosses a bend contour.

difficult to see, but the resolution in this weak beam image is appreciably better, down to of the order of 2 nm. In samples of pure metals and ductile alloys dislocations are often observed to move in the electron microscope, and such behaviour has been used to study the glide process. The phenomenon has been attributed to both thermal stresses, associated with electron beam heating and contrast due to scattering processes. More generally, stresses in the sample are induced by the build up of a carbonaceous contamination film on the surface. When dislocation movement occurs during observation in the microscope, it is often accompanied by the appearance of slip traces in the image at the top and bottom of the film, where the dislocation has disrupted a layer of surface contamination. These traces can be used to determine the glide plane in the crystal or, if the glide plane is known, to derive an accurate value for the specimen thickness.

4.4.6

Point Dilatations and Coherency Strains

Point defects, such as vacant lattice sites and impurity or alloy atoms, cannot be resolved in transmission electron microscopy, but remarkably small clusters of such defects will generate local lattice strains that give rise to detectable diffraction contrast. Such contrast has been observed for three specific types of defect: 1. Radiation damage that is associated with injected ions or with lattice atoms that have been displaced by an energetic incident particle. 2. Defects associated with the condensation of a supersaturation of lattice vacancies, to form, for example, faulted or unfaulted vacancy loops and more complicated defect assemblies, such as stacking fault tetrahedra. 3. The early stages of precipitation, especially the formation of solute rich clusters, generally termed Guinier–Preston zones.

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Microstructural Characterization of Materials

Figure 4.25 Faulted dislocation loops in nickel. Reproduced by permission of Plenum Press from Williams and Carter, Electron Microscopy: A Textbook for Materials Science (1996). Original micrograph from Knut Urban

In all these cases the dominant lattice displacement is usually a hydrostatic strain field which can be approximated by a point defect dilatation. In the case of radiation damage, a wide range of effects has been observed. These include the condensation of interstitial atoms, generated by long-range knock-on collisions, the formation of small vacancy loops associated with the annealing out of vacant lattice sites by diffusion, and the formation of small helium bubbles due to a-particle condensation. The clustering of vacant lattice sites to form a collapsed dislocation loop may result in a planar stacking fault in the crystal lattice (for example in nickel, Figure 4.25), but if the stacking fault energy is high, shear of the crystal across the plane of the defect will reduce the total energy and result in a small vacancy loop that is bounded by a unit lattice dislocation whose Burgers vector lies at an angle to the plane of the defect. These loops can be resolved by weak beam contrast when they exceed 2–5 nm in diameter, and their Burgers vectors analysed using the criteria described previously for dislocations, for example, by noting the change in contrast from inside to outside the loop as the sign of s is changed by tilting the specimen so that a bend contour sweeps across the image. Unresolved, smaller loops behave only as point dilatations, and typically show up as twin black and white crescents, as do larger loops when they are viewed edge on. Again the sign of this contrast, that is, black/white as opposed to white/black, changes depending on the sign of s. However, as we will see below, the situation is usually rather more complicated than can be accounted for by the kinematic contrast approximation. The observed contrast depends on the position of the defect within the thickness of the foil. Point dilatations observed near the top surface of the sample show the same contrast in both bright and dark field, while those near the bottom surface show a reversed contrast for the bright- and darkfield images. It is only the dark-field image that meets the kinematic criterion discussed previously, namely that the contrast is determined by whether or not the phase shift associated with the defect reinforces that due to a deviation from the Bragg condition. In addition to the care required to interpret image contrast resulting from nonkinematic diffraction, there are also potentially confusing effects associated with rotation or inversion of the image in an electromagnetic imaging system. This is particularly the case when diffraction patterns are to be recorded for comparison with bright- or dark-field

Transmission Electron Microscopy 221

lattice images, since any inversion is then equivalent to a sign reversal of the strain. Distinguishing between vacancy and interstitial defects, that is, tensile as opposed to compressive dilatations, may be straightforward in theory, but can be quite confusing in practice. Small precipitates and Guinier–Preston zones behave qualitatively very like vacancy clusters and radiation damage, but now the displacements are determined by lattice misfit that is associated with mismatch in the effective size of the solvent and solute atoms. As the precipitates grow, the strains associated with lattice mismatch accumulate, until these coherency strains are eventually relieved, at least in part, by the nucleation of interfacial, or Van der Merwe, dislocations at the phase boundary. If the orientation of the precipitate and the matrix are uniquely related, so that they have a low-index orientation relationship, usually dictated by interfacial energy terms, then the mismatch in lattice constants of the matrix and precipitate give rise to interference effects in the image which are associated with double diffraction. This doubly diffracted beam will result in a nonlocalized set of interference fringes, termed moir e fringes (Figure 4.26), whose spacing depends on the mismatch between the two operating g vectors for the matrix and the precipitate phase reflections: dmoire ¼

d1 d 2 1

½ðd1
d2 Þ2 þ d1 d2 b2 2

ð4:22Þ

where b is the angle of rotation between the two g vectors and |g| ¼ 1/d. It is not difficult to distinguish between the moire fringes and the interface, Van der Merwe, dislocations. The latter are fully localized and have Burgers vectors which can be analysed by comparing the dark-field images from different reflections in the diffraction pattern, while the former are nonlocalized in the image plane and have a spacing that depends only on the operating Bragg reflection. Moire fringes may be caused by either a difference in lattice spacing, or a rotation between the two diffracting regions. It follows that moire effects may also be observed when dislocation tilt or twist sub-boundaries are present in the thin crystalline film. Once again, diffraction contrast due to the dislocation strain fields must be distinguished from moire contrast due to lattice mismatch either side of the sub-boundary.

4.5

Dynamic Diffraction and Absorption Effects

The kinematic theory of contrast is a poor approximation for all but the thinnest of thin-film crystalline specimens. We will not attempt a full description of the theory of dynamic diffraction, but rather limit ourselves to a simplified physical explanation of its significance. We will also outline some of the more important conclusions that affect diffraction image contrast. One consequence of dynamic diffraction has already been noted: the position of a small defect, equivalent to a point dilatation, within the diffracting column of crystal in the thin foil determines whether or not bright-field image contrast is inverted with respect to dark-field contrast. In the following discussion we assume that only one diffracting plane is active, that is only the direct transmitted beam and one diffracted beam contribute to the image. This condition is termed two-beam diffraction contrast. The concept of only one diffracted beam contributing to the image can sometimes be misleading, and some comments on the geometry for effective two-beam diffraction are in

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Microstructural Characterization of Materials

(a)

(b)

(c)

Figure 4.26 Moir e fringes resulting from the overlap of: (a) two lattices with different d-spacings of the lattice planes; (b) two lattices with the same d-spacing, but with an angle of rotation (b) between them; (c) two lattices that have both different spacings and an angle of rotation between the two sets of planes.

order. The two-beam condition derives from the analytical solution for dynamic electron diffraction. In the transmission electron microscope, the two-beam condition can be approximated by first tilting the crystal so that the incident beam is exactly parallel to a low-index zone axis, and then tilting about an axis perpendicular to this zone axis, so that the Kikuchi line from a diffracting plane is brought to intersect the corresponding diffracted spot (Figure 4.27). Under these conditions, s for this reflection will be zero, while s for all

Transmission Electron Microscopy 223

(242)

(242) (224)

202

022 (202)

(202)

(242)

(242) (224)

202

022 (202)

(202)

Figure 4.27 A Kikuchi pattern from a FCC structure first oriented in a [1 1 1] zone axis, after tilting the specimen such that s ¼ 0 for g ¼ 022. A two-beam, dark-field diffraction contrast image can be formed from the 022 reflection, or a two-beam bright-field image can be formed from the central spot.

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Microstructural Characterization of Materials

the other reflections will have a nonzero value. The selected plane will now diffract far more strongly than the other planes on the same zone-axis. A diffraction contrast image can now be formed from either the central beam (bright field) or the diffracted beam for which s ¼ 0 (dark field). Under these two-beam conditions the solution for the intensity of the diffracted beam is given by: Ig ¼

ðpt=xg Þ2 sin2 ðptseffective Þ ðptseffective Þ2

where seffective ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 þ ð1=x2g Þ

ð4:23Þ

ð4:24Þ

and xg is the extinction distance for the selected reflection g. The extinction distance xg depends on the volume of the unit cell of the crystal, V, the angle of the Bragg reflection yB, the wavelength of the electron beam l, and the structure factor Fg for the selected reflection g: xg ¼

pVcos yB lF g

ð4:25Þ

Table 4.1 compares the extinction distances for the first few prominent reflections calculated for aluminium and gold at 100 kV. For normalized intensities, the intensity of the central spot I0 is 1
Ig. The phenomenon of dynamic electron diffraction is to some extent analogous to the optical wave propagation of polarized light in anisotropic materials that was discussed earlier (Section 3.4.3). The incident electron beam entering the thin-film crystal is split into two propagating beams of slightly different wavelength while passing through the crystal lattice. In the case of electrons, it is the electrical potential inside the crystal that affects the wave vector (the wavelength), and an electron beam incident at the Bragg angle propagates through the crystal as two waves, one with its probability maxima peaked at the atomic positions and one with its maxima between these positions. The small difference in electrical potential experienced by these two waves results in a slightly different electron wavelength, and hence on a difference in their phase angle that increases with increasing path length through the crystal (increasing crystal thickness). For a given reciprocal lattice vector, this phase difference will lead to extinction of the direct transmitted beam at some critical thickness (Figure 4.28). This critical thickness, the extinction thickness xg for that reflection, corresponds to a phase difference of p between the two beams as they exit the crystal. At the same time, the intensity transferred, corresponding to the energy scattered into the diffracted electron beam from the direct transmitted beam, also behaves as two waves of slightly different wave vector propagating through the thickness of the crystal, and will lead to a diffraction maximum at this same extinction thickness, thus conserving the total energy in the diffracted and direct transmitted beams that exit the thin film sample. In other words, it is assumed that no inelastic scattering events have occurred. Within the crystal, solutions for the wave equations for the incident and diffracted beams correspond to four waves. These are termed the Bloch waves, two of which propagate as the

Transmission Electron Microscopy 225

1

Intensity

0

1

Intensity

0

ξg

Thickness t

(1/2) ξ g

(3/2) ξ g

I0

Ig

Figure 4.28 The intensity of the diffracted beam Ig and transmitted beam I0 as a function of crystal thickness. The periodicity is determined by the extinction distance xg.

direct transmitted beam, while the other two propagate as the diffracted beam. These two pairs of waves are approximately p/2 out-of-phase, and between each wave pair there is a small difference in wavelength, caused by the oscillating periodic potential in the crystal lattice. This small difference in wavelength generates a beat pattern of standing waves inside the crystal that result in a thickness-dependent series of complementary maxima and minima in the amplitudes of the direct and diffracted beams. On leaving the crystal the two pairs of four waves recombine into two electron beams, one corresponding to the direct transmitted beam and the other to the diffracted beam (Figure 4.29). Typical values for the extinction thickness at 100 kV vary from of the order of 20 to 100 nm, depending on the atomic number of the material (the electron density) and the reciprocal lattice vector. The most densely packed planes have the strongest scattering power and therefore the shortest extinction distance. In two-beam conditions, and providing the sample can be assumed to be perfectly flat, so that no bending can change the local value of s, then s ¼ 0 and: I g ¼ sin2 ðpt=xg Þ

ð4:26Þ

Thus Ig ¼ 0 at t ¼ 0 and at Ig ¼ xg. Using values of the extinction distance, the thickness of the thin-film sample can be estimated from the dark field or bright-field image (Figure 4.30). If the thickness is constant, but bending of the sample results in local changes of s, then bend contours will be visible in both the bright-field and dark field images (Figure 4.31).

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Microstructural Characterization of Materials

Figure 4.29 The Bloch wave model for dynamic diffraction. Electron energy is transmitted through the crystal as two pairs of waves with a small wavelength difference within each pair. The two wave pairs have their amplitude maxima at and between the atomic positions. Amplitude (energy) is transferred from one pair of Bloch waves to the other as the electrons propagate through the crystal. On leaving the crystal each pair of Bloch waves recombines to form the direct beam and the diffracted beam.

The information that can be derived from two-beam diffraction is limited, and it is often an advantage to tilt the sample so that a high symmetry zone axis is parallel to the incident beam. When this is done accurately, the crystal is aligned parallel to a low-order Laue zone and several diffracting planes will generate reflections simultaneously, a condition termed multi-beam diffraction. Since more than one lattice reflection now contributes to the diffraction contrast from the incident beam, an anomalously low extinction distance will result, leading to especially strong sensitivity to lattice defects. Such conditions are exceptionally favourable to the analysis of the local strains associated with small defect clusters and the early stages of precipitation (Figure 4.32). So far we have assumed that all the scattering interactions of the electrons passing through the thin-film sample are elastic, but for samples whose thickness approaches and exceeds the extinction thickness this is not strictly true. The probability of an inelastic scattering event depends on which of the Bloch waves in the crystal are being considered. Inelastic scattering is associated with those Bloch waves that have their amplitude maxima at the atomic positions, thus maximizing the probability of inelastic scattering, while those Bloch waves whose amplitude minima fall at the atomic positions cannot interact inelastically with the atomic nuclei. It follows that inelastic absorption of the Bloch waves is, to a good approximation, confined to just one member of each of the two pairs of Bloch waves travelling through the thin film. Therefore, in a sufficiently thick crystal, the contrast oscillations associated with thickness gradually fade out, since only

Transmission Electron Microscopy 227

Figure 4.30 Schematic representation of the variation in intensity of a dark-field Ig and brightfield I0 image as a function of specimen thickness and extinction distance.

the waves travelling ‘between’ the atoms are transmitted. It follows that, for thicker samples, the intensity scattered into the diffracted wave by a defect that is near the centre of the foil depends only on the deviation from the Bragg condition and not on the foil thickness. Hence in thick crystals the oscillations in contrast along a dislocation running at an angle through the foil are restricted to the regions at the top and bottom surfaces, and are absent near the centre. 4.5.1

Stacking Faults Revisited

We noted previously, when discussing the contrast from a point dilatation, that this contrast depended on the position of the defect in the foil and whether the contrast was observed in bright or dark field. The case of a stacking fault observed at an inclined angle in a thick foil provides a clear and instructive example of the way that dynamic effects (the extinction thickness) and absorption effects (the removal of one of the Bloch waves from the

228

Microstructural Characterization of Materials

Figure 4.31 Bright-field micrograph of a thin film of sapphire. Residual stresses due to the process used to form the film cause local elastic bending that generates bend contours in the image which reflect the symmetry of sapphire.

transmitted beam) determine the diffraction contrast observed in the microscope (Figure 4.33). When the stacking fault is near the bottom of a thick foil the second Bloch wave (2), with its amplitude peak maxima at the atomic sites, in the diffracted beam from the region of crystal above the fault is absorbed, so that only the first Bloch wave (1), with its peak maxima between the atomic sites, is available to be scattered into the diffracted beam below the fault (Bloch wave 20 ). The two beams exiting the crystal are then the first Bloch wave from the region above the fault (Bloch wave 1) and the second Bloch wave from the region below the fault (Bloch wave 20 ). When the stacking fault is near the top of the foil the first Bloch wave above the fault is scattered (Bloch waves 1 and 2), and both beams are then transmitted, with the appropriate phase shift 2pg  R, through the crystal below the fault (Bloch waves 10 and 20 ). However, both beams 2 and 20 are now absorbed, since both have their maxima at the atomic sites, so that the two beams exiting the crystal are now 1 and 10 . It follows that the bright-field and dark-field images of a stacking fault will be complementary when the fault is near the bottom of the foil (black stripes in the dark-field image corresponding to white stripes in the bright-field image), but identical when the fault is at the top of the foil. Similar arguments can be applied to strain contrast from small

Transmission Electron Microscopy 229

Figure 4.32 Contrast from precipitate nuclei observed in a specimen of uniform curvature in the region of a low order Laue zone (a symmetry axis). Reproduced from Hirsch et al., Electron Microscopy of Thin Crystals, published by Butterworth.

Figure 4.33 The contrast from a stacking fault in bright- and dark-field images results from a combination of extinction interference and absorption (see text).

230

Microstructural Characterization of Materials

precipitates or dislocations and used to rationalize the contrast observed and to identify the relative positions of the lattice defects in the foil thickness. 4.5.2

Quantitative Analysis of Contrast

Enough has been said to make it clear that the qualitative analysis of diffraction contrast requires some special care just to ensure that the sign of the displacement field associated with a defect is correctly identified. It is important to image the same defects in both bright field and dark field, to obtain images from more than one reflection, and to check the effect of reversing the sign of s, the deviation from the Bragg condition, by tilting the sample while under observation. It is also important to verify the calibration of the electron optics in the microscope, in particular any inversion or rotation of the image that may occur with respect to the diffraction pattern. Quantitative diffraction contrast analysis requires considerable additional knowledge, well beyond the scope of this text. However, it should be possible to analyse displacements semi-quantitatively. This includes the sign of the coherency strains around point defect clusters due to vacancies, interstitial defects or solute atoms, as well as any anisotropy associated with these strains. It also includes complex dislocation and partial dislocation interactions. These may be associated with plastic flow mechanisms or observed in subgrain boundaries and semicoherent interfaces between epitaxially related phases. In general, insufficient effort is made to interpret diffraction contrast quantitatively, considering the ability of modern computer modelling to check any analysis of the contrast by computer simulation.

4.6

Lattice Imaging at High Resolution

Phase contrast imaging of crystal lattices and lattice defects is well within the reach of commercial transmission electron microscopes and we have described the main features of this mode of image formation, especially its sensitive dependence on the physical parameters of the microscope, as summarized by the CTF of the microscope. In what follows we now concentrate on the relationship between the observed lattice image and the crystal structure. 4.6.1

The Lattice Image and the Contrast Transfer Function

The CTF summarizes the phase shifts introduced by the imaging system of the microscope into the wave function of the electrons forming the image in the plane of image observation: a fluorescent screen, a photographic recording emulsion or a CCD. The phase shifts x associated with this imaging system is expressed as sinx, and the CTF gives sinx as a function of the wave number or spatial frequency, which is proportional to the scattering angle subtended by that frequency at the objective aperture and commonly measured in reciprocal space as 1/d (in units of nm
1) (Figure 4.18). The final phase and amplitude for each spatial frequency is obtained by convoluting the CTF with the calculated phase and amplitude of the electrons transmitted through the sample. These phase shifts and amplitudes are calculated at the exit plane of the thin-film object using the dynamical

Transmission Electron Microscopy 231

theory of electron diffraction, before being ‘transmitted’ through the imaging system of the microscope, using the CTF, until they interfere in the image plane. The intensity at any position in the image plane is obtained by multiplying the integrated image amplitude by its complex conjugate. The factors that enter into the CTF include parameters that are associated with both the electron beam source and the objective lens. The source parameters are its coherency and energy spread. The relatively large dimensions of a LaB6 crystal electron emitter result in poor coherency for this source. In contrast, a field emission gun has a very small emitting diameter with excellent coherency, approximating a point source, as well as a very small energy spread. The traditional tungsten filament is inferior on both counts, since the effective source diameter is quite large and the operating temperature very high. An additional parameter is the angular spread (the divergence) of the incident beam, which is determined by the condenser system and the beam focusing conditions. The objective lens parameters include the spherical aberration coefficient, as well as the current stability in the lens. Additional corrections are concerned with second- or thirdorder lens defects. Astigmatism (a second-order defect) is readily eliminated by a suitable correction, and modern correction systems can also remove coma (a third-order defect). The single most important parameter limiting the point resolution of the transmission electron microscope is the spherical aberration coefficient of the objective lens. Commercially available electrostatic spherical aberration correctors offer excellent hope that this limitation on the available resolution will soon be removed. At the time of writing, no complete solution is currently available for correcting chromatic aberration. However, the spread in wavelengths (the energy spread) of the electron beam can be significantly reduced by adding a monochromator beneath the electron source. All other performance-limiting factors can be corrected to better than this limit. 4.6.2

Computer Simulation of Lattice Images

The lattice image in phase contrast is often recorded at or close to the Scherzer focus, the under-focus that maximizes the spatial frequency for which the electron microscope imaging system introduces oscillations into the sign of the phase shift (Section 4.3.3). However, the periodicity in the lattice image will be preserved when a series of images are recorded while the focus is changed incrementally, and it is only the intensity at each image point that changes. These intensity changes can include contrast reversal at defocus values below the Scherzer focus, the doubling of characteristic periodicities in the lattice, and apparent changes in the prominence of specific crystallographic directions. It is common to observe rows of bright or dark maxima in one lattice direction being replaced by similar rows in a different direction. Interpretation of such observations is not straightforward. There is a temptation to use both prior knowledge of the crystal structure and the observed electron diffraction patterns, in order to assign specific features of the sample crystallography directly to prominent features in the lattice image. However, some practical experience of the sensitivity of lattice image contrast to the defocus of the objective lens and the sample thickness should quickly convince any scientist of the dangers of ‘recognizing’ features that confirm preconceptions, while ascribing any discrepancies to imaging artifacts.

232

Microstructural Characterization of Materials

The only acceptable scientific procedure is to employ computer simulation to evaluate the image, and several software packages are now available for this purpose. For successful computer analysis of image contrast, three conditions must be satisfied: 1. Accurate calibration of the microscope parameters must have been reliably achieved, especially the determination of the spherical and chromatic aberration coefficients and the beam divergence. 2. A contamination-free, uniformly thinned sample should be available, preferably no thicker than the extinction thickness for the preferred imaging reflections. 3. Accurate alignment of the optic axis of the microscope along a prominent zone axis of the sample must be made, with an angular objective aperture that accepts as many reflections into the imaging system as is compatible with the critical spatial frequencies (compare the Scherzer focus, Section 4.3.3.). If this information can be supplemented by a reasonably accurate estimate of the film thickness, then so much the better. In many cases film thickness is assumed to be a variable during image simulation in order to estimate the error associated with an imperfect knowledge of this parameter. A model for the unit cell of the crystal lattice must be inserted into the computer software program, including the positions of all atoms and their expected occupancy. Lattice defects, such as stacking faults, APBs and dislocations, can be simulated by inserting suitable displacement fields in the model lattice. Boundaries are commonly simulated by placing them accurately parallel to the optic axis, and can also be inserted in the computer model using a transformation matrix for the region of the crystal beyond the boundary plane. Although visual matching of simulated to recorded images has often been employed to decide the ‘best-fit’ simulation, it is far more reliable to derive a ‘difference’ image, in which the simulated intensities for each pixel are compared using a computer program, with the intensities for a through-focus image series recorded by a CCD camera. 4.6.3

Lattice Image Interpretation

A common assumption is that the lattice image ‘shows you where the atoms are’, but this is incorrect and often very misleading. In the first place, as in any magnified image, the lattice image is recorded in two dimensions, as a periodic pattern of varying intensity. The amplitudes leading to these intensity variations can be deduced, but the phase information has been lost. Although the phase information can be partially recovered from one or more images recorded in a through-focus series, this is still a relatively uncommon procedure. Usually it is a single image recorded at or close to Scherzer focus that is published in the literature. A lattice image interference pattern is not localized in space and, as noted previously, is a sensitive function of both the objective lens defocus and the sample thickness, in addition to other, less critical parameters. Provided that the imaging conditions are well-known and that the computer-simulated image is a good match to the observed image, then the conclusion that a model lattice used for computer simulation of atomic positions, occupancies and periodicities may be a good description of the microstructural morphology is justified. However, there are still serious problems, primarily due to the nonlocalized nature of the lattice image. For example, it is common to superimpose a unit cell of the model lattice on a

Transmission Electron Microscopy 233

lattice image and identify ‘channels’ along the optic axis of the microscope with black patches while rows of high atomic number atoms along the same axis are assumed to be ‘seen’ as white patches. Such interpretations will not be accepted by the knowledgeable expert. An excellent example is the apparent structure of many phase boundaries. These are usually complicated by the differences in the lattice potentials of the two phases. This introduces a discontinuity in ‘refractive index’ that results in a shift in the interference pattern of one phase with respect to the other, perpendicular to the boundary plane. This makes it extremely difficult to deduce the actual atomic displacements at the boundary, even though such displacements may be clearly ‘observed’ in the lattice image! Nevertheless, the morphological information that can now be derived from lattice images of both phase and grain boundaries is extraordinary in its atomic detail (Figure 4.34), and the need to make this information quantitative has lead to remarkable improvements in the methods of computer simulation and evaluation of digitally recorded lattice images. In most cases, it is now possible to optimize a model for any crystalline nanostructure observed by lattice imaging in the transmission electron microscope.

Figure 4.34 Some examples of lattice images from grain and phase boundaries: (a) an amorphous glassy layer in equilibrium at the surface of sapphire that had been coated with nickel prior to specimen preparation; (b) a layer of calcium cation segregation at a twin boundary in alumina; (c) a twin boundary in B4C. The inset is a magnified view of the boundary region; (d) The intersection of a stacking fault (SF) with twin boundaries in B4C.

234

Microstructural Characterization of Materials

4.7

Scanning Transmission Electron Microscopy

While conventional TEM makes use of a (nearly) parallel electron beam incident approximately normal to the plane of the sample, it is also possible to focus the electron beam, at a much larger convergence angle, into a focal spot on the sample. We can now collect the transmitted signal as a function of the beam location when it is rastered across the sample. This mode of operation is termed scanning transmission electron microscopy (STEM). The ray optics for STEM are identical to TEM, and can be easily understood by turning the transmission electron microscope ‘upside down’, so that the electron source is at the bottom and the detector at the top. In Figure 4.35 the wave at point B due to a point source at A is identical to the wave at A due to a point source at B. The STEM detector replaces the TEM electron source, and the STEM electron source is placed in the detector plane of the transmission electron microscope. An additional scanning system uses the deflector coils to raster the focused beam across the surface of the specimen.

Figure 4.35 Schematic drawing illustrating the principle of reciprocity in electron optics. The ray diagram for STEM operation is just the inverse of that for TEM.

Transmission Electron Microscopy 235

Converged Incident Electron Beam

Sample

θ3 θ1 HAADF Detector

ADF Detector

BF Detector

θ2 ADF Detector

HAADF Detector

Figure 4.36 Schematic drawing of bright-field (BF), annular dark-field (ADF), and high-angle annular dark-field (HADF) detectors for STEM mode imaging.

There are a number of detectors employed for STEM imaging, and these are shown schematically in Figure 4.36. The first is a solid-state, bright-field detector that counts the number of electrons per unit time as a function of the position of the focused beam on the sample. The scattering angle is relatively small, y1 < 10 mrad. Due to the principle of reciprocity, the diffraction contrast in bright-field TEM and STEM are quite similar. In addition, in STEM mode mass–thickness contrast can provide an important contribution to the image. By adding an annular detector, forward-scattered electrons can be collected over a larger scattering angle, y2 > 10–50 mrad, and we can acquire dark-field STEM images. In addition, electrons scattered to even larger angles (y3 > 50 mrad) can be detected using a high-angle annular dark-field (HAADF) detector. The advantage of HAADF-STEM is that almost none of the elastically diffracted electrons reach the HAADF detector, and the contrast is due to inelastically scattered electrons (see, for example Figure 4.37). Such images are often termed Z-contrast images, since there is now a direct correlation between the local contrast and the local mass–thickness, which depends on the value of the atomic number, Z. The mass concentration may even be estimated by correlating the local concentration using a suitable expression for inelastic scattering. If the STEM includes a field emission gun source with a probe size less than 0.3 nm, atomic resolution Z-contrast is obtainable, with a direct correlation between local intensity and the mass concentration within the atomic columns of the sample.

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Microstructural Characterization of Materials

Figure 4.37 HAADF-STEM micrograph of the interface between a pure gold wire and an aluminium pad, used to electrically connect a microelectronic device. The join is formed by applying heat, pressure, and ultrasonic vibrations, which results in the formation of a 20 atom% Al metastable solid solution adjacent to the pad. Above this a two-phase region forms that contains Al–Au intermetallic phases with a mean concentration of 35 atom% Al. Since the atomic number of Au is significantly larger than that of Al, the Al appears dark and the Au appears brighter.

Summary In the transmission electron microscope high energy electrons are elastically scattered as they penetrate a thin specimen. The transmitted electrons are then focused by electromagnetic lenses to form a well-resolved image that can be viewed on a fluorescent screen or recorded on a photographic emulsion or, more commonly today, a charge-coupled device. The increasing availability of field emission guns for the source of the electron beam has greatly improved the performance of the modern transmission electron microscope. The wavelength of the energetic electrons used in electron microscopy is well below the interatomic spacing in solids, so that atomic resolution is, in principle, possible, although not always easy to achieve. The electromagnetic lenses used to focus the electron beam and form the image in the microscope occupy an appreciable proportion of the total optical path length and suffer from severe lens defects that limit the divergence angle of the beam for a sharp focus to a small fraction of a degree. The most important electromagnetic lens defects are spherical and chromatic aberration, and astigmatism. The diffraction limit on the resolution improves with increasing objective lens aperture angle, while the aberration limit

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on the resolution deteriorates as the aperture angle is increased. The optimum objective aperture for the best compromise between these two limits is typically between 10
2 rad and 10
3 rad, but new spherical aberration correctors should certainly increase this value. The electron microscope requires a vacuum in order to prevent scattering of the electron beam in the microscope column. A high voltage stability for the electron source is also needed in order to ensure a monochromatic beam, and an equally good current stability is required for operation of the electromagnetic lenses, in order to maintain a stable focus. Transmission electron microscope thin-foil specimens must be less than 100 nm in thickness to minimize inelastic scattering of the transmitted beam as it passes through the specimen. Good specimen preparation is critical. Combinations of mechanical, chemical and electrochemical methods are generally used, and the final stage of specimen preparation is most often by ion milling, in which the surface atomic layers are eroded by sputtering, using an incident beam of inert gas ions. To avoid surface charging of thin foils prepared from insulator materials it is often necessary to deposit an electrically conductive coating. The morphology of some specimens may be best studied by making a thin replica of the sample surface and using the replica as the thin-foil transmission specimen. Contrast in the transmission electron image may be associated with differences in the mass–thickness of the sample, coherent diffraction of a portion of the incident beam out of the objective aperture, or phase contrast that results from interference in the image plane between two or more of the coherently diffracted beams, with or without the directly transmitted beam. In biological samples, which are usually amorphous, mass–thickness contrast is the most common source of image information, and a variety of heavy-metal staining agents have been developed to enhance the contrast in soft tissue sections and cellular samples. Crystal lattice defects are generally imaged by diffraction contrast, since lattice defects strongly affect the amplitude that is scattered out of the incident beam. Periodic crystal lattices, however, can be successfully imaged by phase contrast if the microscope can be operated with sufficient resolving power. Changes in lattice periodicity at interfaces can also be detected by phase contrast and interpreted when suitable computer modelling procedures are employed. A simple, qualitative treatment of diffraction contrast is possible using kinematic diffraction theory, in which the dynamic effects of multiple scattering are ignored. The diffracted amplitude of the electron beam in the kinematic theory depends on the sum of just two terms. The first describes the effect of deviations from the Bragg condition on the amplitude that is scattered by a unit cell in a thin column of crystal in the foil sample, while the second describes the effect of displacements from the ideal lattice position, due to the presence of the defect, on the diffraction amplitude. These contrast effects are most conveniently summarized by an amplitude–phase diagram that can be used to explain, qualitatively if not quantitatively, the contrast that is observed from stacking faults and antiphase boundaries, edge and screw dislocations, as well as from the strain fields that are associated with very small precipitate nuclei and radiation damage clusters of point defects. Dynamic diffraction theory, when combined with the inclusion of effects associated with absorption (inelastic scattering), allows the observed diffraction contrast to be interpreted in much more detail, but requires a deeper understanding of electron diffraction theory, considerable expertise in the operation of the microscope and the careful interpretation of the image data.

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The lattice or phase-contrast image is an interference image in which several coherently diffracted electron beams are recombined in the image plane. Since these beams have followed different paths through the electromagnetic lens fields, they experience phase shifts that depend on the electro-optical properties of the electron source and the electromagnetic lenses, and are described by the contrast transfer function for the microscope. The periodicity in a lattice image is essentially independent of the specimen thickness and the focal plane of the lattice image, but these parameters cause gross variations in the recorded image intensity, so that an observed image cannot be directly interpreted as a projection of the periodic crystal lattice of the thin-film sample. Nevertheless a slightly under-focused image, taken at the Scherzer focus, compensates quite effectively for phase shifts that are associated with the lens system. Images taken from very thin specimens at the Scherzer focus therefore correspond approximately to the contrast to be expected from a direct projection of the crystal lattice onto the image plane. However, much better agreement can be achieved by simulating the contrast from a lattice model with the expected crystal periodicity. Computer image simulation has been used successfully to construct models of localized lattice structure to an accuracy that is better than a fraction of the interatomic spacing in the bulk crystal.

Bibliography 1. D.B. Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, Plenum Press, London, 1996. 2. P. Buseck, J. Cowley and L. Eyring (eds), High-Resolution Transmission Electron Microscopy and Associated Techniques, Oxford University Press, Oxford, 1998. 3. J.C.H. Spence, Experimental High-Resolution Transmission Electron Microscopy, Claredon Press, Oxford, 1981.

Worked Examples We now demonstrate the transmission electron microscope techniques that have been discussed, using two quite different material systems: polycrystalline alumina and a thin film of aluminium deposited by chemical vapour deposition (CVD) on a TiN/Ti/SiO2coated silicon substrate. Our first example is polycrystalline alumina. As always, it is important to define the questions we wish to ask before preparing specimens or selecting the microstructural characterization techniques. For our alumina we wish to establish that the grain boundaries are free from secondary phases. To study the detailed morphology of the grain boundaries the most suitable method is thin-foil transmission electron microscopy. For a bulk polycrystalline ceramic sample, the easiest way to prepare a specimen is by first cutting a thin (600 mm) slice from the bulk specimen with a diamond saw. TEM specimens are limited in diameter to 3.0 mm, so a 3.0 mm disc must be trepanned from the 600 mm slice with an annular ultrasonic tool. Alternatively, a hollow diamond bit can be used to extract a rod from the bulk specimen. The circular disc must then be mechanically thinned to 80 mm by using diamond grinding

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media. The region in the centre of the disc should now be thinned to perforation by ion milling. To ensure that the perforation occurs at the disc centre, it is first ‘dimpled’ by further grinding to a thickness of 30 mm. Dimpling from both sides will ensure that a final, electro-optically transparent region will be available near the centre of the disc. Finally, the specimen is ion milled on both the top and bottom surfaces of the specimen for approximately 60 min at 5.0 kV, using argon ions at an incident angle of 6 (Figure 4.38).

Figure 4.38 Schematic drawing of the TEM specimen preparation process for a plan-view bulk specimen. After trepanning the sample is mechanically thinned to 100 mm. The specimen can be ‘dimpled’ on both sides, if additional mounting wax is added to support the specimen before dimpling from the second side.

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Figure 4.39 Bright-field diffraction contrast TEM micrograph of an alumina polycrystal reinforced with SiC particles.

For TEM examination the nonconducting alumina specimens are usually coated on one side with a 10 nm layer of carbon. Figure 4.39 is a bright-field, 200 kV TEM micrograph of an alumina thin foil seen in diffraction contrast using a small objective aperture to select the direct transmitted beam. The polycrystalline alumina contains sub-micron SiC particles, and the Bragg contrast variations are due to the changes in crystallographic orientation for each individual grain, as well as a dislocation network visible in one of the grains. If we use Kikuchi diffraction we can align a particular grain so that a low index zone axis is parallel to the incident beam. The bright-field image will then show this grain in dark contrast. An example is given in Figure 4.40 which shows an AlNb2 particle located within an alumina grain. We can now record an image using a selected diffracted beam from the aligned grain, a dark-field image, in which the diffracting grain will appear light when compared with the neighbouring grains.

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Figure 4.40 Bright-field diffraction contrast TEM micrograph of an AlNb2 particle within an alumina grain. The AlNb2 particle is aligned along a low index zone axis, and thus has a dark contrast in the bright-field image.

The detection of secondary phases at a grain boundary may be achieved using several different methods. In the first, high-magnification bright-field images are used. Secondary phases having a different chemical composition form the matrix will give mass–thickness contrast and appear lighter or darker than the neighbouring regions. Figure 4.41 shows an example for NbO particles at grain boundaries in polycrystalline alumina. An amorphous phase at the grain boundary or at a grain boundary triple junction can be highlighted in a

Figure 4.41 TEM micrograph of a NbO particle located at a grain boundary in polycrystalline alumina. Phase contrast (lattice fringes) and mass–thickness contrast vary from the alumina grain to the NbO grain.

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dark-field image. The dark-field image is recorded using diffusely scattered electrons from the amorphous region. When a glass-containing grain boundary is oriented accurately parallel to the incident beam, the glassy region will appear lighter than the neighbouring crystalline material. If the TEM resolution is sufficient to detect the lattice planes of alumina, we can also use phase contrast to study the grain boundary regions. To form a lattice image of any given grain, the grain should be oriented to diffract with a low index zone axis exactly parallel to the incident beam. To record a lattice image of the grain boundary region or of an interface between two phases, both of the boundary- or interfaceforming grains should be aligned along low index zones, and the boundary must be parallel to the incident beam. This is a very rigid condition, and exceedingly difficult to obtain in a polycrystalline or polyphase sample unless a special orientation relationship exists between the grains. Figure 4.42 shows a lattice image of an interface between an alumina grain and a nickel particle, in a particle reinforced alumina matrix composite. Both phases are in a low index zone axis and the interface is parallel to the incident electron beam. As a result, the thin (0.9 nm) amorphous film that is present at the interface can be detected. In some cases information on the interface faceting and particle size of small secondary phases at grain boundaries is obtainable when the interface is parallel to the incident beam and at least one boundary-forming grain is aligned with a zone axis parallel to the incident beam. Figure 4.43 is a lattice image of a SiC particle located within an alumina grain. A lattice image is visible

Figure 4.42 Lattice image of an interface between an alumina grain and a nickel particle. Both phases are aligned along a low index zone axis with respect to the incident electron beam, and the interface is parallel to the beam. A thin (0.9 nm) amorphous film at the interface is clearly visible.

Transmission Electron Microscopy 243

Figure 4.43 Lattice image of a SiC particle located within an alumina grain. The alumina is aligned with a low index zone axis parallel to the incident electron beam, and is the source of the observed lattice image. A moir e pattern within the SiC particle is due to overlap between the alumina and SiC lattices in the plane of the thin foil.

for the alumina, while only moir e fringes are visible in the area of the SiC particle, since it is not oriented in a low index zone axis. The combination of real-space images and reciprocal space images, that is, diffraction patterns, is one of the main advantages of TEM. Figure 4.44 shows a bright-field diffraction contrast micrograph of a gold particle which was equilibrated on the basal (0 0 0 1) surface of sapphire. Both the particle size and shape can be determined. By taking selected area diffraction (SAD) patterns from the particle and the substrate, the relative orientation of the two phases can be determined. Since there is a direct relationship between the orientation of the SAD pattern and the real space image that is known from calibration, the interface planes observed in the micrograph can be determined. Now we return to the CVD Al system. For this example we wish to characterize the initial deposition conditions, when the first aluminium nuclei form on the surface of the TiN film, the morphology of the complete aluminium film, and the morphologies of the underlying TiN and titanium layers. TEM cross-sections are not easy to prepare, so we use this opportunity to review the specimen preparation process. For TEM we need very thin specimens, with as much as possible of the interface region thin enough for successful TEM investigation. There are a number of methods available to thin bulk specimens, but for a multilayered material chemical thinning is not a good idea, since the different layers will have different chemical

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Figure 4.44 (a) Bright-field TEM micrograph of a gold particle thermally equilibrated on the (0 0 0 1) surface of sapphire (a-Al2O3). (b) The SAD pattern shows the low index orientation relationship between the two phases, and allows the lattice planes parallel to the interface between the two phases to be determined.

potentials and will react differently to a chemical or electrochemical etch. Ion milling is really the only option. To ensure that the largest possible area of the interface region is at or near the perforation made by ion milling, we glue four of our multilayer specimens together; one pair face-toface, and the second pair face-to-back (as shown schematically in Figure 4.11). We need a cross-section for the TEM sample that is at least 3 mm in width, so two more silicon wafers, with a thickness of 300 mm, are glued to the composite specimen. We now use an ultrasonic annular drill to drill down the length of the cross-section, and then glue the resulting ‘rod’ inside a copper tube with an outside diameter of 3 mm. Once the glue has set, slices are cut from the end of the rod with a diamond wafering saw. These slices are mechanically thinned to less than 100 mm, dimpled at their centre to less than 20 mm, and then ion milled to perforation. By gluing the sample wafers face-to-face and face-to-back we are able to locate the interfaces of interest close to the centre of the 3 mm diameter specimen. Combining ion milling with dimpling improves the probability that perforation will occur in the centre of the disc, producing a specimen thin enough for TEM in the region of interest. It is a lot of work for just one specimen, but with some practice the time required to prepare a sample becomes quite reasonable. Figure 4.45 is a bright-field, diffraction contrast TEM micrograph of the as-prepared specimen. The thick aluminium film is clearly visible in cross-section, as are the separate grains within this film. Open and closed voids are easily detected, and statistical analysis of both the grain size and the void density is possible. Higher magnification images (Figure 4.46) show the morphology of the TiN/Ti layers, which have a very small grain size. The dominant crystal defects observable from Figure 4.46 are the grain boundaries, and this image contains important information on the average grain size of both the TiN and Ti. However, Figure 4.46 is a projection of a thick, three-dimensional slice onto a twodimensional image, and the boundaries that are visible come from many different grains within the film. We discuss this problem of image overlap in some detail in Chapter 9.

Transmission Electron Microscopy 245

Figure 4.45 Low-magnification bright-field micrograph of a cross-section specimen of Al on TiN/Ti/SiO2/Si. The individual Al grains, as well as open and closed voids in the Al film, are clearly visible.

Figure 4.46 A higher magnification TEM micrograph of the interface region, showing some phase contrast and the morphology of the TiN and Ti layers.

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Figure 4.47

A very fine lattice image of aluminium aligned along a [110] zone axis.

Figure 4.47 is a high resolution TEM micrograph of the same aluminium grain recorded after the grain had been oriented into a [1 1 0] zone axis. The aluminium film has a strong h1 1 0i texture, and the {1 1 1} planes are predominately parallel to the original interface. There are several reasons for the variations in contrast observed in the aluminium grain in Figure 4.47. First, changes in thickness from the interface region to the edge of the specimen result in changes in contrast that are due to thickness extinction effects. In principle, we can use a computer program to determine the specimen thickness for any region by comparing an experimental image to a simulated image, providing we know the defocus value of the objective lens and have a defocus–thickness ‘map’ in order to determine the two independent variables (Figure 4.48). Additional sources of contrast variations in the lattice image in Figure 4.47 are associated with local bending, due to residual stresses, or to the presence of dislocations. Unusually, stacking faults are also visible in the aluminium grain near the interface. These faults lie on the {1 1 1} planes, some of which are parallel to the interface. The faults were most likely formed during deposition of the film, and are expected to influence the electrical properties. Figure 4.49 is a higher magnification high resolution TEM image of the same interface region shown in Figure 4.46. The lattice image from a [1 1 0] zone axis projection of the aluminium lattice is visible together with individual grains from the TiN layer.

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Figure 4.48 Simulated defocus–thickness map for aluminium aligned on a [110] zone axis. The map is ‘contrast invariant’, meaning that the contrast of each individual simulation has not been changed when the images were ‘pasted’ together. This results in the thinner regions having a lower contrast than the thicker regions, an effect that is not due to absorption.

Problems 4.1. Electromagnetic lens systems employ very small angular apertures, primarily because of their large spherical aberration. Explain why the spherical aberration and other lens defects limit the resolving power of a transmission electron microscope. 4.2. Sketch a graph of the dependence of electron wavelength on the accelerating voltage in the electron microscope. What factors prevent the development of ultra-high voltage (>1 MeV) electron microscopes for ultra-high resolution? 4.3. Distinguish between mass–thickness contrast and diffraction contrast in electron microscopy. When might you expect mass–thickness to dominate the contrast in thinfilm specimens? 4.4. List the experimental and specimen parameters which affect diffraction contrast. Define precisely each parameter given in your list.

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Figure 4.49

Lattice image of the Al–TiN interface region.

4.5. Sketch the kinematic amplitude–phase diagrams for a stacking fault and for a dislocation line in a crystalline thin-film specimen. Show for both cases the effects observed when the defect either increases or decreases the phase shifts between neighbouring atomic columns of crystal. 4.6. Explain the diffraction conditions under which lattice defects will fail to give diffraction contrast in thin-film electron microscopy. Give examples for (a) stacking faults in the FCC crystal lattice and (b) screw dislocations in a BCC metal lattice. 4.7. Under what conditions might a phase contrast, lattice image be expected to mirror the atomic positions in the unit cell of a crystal? What are the problems associated with making such an assumption? 4.8. Outline the steps required to prepare a representative TEM specimen from the following samples: (a) tungsten light bulb filaments; (b) a pinch of talcum powder; (c) large steel bolts, (d) brazed vacuum seals. 4.9. According to the Rayleigh criterion, how will the resolution in TEM change with accelerating voltage? (Neglect aberrations and assume a constant convergence angle.) Explain how the contrast might be expected to change, using the same assumptions.

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4.10. Explain the difference between the structure factors for X-rays and electrons. 4.11. What are the physical principles behind the operation of the vacuum pumps used for electron microscopes? (Try to include rotation, diaphragm, scroll, diffusion, turbo, turbo-drag, and ion pumps in your discussion.) 4.12. Assuming the resolution of a TEM can be expressed as d2Total ¼ d2d þ d2s ; where dd is the Rayleigh diffraction limit and ds is the resolution limit determined by the spherical aberration coefficient, calculate the optimal convergence angle of the objective lens a*. Using this value for a*, what is the expected resolution for a 200 kV microscope with a spherical aberration coefficient of 1.5 mm? 4.13. Compare the point resolution defined by the Rayleigh criterion with the Scherzer resolution. Assume a 200 kV TEM with Cs ¼ 1.5 mm and the optimal value of the convergence angle. Repeat your calculation for Cs ¼ 0.6 mm, and then again for 300 kV. Summarize your conclusions. 4.14. Two-beam diffracting conditions are used during the study of dislocations in a FCC metal structure. Which reflections should give zero contrast for the case of a a=2½ 101 dislocation? 4.15. Figure 4.50 is a dark-field TEM micrograph of the edge of an aluminium sample recorded under two-beam diffracting conditions, and a selected area diffraction (SAD) pattern of the grain which contains the reflection used to form the two-beam image. Solve the diffraction pattern. Use the thickness fringes in the image and the data in Table 4.1 to estimate the thickness profile of the sample. How would your results be changed if the material was gold rather than aluminium? 4.16. A sample made of pure (99.999%) aluminium was thermally annealed, causing grain growth. Figure 4.51 shows a SAD pattern from a grain adjacent to a grain boundary in the sample (SAD1), and a sketch of the bright-field image from the grain boundary region (BF1). After tilting the sample by 18  2 , a second SAD pattern (SAD2) and accompanying bright-field image sketch (BF2) were acquired. Both sketches have been correctly oriented with respect to the SAD patterns. (a) Indicate which grain has been aligned with a low-index zone axis parallel to the incident beam. (b) Solve both SAD patterns. (c) Identify the facet planes that define the grain boundary. (d) Give a rough estimate for the thickness of the sample based on the two sketches and the data given. 4.17. Figure 4.52 shows a series of diffraction patterns acquired from an aluminium grain which contains a single dislocation. After recording the first pattern (SAD1) the sample was tilted 36  2 to acquire the second pattern (SAD2). The sample was then tilted a further 22  2 to obtain the third pattern (SAD3). On each of these

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Figure 4.50 (a) Dark-field micrograph from the edge region of an aluminium thin-film sample. (b) SAD pattern showing the reflection that was used to record the dark-field image under twobeam conditions.

Table 4.1 Extinction distance for prominent lattice planes in aluminium and gold at 100 kV. Material

Al Au

Extinction distance for reflection hkl (nm) 110

111

200

220

400

— —

56.3 18.3

68.5 20.2

47.3 27.8

76.4 43.5

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(a) SAD1

BF1 10 nm

(b) SAD2

BF2 10 nm

Figure 4.51 (a) SAD pattern from a grain that is adjacent to a grain boundary in aluminium (SAD1) and a bright-field image sketch from the grain boundary region (BF1). (b) After tilting the sample by 18  2 , a second SAD pattern (SAD2) and the accompanying bright-field image sketch (BF2) were acquired. Both sketches are in the correct orientation with respect to the two diffraction patterns.

three diffraction patterns a reflection used to form a two-beam image is indicated for which the dislocation contrast was absent. (a) Solve all three diffraction patterns. (b) Determine the Burgers vector of the dislocation. What reasonable conclusion can you reach about the line sense of this dislocation? 4.18. A SAD pattern from a single grain of copper (FCC, a ¼ 0.3615 nm) is given in Figure 4.53. It is possible that a twin boundary exists in the same grain. Assume that the twin boundary plane is {1 1 1}. (a) Given that your goniometer can be tilted up to 35 , find the nearest zone axis for which the twin boundary will be parallel to the incident electron beam. (b) What angle and axis of tilt are required to reach this orientation? (c) Sketch the SAD pattern expected for the zone axis you have chosen.

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(a) SAD1

(b) SAD2

(c)

SAD3

Figure 4.52 A series of diffraction patterns acquired from an aluminium grain which contains a single dislocation in the region imaged. After recording the pattern SAD1 (a), the sample was tilted 36  2 to acquire the pattern SAD2 (b). The sample was then tilted a further 22  2 to generate the pattern SAD3 (c). On each diffraction pattern the reflection is indicated that was used to form a two-beam image in which contrast from the dislocation was absent.

4.19. Five lattice images of aluminium taken from a crystal aligned with a [0 0 1] zone axis parallel to the incident electron beam are given in Figure 4.54. Each image is recorded with a different objective lens defocus value that is indicated in the CTF. In the first image the positions of the atoms in a unit cell are marked. (a) Identify the planes (220) and (200) on the images. (b) Given that the lattice parameter of aluminium is 0.405 nm, determine the image magnification. (c) Explain why the contrast at the positions of the atoms changes from white to black across the defocus series. 4.20. A SAD pattern from a twinned region in copper is shown in Figure 4.55. A sketch of the area from which the SAD pattern was recorded is shown in the correct orientation in Figure 4.56.

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Figure 4.53

SAD pattern from a single grain of copper.

(a) Solve the diffraction pattern and define the orientation relationship between the two grains on either side of the twin boundary. What is the boundary plane? (b) The sample was tilted into a new zone axis, and the resulting SAD pattern is shown in Figure 4.57. A sketch of the bright-field image is shown in Figure 4.58. Estimate the approximate thickness of this sample, explaining any assumptions that you have made. 4.21. A Fe–Ni alloy was investigated by thin-film TEM. A SAD pattern was recorded from a g-FeNi grain in the thin film. This grain is indicated in Figure 4.59 on brightfield and dark-field images recorded with the sample aligned on a low-index zone axis. The SAD pattern for the grain is shown in Figure 4.60. The structure of g-FeNi is an FCC solid solution with four atoms per unit cell (a ¼ 0.35871 nm). (a) Solve the SAD pattern. (b) Which image is bright field and which image is dark field? Why? (c) Explain the changes in contrast that are observed across the two images. 4.22. In a TEM study of polycrystalline molybdenum (BCC, a ¼ 0.3147 nm), a low-angle grain boundary was found. The boundary geometry is sketched in Figure 4.61. In order to identify the dislocations composing the boundary, two-beam diffraction

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Figure 4.54 Five lattice images of aluminium aligned on a [001] zone axis and the CTF at each defocus value used to record the images.

images were recorded. The first image was obtained after orienting the sample to align a [1 0 1] zone axis parallel to the incident electron beam. The SAD pattern is shown in Figure 4.62. Use of reflection g1 resulted in the sub-boundary dislocations going out of contrast. The sample was then tilted by 18.43 into a new zone axis,

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Twin boundary

Figure 4.55 SAD pattern of a twinned region in copper.

Twin

Matrix

Hole

200 nm

Figure 4.56 Sketch of the image from which the SAD pattern in Figure 4.55 was recorded.

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Microstructural Characterization of Materials

SAD pattern of the same sample after tilting into a new zone axis.

Twin boundary

Figure 4.57

Twin

Matrix

Hole

200 nm

Figure 4.58 Sketch of the bright-field image recorded after tilting the sample into the new zone axis shown in Figure 4.57.

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Figure 4.59 Bright-field and dark-field images recorded from an Fe–Ni alloy.

Figure 4.60 SAD pattern recorded from a grain of the g-FeNi solid solution and indicated on the micrographs given in Figure 4.59.

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dislo

catio n

lines

Grain B

Grain A

h = 15.6 nm

Figure 4.61 Schematic drawing of the low-angle grain boundary found by TEM.

from which a two-beam image was recorded using the reflection g2 shown in the second SAD pattern (Figure 4.63). Again, no diffraction contrast was detected from the sub-boundary dislocations under these diffracting conditions. The sample was then tilted to bring the boundary parallel to the incident electron beam.

g1 d = 0.2225 nm

Figure 4.62 SAD pattern recorded from the region adjacent to the grain boundary.

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g2

d = 0.1573nm

Figure 4.63

. (a)

SAD pattern recorded after tilting the sample 18.43 .

Planar defect 1

an Pl

ct efe d ar

Selected area diffraction from the circle around planar defect 1

d = 0.2087 nm

(b)

Planar defect 2

a Pl

ct efe d r na

Selected area diffraction from the circle around planar defect 2

d = 0.2087 nm

Figure 4.64 Sketches of two types of planar defect in thin copper films, and their corresponding SAD patterns shown in the correct orientation: (a) planar defect 1; (b) planar defect 2. The regions from which the SAD patterns were recorded are indicated by the dashed circles.

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(a) Determine the Miller indices for g1 and g2. (b) Determine the Burgers vector of the sub-boundary dislocations. (c) The distance between the dislocations in the sub-boundary (h) depends on the Burgers vector (b) and the relative misorientation of the two crystals (y): b : ð4:27Þ h¼ 2sinðu=2Þ Sketch the SAD pattern that you would expect from the boundary region when aligned with a [0 0 1] zone axis parallel to the electron beam. 4.23. During a TEM study of copper (FCC, a ¼ 0.3615 nm), two types of planar defect were identified in bright-field diffraction contrast. Sketches of each type of planar defect and their corresponding SAD patterns are given in Figure 4.64. (d) Solve the diffraction pattern from the region containing the first type of planar defect and propose an explanation for this type of defect. (e) Solve the diffraction pattern from the region containing the second type of planar defect and propose an explanation for this second type of defect.

5 Scanning Electron Microscopy The scanning electron microscope provides the microscopist with images that closely approximate what the physiology of the eye and brain expect, since the depth of field for resolved detail in the scanning electron microscope is very much greater than the spatial resolution in the field of view. That is precisely how our eyes and the visual cortex of our brains have evolved in order to perceive the three dimensions of the ‘real’ world. The ‘flatness’ of the topological and morphological detail that is observed in the light optical or transmission electron microscope is replaced, in the scanning electron microscope, by an image that appears very similar to the play of light and shade over the hills and valleys of the countryside. These features of the real world look remarkably like the hollows and protrusions of a three-dimensional object viewed in the scanning microscope (Figure 5.1). Only two additional features are needed to complete this ‘optical illusion’. The first is a true representation of the position of the image detail in depth, normal to the image plane. With a little extra time and effort, scanning electron microscopy (SEM) can also provide this threedimensional, depth information by recording two images from slightly different viewpoints; a technique termed stereoscopic imaging (Section 5.7.3). The second feature missing from the scanning electron microscope image when compared with ‘real world’ images is the presence of colour. Again, it is possible to use the capacity of the eye to recognize colour by the introduction of colour coding. Colour may be used to enhance contrast, for example when comparing different images during data processing. Colour can also be used to code for crystallographic information in the morphological image, as we shall see in orientation imaging microscopy (OIM) (Section 5.6.3). This visual impact of scanning electron microscope images and the ability to reveal details that are displaced along the optic axis, in addition to those resolved in the twodimensional field of view of the image plane, has led to the application of SEM to all branches of science and engineering from the time it was first introduced (late 1950s). We first describe the various imaging signals that are detectable in the scanning electron

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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Figure 5.1 A secondary electron scanning electron micrograph of a ductile fracture surface in molybdenum.

microscope due to the interaction of a focused beam of high energy electrons with a solid sample. The interpretation of the information that can be derived from these different signals is then described, together with some of the special techniques that are available to enhance image contrast and assist in image interpretation.

5.1

Components of The Scanning Electron Microscope

The basic structure of the scanning electron microscope was described in Chapter 4 and compared with that of the transmission electron microscope. The main components of the scanning microscope (Figure 4.3) include the microscope column, the various signal detector systems, the computer hardware and software used to process the collected data, and the display and recording systems. As in transmission electron microscopy (TEM), the microscope column is kept under vacuum, and the vacuum system and specimen air-lock are an integral part of the microscope system. If a field emission gun is used to provide the electron source, then the vacuum requirements for this source are very stringent and a separate vacuum pumping and degassing system is required for the electron gun. If very large samples are to be inserted into the specimen chamber, or samples are to be viewed under cryogenic conditions, or in a controlled atmosphere, then a specialized sample chamber is necessary. The variety of specimen stages now available is quite remarkable but the steps that need to be taken to prevent out-size samples or special stages from contaminating the remainder of the column and causing damage to the electron gun, add considerably to the cost of these ‘extras’. The electromagnetic probe lens of the scanning electron microscope behaves in many respects as an inverted transmission electron microscope objective lens. A ‘minified’ image of the electron source is focused onto the specimen surface, in place of the magnified image of the specimen that is focused onto the image plane by a transmission electron microscope

Scanning Electron Microscopy 263

objective lens. However, there are some major differences in the design of the probe lens. Most importantly, since three-dimensional objects are often studied in the scanning electron microscope, the probe lens must have an appreciable ‘working distance’. The working distance is an important parameter in the operation of the probe lens and may vary from 1 or 2 mm up to 50 mm or more. By contrast, in TEM the thin-film specimen usually sits inside the magnetic field of the transmission electron microscope objective lens. Since the electron beam probe in the scanning electron microscope has to be scanned across the sample in a x– y, raster, electromagnetic scanning coils also have to be included in the microscope column. These are positioned above the probe lens. Most of the scanning electron microscope signal detection systems are also built into the column. The only exception is the detector for optical fluorescence of the sample, a rather unusual accessory. Common signal detectors include those for high energy (backscattered) electrons, low energy (secondary) electrons, excited (characteristic) X-rays, and some other signals that will be discussed later (Figure 5.2). The detectors for characteristic X-rays may be either energy-dispersive or wavelength-dispersive. In energy-dispersive spectroscopy (EDS) the excited photons are collected as a function of their energy and the spectrum of energy-dependent, photon intensity is analysed to determine the chemical composition of the region of the sample excited under the electron beam. (Section 6.1.2.2). In wavelengthdispersive spectroscopy (WDS) the intensity of the excited X-radiation is collected as a function of the wavelength (Section 6.1.2.1). EDS detectors collect the excited X-rays simultaneously over a wide energy range, and are therefore highly efficient. However, they have a restricted energy resolution that may sometimes result in unacceptable overlap of the characteristic peaks in the X-ray signal generated by different chemical constituents in the Backscattered Electrons

Incident Electrons

X-Ray (EDS/WDS)

Auger Electrons

Secondary Electrons

Cathodoluminesence

Sample

Absorbed Electrons Transmitted Electrons

Figure 5.2 Schematic drawing of a scanning electron beam incident on a solid sample, showing some of the signals generated that can be used to help characterize the microstructure.

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Microstructural Characterization of Materials

sample. However, WDS detectors have to be rotated around the X-ray goniometer in order to scan across the range of wavelengths collected by the curved crystal detectors. This process can be quite time-consuming. Moreover, the range of wavelengths covered by any single detector crystal is limited. If it were not for the very much better spectral resolution and detection limit of WDS systems, it is doubtful if these would still be in use. Modern scanning electron microscopes now have digital acquisition and storage systems which replace the earlier, high-resolution cathode ray tubes that were mounted in front of a camera. The data processing and display systems are also integrated into the microscope system. Large numbers of SEM images need to be recorded at high resolution, so the computer hardware and software requirements are not trivial. The necessary software for data processing has been developed over nearly half a century and is very reliable. The scanning electron microscope can now routinely integrate morphological data from secondary electron images with compositional information from EDS microanalysis and provides compositional mapping of the chemical constituents in the microstructure. To this microchemical capability has now been added the determination of the crystallographic surface orientation of individual crystalline grains within the microstructure. This is accomplished by analysing electron backscatter diffraction (EBSD) patterns from each individual grain (Section 5.10). EBSD is similar to the Kikuchi diffraction patterns observed in TEM (Section 2.5.3). This crystallographic information can also be mapped onto the morphological secondary electron image in a mode of operation of SEM termed orientation imaging microscopy (OIM, Section 5.6.3). This ability to combine morphological, chemical, and crystallographic information in the output from a single, high resolution instrument has made an impact on research and industry that is in many ways comparable with that due to the introduction of the optical microscope at the end of the nineteenth century. Only a limited number of bulk microstructural parameters can be accurately determined by two-dimensional stereological analysis of sample sections. Deducing additional threedimensional information from a single two-dimensional sample section, for example, the grain-size distribution, is at best inaccurate and may actually be misleading. Serial sectioning is the obvious answer. Successive sections are prepared at known intervals and the recorded data combined to develop a three-dimensional model of the morphology whose resolution is limited primarily by the sectioning interval (Section 9.5). One method for successful high resolution serial sectioning is now available in an important variant of SEM, termed the dual-beam focused ion beam (FIB, Section 5.8.6). Practical quantitative three-dimensional microscopy is increasingly available (Section 9.5.3), and the scanning electron microscope is a key tool for three-dimensional applications.

5.2

Electron Beam–Specimen Interactions

An energetic electron penetrating into a solid sample undergoes both elastic and inelastic scattering, but for thick specimens in SEM it is the inelastic scattering that will predominate, eventually reducing the energy of the electrons in the beam to the kinetic energy of the specimen kT. The various processes that occur along the scattering path are complex, but they are generally well-understood. If we understand the nature of the signals generated by the various beam–specimen interactions and the science behind the operation of the signal

Scanning Electron Microscopy 265

detection systems, then there is usually little ambiguity about the interpretation of image contrast in the scanning electron microscope. 5.2.1

Beam-Focusing Conditions

The probe lens, used to focus the electron beam onto the specimen surface in the scanning electron microscope, has similar characteristics to the objective lens in TEM. The best resolution obtainable cannot be better than the focused probe size on the sample surface. The positions of the source, the condenser system and the probe lens in effect invert the electron path in the scanning electron microscope with respect to that in the transmission electron microscope. That is, in SEM the electron source (the ‘gun’) is where the image would be in TEM. The SEM condenser system reduces the apparent size of the source, rather than magnifying the image, and the SEM probe lens forms the beam probe in the image plane, where the source would be in the geometry of the transmission microscope. The electron beam probe is, effectively, a ‘minified’ image of the electron source. In practice, there are three limitations on the minimum diameter that can be achieved for the probe beam in the plane of the specimen: 1. The spherical and chromatic aberrations of the probe lens, with the spherical aberration being the more important (as for the objective lens in conventional TEM). 2. The maximum beam current that can be focused into a probe of a given diameter. This maximum current is a strong function of the electron gun and a major reason for preferring a field emission gun, despite the cost. 3. The need to allow sufficient working space beneath the probe lens pole-pieces to accommodate large, topographically rough specimens. Sample sizes are typically 20 mm in diameter, but the size may range from 2 to 50 or 100 mm for various applications. This is a long way from the standard 3 mm diameter thin-film foil commonly viewed in TEM. In practice it is the beam current limitation that may prove the most serious, since the beam current varies approximately as the third power of the beam diameter, and hence falls dramatically for a finer probe. Since a reduction in the beam current may result in a poor signal-to-noise ratio in the image, this can be a serious problem. Field emission sources are capable of generating electron probes with very high beam current densities that can be focused to finer probes (as little as 1 nm in diameter). The working distance between the probe and the specimen is also an important factor. Typically, some compromise is needed. The minimum probe diameter is fixed by the required resolution and determines the maximum depth of field. This, in turn, translates into a maximum working distance for the sample beneath the probe lens. The commercial introduction of field emission guns has drastically reduced the minimum size of the primary electron source and increased the available current density in the probe by some four orders of magnitude, allowing for either a much reduced probe size (better ultimate resolution) or, alternatively, the use of appreciably lower incident electron beam energies while maintaining an acceptable signal-to-noise ratio. The best resolution, of the order of 1 nm, is now available at beam energies down to 200 V. This may be compared with the conventional, thermal emission sources that were seldom capable of generating a meaningful signal at beam energies below 5 kV, or achieving a secondary electron image resolution of better than 20 nm.

266

Microstructural Characterization of Materials

The spatial distribution and temporal stability of the electron current within the beam probe is also important. This distribution may be determined experimentally by moving a knife edge across a Faraday cup collector. The beam diameter is usually defined as the width of the experimental current distribution at half the measured maximum beam current, that is, the full width at half-maximum (FWHM) diameter, but this may be a poor guide to the total current. The current distribution commonly includes a long tail that extends from the central spot and can result in a high level of background noise in the signal. For many imaging purposes this may be unimportant, but it does affect X-ray data collection significantly when the quantitative analysis of concentration variations across a phase boundary is being determined. Although the first step in reaching the ultimate resolution in SEM is to reduce the size of the focused electron probe, we need to remember that the resolution in the image also depends on the volume of excited material in the sample that generates the signal being collected. For example, if our focused probe is only 2 nm in diameter, but we are forming an image from backscattered electrons that originate in a region 500 nm in diameter within the sample, then it is this larger dimension that defines the resolution of the backscattered electron image. Increasing the probe diameter by an order of magnitude can increase the backscattered electron signal by several orders of magnitude, without sacrificing the resolution in the backscattered electron image. It follows that, while we may characterize the performance of a scanning electron microscope by determining the probe size using a knife-edge specimen, this is not normally the resolution characteristic of the collected image data. The situation is very different from that in TEM, where the resolution is primarily determined by either the point resolution or the information limit, and these are characteristics of the electro-optical system rather than the sample. A few words should be added concerning the scanning system. Since the data are acquired by scanning the electron probe across the surface of the specimen and collecting one of the signals generated, the rate of data collection is not only limited by the intensity of the probe and the efficiency of signal collection, but also by the scanning speed. A weak signal will require a slower scanning speed to improve the signal-to-noise ratio of the image. In the collection of characteristic X-ray data, when both the inelastic scattering cross-section and the collection efficiency are low, the statistics of data collection (the number of X-ray counts contributing to an intensity measurement) will determine the available detection limit. Under these conditions, the beam-current temporal stability, or current drift, can significantly affect the accuracy of analysis. We will return to these statistical considerations later (Section 5.4). 5.2.2

Inelastic Scattering and Energy Losses

The calculation of inelastic scattering paths for electrons can be simulated quantitatively by Monte-Carlo methods based on random scattering events. These ignore some crystallographic (orientation-dependent) scattering effects, specifically, the effect of lattice anisotropy and channelling processes that scatter the incident high energy electrons into preferred crystallographic directions. The electrons in the beam propagating through the crystal follow an irregular scattering path, losing energy as their integrated path length in the crystal increases (Figure 5.3). It is not possible to calculate the average trajectory for multiply scattered electrons, but it is possible to define and measure two critical, penetration depths in the sample, as well as to estimate the envelope that defines the boundaries of the electron

Scanning Electron Microscopy 267

Figure 5.3 A Monte-Carlo simulation for 200 electron trajectories through aluminium assuming 30 keV incident-energy electrons. The trajectories in red indicate backscattered electrons that eventually escape from the surface of the sample. All other trajectories represent electrons that eventually reach thermal equilibrium (an average energy of kT) and are absorbed into the sample.

trajectories for electrons whose energy exceeds any given average value. Thus the diffusion depth xD is defined as that depth beyond which the electrons can be assumed to be randomly scattered, so that an electron at this depth is equally likely to be moving in any direction in the sample. At depths below xD electrons can continue to diffuse to increasing depths, but with scattering angles that are independent of direction. If a Faraday cup is used to collect the electrons in an incident electron beam that penetrate a thin-film sample, then the diffusion depth will correspond approximately to a critical sample thickness that reduces the transmitted beam current to half its initial value. The penetration depth or range xR of the incident electrons is defined as the depth at which the electron energy is reduced to the thermal energy kT. In terms of the Faraday cup experiment, the penetration depth corresponds to the film thickness that would reduce the transmitted electron current to zero. In practice the Faraday cup also collects secondary electrons, thus increasing the measured transmitted current at all film thicknesses, but in general the experiment works well. Both xD and xR decrease with increasing atomic number Z and decreasing incident beam energy E0. Whereas the change in the shape of the envelope of electron paths with beam energy is more or less similar, that with atomic number is not. The shape of the envelope that defines the scattered electron paths for electrons having a given average

268

Microstructural Characterization of Materials

energy, corresponding to a given energy loss, changes markedly with the atomic number. This is primarily because the lateral spread of the beam is roughly proportional to the difference (xD
xR), while xR is appreciably less sensitive to Z than xD. These effects are summarized schematically in Figure 5.4.

Incident beam

(a)

E>kT

(b)

E=kT

Increasing beam energy xD

xD xR xR

Increasing atomic number Z

xD

xD xR

xR

Figure 5.4 (a) The electron beam is inelastically scattered within an envelope bounded by the condition that the average energy has reached the thermal kinetic value kT. (b) The inelastic scattering envelope for an incident beam of energetic electrons depends on both the incident energy and the atomic number of the target, and may be qualitatively approximated by the two parameters, diffusion depth xD and penetration depth or range xR.

Scanning Electron Microscopy 269

5.3

Electron Excitation of X-Rays

If the incident electron energy exceeds the energy required to eject an electron from an atom in the specimen, ionizing the atom, then there will be a finite probability that ionization will occur. Ionization of the atom is an inelastic scattering event that reduces the energy of the incident electron by an amount that is characteristic of the ionization energy, and raises the energy of the atom above its ground state by an equal amount. The energy of the excited atom can then decay by transition of an electron from some higher energy state into the now vacant state in the atom. All such transitions are accompanied by the emission of a photon. If the excited state of the atom corresponds to the ejection of an electron from one of the inner shells of the atom, then this emitted photon will have an energy that lies in the X-ray region of the electromagnetic spectrum. In general, decay of an excited atom from an excited state to the ground state takes place in several successive stages, with the emission of several photons, each having a different energy and wavelength and each corresponding to a single stage in the transition of the excited atom back to its ground state. It follows that, if a particular ionization state is to be reached, then the energy lost by the incident electron in the inelastic event must always exceed the threshold energy for the creation of that ionization state, while the energy of the most energetic photon that can be emitted will always be less than this threshold energy for excitation. Furthermore, if we consider any specific inner shell of electrons surrounding the atom, for example, the innermost K-shell, then, as the atomic number increases, the ionization energy for the electrons that occupy this shell must also increase (since the electrons in any given shell are closer to the nucleus and hence more deeply embedded in an atom of higher atomic number). This conclusion is illustrated schematically in Figure 5.5. The X-ray spectrum generated when an energetic electron beam is incident on a solid target includes a range of wavelengths, starting from a minimum wavelength that can be hc , where h is Planck’s constant, c is the speed derived from the de Broglie relationship l0 ¼ eV of light, e is the charge on the electron and V is the accelerating voltage applied to the electron source (Section 2.3.1). The wavelengths corresponding to the characteristic X-ray excitation lines that are emitted from the region of the sample beneath the probe constitute a fingerprint for the chemical elements present in the solid and provide a powerful method of identifying these chemical constituents and their spatial distribution. The atomic number dependence of these wavelengths is illustrated approximately in Figure 2.13, but, rather than a single line, a group of spectral lines is commonly emitted, and there may be partial overlap of the lines characteristic of one chemical constituent with those emitted by a second constituent. In parallel to the characteristic X-ray emission spectrum, the X-rays generated by the incident electron beam also have an absorption spectrum, namely the high energy, short wavelength X-ray photons will themselves have a finite probability of exciting the atoms in the sample to higher energy ionization states. Thus a K photon from a higher atomic number element will possess sufficient energy to excite an atom of lower atomic number to the K state, resulting in absorption of the higher energy photon. The excited atom will then decay back towards the ground state, generating a new, lower energy photon that is characteristic of this second atom. This process is termed X-ray fluorescence. X-ray absorption, as noted

270

Microstructural Characterization of Materials K Excitation

(a)

Energy of Atom

K state

Kα

Kβ L states Lα M states

Ground State

Valency Electron Band

(b)

Beam Energy

Ionization Energy

Photon Energy

Atomic Number, Z

Figure 5.5 (a) An inelastic electron scattering event involving ionization of an inner shell electron raises the energy of the atom to the appropriate ionization state. Subsequent decay of the atom to a lower energy state is accompanied by photon emission. The energy of the emitted photon is characteristic of the energy difference between the two energy states for the atom, but must always be less than that needed for the initial ionization event. (b) The ionization energy required to eject an electron from a particular inner shell of the atom increases with its atomic number.

previously (Section 2.3.1), is characterized by an absorption coefficient m which depends on the wavelength of the X-rays and the atomic numbers of the chemical constituents. Critical absorption edges in the X-ray absorption spectrum correspond to those photon energies that have the threshold for excitation of fluorescent radiation of one of the chemical constituents of the solid. Both the excitation and the absorption spectra of either electrons or X-rays can be used to derive information on the chemical composition of the sample. We will return to these topics in much more detail in Chapter 6, where we also consider the quantitative determination of chemical composition from these signals in the scanning electron microscope.

Scanning Electron Microscopy 271

5.3.1

Characteristic X-Ray Images

The X-ray signal generated beneath a focused electron probe comes from a volume element of the sample which is defined by the envelope of electron energies that exceed the energy required to excite the characteristic radiation from any chemical constituent. As the beam voltage is reduced, so the size of this volume element shrinks, improving the potential spatial resolution for the X-ray signal, but also reducing the intensity of this emitted signal. When the energy of the incident beam, determined by the accelerating voltage, falls below the threshold energy for excitation, no characteristic X-ray radiation can be generated. A compromise has to be found which will ensure a statistically significant characteristic X-ray signal that remains spatially localized in the region of interest. The intensity of the X-ray signal that is emitted will at first increase as the electron beam energy is increased above the minimum energy needed to excite the characteristic signal. However, if the incident energy exceeds the critical excitation energy by a factor of about four, then the intensity of the X-ray signal escaping from the specimen starts to decrease. This decrease occurs because of increasing absorption of the excited X-rays within the sample, since the signal generated is now, on average, originating deeper beneath the sample surface. This is a result of the increased diffusion depth of the electrons at the higher energies. It follows that an ‘optimum’ electron beam excitation energy exists which gives maximum excitation of the characteristic X-rays that are emitted. This energy is of the order of four times the excitation energy. In practice the beam energy of the probe should be selected for the highest energy which is of interest, that is, the shortest characteristic X-ray wavelength that is to be detected. The efficiency for characteristic X-ray generation is low and the X-rays are emitted at all angles. A high proportion of the X-ray signal is either absorbed within the sample or fails to reach the detection system because the X-ray collection is itself rather inefficient. Energydispersive collection of the X-rays uses a solid-state detector. The alternative is a crystal spectrometer that employs a series of different curved crystals for each range of wavelengths of interest. The advantage of the curved-crystal diffractometer, a wavelengthdispersive spectrometer, is that the spectral resolution is excellent, considerably reducing the chances of peak overlap ambiguities in ascribing each characteristic X-ray wavelength to a specific chemical constituent in the sample. Solid-state X-ray detectors depend on the energy discrimination capability of a cryogenically cooled semiconductor crystal. The electrical charge that is generated in the detector by each photon absorbed is proportional to the energy of the incident photon. This charge results in a current pulse which is proportional to the energy of the photon that has been captured by the energy-dispersive spectrometer. Signal overlap of characteristic peaks is a frequent problem in EDS, although the problem can usually be solved by selecting one or more alternative characteristic emission lines for analysis from the X-ray energy spectrum. The detector is commonly shielded from contamination by the microscope using a polymer film window that is transparent to all but the longest wavelength, lowest energy X-rays. The detectors are routinely capable of detecting photons of wavelengths exceeding 5 nm, corresponding to X-ray emission that is associated with some of the lightest atomic elements (boron, carbon, nitrogen, and oxygen). All EDS detectors are limited in the rate at which they are able to accept X-ray photon ‘counts’. This corresponds to the time required for each individual charge pulse to decay in the detector. This dead-time is typically somewhat less than 1 ms, during this time no further

Microstructural Characterization of Materials

Dead-Time

Counting Rate

272

Emission Rate Figure 5.6 The count rate as a function of the signal incident on a solid-state detector, illustrating the effect of dead-time on the counting efficiency.

counts can be reliably recorded. It follows that count rates should not exceed 106 s
1. Some counts will inevitably be lost, since the photons are randomly generated in time. The proportion of ‘dead-time’ between counts is registered by the counting system. Acceptable dead-times are of the order of 20 %. Lower values of the dead-time will correspond to lower rates of data accumulation. The effect of increasing dead-time on the collection efficiency of a typical solid-state detector is shown schematically in Figure 5.6. The X-ray signal may be displayed in three distinct formats: 1. An X-ray spectrum. This is used primarily in order to identify the chemical elements present from their characteristic X-ray ‘fingerprints’ (Figure 5.7). Such a spectrum may be collected with the beam stationary, at a specific location on the sample surface (point analysis). Alternatively, to reduce the effects of contamination build-up, the spectrum may be collected while the beam is scanned over a selected area (‘rastered’). Typical signal collection times that are required to ensure EDS detection of all the elements that are present in concentrations exceeding one or two percent are of the order of 100 s. 2. An X-ray line-scan. In this mode the beam is traversed across a selected region of the sample in discrete steps and the signal prerecorded at each step. The spatial resolution for X-ray analysis in a line-scan depends on the number of steps per unit length of line. The best obtainable spatial resolution is limited by the volume element of the sample from which the X-rays are generated while the chemical detection limit will depend on the time of signal acquisition at each step. The signal may be in the form of an entire EDS spectrum, recorded at each point along the line (full-spectrum mapping). Alternatively one or more characteristic X-ray energies may be selected. Full-spectrum mapping requires significantly more digital storage than that needed for individual, selected energies, but no a priori knowledge of the local chemical distribution is required. In either case, the recorded intensity from the characteristic X-ray peaks are displayed as a function of the position of the electron beam (Figure 5.8). The selection of the windows in the energy spectrum for the energies that correspond to the wavelengths of interest are

Scanning Electron Microscopy 273 280000

TaSi

260000 240000 220000

Counts

200000 180000 160000 140000 120000

Al

100000 80000 60000 40000 O 20000

Ta

Cu

Ta

N 0

0

Ta 1

2

3

4

5

6

7

Cu Ta 8

Ta Cu TaTa 9

10

Energy (keV) Figure 5.7

EDS point measurement from a printed circuit board (see micrograph in Figure 5.8).

set to exclude all other photon energies. The number of counts for each selected characteristic energy peak is displayed for each position along the beam traverse. This mode of operation is especially useful for determining concentration gradients and segregation effects at grain boundaries, phase boundaries and interfaces. 3. An X-ray chemical concentration map. In this case, the incident electron beam is rastered across a selected area of the sample, and photon counts are collected for one or more energy windows that are characteristic of the chemical components of interest. The counts are recorded as a function of the incident electron beam coordinates. The photons detected for a particular characteristic X-ray emission line are then displayed as colourcoded dots in a position on the screen that corresponds to the beam coordinates as it is scanned over the area of the specimen surface chosen for analysis. The range of wavelengths to be assigned a particular colour code is pre-selected in an energy ‘window’ which corresponds to a characteristic photon energy of the element of interest. Several elements may be detected and displayed simultaneously, corresponding to separate energy windows that are individually colour coded for the different elements (Figure 5.9). In many SEM instruments these areal X-ray maps can be collected automatically from several successive selected areas whose position on the sample

274

Microstructural Characterization of Materials

Figure 5.8 EDS line-scan across a printed circuit board. The red line in the secondary electron SEM micrograph indicates where the line-scan was acquired.

surface is chosen to optimize the statistical significance or to be representative of different morphological features of the microstructure. The counting times can be adjusted to the required level of statistical significance for the chemical constituents, that is, adjusted to the detection limit for a low concentration element. If full spectra are

Scanning Electron Microscopy 275

Figure 5.9 EDS elemental maps from a printed circuit board for Ta, Al, Cu, O, and Si. The brighter the intensity, the stronger the EDS signal for a specific element. A secondary electron (SE) SEM micrograph is included for comparison.

recorded from each point, then it is also possible to correct for background and secondary fluorescent excitation. Quantitative chemical analysis for the selected region of the specimen surface is then possible. This is a very time-consuming process that presupposes sufficient beam and sample stage stability, together with negligible contamination of the sample surface. Hence, a combination of high instrumental standards with experienced operational competence is required. In most instruments, all three of the above modes of operation involve extended periods of electron-probe irradiation of the sample surface and hence high irradiation doses accompanying the acquisition of the necessary counting statistics. This frequently results in the build-up of a carbonaceous contamination film, on the sample surface. It is always good practice to limit the detection time to the minimum required to collect a statistically significant signal. Contamination films can affect the X-ray spectra, primarily by adding to

276

Microstructural Characterization of Materials

the background radiation and changing the conditions of analysis as the data are acquired. Contamination build-up is a major reason for preferring to raster the probe over a small area of the sample rather than relying on a point-analysis to collect the same total number of counts from a fixed coordinate position that is itself subject to mechanical drift of the specimen stage or electronic drift of the probe. The overriding consideration in recording X-ray data in the scanning electron microscope is always the achievement of statistical significance for the collected signal. The simplest assumption that can be made is that the counting statistics at each point on the sample surface obey a Poisson distribution, so that the counting pffiffiffiffierror that is associated with the characteristic line of each chemical constituent is just 1= N , where N is the number of counts recorded for this constituent from the volume element on the sample surface that is excited by the electron beam. As noted above, point analysis and the collection of a complete spectrum can provide data corresponding either to a fixed beam coordinate or a selected scanned area. For moderate counting times, the statistical significance is usually high and can be determined after first correcting for background counts. These background corrections are measured in two separate windows either side of the selected window for the characteristic line of interest. It is common practice to choose the collection width of the window for each characteristic line using the FWHM of the characteristic energy peak, N ¼ NT
NB, where NT and NB are the total counts within the selected window and the correction for the background counts, respectively. The latter is interpolated from the two background measurements in order to correspond to the same energy as the characteristic line. Line analysis results in some reduced statistical significance, in direct proportion to the square root of the number of pixels (picture elements) along the line selected for analysis on the sample surface. Nevertheless this mode is extremely useful in determining changes in concentration near surfaces, interfaces and grain boundaries, for example, segregation effects, as well as diffusion concentration gradients that are associated with the precipitation kinetics of a second phase or oxidation of a surface. Elemental maps require far longer counting times and are in a different class altogether. They should only be acquired together with a corresponding high resolution secondary or backscattered electron image (see below). A good, high resolution, secondary electron image will usually contain more than 106 resolved pixels, while an X-ray elemental map recorded at the same resolution would require a prohibitive counting time (approximately 106 times longer than that for a single point count). Clearly this is impossible, and not just impractical. Neither the life expectancy of the operator, nor the stability of the electron beam probe, nor the contamination of the sample are anywhere near sufficient for this. Is elemental mapping just a ‘good idea’? What methodology for elemental mapping is ‘practical’? If we assume a counting time of 100 s, as in the case of a point-analysis spectrum, and we require a statistical significance corresponding to a 95 % confidence level, then the minimum number of counts for a resolved X-ray pixel in a mapped X-ray image is about 1000. At a relatively high counting rate of 104 s
1, this corresponds to an acquisition rate for an elemental map of 10 pixels s
1. Hence, to acquire an area on the sample surface of 128  128 pixels would take about 25 min, and it would require well over a day to acquire a statistically significant elemental map with a pixel density approaching that of a secondary electron image. As we will see in Section 5.5, the spatial resolution of a secondary electron image is always better than can be achieved for an elemental X-ray map,

Scanning Electron Microscopy 277

simply due to the differences in size of the volume elements in the sample from which the secondary electron and X-ray signals are emitted. In fact, trying to generate an elemental Xray map with the same density of measured points as a secondary electron image makes no sense at all. Independent of the problems associated with X-ray counting statistics, the dimensions of the X-ray emitting region beneath the probe will limit the spatial resolution of both elemental maps and line-scans. The size of this zone is dictated primarily by the electron diffusion distance in the sample xD, that is, the diameter of the envelope for electrons with energies greater than the critical excitation energy. This diffusion distance depends on the beam voltage and the atomic number, or density of the sample. Typical values of the diffusion distance are between 0.5 mm and 2 mm for a standard scanning instrument. By choosing the typical value of 1 mm, and using the previously estimated, maximum practical number of X-ray counts per point, we conclude that a useful elemental map can be obtained from a 100  100 mm2 region, that is, an area of 104 pixels, in just over 16 min. Attempts to obtain elemental maps from a larger region may be successful if the counting times are significantly increased, but this is usually impractical. Imaging smaller areas, that is employing higher magnifications, will simply result in fewer effective image points and increased ‘blurring’, since the resolution will then be dictated by the dimensions of the excitation envelope. Recognizing that the resolution in an elemental map is inherently worse than that in the secondary or backscattered electron image is the first step to making the best use of the Xray data. By superimposing the X-ray map onto the high resolution, secondary electron or backscattered electron image, it is possible to see the detailed morphology of the sample and its relation to the localized variations in chemical composition. A 128  128 pixel array of the X-ray data can be acquired in less than 30 min, with each pixel representing an area determined by the excitation envelope for X-rays, typically of the order of 1 mm in diameter. Such low resolution elemental maps are an extremely useful guide to local variations in composition near features recorded at high resolution in a secondary electron image, despite the orders of magnitude difference in the density of resolved pixels.

5.4

Backscattered Electrons

A fraction of the incident high energy electrons will be scattered by angles greater than p and these electrons have a finite probability of escape from the surface. The fraction of the incident beam backscattered R depends rather sensitively on the mass density, or, more  increasing with increasing Z  accurately, the average atomic number of the specimen Z, (Figure 5.10). The backscattered electron signal is therefore able to resolve local variations in mass density and results in atomic number contrast. The backscattered fraction is much less dependent on the incident beam energy E0, decreasing as E0 increases. Such atomic number contrast can be very useful, since it offers the possibility of distinguishing between different phases at a far better resolution than can be achieved by X-ray microanalysis. Atomic number contrast is more pronounced at lower voltages. The backscattered electrons originate in a surface layer whose thickness corresponds approximately to the diffusion distance. They come from an area beneath the beam that is proportional to this distance and is significantly less than the range of the inelastically scattered electrons with E > kT.

278

Microstructural Characterization of Materials

η 0.5

Z Figure 5.10 number.

The fraction of backscattered electrons as a function of the average atomic

While the average energy of the backscattered electrons is, of course, less than that of the primary incident beam, it is nevertheless of the same order of magnitude and this backscattered electron signal can be used to acquire crystallographic information. The backscattered electrons are detected over a wide angle in an annular region close to the probe lens pole pieces. As in TEM, the intensity of the inelastically scattered signal falls roughly as cos2a, where a is the angle between the incident beam and the backscattered electron path. As in the transmission case, the backscattered electrons can be diffracted by a crystalline sample, subtracting intensity from the signal at low values of a when the Bragg condition is satisfied and adding intensity at higher angles, to give the backscatter equivalent of the Kikuchi line diffraction observed in the transmission electron microscope (Section 2.5.3). These EBSD patterns can be collected and recorded using a charge-coupled device (CCD) system. The patterns can also be analysed by appropriate computer software, to give the orientation of the crystal surface at each pixel location (Figure 5.11) Improved collection efficiency for backscattered electrons is obtained by tilting the sample to bring the high energy electron diffusion distance closer to the free surface. Suitable computer software corrects the recorded image for the foreshortening associated with the angle between the microscope axis and the normal to the tilted sample surface. Analysis of EBSD patterns collected as a function of position for a rastered incident electron beam can be colour coded for selected ranges of crystal orientation so that the surface orientation of each grain is recorded in the image. (This is more fully discussed in Section 5.6.5 on OIM.) Although the collection efficiency for backscattered electrons is high, the backscattered electron current is only a fraction of the incident beam current. Field emission sources greatly improve the data collection statistics for backscattered electrons.

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Figure 5.11 An experimental EBSD pattern of gold, overlayed with a simulated pattern showing the projections of the possible diffracting planes in red.

5.4.1

Image Contrast in Backscattered Electron Images

Contrast in a backscattered electron image may arise from either of two sources: 1. Regions of the specimen surface that are tilted towards a backscattered electron detector will give an enhanced signal, while the signal will be reduced if the surface is tilted away from the detector. A segmented annular detector can therefore be used to obtain a topographic image of the surface in which the signals collected from detector segments positioned on opposite sides of the specimen surface are subtracted and then amplified. To a good approximation, this enhances the topographic contrast from regions that are tilted in opposite directions, while neutralizing contrast due to differences in density (atomic number). 2. Collecting a backscattered image from a detector that surrounds the probe lens pole pieces, or, equivalently, summing all the signals from a segmented detector, minimizes the contrast associated with surface topography, since the signal is then collected from all possible azimuthal angles. The contrast from features visible in the image is now mainly atomic number contrast which reflects variations in specimen density that are either associated with variations in composition or, in some cases, with regions of fine porosity.

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Figure 5.12 A backscattered electron atomic number contrast image from a polished surface section, showing a niobium-rich, intermetallic phase in bright contrast, dispersed in an alumina matrix in dark contrast. The image was recorded with a 10 kV incident electron beam.

The resolution in the backscattered electron image is usually an order of magnitude better than that of an X-ray elemental map, but still cannot compete with that available in a secondary electron image (discussed below). The direct relationship between the backscattered electron image and the high energy electron diffusion distance in the material typically limits the resolution to between 50 nm and 100 nm when working with beam energies of 10–20 keV. The backscattered electron image provides useful information on the distribution of the phases visible on the surface of a polished sample, providing the phases differ sufficiently in density (as can be seen in Figure 5.12). Commercial scanning electron microscopes can now offer a combination of backscattered electron information on grain morphology with automated EBSD and X-ray elemental mapping of the same grains. This remarkable achievement is fully appreciated by the materials characterization community.

5.5

Secondary Electron Emission

Most of the electron current emitted from a sample excited by a high energy incident electron beam is due to the release of secondary electrons from the sample surface. The secondary electron emission coefficient, that is the number of secondary electrons which are released per incident high energy electron, is always much greater than one and may reach values of several hundred. It is useful to separate the secondary electron signal into two components: first, those secondary electrons that are generated by the

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high energy electrons in the incident beam as they enter the surface; and, secondly, those secondary electrons that are generated by high energy backscattered electrons that have returned to the surface region after several, inelastic scattering events. The former signal comes from a surface area of the order of the beam probe cross-section, and contains information with a resolution only limited by the probe diameter. The secondary electrons that are generated by backscattered electrons come from a surface area which is similar to that responsible for the backscattered electron signal and can only resolve image detail on a scale comparable with the backscattered electron resolution. We will consider, below, how to separate these two signals in order to optimize the resolution in the secondary electron image. A schematic representation of the energy distribution of all the electrons emitted by a SEM sample is given in Figure 5.13, and a comparison between a backscattered and secondary electron micrograph of Al2O3 particles on the surface of Ni are shown in Figure 5.14. While the energy of the backscattered electrons is just below that of the incident beam, E0, the secondary electrons generate a huge peak at the low energy end of the energy spectrum. An Auger excitation signal also exists, at energies that lie just above the secondary electron energy peak. As we shall see in Chapter 8 (Section 8.2), Auger electrons are an important source of information on surface chemistry. However, collection of an Auger spectrum requires extreme vacuum and specimen degassing that are beyond the capability of the scanning electron microscope. Secondary electrons may have energies as high as 100 or

Electron yield

Secondary electrons

Auger electrons Backscattered electrons

Electron energy

E0

Figure 5.13 Schematic illustration of the electron yield as a function of the emitted electron energy. The yield of secondary electrons is orders of magnitude larger, and their energies orders of magnitude less, than that of backscattered electrons. Surface analysis using Auger electrons will be discussed in Chapter 8.

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Figure 5.14 (a) Backscattered electron and (b) secondary electron SEM micrographs of small Al2O3 particles on the surface of a nickel substrate. Both micrographs were recorded using a 25 kV incident electron beam.

200 eV, but their energies usually fall in the range below 10–50 eV (with an energy spread of about 5 eV). The secondary electrons are therefore readily deflected by a low bias voltage and can be collected with very high efficiency (close to 100 %). Moreover, their low kinetic energy severely restricts their mean free path in the sample, so that the secondary electrons that escape to the detector are generated within 1–20 nm of the surface. Consequently these secondary electrons are almost unaffected by beam spreading of the probe beneath the surface. To a good approximation, the secondary electron escape distance is given by: P S ¼ expð
r=LS Þ

ð5:1Þ

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where PS is the probability of escape, r is the distance the secondary electron must travel in order to escape from the surface of the sample, and LS is the mean free path of a secondary electron generated in the bulk solid. For a high atomic number metal, such as platinum, LS  2 nm. For a low density ceramic, such as MgO, LS  23 nm. 5.5.1

Factors Affecting Secondary Electron Emission

Four factors directly affect the secondary electron emission current from a sample surface:

Secondary Electron Yield

1. The work function of the surface, that is the energy barrier that has to be overcome by an electron at the Fermi level in the solid in order to permit it to escape from the sample into the vacuum. Typical work functions are a few eV. The work function depends on both composition and the atomic packing (crystal structure) at the surface, and is sensitive to surface adsorption and contamination films. In SEM, surface contamination by a carbonaceous layer is generally sufficient to obscure all effects attributable to the work function of the substrate. 2. The incident electron beam energy and beam current. As the beam energy is increased, more secondary electrons are expected to be created beneath the probe, but since a higher energy beam is inelastically scattered at depths which are further beneath the surface, the proportion of secondary electrons that can escape from the sample is eventually reduced. In practice, the yield of secondary electrons rises rapidly as the probe energy increases from zero up to several kilovolts. The yield then goes through a shallow maximum, before decreasing slowly above 5–10 kV depending upon the material (Figure 5.15). However, the secondary electron current is directly proportional to the current in the incident beam, which decreases as the accelerating voltage for the incident beam is decreased.

1

2

3 4 5 6 Incident Electron Energy (kV)

7

Figure 5.15 Schematic illustration of the expected secondary electron yield as a function of the incident electron beam energy.

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3. The density of the sample also has an influence on the secondary electron yield. At higher values of the atomic number Z, the backscatter coefficient R is higher, so as Z increases more secondary electrons are created by the backscattered electrons. At the same time, the secondary electrons in higher atomic number materials have a smaller diffusion distance, while, at a given beam intensity, the number of inelastic scattering events in high Z materials is higher in the surface region than for low Z materials. It follows that, for a given excitation energy, larger numbers of secondary electrons should be collected from higher atomic number samples. The Z dependence of secondary electron yield is more pronounced at lower beam energies, when the diffusion distance of the incident electron probe becomes comparable with the mean free path of the secondary electrons in the solid. 4. The most pronounced contrast effects in the secondary electron image are due to surface topography, or, more precisely, to the local curvature of the surface and the angle of incidence of the electron probe. In general, changes in local curvature change the probability that a secondary electron that has been generated near the surface can escape, while the angle of incidence of the probe determines the path length of the incident high energy electron within the surface region (Figure 5.16). A region protruding from the surface, that is, a region having a positive radius of curvature increases the chances of secondary electrons escaping, while any recessed region, having a negative radius of curvature, will reduce the secondary electron current by local trapping of the secondary electrons. The secondary electrons are commonly collected using a bias voltage applied to the collector, so even though some regions of the sample are out of the line of sight (Figure 5.17), we may conclude that the secondary electron image should provide topographic images of rough surfaces having both high resolution and excellent contrast (Figure 5.18). The introduction of field emission guns has made an additional, dramatic improvement in secondary electron image resolution. This has been further improved by reducing the working distance of the probe lens to just a few millimetres and placing a secondary electron detector inside the magnetic field of the pole pieces (Figure 5.19). The secondary electrons emitted in the forward direction due to the initial inelastic

r < Ls

r < Ls r > Ls

r < Ls

Z

Sample Figure 5.16 A rough region on a sample surface affects the probability of escape for secondary electrons. If the distance needed to escape from the surface r is greater than the mean free path of the secondary electrons, Ls, then the secondary electrons cannot escape from the sample.

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Secondary electron trajectories

Collector (~+200 V)

Specimen Figure 5.17 By applying a small bias voltage, the secondary electrons can be collected with high efficiency from regions of the surface that are not in the direct line of sight of the collector.

scattering events in the primary beam, are then trapped by the magnetic field of the pole pieces and collected, while those generated by the backscattered electrons at wider angles cannot reach the in-lens detector placed within the electro-optical column. The secondary electron signal detected by this in-lens collector is very much weaker than the secondary electron signal normally detected, but this does not matter at the beam intensities generated by a field emission gun. The in-lens detection system improves the secondary electron image resolution (from 20 to 50 nm to only 1 or 2 nm).

Figure 5.18 High resolution secondary electron SEM micrograph of nanometre-sized TiCN particles, recorded with a 5 kV incident electron beam at a working distance of 4 mm.

286

Microstructural Characterization of Materials BSE Detector SE Detector BSE1

SE1

Column

SE3

BSE Detector

ET Detector

BSE Detector

BS

1 SE

E2

2

SE

2 SE

Backscattered Electrons

Sample

Figure 5.19 The location of secondary electron (SE) and backscattered electron (BSE) detectors. Secondary electrons of the first type (SE1) generated by the primary incident beam, are preferentially acquired by a detector placed within the electro-optical column to form a very high resolution image when compared with secondary electrons of the second type (SE2) that are acquired by an Everhart–Thornley (ET ) detector. Similarly, BSE1 electrons, defined as primary beam electrons that have undergone very few inelastic scattering events, can be acquired by a BSE detector placed within the column, to improve the resolution for BSE micrographs when compared with those acquired with a conventional annular BSE detector that collects all the multiply scattered backscattered electrons.

5.5.2

Secondary Electron Image Contrast

Some complications in the interpretation of secondary electron image contrast arise from the two major factors affecting signal generation: namely, the initial production of the secondaries by the primary electrons followed by their subsequent escape from the sample surface. As long as the sample is reasonably planar, the escape of secondaries is only

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restricted by variations in work function, which is why the intensity of the signal falls rapidly to zero at secondary energies below a few eV (Figure 5.13). The generation of secondaries depends on the number of inelastic scattering events that occur in the surface region due to either the incident primary beam or the backscattered signal. In general, reducing the beam voltage will enhance the secondary electron image contrast, and this does not usually cause any sacrifice of resolution. High resolution scanning instruments that are equipped with a field emission gun are capable of 1–2 nm resolution by combining an in-lens detector with electron-probe beam energies that are as low as 200 eV. These conditions give excellent contrast and resolution that reflect either variations in atomic number or surface topography. At these low beam voltages there is usually no problem of surface conductivity and no need to conductively coat a nonconducting sample. The beam penetration into the sample (the diffusion distance) is very limited and any electrostatic charging (Figure 5.20) is neutralized by the large yield of secondary electrons. For such samples, the features showing contrast may be confidently associated with the sample material, and not with any conducting layer that has been deposited to avoid electrostatic charging. Topographic contrast also presents some problems of image interpretation. Protruding regions of the specimen may trap secondary electrons and screen recessed regions from the

Escaping Secondary Electron Current

Secondary Electron Yield

Incident Primary Electron Current

Zero Net Current Flow

1

2

3

4

5

6

7

Incident Electron Energy (kV) Figure 5.20 Schematic illustration demonstrating charge compensation by selecting an optimal accelerating voltage. The total incident current decreases as the accelerating voltage is decreased (black curve). This curve intersects the characteristic secondary electron yield at two points, where the total net current of electrons to the sample is zero and electrostatic charging of the surface is prevented.

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Microstructural Characterization of Materials

detector, despite the high secondary electron collector efficiency. More commonly, enhanced emission from a region of high positive curvature may extend some distance beyond the high curvature region. This enhanced emission occurs over a distance that corresponds approximately to the electron range in the sample xR. This is because the inelastically scattered, high energy electrons will generate secondary electrons that are able to escape whenever a high energy electron approaches the surface. Similarly, a darkened area in the image, associated with a feature of negative curvature, may also extend for a similar distance beyond the feature. It follows that there is a ‘shadowing’ of hollows and a ‘highlighting’ of protrusions that looks very similar to that produced by sunlight falling on hilly country, but is associated with a completely different mechanism, since the secondary electrons are being collected from all directions (Figure 5.16). This can be particularly misleading when operating at high magnifications and close to the resolution limit.

5.6

Alternative Imaging Modes

In addition to the three imaging signals which we have discussed so far (X-rays, back scattered electrons and secondary electrons), there are some other modes of operation for the scanning electron microscope that have useful applications. We shall briefly discuss just two of them, namely cathodoluminescence and electron beam image current (EBIC). 5.6.1

Cathodoluminescence

Many optically-active materials will emit electromagnetic radiation in the visible range when suitably excited. Under an energetic electron beam, optically-active regions of the SEM sample will glow, emitting visible light at a wavelength that is characteristic of the energy levels at the surface of the sample that have been excited by the incident beam. The fluorescing sample may be observed with an optical microscope, in which case the resolution is limited to that characteristic of the optical microscope objective, or the light emitted may be collected by a photoelectron detector, and then amplified, recorded and displayed using the same time-base as the beam scanning coils. In this case the resolution is not limited by the wavelength of the emitted radiation but rather by the diffusion distance of the electron probe in the sample. Biological samples can be labeled by suitable fluorescing ‘stains’, and cathodoluminescence used to image soft tissue structures in which different features are identified by different stains, each of which fluoresces at a characteristic wavelength in the optical range. 5.6.2

Electron Beam Induced Current

If the specimen is electrically isolated from its surroundings and the current flowing through the specimen can be monitored then it is possible to form an image that displays this specimen current signal synchronously with the scanning of the beam over the sample. Several variants of this mode of operation have been used to study defects in semiconductors and solid-state semiconductor devices. For example, the electrical conductivity of a semiconductor is often a sensitive function of its defect structure and dopant or impurity concentrations, and this conductivity can be monitored as the beam is scanned across the

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surface, either with or without a bias voltage applied to the sample. This form of operation of the scanning electron microscope is referred to as the electron beam induced current mode. A complete solid-state device may be inserted in the specimen chamber and operated in situ while under observation using this mode. The electric field that is developed in the various regions of the device will modify both the secondary electron signal and the specimen current flowing through the various components of the device. Devices can be imaged at high resolution and the method has been used by industry for device development and the analysis of process defects that are associated with device operation. This topic is well beyond the scope of the present text. 5.6.3

Orientation Imaging Microscopy

OIM is a relatively recent development, made feasible by the increasing availability of field emission electron sources and high resolution CCD cameras, together with the computer data processing and storage capacity required to handle gigabyte quantities of image data. When first developed some 50 years ago, the scanning electron microscope and the microprobe analyser were separate instruments, the first designed to provide submicrometre resolution and high depth of field in secondary electron images, and the second to perform highly localized chemical analyses using characteristic X-ray excitation under a focused electron probe. Combining the detection of these two separate signals, the secondary electron and the X-ray, in a single, commercial instrument was accomplished as early as 1960, but no one dreamed at that time that it would also be possible to extract crystallographic information from the same sample area and at the same time in a single instrument. Three major problems stood in the way of the successful development of an orientation imaging microscope. The first was the lack of a sufficiently sensitive signal detection system for acquiring the angular distribution of the EBSD pattern from a single crystal grain irradiated with a high energy electron probe. The second was the detection of the very weak signal in the EBSD pattern that was generated using a conventional, tungsten hairpin, thermionic electron source. The third problem was the need to analyse the massive amounts of data that were required to determine the crystal orientation for a large number of pixel locations on the surface of a polished, polycrystalline sample, and the conversion of this information into an orientation distribution map that could display regions of similar orientation on the sample surface. The most recent publications reporting the application of OIM to materials research have now combined local chemical information, derived from X-ray elemental mapping, with morphological and crystallographic analysis, and have begun to report the serial sectioning of samples in order to develop a complete, three-dimensional characterization of a sample that has been taken from a specific location in an engineering component. 5.6.4

Electron Backscattered Diffraction Patterns

The geometry of EBSD is very similar to that of Kikuchi diffraction in TEM (Section 2.5.3), and is shown schematically in Figure 5.21. When the high energy electron probe strikes a crystal in a solid sample, inelastic scattering events scatter the incident electrons so that a fraction acquire velocity vectors that allow them to escape from the

290

Microstructural Characterization of Materials

+θ

Incident electron beam

−θ

Pattern centre

Specimen

Figure 5.21 Schematic illustration of the formation of an EBSD pattern from a crystalline sample in the scanning electron microscope.

specimen surface. The angular distribution for these backscattered electrons, most of which have been multiply scattered but nevertheless still have energies close to that of the incident electron beam, is determined by diffraction from the crystal planes. Intensity is subtracted from regions of the diffuse backscattered electron distribution close to the electro-optical axis of the microscope and intensity is added to the darker, high angle scattering regions. The result is a single pair of light and dark lines in the EBSD pattern for

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each reflecting crystal plane. These lines correspond to the intersection of hyperbolae with the plane of observation of the pattern, but since all the scattering angles are small, the light and dark lines appear sensibly straight and parallel. The distance between the lines is proportional to 2y, where y is the Bragg angle for each family of diffracting planes, and hence inversely proportional to the interplanar spacing, since at small values of y, sin y  y. Analysing the geometry of the crystal surface causing the EBSD pattern is therefore quite straightforward. The bisector of each pair of parallel light and dark diffraction lines is the projection of the diffracting plane. With a little practice, it is possible to recognize prominent symmetry zones directly from the EBSD pattern, even before the crystallographic orientation has been analysed by the computer software. To improve the intensity of the signal we can increase the probability that backscattered electrons will escape from the surface by tilting the sample through a large angle, leading to considerable foreshortening of the backscattered electron image. This foreshortening is proportional to the angle between the optic axis of the microscope and the normal to the surface of the diffracting crystal. Foreshortening is usually corrected automatically when displaying the image data, but the image contrast is unchanged, and usually reflects the ‘shadowing’ of topographical features by the inclined incident beam. The incident electrons in the inclined beam have a longer projected path length through the sample perpendicular to the axis of tilt, so the distribution of back scattered electrons escaping from the surface is also elongated in this direction. The resolution in this direction is therefore reduced and grain boundaries are partially ‘blurred’ in the topological image. It follows that crystallographic analysis close to a boundary can be rather unreliable. Nevertheless, EBSD analysis has been successfully applied to individual grains with particle sizes below 1 mm, allowing the orientation relationship between small crystalline precipitates and a matrix phase to be determined. 5.6.5

OIM Resolution and Sensitivity

So far, we have only discussed the determination of crystal orientation beneath a stationary probe (as in point microanalysis), that is, a probe positioned on a selected image pixel. When applied to a polycrystal, we usually wish to know the distribution of crystal orientations for the grains intersected by the plane of the sample section. This involves collecting very large sets of EBSD data and presenting the results both quantitatively and in an image format. The first step is to define a grid of test pixels across the region of interest on the sample surface. The total number of pixels in the image grid is critical, since it determines, with the dwell-time per pixel, the time required to collect the data. Even with a field emission source, the required image scan times are long, both with respect to the stability of the microscope and the patience of the operator. The second step is to bin the crystallographic data from the solved EBSD patterns acquired from each pixel, usually on a stereographic projection in which all the crystal orientations are mapped onto a single unit stereographic triangle (Section 1.2.3.4) (Figure 5.22). This information can be converted into an image of the microstructure, based on the change in orientation (or, sometimes, crystal structure) from one image region to another. Colour coding is commonly used for such micrographs, where each colour defines either an orientation range for the grains, or a variation in crystallographic structure (Figure 5.23).

292

Microstructural Characterization of Materials 1/2 5 10 20 50 100

[001]

h h h h h h

[111]

[101]

Figure 5.22 Orientation of gold particles equilibrated on a ð10 10Þ surface of sapphire as a function of annealing time.

5.6.6

Localized Preferred Orientation and Residual Stress

OIM starts to reveal its full potential when preferred orientation is present (Figure 5.24) and selected surface orientations are clearly displayed, both in the orientation stereogram and in the corresponding colour-coded image, but once this information is available, it also becomes possible to answer more complex microstructural questions. For example, it is

Figure 5.23 OIM micrograph of a polycrystalline In2O3 film grown on a (001) MgO substrate. The colour assigned to the In2O3 grains is correlated with their orientation relative to the substrate surface. Reproduced from J.K. Farrer and C.B. Carter, Texture in Solid-State Reactions, Journal of Materials Science, 41(16), 5169–5184, 2006. Copyright 2006 with permission from Springer Science and Business Media. (See colour plate section)

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Figure 5.24 OIM used to study preferred orientation in polycrystalline alumina. (a) A sample prepared from conventional alumina powder has nearly random crystal orientations. (b) A sample prepared from aligned platelets is highly oriented. The stereographic colour coding for both micrographs is the same and is shown in (c). Reprinted with permission from V.R. Vedula, S J. Glass, D.M. Saylor, G.S. Rohrer, W.C. Carter, S.A. Langer and E.R. Fuller, Residual-Stress Predictions in Polycrystalline Alumina, Journal of The American Ceramic Society, 84(12), 2947–2954, 2001. Copyright (2001), with permission from Blackwell Publishing Ltd. (See colour plate section)

well-known that certain orientation relationships may be preferred when neighbouring grains are separated by so-called special boundaries. Using OIM it is possible to detect these boundaries and explore the statistics of special boundary formation. In some cases, the fraction of special boundaries can be controlled by thermomechanical treatment of the material to yield improved engineering performance. A closely related question concerns the spatial distribution of preferred orientation in a polycrystalline material. X-ray diffraction can only determine the average distribution of grain orientations in a sample with respect to the sample geometry, but with OIM it is possible to explore the local preferred orientation and its distribution in the material. These local variations are expected to be important in controlling plastic flow through a die or during a forging operation. Although EBSD is considerably less sensitive to changes in lattice spacing than X-ray diffraction, which can detect lattice strains of the order of 10
5, the EBSD data can be combined with finite element analysis based on the known thermal anisotropy and elastic constants of alumina in order to map the calculated residual stress distribution as the stress invariant (s11 þ s22) and maximum principal stress (s11). Figure 5.25 shows the calculated spatial distribution of the maximum principal stress (s11) for the same regions of the alumina samples in the OIM images of Figure 5.24. The stresses are strongly localized to the boundary

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Microstructural Characterization of Materials

Figure 5.25 Distribution of the maximum principal stress calculated for the same regions of the alumina samples shown in Figure 5.24 by inserting the OIM EBSD data into a finite element program. (a) Note the stress scale and the strong localization of the stress near the boundaries. (b) In the highly oriented sample the residual stress is dramatically reduced. Reprinted with permission from V.R. Vedula, S.J. Glass, D.M. Saylor, G.S. Rohrer, W.C. Carter, S.A. Langer and E. R. Fuller, Residual-Stress Predictions in Polycrystalline Alumina, Journal of The American Ceramic Society, 84(12), 2947–2954, 2001. Copyright (2001), with permission from Blackwell Publishing Ltd. (See colour plate section)

regionsand dramatically reduced by the preferred orientation of the second sample. While the OIM data make this analysis possible, the analysis is far from straightforward. Direct information on the tensile strains is not available and the annealing history of the samples has been assumed. Nevertheless, the results clearly illustrate the anisotropic distribution of the residual thermal stresses and the strong effect that the microstructure can have. Most OIM results that have been published are from single-phase materials, where all the grains possess the same crystal structure. There is no reason, in principle, why other microstructural morphologies should not be studied by OIM. For example, two-phase eutectics might be expected to exhibit coupled growth of the two constituent phases. OIM research of such systems would be expected to throw some light on the effect of the rate of heat transfer and casting additions on the crystallography of such coupled growth.

5.7

Specimen Preparation and Topology

The most important specimen requirement in SEM is that electrostatic charging of the surface should be avoided, since the associated charge instability will lead to unstable

Scanning Electron Microscopy 295

secondary emission, destroying both the resolution and the image stability. Charging of nonconducting samples can be prevented or inhibited, either by operating at low beam voltages or by coating the samples with a thin layer of an electrically conducting film. The specimen must also fit into the sample chamber without seriously constraining the geometrical freedom to manipulate the sample in the microscope column. This is needed both to select any area of interest for examination and to tilt the specimen surface at any required angle with respect to the optic axis. As noted previously, most SEM specimen chambers readily accept large specimens (some well over 10 cm in diameter). Apart from these two requirements, the specimens must also be stable in the vacuum system and under the electron beam. They should be free of any organic residues, such as oil and grease, which might lead to the build up of carbonaceous contamination, either on the specimen, or in the electro-optical system, or within a wavelength dispersive spectrometer, or elsewhere in the sample chamber.Loose particlesneed to be removed from the sample surface before insertion in the microscope, usually by ultrasonic cleaning in a suitable solvent, followed by rinsing and drying in warm air. These precautions are especially important for low voltage, high resolution SEM. At probe energies below 1 keV, all the secondary electrons come from a region that is very close to the sample surface. Electron beam induced carbonaceous contamination then becomes the primary source of secondary electrons. 5.7.1

Sputter Coating and Contrast Enhancement

Coating of the samples to enhance contrast and improve electrical conductivity is usually performed in a sputtering unit, as discussed previously for nonconducting transmission electron microscope specimens (Section 4.2.4). Two types of coating are commonly used, either a heavy metal or an amorphous carbon film. Heavy metal coatings of a gold–palladium alloy that have been deposited with a 5 nm particle size on the sample surface only interferes with the resolution at the highest magnifications and these coatings improve the contrast appreciably. However, metal coatings do interfere with chemical microanalysis, and may not be suitable if the best resolution is to be achieved. Carbon coatings can be deposited with a much smaller particle size (2 nm); this is usually below the resolution limit of the instrument. Carbon coatings do not improve the contrast, but they are mandatory for nonconducting samples intended for microanalysis at accelerating voltages above 5 keV. The best solution to electrostatic charging of the specimen is to reduce the beam voltage, but of course this cannot be done if microanalysis is required and the excitation energy for the characteristic lines of interest is more than a few keV. 5.7.2

Fractography and Failure Analysis

A major area of application for the scanning electron microscope is in fractography (that is, the imaging of fracture surfaces) and failure analysis, not only for engineering metals and alloys, but also for plastics and composites (polymer, ceramic or metal matrix), as well as engineering ceramics and semiconductor devices (Figure 5.26). The morphology of many other classes of materials has also been studied using the same scanning electron microscope techniques: natural and artificial foams, textiles and fibres, as well as systems that are unstable at ambient temperature and require cryogenic cooling by liquid nitrogen. Figure 5.27 shows a few examples.

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Microstructural Characterization of Materials

Figure 5.26 Some examples of failures in engineering materials imaged by SEM: (a) mechanical fatigue failure in steel (from the Metals Handbook, American Society for Metals); (b) brittle failure in porous TiCN; (c) failure of a fibre-reinforced polymer matrix composite (courtesy of A. Siegmann).

Scanning Electron Microscopy 297

Figure 5.27 Some other materials studied by SEM: (a) paper; (b) bone; (c) wood. (From Cellular Solids; Structure and Properties, L.J. Gibson and M.F. Ashby, Pergamon Press).

There are some simple but important guidelines to be followed if the maximum information is to be extracted from scanning electron fractography. Although these have been covered to some extent in previous sections, they will bear repeating: 1. The sample selected for examination in the microscope must bear a known geometrical, spatial and orientation relation to the original engineering system or component from which it was taken. Without this information it may be difficult to evaluate the significance of any fractographic observations. 2. The surface should not be damaged or altered in any way by the specimen preparation procedure. It is unbelievable how often the two halves of a failed component are ‘fitted together’ by an unthinking investigator prior to microstructural examination. The result is superficial damage to the failure surface and, too often, the destruction of important evidence of the cause of failure. 3. The specimen should be mounted in the microscope according to the specimen stage x–y coordinates and the axes of specimen tilt, for example, parallel and perpendicular to a known direction of crack propagation. A little forethought can save a lot of frustration, not to mention microscope time, and greatly simplify the process of investigation. 4. Images should be recorded over the full range of magnifications that are found to show any significant microstructural features. It is especially important to be able to relate the surface topography of the microstructural features to the results of any other observations. If these initial observations were purely visual, then an initial magnification in the scanning microscope of ·20 is not too little. A good procedure is to locate features of interest by first scanning rapidly over the sample, experimenting with different magnifications, and only then starting to record a series of magnified images, identifying

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features in the magnified images by their geometrical relation to details that can be observed with the naked eye. A factor of ·3 between image magnifications ensures that about 10 % of the surface area that is recorded at the lower magnification will always be visible in the image taken at the next higher magnification (Figure 5.28).

5.7.3

Stereoscopic Imaging

As noted at the beginning of this chapter, the scanning electron microscope is unique in its ability to focus and resolve detail over a large depth of field, parallel to the optic axis of the incident beam. This information can be extracted by recording a pair of images at the same magnification and from the same area of the sample, but at different angles of tilt of the sample with respect to the optic axis of the microscope. The two images, a stereo pair, correspond to two observations of the specimen surface taken from two different points of view. If the angle of tilt is well-chosen, then the geometry is equivalent to the stereoscopic, three-dimensional visual image observed by the superposition of the retinal data transmitted to the brain by the left and right eyes (Figure 5.29). Several commercial systems have been developed for viewing stereoscopic images directly in the scanning microscope, for example by a lateral displacement of the axis of scan of the probe, either for two sequential scan sequences or for alternate lines of the xsweep in a single scan. These commercial systems have never been particularly popular, since it is equally easy to record a pair of images without disturbing the scanning coil settings, but rather by mechanically tilting the specimen itself. Given that the microscope screen is commonly viewed at a distance of some 30 cm (a comfortable reading distance) and that the eyes are set some 5 or 6 cm apart, the required angle of specimen tilt needed to give an impression of depth identical to that received by the human eye is 12 . This is equivalent to setting the depth ‘magnification’ equal to the lateral magnification in the image. If the tilt angle is less than 12 then this will reduce the sensitivity to depth, and this may be necessary for very rough surfaces. Tilt angles greater than 12 will amplify the impression of depth and this may be useful for recording shallow features. An example of a stereo-pair is given in Figure 5.30. Pairs of stereo images, having their axis of tilt aligned accurately and placed with corresponding points on the images at the approximate separation of the observers’ eyes, may, with a little practice, be viewed by most people at the normal reading distance, just by focusing the eyes on infinity and without additional optical aids. More commonly, commercial stereo viewing systems are available, and can provide striking in-depth information that is often unobservable in a two-dimensional recorded image. 5.7.4

Parallax Measurements

In addition to the visual impact of a stereo image, it is also possible to extract quantitative information on the vertical distribution of features along the axis of the incident beam. This is accomplished by measuring the horizontal displacement, perpendicular to the axis of stereo tilt, for the same features recorded perpendicular to the tilt axis in the two images of the stereo pairs, (Figure 5.31). This displacement is termed parallax and its value is given by: x L
x R ¼ 2h cos y

ð5:2Þ

Scanning Electron Microscopy 299

Figure 5.28 Ductile failure in molybdenum at increasing magnification reveals both the general topographical features and the fine details of the fracture surface.

300

Microstructural Characterization of Materials 2θ

Left-Eye Viewpoint

2θ

Right-Eye Viewpoint

Left-Eye Equivalent Right-Eye Equivalent

Imaging System

Figure 5.29 The twin images observed by the left and right eyes are equivalent to the image pair recorded before and after tilting a specimen by a known angle about a direction perpendicular to the optic axis.

where xL and xR are the projected distances in the two images measured from any fixed position perpendicular to the axis of tilt y, h is the ‘height’ difference between the two features measured along the optic axis of the incident electron beam, and the stereo pair have been recorded at a tilt angle of y with respect to the incident beam normal. Parallax measurements are useful for checking the thickness of surface films, measuring the height of growth steps, or determining plastic slip displacements. They have also been used to estimate the roughness of ground and machined surfaces, and to determine the fractal dimensions of a fracture surface.

Figure 5.30 A stereo pair of micrographs from the surface of a brittle fracture. Hold the book at a comfortable reading distance and then focus the eyes at a distant object above the page. It should be possible to fuse the two images into a single three-dimensional view of the surface topography (most observers fail at the first attempt while some are never able to view a stereo pair without a suitable viewing system).

Scanning Electron Microscopy 301 Surface trace

Incident beam

h Surface trace

O

2θ Tilt angle

xL – xR Figure 5.31 The parallax geometry. The difference in separation of two features seen in a stereo pair and projected perpendicular to the tilt axis is directly related to the difference in height of the two features, measured along an axis parallel to the incident beam.

5.8

Focused Ion Beam Microscopy

In FIB microscopy the specimen probe is a beam of high energy ions focused onto the specimen, as the name implies. It follows that this instrument is not really an electron microscope. Nevertheless, the instrument has many structural features in common with the scanning electron microscope, so that it is appropriate to discuss the principles and applications of FIB microscopy. In addition to not really being an electron microscope, the FIB is also unusual in that it has, until recently, been primarily a tool for the electronics industry, employed for the micromachining of electronic and optronic device components. The first commercial units were introduced in the 1980s, primarily for device development as well as for industrial research. Today, FIB facilities are increasingly common in research institutes and some universities, where they are able to provide unique materials characterization information. In addition to this role in microstructural characterization, they are also able to function as platforms for micromachining, with tolerances that are in the nanometre range. The key enabling technology for the FIB is the liquid metal ion source, which operates by field evaporation of liquid metal at a tungsten tip that is maintained at a high positive potential. These sources were originally developed for ion propulsion in space at the beginning of the 1960s. The theory was developed by Taylor in 1964, who demonstrated that an electrically conducting liquid drop should be drawn into a sharp cone whose tip radius was small enough to generate the very high electric fields required for field evaporation of a liquid metal without thermal activation. As the accelerating voltage applied to the source is increased, the ion current also increases, increasing the source radius but leaving the electric field at the cone tip unchanged and equal to the evaporation field for the dominant ion species (which are usually doubly charged). Two other technologies preceded the development of the FIB microscope. In 1955 Mueller and his co-workers studied both field ionization and field evaporation and were able

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to demonstrate atomic resolution in the field-ion microscope. The source of the imaging ions in field ion microscopy was an inert gas (usually helium). The sharpened tip of a metal needle was the ion source in the case of field evaporation from the solid state. Only 5 years later, Castaing and his co-workers, who had previously developed the first electron probe microanalyser, subsequently demonstrated microanalysis of polished metallographic sections by the mass spectroscopy of ions that were sputtered from a sample surface under bombardment by a focused beam of inert gas ions. The secondary ion mass spectrometer (Section 8.3) has a remarkable mass sensitivity, but this is not easy to calibrate. The field ion microscope has since developed into the atom probe tomograph (Section 7.3). The atom probe has improved both the mass sensitivity and the resolution for microanalytical studies to nanometric dimensions, but is limited in the range of engineering materials that can be studied by the need for adequate electrical conductivity and mechanical strength. 5.8.1

Principles of Operation and Microscope Construction

Figure 5.32 shows schematically the geometry of the FIB ion source. The first liquid metal ion sources were fed by the flow of the liquid metal through a capillary, but much better control is achieved by allowing the metal from a heated reservoir to wet a tungsten needle and then be sucked into a cone source of ions at the tip of the charged needle. The ion

Filament Current

Reservoir of Melted Metal

Tungsten Needle Liquid Metal Film

Ion Beam Taylor Cone

Figure 5.32

Schematic drawing of the FIB liquid metal ion source.

Scanning Electron Microscopy 303

currents from a liquid metal source can far exceed those generated by field ionization of a gas at a sharp metal tip, while the continuous supply of liquid ensures that the source size remains effectively constant during operation. By comparison, field evaporation from a solid metallic needle would rapidly blunt the tip radius. The choice for the liquid metal is limited. The most successful sources so far employ gallium (atomic number 31, melting point 29  C). Liquid gallium partially wets tungsten and the comparatively heavy gallium ions sputter most solid surfaces efficiently. With a tungsten tip radius of about 10 mm (rather larger than that used for a field emission source), the liquid is drawn into a cone of similar length by the applied electric field. The design of the microscope column (Figure 5.33) differs substantially from that of a conventional electron microscope. For example, electromagnetic focusing is impractical because of the large mass of the ions, and electrostatic lenses must be used to focus the probe. Sputtering damage within the microscope also needs to be minimized. This is achieved by a sequence of beam-blanking plates with both variable and fixed apertures. Electrostatic quadrupole and octopole assemblies are used to position and scan the ion beam. The position and orientation of the sample is accurately controlled on a eucentric stage that permits tilt about all three coordinate axes, as well as full Cartesian, x, y and z, control of the sample position with respect to the optical axis of the ion beam. This specimen stage also ensures that the axes of specimen tilt remain perpendicular to, and pass through, Liquid Ion Source Suppressor Assembly Extractor Cap Beam Acceptance Aperture

Asymmetric Electrostatic Lens

Variable Aperture Quadrupole Beam Blanking Plates Blanking Aperture Octopole Asymmetric Electrostatic Lens

Detector Sample Figure 5.33

Schematic drawing showing the components of the FIB column.

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the optic axis. A wide range of FIB accessories are available, including an option for scanning of the sample by a beam of high energy electrons, as in the scanning electron microscope. Microscopes that include both ion beam and electron beam probes are often termed ‘dual beam’ instruments (Section 5.8.3). Gas injection systems are also available for chemically-assisted etching of the surface or for deposition of thin films in selected areas (see below). 5.8.2

Ion Beam – Specimen Interactions

The interaction between the focused high energy beam of metal ions and the specimen comes under three headings that are sometimes difficult to separate during operation: 1. Micro-machining by sputtering of matrix atoms from the surface. This process can be used to mill and section the sample to an accuracy of approximately 1 or 2 nm. 2. Secondary excitation, primarily by the generation of secondary electrons by the incident ion beam. The secondary electrons can be collected and used to image the sample surface with a resolution down to about 2 nm. 3. Sub-surface radiation damage, associated with lattice damage, primarily in the form of point defects (vacancies and self-interstitials), but also due to ions from the ion beam that have been injected into the solid. Figure 5.34 summarizes these effects. In general, an impinging high energy ion will generate so-called “knock-on” damage when it collides with an atom of the solid. The maximum kinetic energy E that can be transferred in a Newtonian model to an atom of the matrix is given by: E 4m1 m2 ¼ E 0 ðm1 þm2 Þ2

ð5:3Þ

where E0 is the energy of the impinging ion and m1 and m2 are the mass of the impinging ion and the struck atom, respectively. In electron irradiation damage m1  m2, and the electron beam energy must exceed 150 kV before individual atoms can be displaced in a thin-film specimen viewed by TEM, even for low atomic number specimens. In a FIB instrument every high energy ion generates a cascade of multiple atomic displacements, but only a small fraction of these result in sputtering of atoms from the specimen surface. If the threshold for atomic displacement is assumed to be 25 eV (a commonly accepted value), then a 25 kV impinging ion may displace up to 1000 of the sample atoms in a collision cascade. Only a small proportion of these will be sputtered. The remaining point defects may either anneal out or condense as radiation damage. Of course, the original impinging ion may also be trapped beneath the surface. As in ion milling (Section 4.2.3), the microfinish of the surface can be improved by tilting the sample so that the incident beam strikes at a glancing angle. Sub-surface radiation damage is a serious problem and cannot be ignored. Frenkel defects are generated consisting of vacancy–interstitial pairs. Self-interstitials in metals have very low activation energies for diffusion and may be expected either to anneal out to a free surface or to annihilate with vacancies before their concentration becomes high enough to cause them to condense as more extended lattice defects. Vacancies are another matter: in most materials these point defects diffuse rather slowly at room temperature and their

Scanning Electron Microscopy 305

Surface

Figure 5.34 Schematic drawing illustrating the damage collision sequence resulting from a primary ion incident on a sample surface. Squares indicate vacancies.

concentration can increase rapidly before they condense into immobile clusters or small dislocation loops. The concentration of injected ions is much less than the number of Frenkel defects formed, but these injected ions can also condense and form defect clusters which are visible as ‘radiation damage’ strain fields in thin-film samples that have been prepared for the transmission electron microscope by FIB milling. The high energy injected ions incident on a crystalline sample also give rise to an additional phenomenon termed ion channelling. If the ion beam is aligned along a prominent zone axis of the crystal, then the atomic mass distribution can focus the velocity vector of an incident ion along the crystal zone axis. This mass–lens focusing allows ions to tunnel for distances of several tens of nanometres along the zone axes, depending on the atomic mass and the elastic constants of

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Figure 5.35 Secondary electron micrographs from a gold wire-bond generated using high energy incident electrons or ions. (a) A 5 kV electron beam reveals the surface topology and roughness at high resolution. (b) A 30 kV gallium ion beam provides crystallographic channelling contrast from the individual grains of the bond.

the target material. This results in occlusion of the incident ions well below the surface of the specimen, as well as a strong crystallographic dependence for the escape of secondary electrons generated by the incident ion beam. Since secondary electrons can only escape and contribute to a secondary electron signal if they are generated within a few nanometres of the surface, very few secondary electrons will be emitted when the ion beam is aligned with a prominent zone axis. This leads to very strong crystallographic contrast in the secondary electron image that has been generated by the primary ion beam. Figure 5.35 shows this effect in a gold wire-bond. The secondary electron image in Figure 5.35(a) has been excited by a high energy electron beam, while that in Figure 5.35(b) has been excited by a FIB. 5.8.3

Dual-Beam FIB Systems

The FIB is a powerful tool for both characterization and micromachining when it is combined with a high-energy scanning electron beam. In these dual-beam instruments, the scanning electron microscope column is usually mounted vertically above the sample, and the ion beam source and column are attached at an inclined angle to the sample chamber. Both beams are focused onto the sample surface positioned at a set working distance (Figure 5.36). The electron beam is used to monitor the progress of ion beam milling or deposition, while the two types of secondary image can be recorded from either the ion beam probe or the electron probe, to provide a wealth of morphological information. 5.8.4

Machining and Deposition

The power of the FIB technology is best demonstrated when it is used to prepare samples in situ, using one of several techniques that we will list and then discuss in more detail: 1. Micromilling can be used to remove appreciable quantities of material, mimicking conventional mechanical machining in what amounts to turning, milling and trepanning

Scanning Electron Microscopy 307

Figure 5.36 stage.

Schematic drawing of the dual-beam configuration with respect to the sample

operations. In dual-beam systems, the electron beam is used to monitor the milling process. Perhaps surprisingly, contamination of the FIB is not a serious problem, primarily because the total volume of material removed during micromachining is actually quite small, even though it may be substantial when compared with the size of the microcomponent that is produced by the milling process. 2. Gas-assisted etching is possible by injecting small amounts of an active gas through a capillary needle placed close to the sample surface. During ion beam etching, the gas reacts with the sample surface to form volatile species which are then pumped away by the vacuum system. This is particularly useful if the sample contains atomic species with low vapour pressures, since such materials tend to redeposit on or around the sample. Some materials such as copper, tend to ion mill anisotropically resulting in a rough surface finish. Ion milling can then be improved using an appropriate etchant gas that yields a planar, smooth surface. 3. Gas deposition of thin layers is performed by admitting a reactant gas through the capillary needle. On exposure to either the ion or the electron beam the gas reacts at the sample surface (compare chemical vapour deposition, CVD). The deposition only occurs under the electron or ion beam probe, so that quite complicated shapes can be selectively deposited with the help of a computer control program to manipulate the specimen stage and determine the beam scans. In the course of ion beam assisted deposition, both the deposit and specimen may be partially milled during operation. Even so, ion beam assisted deposition is significantly faster than electron beam assisted deposition. 4. Cross-sectioning a sample is when the ion beam is used to prepare a section either for subsequent SEM characterization, or for transfer to another experimental platform (Figure 5.37).This method, when employed on a dual-beam system, allows the user to scan the surface in SEM mode and select a region for sectioning before examining the

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Figure 5.37 Secondary electron scanning microscope micrograph of a cross-section from an ion-milled microelectronic device. A ‘FIB Box’ has been cut into the surface of the specimen.

sub-surface morphology, all without removing the sample. This approach can be used to prepare specimens for TEM (as discussed in Section 5.8.5). 5. Serial sectioning can be done by repeatedly micromachining away layers from a solid sample and then assembling a composite three-dimensional image from the twodimensional image data recorded from each of the layers separately. This process is discussed in more detail in Section 5.8.6. Micromilling using FIB technology is possible to near-nanometre accuracy and has been especially successful in preparing sub-micrometre samples for mechanical testing and components for microelectronic mechanical systems (MEMS). Fabrication for microelectronic device development is also often dependent on FIB technology. In the microelectronics industry, circuit editing is a major task that uses the FIB to cut through metallization lines between the individual device components of a microelectronic circuit, or to deposit new metallic conductor lines (usually tungsten) between devices that were previously unconnected. Repair of lithography masks is also often possible in the FIB beam. With the dual-beam FIB, the options for fabrication and testing of sub-micrometre components is limited only by the initiative of the user (Figure 5.38). Gas-assisted etching and deposition rely on gas injection from a capillary tube sited close to the sample surface (Figure 5.39). Many systems are designed with two or more capillary

Scanning Electron Microscopy 309

Figure 5.38 Secondary electron scanning micrograph of a micrometre-scale ‘etching’ showing the authors in a good mood. The specimen was prepared by FIB milling of a platinum-coated surface using data from a bit-mapped digital photograph.

Figure 5.39 Scanning ion generated secondary electron micrograph of a TEM specimen mount placed adjacent to a gas injector. This assembly is used to admit metallo-organic or etch gases across the sample surface. A TEM specimen attached to the end of a tungsten nanomanipulator needle is just visible.

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Microstructural Characterization of Materials

gas injectors, allowing CVD (or etching) reactions to be performed with different gases and hence the deposition of components having different compositions. The processes described above require assemblies that go beyond the dual-beam combination of ion beam and electron beam columns. The gas injectors have to be carefully manoeuvred above the specimen, without coming into contact with the specimen’s surface. Several stages are needed to mill a thin-film transmission electron microscope specimen from a sample in the FIB. One option uses a nanomanipulator consisting of a sharp tungsten needle mounted at the end of a piezoelectric drive (Figure 5.39). Computer control of the specimen location on the specimen stage is critical. The stepping motors that are commonly used for SEM stages are not sufficiently accurate. Piezoelectric-driven sample stages are preferred, and computer-mapped positioning of the sample, on the stage, is essential to avoid driving the specimen into the very expensive detectors and accessories. 5.8.5

TEM Specimen Preparation

The use of FIB technology to prepare thin-film sections for TEM has proved revolutionary for many laboratories engaged in high resolution microstructural studies. One needs to keep in mind that, as in conventional ion milling, FIB milling introduces beam damage and gallium ion implantation. The artifacts associated with specimen preparation must be understood in order to interpret the microstructure seen in the thin-film section. The major advantage of FIB for TEM thin-film preparation is that a specific region can be pre-selected from a component before ion milling the specimen. If a dual-beam system is used, then the quality of the TEM specimen can be checked using the scanning transmission electron microscope mode (Section 4.7) in the dual-beam system. Two methods commonly used for preparing thin sections are discussed below. 5.8.5.1 The H-bar Method. This method has the advantage that it does not require a dualbeam FIB system, although it is easier to implement in a dual-beam FIB. The process begins with the mechanical slicing and polishing of a cross-section of the material of interest. Mechanical reduction of the sample thickness by grinding saves time in the FIB, and is always a good idea. After mounting the mechanically thinned specimen in the FIB, a protective coating, such as platinum, should be deposited on the surface in the region to be thinned (Figure 5.40). This coating is to prevent high energy ion beam surface damage and is critical if the surface region of the specimen is to be characterized by TEM. In general, it is better to first deposit the protective coating using the electron beam, and then only later micromill with the ion beam. This will avoid damage to the surface region during ion beam deposition. The ion beam is now rastered over two regions of the specimen, separated by a 1 mm thick region in the centre (the beam of the H-bar) (Figure 5.40). In this way the section is thinned until a 1 mm lamella is left at the centre of the chosen region, supported by the relatively thick material at the edges. Final thinning of the lamella is then done using a much lower ion beam current, until the section is thin enough for TEM. In a standard FIB this is judged by measurement of the lamella thickness from a secondary electron image made using a rapid ion beam scan. In a dual-beam system the milling of the thin section can be completed while an electron beam is scanned independently over the sample surface. Forwardscattered electron images acquired using a STEM detector (Section 4.7), are then

Scanning Electron Microscopy 311

Incident Ions

In ci de nt El ec tr on s

Pt

Figure 5.40 Schematic illustration of the H-bar method for transmission electron microscope specimen preparation. Prior to ion milling, a thin platinum coating should be deposited on the surface of the sample in order to protect the specimen. Two ‘FIB boxes’ are cut either side of the lamella with the FIB. The arrow indicates the optic axis of the incident electron beam when the specimen is subsequently inserted in the transmission electron microscope.

used to determine if the specimen is sufficiently thin. High-angle annular dark-field (HAADF) STEM (Section 4.7) is an even better indicator of the success of the preparation process, since regions of high average atomic number will appear in dark contrast when the specimen is thick, but then appear brighter when the specimen is thin enough for the electron beam to be transmitted. A final ion beam ‘polish’ at a low ion energy (2 kV) is an effective method of removing surface layers that have been damaged by the previous milling stages and eliminating any surface contamination. An example of an H-bar specimen is shown in Figure 5.41. 5.8.5.2 The Lift-out method. The lift-out method is more precise than the H-bar method, since the exact sample region can be selected from which a thin-film specimen is to be prepared without the need for preliminary mechanical thinning. However, this method does require a dual-beam FIB that is equipped with a nanomanipulator. The sample is first viewed using the SEM facility in the dual-beam instrument, and a region of the sample is selected for thinning. As in the H-bar method, a protective coating should be deposited on the sample surface over the region from which the specimen is to be

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Microstructural Characterization of Materials

Figure 5.41 Secondary electron SEM micrograph of an H-bar specimen prepared using a dualbeam FIB. Prior to ion milling, recognition markers were ion milled into the platinum surface. These markers allow automatic image recognition by the computer control software during the milling process. The highly anisotropic milling adjacent to the lamella is termed the ‘curtaining effect’, and is associated with rapid removal of material at high beam energies and incident angles in these regions.

cut, in order to avoid surface damage by the high energy ions. Two depressions, or ‘FIB boxes’, are milled into the sample on either side of the selected region, leaving the boxes separated by a 1 mm lamella (Figure 5.42). The boxes are not cut exactly normal to the sample surface, but have a small taper angle running from the surface to the bottom of the lamella. The lamella is then cut almost entirely free of the surface by the ion beam, in the shape of a ‘U’ (Figure 5.43). The nanomanipulator is now inserted. This consists of a sharp tungsten needle attached to piezoelectric drive motors. The more advanced manipulators are computer-controlled and their movement calibrated for the viewing direction on the SEM monitor, so that the tungsten needle can be brought gently into contact with the lamella. A gas injector is now introduced, and either platinum or tungsten is deposited by electron or ion beam excitation to reinforce the point of contact of the tungsten needle with the thin-film lamella (Figure 5.44). The lamella is then completely separated from the substrate by the ion beam, and transferred to a TEM specimen mount that is located within the vacuum chamber but elsewhere on the stage. The TEM specimen mount, (or ‘grid’), is usually of a 3 mm diameter, half-ring (Figure 5.45). The lamella is gently moved next to the grid, and platinum or tungsten is now deposited to join the lamella to the grid. Ion milling is then used to cut the join between the sample and the nanomanipulator. Final ion thinning of the 1 mm thick lamella is performed to reach the required specimen thickness, exactly as for the H-bar technique. The

Scanning Electron Microscopy 313

Pt

Figure 5.42 ‘FIB boxes’ cut into the surface of a sample, in the first stage of the lift-out TEM specimen preparation procedure.

U-Cut Pt

Figure 5.43 A ‘U-Cut’ milled into the lamella that almost entirely separates the lamella from the substrate.

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Figure 5.44 The nanomanipulator is attached to the surface of the lamella TEM specimen by depositing a small amount of platinum or tungsten at the point of contact.

ability to tilt the TEM specimen towards the ion-beam and monitor the microstructure using STEM imaging in the electron beam makes this rather complex process completely feasible, flexible and efficient. 5.8.6

Serial Sectioning

A final option, that has only recently begun to make an impact in materials characterization, is the serial sectioning of microstructural samples using the dual-beam FIB. The concept of serial sectioning dates back to the early days of optical metallography (the first half of the twentieth century). Repeated mechanical grinding and polishing was employed to prepare a series of micrographs as a function of the depth of material removed from the sample. This process was slow, inefficient and inaccurate. Just as serious, the sectioning interval was seldom less than 20 mm. The results seldom justified the effort. While serial sectioning in the FIB is essentially the same concept, the sectioning accuracy is in the range of 10–20 nm and the full resolution of the secondary electron scanning image is now available to record the results. The ion beam is used to mill and polish the sections, while either the ion or electron beam is used to acquire scanning electron micrographs of the same area as a function of depth. The process is still time-consuming and is confined to a region of the

Scanning Electron Microscopy 315

Figure 5.45 Schematic drawing of a 3 mm diameter half-ring, used for mounting a thin TEM specimen lamella in the lift-out method.

sample surface only a few square micrometres in area, but the information gained from the serial sectioning of a complex, solid-state device can be impressive. The FIB technology removes layers from a solid sample whose thickness can be controlled from several tenths of a micrometre to just a few nanometres (Figure 5.46) Possible applications in materials science and engineering could include: 1. the removal of successive layers from a microelectronic device to reconstruct a threedimensional image from the two-dimensional projected images; 2. determining the connectivity and contiguity in a mesoporous catalyst substrate; 3. quantitative analysis of the branching morphology of dendritic cast structures; 4. analysing the interconnectivity of the phases in a spinodal alloy.

Summary In the scanning electron microscope a high energy electron beam is focused into a fine probe that is inelastically scattered when it strikes the surface of a solid sample. The inelastically scattered electrons generate several signals from the sample that can be collected and amplified. An image is formed by scanning the probe beam across the sample surface in a digitized television raster and displaying one or more of the collected signals on a monitor that has the same time-base as the probe scan. The most commonly used signal is from secondary electrons, but characteristic X-rays, high energy backscattered electrons, visible cathodoluminescence and the net specimen current have all been used to acquire microstructural information from samples examined in the scanning electron microscope. The increasing availability of field emission sources for the electron beam has greatly improved the performance of both TEM and SEM. In the scanning electron microscope the beam is focused by an electromagnetic probe lens to a diameter that may be as little as 2 nm. However, the probe current decreases rapidly as the probe diameter is reduced, and some signals used to acquire microstructural data require much larger probe diameters (up to 1 mm). This is especially the case for the

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Figure 5.46 Reconstructed three-dimensional morphology of a dendritic microstructure in a Pb–Sn alloy, produced by serial sectioning in a FIB. From J. Alkemper and P.W. Voorhees, ThreeDimensional Characterization of Dendritic Microstructures, Acta Materialia, 49(5), 897–902, 2001. Copyright (2001), with permission from Elsevier.

characteristic X-rays coming from the individual chemical constituents of a sample. These are excited by the incident beam with rather low efficiency, since the excitation crosssection is small. The resolution in an X-ray image is not only limited by the photon counting statistics, but also by the size of the excitation volume beneath the sample surface. The characteristic X-ray signal is an important source of information for the quantitative determination of the microchemistry of the sample (Chapter 6), but can also be used simply to demonstrate the presence of one or other constituent in any specific region of the sample. The characteristic X-rays are generated in a volume of material beneath the probe that is of the order of 1 mm in diameter, corresponding roughly to the depth of penetration of the energetic electrons into the sample and their lateral dispersion by inelastic scattering. The X-ray signal can be displayed in three distinct formats: 1. An X-ray spectrum, in which the intensity of the signal from a selected region is displayed as a function of either the X-ray energy or its wavelength. 2. An X-ray line-scan, in which the intensity of the characteristic signal from one or more elements are collected and displayed as a function of probe position along the scan line. (An example would be a scan across an interface or a second-phase particle.) 3. The X-ray data can be viewed as an elemental image map in which all photons arriving within a given energy window are displayed and recorded as colour-coded dots whose position in the image is correlated with the position of the beam at the time of detection.

Scanning Electron Microscopy 317

High-energy backscattered electrons are useful because the intensity of the backscattered signal reflects the mass density or average atomic number of the sample and not just the surface topology. The brighter regions are therefore a clear indication of denser material with a higher average atomic number. Nevertheless, the secondary electron signal is very often the most useful of the several signals that may be collected. There are two reasons for this. First, the number of secondary electrons emitted per incident high energy electron exceeds by orders of magnitude the electron current in the primary beam, and, secondly, these secondary electrons can be collected with close to 100 % efficiency. The secondary electron signal originates in the surface layers of the specimen, since these low energy electrons have a very limited mean free path in the solid sample. The secondary electrons can be generated both by the incident beam, as it enters the sample and by backscattered, high energy electrons that are leaving the sample. The secondary electron resolution need not necessarily be degraded by inelastic scattering of the primary beam and because of the high secondary electron flux, there are no significant statistical limitations on the resolution in the secondary electron image. It follows that the ultimate resolution in a secondary electron image should be determined primarily by the ability to focus the probe beam, and is typically 2 nm. If the secondary electrons generated by the flux of backscattered electrons are also collected, then the resolution will be degraded to of the order of 20–50 nm. Other types of signal are also available in the scanning electron microscope: the beam current to the sample or cathodoluminescence, but these are usually of secondary importance. Specimen preparation for the scanning electron microscope is straightforward, although it must be remembered that less stable specimens, such as polymers and biological tissues, may be degraded by the high energy electron beam and give rise to contamination of both the sample and the microscope column. For nonconductive specimens a conductive coating is commonly required to prevent charging, unless very low beam energies are used. Such coatings often enhance the image contrast. In general, contrast in the scanning electron microscope may be associated with both surface topology and variations in mass density or atomic number. It is not always easy to separate these two sources of contrast. Insufficient use is made of stereoscopic analysis. By tilting the sample about a known axis and recording two images at different tilt angles, a stereo image can be observed in which the depth distribution of the microstructural features is clearly evident. In addition, accurate measurements of the lateral displacement of features in the two components of the stereo image, that is, parallax permits their displacement along the optic axis, normal to the plane of observation, to be determined. Stereo imaging in the scanning electron microscope allows the surface of rough samples to be viewed in three dimensions. The backscattered electrons also contain diffraction information on the crystallography of the sample. This information can be extracted by using a charge-coupled device (CCD) camera and the electron backscatter diffraction (EBSD) pattern interpreted automatically using appropriate computer software. In a polished, polycrystalline sample the orientations of the different grains can be colour-coded and displayed in a digitized image of the grain morphology, a mode termed orientation imaging microscopy (OIM). Finally, a focused ion beam FIB can also be used as the specimen probe in the FIB microscope. In this instrument the ion beam from a liquid-metal field-evaporation source can be used, either to generate a secondary electron signal, or to micromachine the specimen surface. If the incident ion beam is aligned along a zone axis of a crystal, the ions may ‘channel’ beneath the surface. Channelling greatly reduces the secondary electron yield from the

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surface, resulting in strong crystallographic contrast in the secondary electron image generated by the ion beam that is observed. Such contrast is absent if a high energy electron beam is used to excite the secondary electron signal. Dual-beam FIB instruments are available that incorporate, in separate optical columns, a field emission electron source in addition to the ion beam source. Such an instrument can be used to micromachine thin-film transmission samples in situ and examine them in the same FIB instrument operated in a scanning transmission electron microscope mode.

Bibliography 1. J.I. Goldstein and H. Yakowitz(eds), Practical Scanning Electron Microscopy, Plenum Press, London, (1975). 2. O.C. Wells, Scanning Electron Microscopy, McGraw-Hill Book Company, London, (1974). 3. J.J. Hren, J.I. Goldstein and D.C. Joy (eds), Introduction to Analytical Electron Microscopy, Plenum Press, London, (1979). 4. L. Reimer, Scanning Electron Microscopy: Physics of Image Formation and Microanalysis, Springer, Berlin, (1998). 5. L. Reimer, Image Formation in Low-Voltage Scanning Electron Microscopy, SPIE Optical Engineering Press, Bellingham, WA, (1993).

Worked Examples Once again we demonstrate the techniques we have discussed. We use SEM for two quite different material systems: polycrystalline alumina and a thin film of aluminium deposited by CVD on a TiN/Ti/SiO2-coated silicon substrate. We also examine a wire-bond from a microelectronic device using FIB. The first example is polycrystalline alumina. As always, it is important to define the questions we want answered before preparing our specimens or specifying the characterization techniques. For our alumina we wish to check for residual porosity, determine the grain size, and establish that the grain boundaries are free from secondary phases. The presence of residual porosity and grain size measurements can best be determined by SEM. However, it is difficult to detect small secondary phase particles or glass at grain boundaries by SEM, so TEM would be required for this (Chapter 4). We can prepare SEM specimens from alumina in two ways: either by mechanical polishing, down to a sub-micrometre diamond grit polish, followed by thermal etching to form grain boundary grooves on the polished surface, or by breaking a suitable mechanical specimen, so that we can examine the fracture surface (and learn something about the features determining the fracture strength). Alumina is an electrical insulator, so we would usually coat the surface of the specimen with a conducting layer prior to SEM. However, we have access to a low voltage SEM with a field emission gun, so we can use a low accelerating voltage to minimize charging of the insulating surface.

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Figure 5.47 shows SEM micrographs of a polished and thermally etched alumina sample. Two micrographs of the same region are shown. The first was recorded using secondary electrons at an accelerating voltage of 5 kV, while the second was recorded at 20 kV, also using secondary electrons. The higher accelerating voltage leads to unstable surface charging of the specimen, which affects both the secondary electron emission coefficient and the trajectories of the secondary electrons that are collected. The image is blurred, and unsatisfactory for morphological analysis. When using 5 kVelectrons, the incident electron current is compensated by the exiting secondary and backscattered electrons, so charging is reduced, and the image is free of these distortions. Secondary electrons are generated throughout the depth of penetration of the incident beam, but lowering the accelerating voltage helps to ensure that the secondary electrons are only generated in regions very close to the surface. These have been termed type 1 secondary electrons (SE1). With a field emission gun and a specialized secondary electron detector (discussed below), it is possible to detect the relatively few SE1 electrons released

Figure 5.47 SEM micrographs of thermally etched, polycrystalline alumina, recorded using 5 kVelectrons to minimize charging (a), and then 20 kVelectrons (b). No conductive coating was present on the specimen.

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Figure 5.48 SEM micrograph clearly showing the fine surface facets on a thermally etched alumina. The image was recorded using a field emission gun and a specialized secondary electron detector. Most of the contrast comes from SE1 electrons generated by the primary beam.

at low accelerating voltages (Figure 5.48). The resolution in Figure 5.48 is sufficient to resolve very fine surface facets formed during thermal etching. Secondary electrons are generated from the near-surface region by the primary beam (SE1 electrons), but they can also escape from the sub-surface if the voltage is more than a few hundred volts (SE2 electrons). In addition, a secondary electron current is generated by the backscattered electrons in a zone well beyond the diameter of the incident probe, as well as by stray electrons striking the microscope chamber, the column or the specimen holder (SE3) electrons. The secondary electron detector is usually located within the chamber, above and to the side of the specimen, so the SE3 electrons can contribute appreciably to signal noise. An annular secondary electron detector can be placed above the specimen and in the microscope column to screen out SE3 electrons, and improve the contrast in the image. However, since the detector is above the specimen, rather than to the side, we lose some of the three-dimensional shadow effect that is associated with the surface topology and is observed using the conventional secondary electron detector (Figure 5.49). Some scanning electron microscopes allow on-line mixing of signals from both types of detector in order to optimize both resolution and contrast. Figure 5.50 shows a 5 kV SE1 micrograph of the thermally etched alumina specimen. Grain boundaries are easily identified, since the grain boundary grooves emit fewer secondary electrons than the groove shoulders. It is straightforward to determine the average grain size to any required accuracy (see Chapter 9). Figure 5.51 shows a fracture surface of the same grade of alumina. Two fracture modes are visible: intergranular (grain boundary) fracture and transgranular (cleavage) fracture. Transgranular fracture is characterized by the appearance of planar, crystallographic

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Figure 5.49 Fracture surface of alumina showing cleavage facets and intergranular failure. The images were recorded using both a conventional secondary electron detector (a) and an in-lens secondary electron detector (b). More detailed features of the fracture surface are visible in the image from the in-lens detector.

cleavage planes, seen in sharp contrast. Residual porosity is also readily visible, both at the grain boundaries, and within the grains. Now we return to the CVD aluminium system. For this example we wish to characterize: 1. the initial deposition conditions, when the first aluminium nuclei form on the surface of the TiN; 2. the morphology of the final aluminium film; 3. the morphologies of the underlying TiN and titanium films. Preparation of a suitable sample for SEM is straightforward, since the sample is a metallic conductor and we only need to investigate the surface of the sample. However, the aluminium nuclei are very small, requiring the highest possible resolution from a low-voltage scanning electron microscope, generating secondary electrons only from the surface layer. Contamination of the surface can be a critical factor limiting image contrast and resolution. Figure 5.52 shows a secondary electron SEM image of the surface imaged using an in-lens secondary

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Figure 5.50 Secondary electron SEM micrograph of a thermally etched alumina, recorded at an accelerating voltage of 5 kV.

Figure 5.51 Secondary electron SEM micrograph of the fracture surface of alumina, showing transgranular fracture (cleavage), intergranular fracture, and pores, both at grain boundaries and within the alumina grains.

electron detector. The central region of Figure 5.52 was exposed to the electron beam for approximately 30 s, and then the magnification was reduced and the micrograph was recorded. Thus the outer region of Figure 5.52 was recorded immediately after focusing on the area of interest, reducing the contamination in this area to a minimum. The build-up of contamination

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Figure 5.52 Secondary electron (in-lens detector) SEM micrograph of aluminium on TiN, recorded with a 1 kV incident electron beam. The beam was first focused on the central region of the micrograph for approximately 30 s, and then the magnification was reduced and the micrograph was recorded immediately. The build-up of contamination in the central region strongly affects the contrast and resolution of the image.

in the central region of the image is evident. The reason we need an uncontaminated, clean specimen is clear. There are three options commonly used to clean samples: 1. plasma etching before inserting the specimen in the microscope; 2. heating the specimen above 100  C, to drive off absorbed gases and water; 3. swabbing the sample with a volatile organic solvent (acetone, ethanol or methanol) to remove grease and oil. (This last option will only remove soluble contamination.) Since any specimen heating of the surface by the electron beam might cause changes in the aluminium morphology, we should acquire the images as quickly as possible to minimize the electron dose. The final, thick aluminium film has a different morphology, and SEM shows that open voids have formed during the deposition process (Figure 5.53). These voids are process defects which change both the electrical and optical properties of the aluminium. SEM can also be used to examine specimens in cross-section. Figure 5.54 is a secondary electron SEM micrograph of a cross-section from a cleaved sample. The aluminium film is clearly visible at the surface of the TiN/Ti/SiO2/Si stack. Open voids are also just visible, as well as possible closed voids that could not have been detected from the plan-view of the specimen surface. Further details of the film morphology would require TEM (discussed in Chapter 4). Now for another example: wire-bonding is a commercial process used to connect a semiconductor device to the main board on which it is mounted. The wires are usually gold, although copper wires are now being commercially introduced. The bonding process commonly uses a 20 mm diameter wire. The wire is fed through an alumina capillary tube, and the end of the wire is melted using a spark generated across an electrode gap. Figure 5.55 is an ion-induced secondary electron micrograph taken in a dual beam FIB and shows the melted and solidified end of a gold wire, just prior to bonding to the connecting pad on the

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Figure 5.53 Secondary electron SEM micrograph of a thick aluminium film, showing a relatively rough surface and some open voids.

device. Ion channelling orientation contrast reveals the very different grain morphologies in the wire ball and the parent wire. FIB can also be used, to prepare H-bar specimens from these thin wires for SEM and TEM (Figure 5.56). The anisotropic shape of the gold grains in the wire is immediately evident on account of the ion channelling contrast.

Figure 5.54 Secondary electron SEM micrograph of a cross-section prepared by cleaving the silicon wafer together with the Al/TiN/Ti stack of films. The ductile aluminium shows considerable plastic deformation compared with the brittle underlying layers and a void is visible in the aluminium film.

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Figure 5.55 Ion-induced secondary electron micrograph of a gold wire used for wire-bonding, showing orientation contrast associated with ion-channelling.

A more serious challenge is to characterize the gold wire after it has been bonded to the connecting pad on the device. Before the FIB technology was introduced, the only way to cross-section such a join would have been to encapsulate the entire device (Figure 5.57) in a moulding compound, and then to mechanically grind and polish the mounted sample. A talented experimentalist with steady hands might hope to stop the polishing process when the central region of the 40 mm diameter bond was sectioned. Not a very promising

Figure 5.56 Ion-induced secondary electron micrograph of an H-bar section ion milled from a gold wire. Note the magnification scale.

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Figure 5.57 device.

Low magnification SEM micrograph of a series of wire bonds on a microelectronic

procedure! With a dual-beam FIB, the ion beam is used to cut away any remaining length of wire and then section the bond itself. Figure 5.58 shows an example from a copper wire connected to an aluminium pad on a device. The bond has been sectioned by FIB for SEM characterization. There is no reason to restrict characterization to SEM, and the FIB can be used to prepare a thin-film TEM specimen from the same wire-bond. Figure 5.59 shows a thin lamella section after two neighbouring ‘FIB boxes’ have been ion milled and a U-cut made to almost free the lamella from the surface. The sample is then attached to the nanomanipulator, cut free from the substrate and attached to the TEM specimen mount (Figure 5.60). After freeing the nanomanipulator from the sample, the ion beam can be used to thin the lamella until it is transparent to an electron beam (Figure 5.61). At this point, the sample can be characterized in the dual-beam FIB, using STEM, or transferred to the transmission electron microscope.

Problems 5.1. How does the working distance of the probe lens from the sample surface affect the minimum probe size in a scanning microscope? 5.2. Given that the signal collected in a scanning system is determined by inelastic scattering and secondary excitation processes, discuss the effect that these probe– specimen interactions have on the scanning resolution. Compare especially characteristic X-ray and secondary electron excitation.

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Figure 5.58 (a) Secondary electron SEM micrograph of a copper wire bonded to an aluminium pad. In (b) the wire has been cut and removed using the FIB ion beam, and the bond has been partially sectioned to expose the interface. In (c) the cross-section of the bond is exposed, and SEM can be used to characterize any defects or intermetallic compounds that are present in the interface region.

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Figure 5.59 (a) Ion-induced and (b) electron-induced secondary electron micrographs of a thin lamella attached to the substrate, prior to lift-out.

5.3. Both surface topology and local mass density can influence the scanning electron image contrast. How would you expect the beam voltage to affect the atomic number (mass density) contrast in a backscattered electron image? 5.4. The secondary electron image is that most commonly used for routine examination in the scanning electron microscope. Why? 5.5. Discuss some ways in which samples that are sensitive to degradation in a vacuum or under an electron beam could nevertheless be imaged in the scanning electron microscope. 5.6. What minimum angle of tilt should be used to distinguish two features by stereoimaging when they are separated by a vertical distance h and the lateral resolution is d?

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Figure 5.60 Ion-induced secondary electron micrographs: (a) the nanomanipulator attached to the lamella with a platinum join; (b) immediately after the lamella has been fully separated from the substrate; (c) the lamella is then moved next to a TEM sample mount; (d) the lamella is attached to the TEM mount by platinum deposition.

5.7. How does the depth of field of an optical microscope compare with that of a scanning electron microscope at a magnification of ·200 when the scanning electron microscope convergence angle 2a ¼ 4 · 10
2 rad? How would a change in the working distance affect the depth of field and probe size in the scanning electron microscope? Explain any assumptions that you make. 5.8. What signals are generated from a solid sample by an incident beam of high energy electrons? Which signals are used in TEM, and which are used in SEM? How does the resolution in a SEM micrograph depend on the type of signal that is collected? 5.9. Are secondary electrons or backscattered electrons to be preferred for imaging and analysing a fracture surface? Explain your reasoning. 5.10. Are secondary electrons or backscattered electrons to be preferred for imaging and analysing variations in the local chemical distribution on a polished sample containing aluminium and gold? Give your reasons. 5.11. Are secondary electrons or backscattered electrons to be preferred for imaging and analysing the local chemical distribution on a polished sample composed of aluminium and magnesium? Explain your reasoning.

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Figure 5.61 (a) Secondary electron SEM micrograph (5 kV incident electrons) of the thinned lamella. (b) Bright-field STEM micrograph of the microstructure recorded in the dual-beam FIB (30 kV incident electrons).

5.12. Assuming that the standard scanning electron microscope viewing screen is 10 · 10 cm2 and based on 256 · 256 pixels, what magnification is required to resolve round particles with a radius of 10 nm? (Assume that reasonable resolution is achieved when an image feature covers 3 · 3 pixels.) 5.13. Sketch the intensity distribution for secondary electrons that are collected by scanning the incident beam across a protrusion on a flat surface. Assuming the same detector geometry and feature size, sketch the intensity distribution expected when the incident beam is scanned across an indentation on the same surface. Discuss the dependence of the contrast on the location of the detector and the dimensions of the features.

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Figure 5.62 (a) Secondary electron (SE) and (b) backscattered electron (BSE) micrographs of Ybased oxides in an AIN matrix.

5.14. Given an aluminium sample that contains precipitates of Al–Au intermetallic compounds, how should the backscattered electron contrast change when the detector is located (a) at an angle of 170 and (b) an angle of 105 relative to the incident electron beam? Assume a flat surface normal to the incident electron beam. 5.15. Sketch the expected secondary electron yield as a function of the accelerating voltage (electron beam energy) for a nonconducting specimen. Indicate the accelerating voltages that will minimize electrostatic charging of the specimen surface. 5.16. How is the magnification controlled in SEM? What image distortions might be expected? 5.17. In SEM, unlike TEM, the image can never rotate when the magnification is increased. Explain why. 5.18. The secondary electron and backscattered electron SEM micrographs shown in Figure 5.62 were acquired from the same area of an AION sample that contained Ybased oxide particles. Explain the difference in contrast between the two micrographs.

6 Microanalysis in Electron Microscopy We have already noted (Section 5.3.2) that a proportion of the X-rays emitted under electron excitation, together with the corresponding energy losses in the primary electron beam, are characteristic of the chemical constituents of a solid sample. These characteristic X-rays may be selected from the observed spectrum of emitted electromagnetic radiation, according to either their energy or their wavelength, and the signal distribution from the sample can be displayed in a line-scan or elemental map to provide both qualitative and quantitative information on the morphological relationship between the microstructure and the chemical composition. In this chapter we explore several ways in which the qualitative chemical information contained in the characteristic X-ray signal can be made quantitative. In all the methods we present, inelastic scattering of the probe (either electrons or X-rays) excites a signal that depends on the chemical composition of the material beneath the probe, and the challenge is to interpret this signal as quantitatively as possible. The chemical sensitivity of each method (the minimum detectable concentration of a selected constituent) will be different, as will the analytical accuracy (the errors involved in quantitative analysis). We will also be concerned with the spatial resolution obtainable for the chemical composition, both in the image plane of the sample surface and within the depth beneath the surface that is sampled by the incident high-energy probe. In this chapter we only consider those methods that are commonly available as microanalytical facilities, attached to either the transmission or the scanning electron microscope, but in Chapter 7 we will also describe some additional methods that may be used to characterize the composition of the near-surface layer, thin surface films and adsorbates. We first discuss qualitative and quantitative microanalysis using the excited, characteristic X-rays before considering the microanalytical information that can be derived from the electron energy loss spectra.

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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6.1

X-Ray Microanalysis

In X-ray microanalysis, the characteristic X-rays emitted from a sample viewed in the electron microscope (usually the scanning electron microscope, but also in the transmission electron microscope) are analysed both qualitatively and quantitatively in order to relate the local chemical composition of the sample to the morphological features visible on the sample surface. The physical basis for the excitation of X-ray emission by a high energy electron beam that is incident on a solid target has already been introduced (Section 5.3.2), but we still need to explore the factors that influence the sensitivity of microanalysis (the minimum detectable concentration of a constituent) and its accuracy (the cumulative microanalytical errors). These factors involve both the properties of the X-ray detection system and the computer software that is used to convert the collected X-ray data to a quantitative estimate of composition. They also include the specimen preparation procedures that are necessary to ensure reproducible and accurate results, the geometry of the sample surface in relation to the electron beam and detector system, together with some features of the composition distribution and microstructural morphology of the sample that can also affect the results. To simplify the discussion, we will assume that the sample has been polished but not etched, and that the specimen surface is smooth and planar, so that the only significant geometrical parameters are the angle at which the electron beam strikes the specimen surface and the angle subtended by the detection system at the sample surface. If the surface roughness is on a scale that is small compared with the diffusion depth for the incident highenergy electrons in the sample (typically <1 mm, at 20–30 kV, even for low atomic number materials), then there will be no significant errors in assuming that a polished and etched surface is planar. Fracture surfaces, however, and surfaces that have been machined, heavily corroded or etched are certainly not planar. The computer software programs for quantitative X-ray microanalysis are not intended to be used for the analysis of rough surfaces. The same applies to powder samples, fibres and grits, and the results of X-ray microanalysis on these materials must be regarded as only qualitative. In addition, all computer correction procedures for quantitative microanalysis assume that the composition of the sample in the surface region being analysed is homogeneous. It follows that quantitative microanalysis is only reliable for regions that are reasonably far from phase boundaries and in the absence of strong concentration gradients. This applies both to phase boundaries that are sensibly perpendicular to the surface being analysed and also to thin films deposited on a substrate. Even so, it is often possible to extract qualitative information on the elements present in a thin, sub-micrometre layer, and this information can often be made semi-quantitative by reducing the energy of the primary beam, so that the diffusion distance for the high energy electron probe is of the order of the film thickness. Nevertheless, the primary purpose of quantitative microanalysis is the determination of the bulk composition, albeit in a very small, micrometre-sized, volume element of the specimen. 6.1.1

Excitation of Characteristic X-Rays

As we noted in the previous chapter, the characteristic X-ray signals and the corresponding energy loss spectra (Section 5.2) that are generated by an incident beam constitute fingerprints of the local chemistry. To carry the fingerprint analogy further, qualitative

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analysis consists in identifying the origin of the print, as from the index finger or the thumb, while quantitative analysis has as its goal the positive identification of the perpetrator; a rather more difficult task. The characteristic X-rays are generated within a region of the envelope of scattered electrons for which the average electron energy exceeds the threshold energy for X-ray excitation. The volume of this region depends on the incident beam energy, the atomic number of the elements present and the mass-density of the sample. The mass-density and the beam energy determine both the diffusion distance (that is, the spread of the scattered electron beam, which is dependent on the rate of energy loss) and the range of the electrons in the sample (the thickness required to reduce the electron energy to the thermal level, kT). For example, a copper sample exposed at high beam energies will generate both K and L characteristic radiation (Figure 6.1). The L radiation will originate from a region that extends to just within the thermal energy envelope, while the K radiation is generated within a much smaller volume. At lower beam energies that are below the threshold for K excitation, only the L radiation can be excited, but since the range of these lower energy electrons is much reduced, the source of the L radiation will have a correspondingly smaller volume. In principle, the spatial resolution for the identification of copper in the sample should be better when using the L characteristic lines at low electron beam energies. However, better sensitivity and more accurate quantitative analysis is possible with K excitation at higher probe voltages, and in many cases, the spatial resolution may be sacrificed to the analytical accuracy. Increasing the energy of the incident electron beam generally increases the total X-ray signal and ensures that most, if not all, characteristic lines are excited. However, the total amount of white, background radiation or ‘Bremsstrahlung’, due to inelastic scattering events that do not involve the ejection of an inner shell electron, is also increased. If the

(a)

(b) E0> E K

E0< E K

E>EK

E>EL

E>EL

E=k T

E=k T

Figure 6.1 Schematic representation of the volume elements beneath an electron probe that generate characteristic X-ray radiation in a copper sample (a) well above and (b) below the energy threshold for K excitation.

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beam energy exceeds approximately four times the threshold energy for X-ray excitation, then the ratio of the characteristic line intensity to that of the background intensity (the signal-to-noise ratio) starts to decrease. In addition, the size of the volume element generating the signal, dependent on the diffusion distance, is also increased. This reduces the spatial discrimination, that is, the resolution, and is therefore a good reason for limiting the energy of the incident beam. Although reducing the beam energy both improves the spatial resolution and limits the background level of white radiation, as the beam energy approaches the critical excitation energy, the intensity of the characteristic line decreases dramatically. As a general rule, there is no advantage to be gained by reducing the beam energy to less than three times the excitation energy for the shortest wavelength, that is, the highest energy characteristic lines which are of interest. In fact, light element analysis, for which only characteristic lines of long wavelength are available, is best combined with analysis of the heavier, higher atomic number constituents by using the characteristic wavelength from the L or even M shells of the high atomic number elements. Before discussing the statistics of signal collection and the ‘white’ noise background in the spectrum which is associated with emission of X-rays that are not related to characteristic excitation, we should note the existence of some ‘false’ peaks associated with the method of detection of the photon energy. The first of these peaks are ‘escape’ peaks, below the principal peaks observed in the spectrum. These peaks are due to fluorescence of silicon in the detector that has been excited by high energy impinging photons. The Si-K absorption edge, at 1.74 keV, can reduce the photon energy of any incident radiation that excites the silicon in the detector by exactly this amount and can lead to a small ‘escape peak’ that is just 1.74 keV below any main, characteristic peak. Most software now available will correct automatically for the presence of an escape peak by first subtracting the background (see below) and then adding the escape counts to those registered for the main peak of the element in question. The second type of ‘false’ peak that may be important can arise when two photons arrive ‘simultaneously’ at the detector. This is only observed at very high counting rates, and the available software is set to reject ‘second’ counts that arrive before the current pulse due to an impinging photon has been cleared and registered (typically this takes >1 ms). Nevertheless, there is a finite probability that two photons may impinge on the detector sufficiently close in time to be registered as a single count of energy equal to the sum of the energies of the two impinging photons. Such events are rarely a problem. Finally, characteristic X-rays from low atomic number (low Z) constituents may appear at an energy that depends slightly on the chemical bond energy of the atom in the material, since the decay process responsible for emission involves an electron from the valence band. Carbon is an excellent example, and the carbon peaks from diamond, polymers and pyrolytic (amorphous) graphite show a clearly detectable shift in peak position. These are not ‘false’ peaks but they do make quantitative analysis more difficult. 6.1.1.1 Signal-to-Noise Ratio. Before any quantitative analysis is possible, the background ‘noise’ must be subtracted from the intensity detected at the position of a characteristic ‘line’ by fitting a function to the ‘white’ background signal (Figure 6.2) and calculating the integrated intensity difference. The simplest function is a straight line that is

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Figure 6.2 If a background correction can be calculated, then the intensity in a characteristic signal can be obtained by subtracting this background from the total number of counts registered over the full range of the detected intensity peaks. In this figure the background is calculated for a spectrum recorded from NaFe(Si2O6).

based on background counts that are summed on either side of the selected intensity peak and at a sufficient distance from the peak to ensure that the main peak does not interfere with these background measurements. A far better procedure is to use a fitted curve that includes, to a first-order approximation, corrections for the absorption edges that are associated with excitation of the main peaks in the spectrum. Such fitted curves are commonly included as the first step in computer software correction routines. The available spectral resolution in wavelength-dispersive spectrometry (WDS) is far better than that available in energy-dispersive spectrometry (EDS). In EDS the energy resolution is usually determined for MnKa radiation (5.9 keV), and is typically in the region of 140 eV for this radiation. The resolution is a monotonic function of the photon energy, but the limited energy resolution of the spectrometer, together with the potential for overlap between characteristic intensity peaks, makes it impractical to set the acceptance channel for a characteristic line to the full energy width of the peak. A satisfactory compromise is to set the width of the channel to coincide with the intensities at those energies either side of the maximum characteristic intensity that correspond to half the observed peak height, a condition termed full- width at half-maximum (FWHM). There are, unfortunately, many cases for which the characteristic signals from different elements give spectral peaks that overlap significantly. This is especially common in EDS

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spectra, but is by no means impossible for WDS results. In such cases, removing the background is only the first step in preparing the data, and it is necessary to deconvolute the overlapping peaks. The simplest way of achieving this is by assuming that the overlapping peaks are symmetrical and by using the data from the major peak to estimate the additional counts in each channel that are due to the minor peak. The errors can be large and some more accurate software algorithms are available. Peak overlap is a problem that often needs to be solved, but is seldom an insoluble problem. 6.1.1.2 Resolution and Detection Limit. The spatial resolution for microanalysis in scanning electron microscopy (SEM) ranges from 0.2 mm to several micrometres, limited primarily by the diffusion distance that determines lateral spreading of the beam and the range of the electrons. In transmission electron microscopy (TEM) (see Section 4.1.3.3) the very limited thickness of the thin-film sample limits lateral spreading of the beam by inelastic scattering, and resolutions of the order of 1nm can be obtained when a field emission source is available. In effect, the chemical spatial resolution and the concentration sensitivity are in competition, as we noted when discussing the size of the envelope that defines the volume element generating the characteristic X-ray signal. If detection sensitivity could be improved for longer wavelength X-rays, then the spatial resolution for microanalysis at lower beam energies could be better exploited. This is made possible by using a field emission gun which increases the beam current in the probe by some two orders of magnitude and can be used at incident beam energies down to below 1 kV, localizing the excitation volume to the initial probe size and film thickness. The analysis of long wavelength radiation (1 kV
1.24 nm) can also provide information on the nature of the chemical bonding, and we have already noted the shifts in peak position for carbon in different bonding states. The detection limit in X-ray microanalysis at the beam energies more commonly used (5–20 kV) is of the order of 0.5 atom %, but in some cases may be as low as 0.1% for EDS. The correction errors in quantitative analysis for a well-calibrated system are about 2 % of the measured concentration, providing the concentration exceeds a few atom per cent. However, it is important to be aware of the limitations of quantitative analysis and not to be misled by results obtained from software procedures that introduce statistical bias by normalizing the composition or assuming stoichiometry. This is a common procedure when data for one component in a compound are unavailable. 6.1.2

Detection of Characteristic X-Rays

The detection of characteristic X-rays requires both good discrimination and high detection efficiency. To achieve 100 % detection efficiency, every photon emitted by the sample would have to be recorded. This is impossible for two reasons: first, the detector always subtends a limited solid angle at the sample, and only those photons that reach the detector have any chance of being recorded. Secondly, the detector itself has a limited detection efficiency that depends on the energy (that is, the wavelength) of the incident photon. Figure 6.3(a) shows the structure of a typical lithium-drifted silicon detector and Figure 6.3(b) and (c) shows the detection efficiency as a function of the energy of the incident photon and the pulseprocessing time. At low energies the photon may be absorbed by a beryllium foil window

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P-Type Li-Drifted N-Type Region IntrinsicRegion Region

Holes

Electrons FET Pre-Amplifier

(a)

ThinWindow

-500V

Au Contact ~20 nm

Au Contact ~200 nm 100 000

(b)

80 000

Counts

Resolution FWHM (eV)

200

150

100

(c)

Mg Kα Al Kα Si Kα

60 000

40 000 50

20 000

0

0 0

2

4

6

Energy (keV)

8

10

1.0

1.2

1.4

1.6

1.8

2.0

Energy (keV)

Figure 6.3 (a) Schematic drawing of the Si-based Li-doped (drifted) EDS detector. Photons passing through the thin window and gold contact interact with the silicon, generating both holes and electrons. The accumulated charge is then measured by a field-effect transistor (FET) pre-amplifier, which feeds the signal into the EDS pulse processor. (b) The energy resolution of a typical EDS detector as a function of energy. (c) Short process times (red) degrade the peak resolution compared with long process times (black).

that protects the detector form contamination in the microscope. Better detection performance is obtained using thinner but more fragile, polymer-based windows. At very high photon energies the photon may pass through the detector (which is typically 3 mm thick) without being absorbed and detected.

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Perfect discrimination, or energy resolution, would require that the wavelength of every photon which is detected should be accurately determined and separated from other activation events that are associated with photons of different wavelength. This also is impossible. The background radiation that overlaps a characteristic peak implies some uncertainty in the source of the photon, while the detector can only identify photon energy to an accuracy that is determined by its spectral resolution. This spectral resolution is roughly proportional to the square root of the incident photon energy and is usually quoted in the literature as the pulse width at FWHM for MnKa radiation at 5.895 pkeV. ffiffiffiffiffiffiffiffiffiffiffiffiThe energy resolution for any other characteristic line energy is then DEE  DE5:9 E=5:9, where DE is in eV and E is in keV. 6.1.2.1 Wavelength-Dispersive Spectrometers. Far better discrimination of the X-ray signal emitted by the region excited under the electron probe is achieved in a wavelengthdispersive spectrometer. In this system (Figure 6.4), a series of bent single crystals of different lattice spacings covers the range of wavelengths that are of interest. The wavelengths within the range of each spectrometer crystal are scanned by rotating the crystal to scan the Bragg angle 2y and synchronously moving the detector, while keeping the position of the bent crystal fixed. Note that the radius of curvature of the bent crystal is set to an ‘average’ diameter for the focusing circle. Since energy discrimination is not an issue, an argon gas proportional counter can be used to collect the photons selected by the diffracting crystal. These gas counters have sufficient energy discrimination to ensure that second- and higher-order reflections (wavelengths that are multiples of that selected) are rejected. At the same time, the width of the current pulse generated by a photon in the gas Bent and Polished Crystal (radius=2R)

Focal Circle

R

Receiving Slits

Specimen X-Ray Detectors

Figure 6.4 The wavelength dispersive spectrometer is usually a semi-focusing system in which a curved crystal reflects the radiation emitted from the specimen surface over a specific solid collection angle. The distance R from the source to the crystal, the average take-off angle for the X-rays and the Bragg diffraction angle 2y for the wavelength of interest are adjustable. Note that this figure shows a fully focusing geometry.

Microanalysis in Electron Microscopy 341

proportional counter is much narrower than can be achieved in the Si(Li) solid-state detectors, so that the ‘dead time’ of the gas proportional counter is not a limitation on the counting rate. In the WDS geometry, the angle at which the X-rays are collected from the sample is fixed. The angle subtended at the collecting crystal will vary with 2y, while the diameter of the focusing circle will change (hence the system is designated ‘semi-focusing’, since the radius of the collecting crystal is constant and only ‘fully focusing’ for one specific focusing circle). There is an additional reason why the system can only be ‘semi-focusing’, and that is because the X-rays are generated over a finite depth of the sample, which depends on both the incident electron energy and the sample density. This effect is not usually significant; however, it does depend on the position of the plane of the spectrometer with respect to the axis of the microscope. The bent diffractometer crystal focuses X-rays which originate from an elongated region parallel to the axis of bending of the crystal, and is therefore perpendicular to the plane of the spectrometer. If the spectrometer is horizontal, that is perpendicular to the microscope column, then the focus will not be sensitive to the position of the sample along the axis, but now the area in focus in the sample plane will be rather small, and this could be a problem for areal analysis at low magnifications. If the spectrometer is vertical (parallel to the microscope column), then the sensitivity to vertical displacement of the sample will be much greater, but there will now be an elongated area of the sample (a few millimetres) that is in focus. Some spectrometers are mounted at an angle to the microscope column to achieve a compromise of relative insensitivity to z-axis displacements of the specimen combined with good areal coverage for analyses performed at low magnifications. An important consequence of the geometrical discrimination provided by the WDS system is that all the characteristic emission peaks must be scanned sequentially and the only way to record more than one characteristic line at a time is by using more than one spectrometer. In practice, a wavelength-dispersive spectrometer can be programmed to scan through a series of characteristic peaks and settings for background measurement, in order to maximize the efficiency of data acquisition. Nevertheless, the improved discrimination, usually at least an order of magnitude in energy resolution when using WDS instead of EDS carries a heavy penalty in data collection time associated with the sequential (WDS) rather than a parallel (EDS) data collection mode. 6.1.2.2 Energy-Dispersive Spectrometry. In the energy-dispersive spectrometer the pulse height recorded for an incident photon by a detector is directly proportional to the energy of the photon responsible for the pulse. The detectors used for this purpose are lithium-drifted silicon, Si(Li), solid-state detectors. An incident photon absorbed by the silicon crystal creates ionization events in the active thickness of the detector. The total charge developed is proportional to the incident photon energy and is detected as a current pulse that is shaped, digitized and counted in a multi-channel analyser. There are two problems associated with EDS, as opposed to WDS systems. The first concerns their relatively poor energy resolution. Good WDS systems have better energy resolution, by at least an order of magnitude, especially for the detection of longwavelength, low-energy radiation. The better energy resolution is also important in cases where the characteristic lines from different elements overlap. WDS systems can also resolve the multiple lines of L and M spectra unambiguously, improving the accuracy of quantitative microanalysis using L and M radiation (Figure 6.5).

342

Microstructural Characterization of Materials Ta

W

Mα

Mα

W Ta

Mβ

Mβ

Re Mα Re Mβ

Si Kα

1.6 keV

2.0 keV

Figure 6.5 Resolution of M-lines in a wavelength-dispersive spectrum of a super alloy. (Courtesy of Oxford Instruments).

The second problem occurs with low-energy (long-wavelength) photon detection (Figure 6.6), that requires either a windowless detector, or a detector protected from the rest of the system by only a very thin and fragile window. Care is then needed to ensure that the detector retains its long-wavelength sensitivity, and is not degraded nor contaminated in the vacuum environment of the microscope. All solid–state detectors are cooled by liquid F Counts 6000

4000

Ba 2000

C Ba Ba Ba

0 2

4

Ba BaBa 6

Energy (keV)

Figure 6.6 Energy-dispersive spectrum of BaF2 showing the resolution of the characteristic lines of low atomic number elements. The carbon signal is from surface contamination of the sample.

Microanalysis in Electron Microscopy 343

nitrogen, so that the detector is often the coldest region in the system. Cryogenic condensation of contamination on the surface of the detector is then a serious life-limiting concern. 6.1.2.3 Detection of Long-Wavelength X-Radiation. The wavelength of X-radiation commonly used for crystallographic structure analysis is usually <0.2 nm, and it was not until electron-probe microanalysis was developed (in the late 1950s) that any commercial need existed for the detection of longer wavelength, ‘soft’ X-radiation. The past decades have seen a steady improvement in the reliability, the sensitivity and the discrimination available for the detection of wavelengths longer than 1 nm, corresponding to critical excitation energies of 1 keV or less. There are two major problems to be solved: the first concerns the absorption of soft Xrays in the sample itself. Even for low-density specimens, the absorption coefficient for soft X-rays is large. Signal detection is significantly improved when the specimen is inclined towards the detector, in order to reduce the absorption path for X-rays in the sample. Strong absorption also occurs in the detector. Ultra-thin-window or windowless detectors are common solutions for the detection of the longest wavelengths. The second problem is the loss of wavelength discrimination at long wavelengths. In WDS systems, the crystals used for soft X-ray applications have long periodicities, such as those developed by repeated deposition of long-chain molecular monolayers from the surface of a Langmuir trough. In this process the molecules segregate to the surface of a liquid in a selfordered array that can be collected on a suitable curved substrate. Subsequent ordered layers are then deposited in the same way, one upon another. These WDS crystals are very fragile, easily damaged and may contain defects introduced during deposition. Their ability to resolve the long wavelengths of the incident X-radiation is generally less satisfactory than for the more perfect, ionic crystals used to analyse the shorter wavelengths. In spite of these problems, both EDS and WDS systems are easily capable of detecting light elements down to boron (Z ¼ 5), even for quantitative analysis, as long as the specimen is kept free of carbonaceous contamination. This may be compared with the early years of microanalysis, when quantitative analysis of elements below magnesium (Z ¼ 12) was judged impossible, and the qualitative detection of the light elements below carbon (Z ¼ 6) seemed unrealistic. Today, it is more often specimen contamination from the sample exposed to the electron beam that restricts quantitative analysis of the light elements, rather than any equipment short comings.

6.1.3

Quantitative Analysis of Composition

From the point of view of the microscopist, it is the composition of a given region on the sample that has to be derived from the recorded spectrum from the same region. As we will see below, this calculation is an iterative process. Before beginning ‘quantification’, the measured characteristic intensities from the constituents of the region to be analysed must first be corrected for ‘white’ background noise and spurious ‘escape’ and other peaks. Any peak-overlap needs to be deconvoluted before determining the area beneath each peak, using the FWHM criterion. Once the integrated peak intensities have been determined, the concentration of the region in the sample can be evaluated using measured (or calculated) intensities from

344

Microstructural Characterization of Materials

standard samples of known composition. This is done by taking separate account of corrections due to fluorescence, absorption and atomic number effects. These correction parameters themselves depend on the chemical concentration of the sample, which is to be determined. A numerical solution has to be sought. Note that in making these corrections to the measured relative peak intensities, it is the fluorescence (F) correction that should be applied first, followed by the absorption (A) correction, and then the atomic number (Z, or density) correction, before repeating the cycle iteratively until the results numerically converge. That is, the order of the corrections is F, then A, then Z. This is just the reverse order of the effects that were considered during generation of the characteristic X-ray signal from the sample. In X-ray generation we first follow the path of an energetic electron as it is slowed down by the atomic number effect (Z). We then consider the absorption of X-rays before they escape from the sample (A), before correcting for secondary fluorescence (F). Fluorescence can be a problem, since the range of X-rays in the sample is an order of magnitude greater than that of the incident high energy electrons. The volume of the sample that may fluoresce is therefore of the order of ·1000 the volume of the region we are trying to analyse. It follows that, in a polyphase sample, secondary fluorescence may be responsible for some unexpected effects that cannot readily be removed by a routine microanalysis correction procedure. In all software correction protocols, the relative characteristic intensity is defined as the integrated intensity beneath the peak minus the background counts and corrected for spurious peaks and peak overlap. These adjusted measured intensities are used as a ‘first guess’ for the relative concentrations of the corresponding elements in order to estimate the magnitude of the required corrections. This first estimate of the corrections is then used to calculate a ‘second guess’ in an iterative procedure for estimating the relative concentrations. In practice, few iteration cycles are needed before the differences between successive iterations converge to an approximately constant value, which is taken as the ‘best’ estimate of the relative concentrations. The residual error is typically better than 2 % of the calculated, corrected concentrations. However, there is no guarantee that convergence of the iterative series of correction calculations implies accuracy of analysis, not least because morphological features of the sample surrounding the nominal volume element being analysed may vitiate the correction procedure, either because of the surface topology or because of long-range fluorescence effects. The computer software available for performing these corrections varies considerably, but all such codes follow the same iterative logic. The geometrical parameters must be entered into the program, including the angle between the specimen surface and the incident beam, and the angle subtended by the detector at the specimen surface (termed the take-off angle for the X-ray signal). The accelerating voltage (the incident beam energy) must also be entered and the calculated standards for the background and characteristic line intensities need to be adjusted whenever the beam voltage is changed. Finally, the program will generally ask the operator if the results are to be normalized, that is whether the calculated concentration values should be adjusted so that they sum to 100 %. This is not usually a good idea, since if the sum of the calculated concentrations differs seriously from 100 %, then this is a strong indicator of a serious analytical problem: either one element has not been detected at all, or the sample density differs significantly from the expected value (possibly because of sub-surface porosity), or because some other feature of the microstructural morphology is affecting the results. Alternatively, the take-off angle, the operating voltage,

Microanalysis in Electron Microscopy 345

or some other operating parameter may have been mis-entered into the correction program. It is best to regard deviations from a summation to 100 % as a measure of the accuracy of analysis, and to take a very close look at the measurements and the calculation if the sum deviates from 100 % by more than a few percentage points. The computer software may also allow you to omit data from one element that is known to be present. When this element is identified, the concentration of the missing element will then be calculated by assuming that the total of the estimated concentrations is always 100 %. This may sometimes be useful for light elements that are hard to detect accurately, especially in nonstoichiometric materials, for example, for lithium in an Al–Li casting alloy. An additional computer software option that is commonly available is to assume stoichiometry during the calculation for compounds containing an undetected element. Typically, the missing element will be oxygen, carbon, boron, hydrogen or nitrogen, that is a low atomic number constituent. This procedure significantly improves the analytical accuracy when working with known stoichiometric compounds, such as ceramics, but it may be misleading when two phases with more than one valency are possible, for example TiN (Ti3þ) and TiO2 (Ti4þ), or where a compound is known not to be stoichiometric, such as Fe1-xO. In the present text we will describe the various corrections for quantitative analysis, not as FAZ corrections, in the sequence that would be performed in an iterative, quantitative analysis, computer correction program, but rather in the order that these corrections affect the X-ray signal as it is generated by the incident beam and subsequently emitted from the sample (the ZAF sequence). Therefore, we first consider the size of the volume element of the material in which the X-rays are generated, then we describe the absorption losses as the X-rays travel through the sample before escaping, and finally we analyse the incidence of fluorescent excitation for a characteristic excitation line by X-rays of higher energy (shorter wavelength). We also include a short discussion of the applications of X-ray microanalysis in TEM, for the case of thin-film samples analysed in transmission. Note that while the incident beam energy in the scanning electron microscope is typically less than 30 kV, and may be less than 3 kV, that in the transmission electron microscope is often in the range 80–300 kV. 6.1.3.1 Atomic Number and Absorption Corrections. For the most part, we shall restrict ourselves to K excitation in a two-component alloy, and ignore the more complicated situations in which the L and M spectra are excited and analysed, although the excitation process does not differ in principle. We first discuss the factors affecting the excitation efficiency, which depend primarily on atomic number, and then treat the subsequent process of X-ray absorption within the volume of the specimen. The foundations for quantitative microanalysis date back to the work of Castaing (1951), who first introduced the concept of measuring the ratio between the intensity I i of the characteristic X-rays produced from the element i in a specimen of unknown composition to the intensity of those produced from the same element in a standard sample of known composition I stnd i . In order to understand this relation, we first need to examine the number of atomic ionization events n that the incident electrons can induce in the sample for a particular

Atomic Number Correction

346

Microstructural Characterization of Materials

ionization edge of an element (K, L, or M). Following Scott, Love, and Reed, the number of ionization events will be directly proportional to the number of atoms per unit volume of the material and can be derived from the following relationship:   N Av r Q dE ð6:1Þ dn ¼ A ðdE=dxÞ where NAv is Avogadro’s number, r is the density and A is the atomic weight. Q is the ionization cross-section for an excitation event, and is a measure of the probability that an incident electron will ionize a constituent atom in the sample. dE/dx is termed the stopping power, and measures the deceleration of the incident electrons as they penetrate the sample. Ignoring for the moment backscattered electrons, and assuming a form for the stopping power that is based on the velocity of electrons in the sample v relative to the speed of light c: c1:4 ð6:2Þ dE=dx ¼ const·r· v Then the number of ionization events can be calculated as: ZEc  n¼

 N Av v1:4 const·Q·dE A c

ð6:3Þ

E0

where E0 is the energy of the incident electron beam, and Ec is the minimum energy required to ionized the electron shell of interest (K, L or M), that is, the critical excitation energy. Remembering that absorption and fluorescence are not yet included, n will then be proportional to the generated X-ray intensity I, and we can compare the total number of ionization events in a sample of unknown composition to that occurring in a standard sample examined under identical experimental conditions to yield the concentration of the element I in the unknown. Ki ¼

Ii ¼ f ðC i Þ I stnd i

ð6:4Þ

where Ki is a ‘sensitivity’ factor and is not a constant. In order to make the analysis quantitative, we must now take into account absorption and fluorescence, which we have ignored so far. The first step is to understand the depth dependence of X-ray generation in the sample from which the photons are collected. The depth distribution of photon generation can be calculated, and has also been measured experimentally. An example is shown in Figure 6.7 in which the distribution f(rZ) is plotted as a function of depth using the mass–thickness parameter, rZ. A number of important conclusions can be derived from this distribution. First, characteristic photon production in the surface region of the sample is large, and greater than unity on the f(rZ) scale. This is due to backscattered electrons that originate from the depth of the sample and generate photons in this surface layer. Second, the initial rise in the curve is associated with a progressive increase in electron scattering as the high

Microanalysis in Electron Microscopy 347 φ ((ρ ρZ) 3

Al Kα 2

1

0

5 ρZ (10 –4 gcm –2 )

10

Figure 6.7 Mass–thickness depth distribution f(rZ) for characteristic X-ray generation in an aluminium sample under a 20 kV incident electron beam. The depth is given in terms of the mass–thickness (rZ).

energy electrons penetrate further into the sample and their energy is reduced to the critical excitation energy. Finally, fewer photons are generated deep in the sample because few electrons reach these depths, and many of those that do have an energy that is below the critical excitation energy. Absorption Correction We define the total characteristic X-ray intensity that is generated in a specimen of pure A in the direction of the spectrometer by integrating fA(rZ):

Z1 I0 ¼

fA ðrZ ÞdðrZ Þ

ð6:5Þ

0

Since the photons must travel through the specimen in the direction of the detector, some of the X-rays will be absorbed before they can escape from the surface of the specimen. As discussed in Chapter 2, the reduction in intensity depends on the mass absorption coefficient for each specific wavelength passing through the specimen (m/r) and the path length x of these X-rays within the specimen:     m rx ð6:6Þ I ¼ I 0 exp  r The path length depends on the relative position of the detector with respect to the specimen surface, or the ‘collection angle’, often called the take-off angle (Figure 6.8). We can now redefine the reduction in intensity due to absorption for our specific microanalysis geometry as:     m rZcosecðaÞ ð6:7Þ I ¼ I 0 exp  r

348

Microstructural Characterization of Materials Incident electron beam

Exiting X-ray beam α

Specimen x

Z

dZ

Figure 6.8 X-ray path length within the sample x for a take-off angle a of X-rays emitted from a sample in the direction of the detector.

where the incident beam is assumed normal to the surface and a is the take-off angle. The emitted intensity is therefore reduced to: Z1 fA ðrZ Þexp½ðm=rÞrZcosec adðrZ Þ ð6:8Þ I¼ 0

The absorption factor is simply the ratio of the number of photons emitted in the direction of the detector to the number generated in the sample under the beam: R1 f ð wÞ ¼

fA ðrZ Þexp½ðm=rÞrZcosec adðrZ Þ

0

R1

ð6:9Þ

fA ðrZ ÞdðrZ Þ

0

This ratio of course depends on the energy of the incident electron beam (Figure 6.9).

1.05

E K(Cu)=8.048 keV

f (χ) 1.00 0.95 0.90

α=5° α=10° α=15° α=20° α=25° α=30°

0.85 0.80 0.75 0.70 0

5

10

15

20

25

30

E0 (keV)

Figure 6.9 Absorption correction function f(w) as a function of the over-voltage (E0  EK) and the absorption parameter w ¼ (m/r)coseca and for different values of the take-off angle.

Microanalysis in Electron Microscopy 349

k/c Cu

1.2

1.1

Without Absorption

1.0 WithAbsorption 0.9 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1

0.0

Concentration of Cu

k/c Au

1.0

Without Absorption

0.9 With Absorption 0.8

0.7 1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Concentration of Au

Figure 6.10 Calculated ratios of k/c for the two constituents in a Cu–Au alloy with and without an absorption correction.

An example of the expected effect of the absorption correction on the relation between the K intensity ratio and the concentration for Cu–Au binary alloys is shown in Figure 6.10. As expected, the correction exceeds unity for the lower atomic number copper, but is less than unity for the higher atomic number gold. The magnitude of the correction varies inversely as the concentration of the component. In different software correction packages, the ‘standard’ data employed in calculating the corrections, as well as the correction algorithm for the calculations, vary appreciably. In most cases the supplier of the software package is more than willing to discuss and explain the assumptions and approximations involved. All commercial software packages should be able to convert the relative characteristic intensity measurements to quantitative estimates of the elemental concentration to an accuracy of better than 2 % of the true concentration, providing the constituents are present at concentration levels of at least a few percent and characteristic excitation lines of all the elements are detected. However, this accuracy can only be achieved for flat, polished specimens, and providing the beam energy is of the order of 3EK for all the constituents with adequate counting statistics for all the measured characteristic lines. 6.1.3.2 The Fluorescence Correction. The fluorescence correction is, in a sense, the inverse of the absorption correction, since strong absorption in the sample for the X-rays generated by the incident beam implies that secondary excitation of lower energy X-rays is also taking place. The case of nickel and iron is instructive. Figure 6.11 shows the absorption coefficient for both elements as a function of the photon energy. Since nickel has the higher atomic number, its X-ray absorption coefficient is generally higher than that

Mass absorption coefficient

E(Kα)=7.47 E(Kβ)=8.26

Microstructural Characterization of Materials E(Kα)=6.39 E(Kβ)=7.04

350

EK (Fe)=7.109

EK (Ni)=8.329 Energy (keV)

Figure 6.11 Energy dependence of the mass absorption coefficients of iron and nickel and the positions of the characteristic Ka and Kb lines for these elements. The absorption edges correspond to the critical K-excitation thresholds of the two elements.

of iron for any given wavelength, but the excitation threshold for displacing a K-shell electron in nickel comes at a shorter wavelength (higher energy), so that the K absorption edge for nickel appears at a lower wavelength. It follows that, for a band of critical wavelengths between the two absorption edges, iron has a very much larger absorption coefficient than nickel. The characteristic K-lines of any element that is fluorescing must lie to the longer wavelength (lower energy) side of the excitation threshold (the absorption edge), and in the region of low absorption by the fluorescing element. By contrast, the characteristic K-lines for nickel lie in the high absorption region for iron, so that nickel radiation will be strongly absorbed in an iron alloy, and result in fluorescent excitation of the iron matrix, enhancing the characteristic signal for iron. More seriously, in a nickel alloy the strong excitation of iron will result in a fluorescence correction for iron that may amount to 30 % of the total signal, and nearly all this fluorescent radiation is generated in a volume well away from the region beneath the incident beam in which the primary radiation is excited. The major problem in the quantitative microanalysis of samples where strong fluorescence is to be expected arises from this large volume in which the fluorescent excitation occurs. The penetration depth for X-rays in the sample is typically an order of magnitude greater than the penetration depth for the incident beam, although it decreases with increasing X-ray wavelength and sample density. Thus the primary excitation events occur in a volume of the order of 1 mm3 in diameter, but secondary fluorescent excitation will occur in a volume of the order of 103 mm3 or even larger. The situation is illustrated schematically in Figure 6.12. Consider as an example, a phase boundary between a nickel-rich and an iron-rich phase, in which the X-ray detection system is placed perpendicular to the boundary, either on the

Microanalysis in Electron Microscopy 351 Incident Beam

Primary Excitation Envelope

Fluorescent X-Ray

X-Ray

Secondary Event

Secondary Excitation Envelope

Figure 6.12 Secondary, fluorescent excitation is expected to occur well outside the envelope of excitation events for the primary characteristic X-radiation.

side of the nickel-rich phase or on the side of the iron-rich phase. Two quite different excitation spectra are to be expected from the boundary region (Figure 6.13): 1. When the boundary is perpendicular to the X-ray take-off direction and the detector is positioned on the nickel-rich side of the boundary, thus maximizing absorption by nickel, then neither the iron nor the nickel signals will be strongly absorbed. However, when the probe is on the nickel-rich side of the boundary, the nickel radiation penetrating into the iron-rich region will generate a strong fluorescent iron signal that cannot be corrected, even when the electron beam is positioned up to 10 mm or more from the boundary. 2. Rotating the specimen by 180 so that absorption is for the most part through the ironrich region, will generate roughly the same fluorescent excitation of iron by nickel when the probe is on the nickel-rich side of the boundary. However, now strong absorption of the primary nickel radiation will also occur, since the nickel radiation must pass through the iron-rich region to reach the detector. The only sensible course of action with this sample is to position the detector first on one side of the boundary and then on the other, by rotating the sample through 180 , to check for the significance of these artifacts. Note the strong dependence of these effects on the X-ray take-off angle. In such a case, only semi-quantitative analysis near the boundary is possible. This may be achieved by determining the apparent concentrations as a function of the takeoff angle, tilting the specimen towards the detector, and then extrapolating the results to a

352

Microstructural Characterization of Materials IFe 1

INi 2

IFe 2

INi 1

2 µm

30 mm

Probe scan Detector 1

Detector 2

Fe

Ni

Figure 6.13 Iron and nickel intensities perpendicular to a Fe–Ni interface measured by two different detectors; detector 1 to the left of the sample, and detector 2 to the right. Using a 1 mm diameter probe and ignoring fluorescence, the iron signal will drop rapidly away from the interface over a distance of approximately 2 mm. Fluorescence will result in an iron signal from the nickel side of the couple, no matter which detector is used. The spurious ‘iron’ intensity will be approximately 6 % of the signal from pure iron and decay to zero some 30 mm from the interface. The nickel signal collected by detector 1 will show a sharp decrease due to absorption by the iron, while the nickel intensity collected by detector 2 will be unaffected by absorption.

p/2 take-off angle. Bearing in mind that these changes in fluorescence may be significant at distances of 10 mm from a phase boundary, concentration measurements near such interfaces should be treated with considerable caution. 6.1.3.3 The ZAF Equation. Let us now return to the quantitative correlation of intensity with concentration, we first consider a binary solid solution containing elements A and B in which we wish to determine the concentration of A. Corrections for the background radiation and for any escape or other spurious peaks are assumed to have been made. We now include a correction factor for fluorescence and describe the total emitted X-ray intensity as: Z1 ð6:10Þ I ¼ fðDrZ Þ fðrZ ÞdðrZ Þf ðwÞð1þgþdÞ 0

Microanalysis in Electron Microscopy 353

where fDrZ corresponds to the emission from a thin layer of mass thickness DrZ, and (1 þ g þ d) is the correction that has been inserted for fluorescence. Specifically, g is the ratio of the intensity of the fluorescent emission to the primary characteristic X-ray emission, while d is the corresponding ratio for the continuum fluorescence contribution. This equation for the characteristic intensity generated by element A from our sample of unknown composition containing the elements A and B, can now be expressed relative to the corresponding intensity from a standard sample of pure A: 1 AB R f ð rZ Þd ð rZ Þ AB I AB fðDrZ ÞAB f ðwÞAB 0 A A A ð1þgþdÞA A ¼ · · ð6:11Þ   A A R1 IA fðDrZ ÞA f ð wÞ A A A A ð1þgþdÞA fðrZ Þd ðrZ Þ 0

Z In this equation we can substitute: CAB A ¼

A

A fðDrZ ÞAB A fðDrZ ÞA A

F ð6:12Þ

So that: I AB A ¼ CAB A ðZAF Þ IA A

ð6:13Þ

6.1.3.4 Errors, Detection Limits, and Spatial Resolution. Error analysis for quantitative microanalysis using both EDS and WDS follows Poisson statistics, so that if we collect N counts for any specific, characteristic X-ray energy, then the standard deviation for the measurement, s, is N1/2. The relative standard error, e, is then s/N ¼ N1/2. It follows that, for a standard error of 1% some 104 counts are required. This value can be easily reached in WDS, but may require quite long integration times in EDS. As noted earlier, is important to separate the statistics of the measured peak from that of the background before making the FAZ corrections. The intensities are expressed as the net peak count rate (IPIB), where IP and IB are the number of counts for the characteristic peak (NP) and the overlapping background (NB), divided by the signal integration times used to acquire the two sets of counts, tP and tB. The standard deviation of this net peak count rate is then: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NP NB þ 2 ð6:14Þ sPB ¼ t2P tB while the standard error for the net peak count-rate is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NP NB þ 2 t2P tB  ePB ¼  NP NB  tP tB

ð6:15Þ

The detection limit, sometimes (redundantly) termed the minimum detection limit, depends directly on the signal-to-noise ratio of the spectrum, that is, the net counts of the peak versus

354

Microstructural Characterization of Materials

the net counts in the background. The detection limit corresponding to a 95 % confidence level can be expressed as: C DL;0:95

pffiffiffiffiffiffiffi 2I B ¼ ðZAF Þ pffi  Stnd Stnd t I P I B

ð6:16Þ

The detection limit for heavy elements in a light element matrix can be as low as a few parts per million when using WDS. It is always important to understand the source of the measured signal, which depends on the accelerating voltage of the incident, high energy electron beam, the extent of absorption for the characteristic primary radiation and fluorescence or secondary radiation. The maximum depth in the sample from which the characteristic X-ray signal originates, ignoring fluorescence, can be approximated by:  A ½mm  ð6:17Þ Z r  0:033 ð E1:70 E1:7 C rZ where A and Z are the average atomic weight and average atomic number in the volume of material being excited by the incident beam. The maximum diameter of the excited volume generating the X-ray signal is usually approximated by: D

0:231  1:5 1:5 E0 EC r

½mm 

ð6:18Þ

6.1.3.5 Microanalysis of Thin Films. The maximum useful specimen thickness in thinfilm TEM is often determined by the onset of inelastic scattering processes in the sample, although some inelastic scattering always occurs in addition to the dominant, elasticallyscattered, transmitted signal. It follows that a characteristic X-ray signal should be available in thin-film TEM. Since the specimen is very thin and the electron energy is very high compared with the beam energies used in SEM, the signal will be weak and very few characteristic X-ray photons are generated. As a consequence, it is not generally possible to detect constituents that are present in concentrations of less than 5 %. The signal is improved if a field emission source can be used to generate a highly focused electron probe which is scanned over a small area of the thin-film sample. That said, the excellent spatial resolution of the TEM and the negligible lateral spread of the electron beam transmitted through the thin-film specimen, has motivated the development of X-ray detection systems and software for quantitative microanalysis designed specifically for TEM. These systems depend heavily on the availability of a field-emission source and are combined with EDS, since only this combination provides an adequate X-ray signal intensity for quantitative analysis at low concentrations. As we shall see below, EDS of X-ray signals generated by the exceptionally small volumes in a TEM thin-film specimen is best accomplished in scanning TEM (STEM) mode, rather than by conventional TEM. The principle factors that have to be considered when attempting quantitative thin-film microanalysis by using characteristic X-rays generated in the transmission electron microscope are the following:

Microanalysis in Electron Microscopy 355

1. The high background noise that is associated with stray high energy electrons in the microscope column. These generate both white radiation and spurious characteristic Xrays, especially from copper, which is the major alloy constituent of the specimen stage. 2. Maximizing the X-ray collection efficiency by ensuring the maximum solid angle for detection, achieved by placing the solid-state detector as close as possible to the sample. 3. The thickness dependence of the characteristic X-ray signal and the use of thicknessdependent intensity measurements in place of standard correction procedures. 4. The reduction of the TEM accelerating voltage for thin-film microanalysis in order to improve the X-ray excitation probability and the counting statistics. 5. Correlation of the source of X-ray excitation in the sample with microstructural features that may be observed under different beam conditions. The problem of excitation from physical components of the microscope column is exacerbated by the electro-optical limitations on beam-probe focusing in TEM. It is not difficult to focus a fine probe, of the order of 2 nm diameter at FWHM, by using a standard magnetic lens condenser system, but much of the current in the beam remains in the tail of the distribution, outside the central, focused probe, and leads to characteristic X-ray excitation well beyond the FWHM diameter (Figure 6.14). This effect has been successfully reduced by aberration correctors placed in the condenser assembly of current STEM systems. A significant improvement is also possible by manufacturing the components of the specimen holder and specimen mount from a low atomic number element such as graphite and especially beryllium. Both these elements are electrical conductors and are free of electrostatic charging. Specimens that do not require a support grid are also to be preferred. Solid-state Si(Li) X-ray detectors are used for energy-dispersive microanalysis in TEM. They can be placed within 10 cm of the specimen in an inclined geometry with a high takeoff angle (30–35 ) giving a wide solid angle for data collection. Some systems offer a horizontal take-off angle, designed for a sample tilted towards the detector. This results in a significant increase in the available X-ray signal, but such a geometry is problematic if the goal is to measure the concentration associated with an interface or grain boundary, since such interfaces must necessarily be aligned parallel to the incident electron beam. Recently, sample holders, made of beryllium have become available. These are cut to allow the X-rays to reach a horizontal detector. Such holders improve the counting statistics without tilting the specimen, which may be critical for planar defect characterization. Variations in the specimen thickness may be an advantage for some samples: assuming a wedge-shaped specimen, the relative intensities of the characteristic lines can be measured as a function of distance from the edge of the hole in the sample prepared by dimpling and ion milling. The measured intensity ratios can then be extrapolated to zero thickness, when the absorption correction for the generated X-rays extrapolates to unity. In addition X-ray fluorescence can only occur in a region that is well outside the region of primary excitation, so that, thanks to the thin section of the sample, all X-ray microanalysis measurements made in the transmission electron microscope are free of fluorescence effects. As the accelerating voltage decreases, the probability of an inelastic interaction that generates characteristic X-rays will increase, so a decrease in the accelerating voltage of the transmission electron microscope should improve the yield. Unfortunately, any decrease in the accelerating voltage also decreases the brightness, that is, the current density of the source per unit solid angle. In practice the ‘best’, statistically significant, spectra may be

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Microstructural Characterization of Materials

(a)

FWHM Probe Tail

(b)

FWHM

Probe

Tail

Figure 6.14 (a) Distribution of the beam current in a focused probe includes a significant component in the tail of the electron spatial distribution. (b) Aberration correctors for condenser systems can reduce the tail and the FWHM, resulting in a much sharper focused probe.

recorded at the highest available accelerating voltage. Results obtained at the highest accelerating voltage should also correspond to the highest spatial resolution for microanalysis, since beam spreading in the thin foil decreases with increasing accelerating voltage, reducing the diameter of the region in the thin film from which the X-rays are generated. A primary motivation for using energy-dispersive X-ray microanalysis in thin-film transmission microscopy is the improvement in the spatial resolution for analysis, due to the smaller high energy electron probe size available in TEM or STEM. (In, Figure 6.15 two spectra from two neighbouring points are shown). When working with very small probe sizes, it can be difficult to know the exact position of the beam relative to a microstructural feature. Both specimen drift and beam drift commonly occur during the long acquisition times that may be required to obtain good counting statistics. This is where a STEM system has an advantage over conventional TEM, since the beam can be rastered to produce either a line-scan or elemental map, as in SEM, while bright-field STEM, annular dark-field (ADF) STEM, or high-angle annular dark-field (HAADF) STEM images can be acquired concurrently, fully synchronizing the EDS measurements with the features of interest in the microstructure.

Microanalysis in Electron Microscopy 357

Figure 6.15 EDS X-ray spectrum from a small titanium carbon nitride (TiCN) particle in an aluminium matrix, illustrating the power of the transmission microscope to overcome the resolution limitations for X-ray detection in SEM. Two spectra are shown: (a) from aluminium metal and (b) from the TiCN particle. Note the change in relative intensities (Al/Ti) between the two spectra.

6.2

Electron Energy Loss Spectroscopy

Microanalysis that is based on characteristic X-ray excitation faces two practical problems. The first is the very low collection efficiency for X-rays (no better than 103 for WDS systems and of the order of 102 for EDS detectors). The second concerns the inefficient characteristic X-ray excitation and poor detection resolution for radiation generated from the light elements. Although the characteristic spectra from the elements below magnesium (Z ¼ 12) are readily detectable down to lithium (Z ¼ 3), these elements are difficult to analyse quantitatively, for three reasons. First, due to absorption by the solid-state detector window and loss of detection efficiency at low photon energies. Second, absorption by sample surface contamination and surface films distorts any quantitative analysis. Finally, only a small fraction of the low energy, primary excitation events lead to characteristic Xray emission, while the remainder generate Auger electrons (Section 7.2). For example, it is estimated that the chance of a K-shell excitation of carbon (Z ¼ 6), yielding a photon is only 1:400, and this yield increases only slowly with atomic number, rising to 1:40 for sodium (Z ¼ 11). Analysis of the energy spectrum of the inelastically forward scattered electrons, overcomes all these limitations, since very high electron collection efficiencies, of better than 50 %, are possible at the energy loss detector while maximum analysis sensitivity is achieved precisely for the low energy losses that are characteristic of the low atomic number elements. Moreover, the electron energy loss spectrum is usually measured by thin-film,

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Microstructural Characterization of Materials

transmission microscopy, using forward scattered electrons, precisely the geometry for which characteristic X-ray microanalysis by EDS is most difficult, primarily due to the very poor counting statistics (Section 1.3.5). Two primary modes of operation are possible for electron energy loss spectroscopy (EELS) in the conventional transmission electron microscope. If an image is focused on the screen, then the back focal plane of the projector lens will contain a diffraction pattern that can serve as the signal for an EELS spectrometer. The selected area aperture in this plane determines the origin of the signal and the image formed in the back focal plane of the projector lens is the image ‘seen’ by the spectrometer. Alternatively, if a diffraction pattern is focused on the screen, then the back focal plane contains an image that serves as the signal source for the EELS spectrometer, and the source of the signal is then the area of the specimen within the selected area aperture that is illuminated by the incident beam. Focusing the incident beam, to form a convergent-beam diffraction pattern on the viewing screen can also define the region from which an EELS signal is acquired. If the incident electron beam is fully focused to a fine probe, as in STEM mode, then this will be analogous to a diffraction pattern that is focused on the viewing screen. The STEM probe will then also define the sample area from which the signal is collected. In effect, it is the electrons focused in the back focal plane of the microscope that determine the source of the EELS spectrum. It follows that in TEM mode we can acquire a spectrum from either an image, or a diffraction pattern, while in STEM mode we will always acquire a spectrum from a diffraction pattern. The advantage of STEM mode is that the electron beam can be rastered across the sample, to acquire EELS data as a function of the position of the incident electron probe, in order to form an EELS line-scan or composition map, just as we discussed for EDS. In addition, when using a field emission gun source, the spatial resolution of the EELS signal is actually better in STEM mode than it is in TEM mode, and sub-nanometre spatial resolution of EELS results for chemical composition has been demonstrated. If a HAADF or ADF detector is available for the STEM system, then the pixel intensity in the image can be measured at the same time as the EELS spectrum, and this image intensity can be correlated directly with the local EELS signal. The electron energy loss spectrometer is a magnetic prism, positioned beneath the main microscope column (Figure 6.16). The magnetic spectrometer is located below the primary image plane of the microscope and accepts electrons that pass through an aperture positioned on the optic axis. The electrons deflected by the spectrometer will have an angular spread, with those electrons of energy E0 that have experienced no energy loss being the least deflected. All those electrons that have experienced an energy loss will be deflected by an additional angle that is dependent on the extent of this energy loss. Instrumental developments in EELS have now improved the energy resolution to much better than 1 eV, as compared with, perhaps, 50 eV for the light element, characteristic X-ray energy resolution in the best EDS systems. Parallel EELS detectors have replaced serial detection systems, providing simultaneous data collection across a large range of the energy loss spectrum. By adding an extra set of lenses after the standard EELS spectrometer, it is possible to convert the signal from reciprocal space (the original energy spectrum) to real space (to form an image). This allows us to collect images corresponding to specific values of energy loss that are characteristic of one or other of the chemical constituents of the sample. Such energy loss imaging filters can now provide

Microanalysis in Electron Microscopy 359

Figure 6.16 Schematic drawing of an EELS spectrometer, positioned below the main transmission electron microscope column.

quantitative chemical information with a spatial resolution of the order of 0.5 nm and an absolute volume detection limit of less than 100 atoms. The development of energyfiltered transmission electron microscopy (EFTEM) (Section 6.2.6) has had a major impact on materials research at the nanometre level. One major caveat is in order. In both crystalline and amorphous thin-film specimens, elastic scattering events are far more probable than inelastic events and much of the inelastic energy absorption spectrum may be generated by electrons which have first been elastically scattered. It follows that inelastic scattering of the electrons in a diffracted beam is a common occurrence. However, it is the energy spectrum in the direct transmitted beam that is usually collected, while those electrons in the diffracted beams that have been inelastically scattered are excluded because they are prevented by an aperture from entering the spectrometer. Relatively little work has been done to compare the energy loss spectrum from the direct transmitted beam with that from a diffracted beam, that is, the energy losses corresponding to a dark-field image.

360

6.2.1

Microstructural Characterization of Materials

The Electron Energy-Loss Spectrum

A typical energy loss spectrum is shown in Figure 6.17. The y-axis is the detected intensity at a given energy, measured in arbitrary units (a.u.). The x-axis is the energy loss (E0E), measured in eV. Energy losses as high as 2000 eV can be detected, although the count rates at high energy losses are very low. Note that the scale for the y-axis has been changed by a factor of ·500 above the low-loss peak labelled plasmon. The spectrum consists of a series of sharp intensity changes superimposed on an exponential decay in the electron current with increasing energy loss. Four processes contribute to the energy losses of the electron beam, although only two of these can be detected and measured by the magnetic spectrometers that are used to analyse the energy loss spectrum: .

.

Phonon excitations. These excitations result in very small energy losses, typically less than the thermal energy spread in the incident beam. These peaks are within the zero-loss peak of the energy loss spectrum, and are not resolvable. Nevertheless, the zero-loss peak can provide important information on the performance of the electron source and the stability of the high-voltage supply. A field emission gun is capable of limiting the thermal (kT) spread of the primary beam energy to less than 0.3 eV. By comparison, a LaB6 source may have a thermal spread of the order of 1.5 eVand a conventional, thermionic emission tungsten filament, will increase this spread to about 3 eV. Electron transitions. These transitions occur both within and between the different electron shells of the atom, and commonly correspond to energy changes in the range 1–50 eV. These very low-loss peaks can be detected in the spectrum, and may be used to identify a phase containing a specific element by a comparison with a known, standard

Plasmon x500 Gain

Intensity (a.u.)

Zero-loss peak

Energy loss near-edge structure (ELNES)

valence loss coreloss (>50 eV) 0 500 Energy loss (eV)

1000

Figure 6.17 Example of an EELS spectrum showing the zero-loss peak, a plasmon peak in the low energy loss region, and a complex excitation absorption edge in the core loss region associated with inelastic interaction with an inner (core) electron shell.

Microanalysis in Electron Microscopy 361

.

.

spectrum. Analysis of the low energy loss region, in the 10 eV range, can, in principle, be used to determine the localized dielectric properties of a material. Plasmon excitation. The plasmon phenomenon is associated with quantized oscillations in the conduction band of a metallic conductor. These typically result in energy-loss peaks in the range 5–50 eV. The plasmon effect is roughly analogous to the ripples created on the surface when a small stone is dropped into a pond. Plasmon peaks have been interpreted in chemical terms, since they are concentration-sensitive, but the interpretation is controversial. The one or more plasmon peaks from metallic conductors are quite sharp, while those from nonconductors are rather diffuse. If the mean free path of the plasmons lp is known, and if the sample is thin enough to ensure that only a single plasmon peak is excited, then we can use the total intensity in the low-loss region of the spectrum relative to the zero-loss intensity to estimate the specimen thickness t from the relationship t ¼ lp lnðI T =I 0 Þ, where IT is the total integrated intensity that includes the plasmon and zero-loss peaks, and I0 is the intensity of the zero-loss peak alone. Note that relative values of specimen thickness are always proportional to the plasmon losses, and these can therefore be used to check for any thickness dependence of the EELS analysis results. Higher energy losses. The high energy loss region of the spectrum (DE>50 eV) contains the inner shell absorption edges that are associated with atomic ionization and are accessible for chemical analysis in EELS. Because the spectral resolution in EELS is so much better than that available by EDS (some two orders of magnitude) or WDS (about an order of magnitude), more detailed information is available in the excitation edge associated with each individual atomic species. Moreover, since EELS is particularly sensitive to low energy excitations, this signal contains considerable chemical information. However, the quantitative analysis of chemical composition by EELS is usually less accurate than can be achieved by X-ray microanalysis using EDS or WDS. It follows that the major application of EELS is for the detection of the atomic species present on the nanometre scale and in the study of localized chemical bonding states, rather than for accurate, quantitative microanalysis. The remainder of this account is limited to the EELS signal that is specific to the atomic species present.

6.2.2

Limits of Detection and Resolution in EELS

The energy resolution of the magnetic spectrometers commonly available for EELS is of the order of 0.1 eV, but it requires a monochromatic field emission source to exploit this resolution. In most cases it is the kinetic energy spread in the beam, not the characteristics of the EELS detector, which limits the energy resolution. It is therefore common practice to record the zero-loss peak, corresponding to the primary Gaussian peak for the beam exiting the specimen. This primary peak is then used to calibrate both the absolute zero for the energy loss spectrum and to estimate the available energy resolution, usually from the FWHM of the zero-loss peak. The intensity of the first plasmon peak relative to the zero-loss peak is also a good benchmark for judging the suitability of a thin-film specimen for EELS analysis. As a rough guide, if the intensity in the first plasmon peak is less than one-tenth of the zero-loss peak, then the specimen should be thin enough for EELS. The inelastic signal from thin samples is very weak, but increases as the thickness increases. However, if the sample is too thick, then

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Microstructural Characterization of Materials

multiply scattered electrons obscure the edge structure of the energy loss spectrum. In addition, the analysis of multiple scattering, more than one scattering event per electron, requires an error-prone deconvolution correction for quantitative EELS microanalysis. In practice an optimum specimen thickness for EELS exists for which the counting statistics are adequate, while the probability of multiple scattering is small. Not too surprisingly, the optimum results are obtained at thin-film thicknesses comparable with the extinction thickness of the sample for dynamic diffraction. These are typically less than 10 nm, but depend on atomic number (Z), accelerating voltage (E0) and the diffracting conditions. The analytical sensitivity of an EELS system can be discussed in terms of either the minimum detectable mass, the minimum signal that can be identified from a given constituent, or the minimum mass fraction, the minimum detectable concentration of a given element in the sample. The analytical sensitivity of EELS is far better than can be achieved by X-ray microanalysis and can correspond to a signal from a few hundred atoms. The detection limit varies with atomic number and the position of the edge in the loss spectrum. In many cases the best spatial resolution for analysis, that corresponding to the minimum focused probe size, is not required and it is possible to work with much higher incident beam currents, either by increasing the probe size in STEM mode, or by using a larger selected area aperture in TEM mode. The sensitivity is then no longer limited by the signal statistics but rather by errors of extrapolation in subtracting the background corrections from the spectrum and the relative contributions to the signal from the different element-specific, inner-shell absorption edges. Unfortunately, these errors are of the order of 10 % for K-excitation, while data from L-edges are even less accurate, primarily because the L-edge is much broader than the K-edge. M-edge data are useful for confirming that a constituent is present, but not for estimating concentration. The energy loss spectra can be displayed for any energy range and at any gain (Figure 6.17). The ability to display change in gain is especially important, given the exponential decay of the signal with increasing energy loss. Identification of the constituents responsible for the absorption edges observed in the energy loss spectrum depends on accurate calibration of the energy scale for the magnetic spectrometer. The zero-loss peak defines the origin for zero energy by the centre of mass for this Gaussian peak, while, as noted previously, the width of the zero-loss peak at FWHM defines the available energy resolution. A carbon K-edge (Figure 6.18) is often selected to define the linear energy scale, since carbonaceous contamination is a common feature of many acquired spectra. The carbon edge from either diamond or cementite (Fe3C) will have a different shape from the amorphous carbon contamination peak and will appear in a slightly different position on the energy scale, so one has to be careful when calibrating the energy scale from this ionization edge. The position of each edge is usually defined as the position of maximum slope. Energy absorption events corresponding to an excited state of the atom will also result in higher energy losses that lie beyond the initial absorption edge and, in principle, extend to the maximum incident beam energy. The total number of events associated with any particular absorption edge can only be estimated by fitting an empirical function to the preedge data and then extrapolating this estimated background curve to the high-energy loss region. The estimated background is then subtracted from the measured data to yield a value for the true number of characteristic absorption events (Figure 6.19). Typical empirical functions for the background are of the form I ¼ AEr, where A and r are empirical constants

Intensity (a.u.)

Microanalysis in Electron Microscopy 363

C-K

280

300

290

310

Energy loss (eV) Figure 6.18

A measured carbon K-edge at 284 eV.

that have to be determined separately for each pre-edge energy loss region. The signal from each edge decreases rapidly at higher energy loss values and it is neither necessary nor desirable to sum the counts derived for a particular edge over all energies above that edge, because the extrapolation errors above the edge increase with energy. However, it is important that the data for all edges should be summed over the same range of energy loss.

80

NiO

Spectrum Background fit Ni edge after background substraction

O 70 60

x 1000

50 40 30 20

Ni

10 0 600

800

1000

1200

1400

1600

Energy loss (eV)

Figure 6.19 Determination of the number of excitation events associated with a specific absorption edge for nickel in NiO. The background is fitted to the energy loss curve before the nickel edge, and then extrapolated to the edge region and beyond before being subtracted.

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Microstructural Characterization of Materials

This ‘energy loss window’ D is typically 50–100 eV, and the same window can be used to derive an absolute mass data scale by applying this window to the zero-loss and low energy, plasmon loss region. The assumption is that the data are collected over a fixed solid angle, the collection semi-angle b, determined by the spectrometer aperture, and that the inelastic scattering cross-section for this solid angle and energy range s(bD) is known, so that good estimated values for K excitation are assumed to be available. When this is the case, then the number of atoms per unit area N is given by: N¼

I K ðbDÞ 1 : I 0 ðbDÞ sK ðbDÞ

ð6:19Þ

where IK refers to the K-edge signal for a particular element and I0 is the zero-loss and low energy loss signal collected for the same energy interval and collection angle.

6.2.3

Quantitative Electron Energy Loss Analysis

Quantitative estimates of the absolute mass of elements present in the excited region for an EELS spectrum are generally less useful than determinations of their relative concentrations. The simplest method for obtaining a quantitative estimate of relative concentration is by calculating the ratios of the data derived from each K-edge. In this case, the zero-loss intensity data cancel out: NA IA ðbDÞ sBK ðbDÞ ¼ K · N B I BK ðbDÞ sA K ðbDÞ

ð6:20Þ

The calculation assumes that good estimates are available for all the relevant values of the partial inelastic scattering cross-sections s for the constituents. If two separate edges come within the range D then it is important to subtract the extrapolated energy loss curve for the higher energy loss edge to obtain the excitation intensity for the lower atomic number constituent before subtracting that from the lower energy-loss absorption edge. Again, any such overlap will further limit the accuracy of the analysis. A major problem in applying EELS to quantitative microanalysis is in selecting an optimum sampling thickness from a wedge-shaped thin-film specimen, prepared by dimpling and ion milling. Figure 6.20 shows the normalized, calculated intensity ratios for three different binary element combinations (B/N, Al/O and Al/Ni) as a function of the relative intensity of the first plasmon, low energy loss peak, to the zero-loss peak. As noted previously, the relative height of the first plasmon peak is proportional to the absorption thickness in the thin-film sample. From Figure 6.20, it is clearly an advantage to choose a region that is as thin as possible, commensurate with obtaining adequate counting statistics. Even so, comparing results acquired at different sample thickness is probably the best way to ensure confidence in the analysis. Characteristic X-ray microanalysis, using the EDS or WDS method, remains far more accurate for quantitative microanalysis, but cannot compete with EELS in mass sensitivity. EELS methods are certainly less valuable for high energy loss peaks, but EELS retains its superiority for low atomic number materials that result in low energy loss values. Both techniques can overlap over the full range of atomic number in the periodic table. Although

Microanalysis in Electron Microscopy 365

I A /I B

IB(K) /IN(K) IO(K) /IAl(K) IAl(K) /INi(L2,3)

0.01

0.1

1.0

I P /I 0

Figure 6.20 Intensity ratio for two ionization edges in three different materials as a function of the thickness. The results are plotted as the ratio of the intensities of the first plasmon, low energy loss peak to the zero-loss peak. (Courtesy of David B. Williams and Philips Electronic Instruments, Inc.).

EDS possesses greater analytical accuracy, EELS has far better spatial resolution and detection limits, while the better spectroscopic energy resolution in EELS can also provide information on the nature of the chemical bonding in the sample. This is our next topic. 6.2.4

Near-Edge Fine Structure Information

The 0.1 eVenergy resolution of EELS spectrometers has revealed a wealth of fine structure in the region of the absorption edge that is still only partially understood, even though the general physical principles are clear. This energy loss near edge structure (ELNES) reflects both the chemical state of the atom and, to some extent, the atomic coordination and symmetry with respect to the neighbouring atoms. Changes in the chemical bonding state result in energy shifts of both the primary edge and the fine-structure peaks, while peak splitting results from a reduction in coordination symmetry. In alumina, cation disorder that is present in the higher symmetry, transient crystal structures generates a few broad, comparatively simple peaks at the adsorption edge, while for the stable, less symmetric corundum phase several sharp peaks are visible in the absorption-edge spectrum (Figure 6.21). Calculations of the energy shifts that are associated with resonant energy exchange processes in the nearest neighbour coordination sphere are consistent with differences in the cation coordination symmetry of the g and a alumina phases (as indicated for the aluminium L2,3-edge in Figure 6.22). This high resolution EELS information is not related to chemical composition, but rather to the localized atomic bonding in the solid. The ability to distinguish bonding states and

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Microstructural Characterization of Materials

Intensity

(a)

60

70

80

90

100

110

120

130

140

120

130

140

Energy loss (eV)

Intensity

(b)

60

70

80

90

100

110

Energy loss (eV) Figure 6.21 Aluminium L2,3-edge in (a) the cubic, g-Al2O3 metastable phase and (b) the rhombohedral, a-Al2O3 stable phase. (Courtesy of Igor Levin).

coordination symmetry provides a major motivation for expanding existing EELS applications. 6.2.5

Far-Edge Fine Structure Information

Oscillations in the energy loss signal are also observed at some distance above the position of an edge and are referred to as extended energy loss fine structure (EXELFS). The wavelength of these oscillations in the energy spectrum is of the order of 20–50 eV, while the amplitude may amount to 5 % of the edge signal. The effect is analogous to an effect in X-ray absorption spectra referred to as extended X-ray absorption fine structure (EXAFS). EXELFS is due to elastic scattering of an inelastically scattered electron by the periodic crystal structure. The oscillations may persist for several hundred eV above the edge. The information contained in the EXELFS signal is related to the local atomic density and could, in principle, be extracted to give the radial distribution function (RDF) for the chemical component responsible for the absorption edge. The RDF records the probability

Microanalysis in Electron Microscopy 367 Al L 2,3

Intensity

Tetrahedral AlO4

Octahedral AlO 6

5

10

15

20 25 30 35 Relative energy loss (eV)

40

45

Figure 6.22 Calculated ELNES L2,3-edge spectra for tetrahedrally and octahedrally coordinated Al cations. (Courtesy of Rik Brydson).

that an atom occupies a specific coordination sphere located at any given distance from the source atom. To extract this RDF information, we must first subtract the background from the signal to reveal the oscillations, and then perform a Fourier transform of the oscillations to convert them into the radial distribution function. In principle, this technique should supply information on the near neighbour atomic environment, dramatically extending the usual concept of microanalytical information. However, the results are rather sensitive to the correction procedure and are controversial. 6.2.6

Energy-Filtered Transmission Electron Microscopy

The addition of an adjustable aperture followed by an additional set of lenses after the EELS magnetic spectrometer (Figure 6.23) makes it possible to select a part or all of the energy loss spectra to form an image in real space on a charge-coupled device (CCD) camera. This system configuration is termed energy-filtered transmission electron microscopy (EFTEM).

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Microstructural Characterization of Materials

Figure 6.23 Additional lenses placed behind an adjustable aperture or slit in a parallel EELS system can be used to convert the energy loss signal from reciprocal space to real space, so that an image can be acquired by a CCD camera.

This extension of EELS to EFTEM allows us to filter the electron energy loss signal so as to select a specific energy loss range to form an image. The energy loss range selected can be from any part of the EELS. In the simplest case, only the zero-loss peak of the EELS is selected in order to eliminate the chromatic aberration associated with inelastic scattering of the transmitted electron beam. An example of a zeroloss diffraction pattern is compared with a conventional pattern in Figure 6.24. The improved sharpness of the diffraction spots and the readily resolved secondary spots demonstrate clearly the additional diffraction information that is obtained by excluding the inelastically scattered electrons present in the conventional diffraction pattern. Similar improved

Microanalysis in Electron Microscopy 369

Figure 6.24 Comparison of a conventional selected area diffraction pattern (a) with an energyfiltered selected area diffraction pattern (b) of flagellin. Reproduced from K. Yonekura, S. MakiYonekura and K. Namba, Quantitative Comparison of Zero-Loss and Conventional Electron Diffraction from Two-Dimensional and Thin Three-Dimensional Protein Crystals, Biophysical Journal, 82(5), 2784–2797, 2002. Copyright (2002), with permission from the Biophysical Society.

sharpness is also often observed in the image formed by diffraction contrast when the inelastically scattered electrons are excluded from a bright field micrograph (Figure 6.25). However, the full potential of energy filtered TEM is only realized when energy loss signals from two or more characteristic absorption edges are used to form the image, by combining different regions of the energy-loss spectra into a single image. One way to picture this is shown in Figure 6.26, where the energy loss spectra is plotted vertically and the energy ranges (DEi) characteristic of each chemical species form sequential images. Images formed from a selected energy loss range can be viewed directly, and the contrast is then associated with areas in the microstructure that have resulted in the selected characteristic energy loss. Alternatively, the background subtraction corrections applied to the energy loss spectra can also be applied to a series of energy loss images, and local contrast in the images can then be qualitatively related to local concentration variations. The resultant elemental map is analogous to that obtained by X-ray mapping, but the spatial resolution for concentration differences is very much better. As an example, we return to the Cu–Co alloy shown in Figure 6.25. Copper and cobalt are immiscible in the solid state, and the alloy is expected to be a two phase dispersion of these constituents. However, copper and cobalt have similar atomic number and the difference in lattice parameter between the two phases is also small. Neither diffraction contrast in TEM nor STEM imaging can be used to

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Microstructural Characterization of Materials

Figure 6.25 Bright-field TEM micrograph of a cross-section from a Cu–Co alloy, deposited on a silicon wafer that was coated with a composite SiO2/TaN/Cu film. Only the zero-loss peak in the electron energy loss spectrum was used to form this image.

locate the two phases with any confidence. By using the characteristic energy losses for copper and cobalt in the energy loss spectra and following the schema indicated in Figure 6.26, it is possible to build up a composite EFTEM image that clearly displays the location and size of the cobalt particles within the copper matrix (Figure 6.27).

Summary Information on the chemical composition of individual microstructural features is often of crucial importance, and considerable effort has been expended on microanalysis to complement morphological and crystallographic data. Beyond the primary, qualitative requirement to identify the chemical elements present in a specific region, microanalysis has two quantitative aspects. The first is the spatial resolution of the analysis in the recorded image, while the second is the spectral sensitivity in the collected spectrum. The characteristic X-ray signal generated in the scanning electron microscope is the most frequently used of the microanalytical spectral tools that are available. This signal comes from a volume element of material near the surface of the solid sample. This volume is usually at least 1 mm3. The limiting spatial resolution for microanalysis is therefore of the order of 1 mm. Much better spatial resolution is available in the transmission electron

Microanalysis in Electron Microscopy 371 x y

∆ Ei Image at ∆ Ei

∆E

Figure 6.26 Schematic illustration of the concept used for energy-filtered imaging. A selected range DE of the energy loss spectra is used for each image.

Figure 6.27 Energy-filtered image of the micrograph shown in Figure 6.25. Green indicates cobalt, and red indicates copper. Blue indicates oxygen-rich areas located in the SiO2 layer. (See colour plate section)

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microscope using the energy loss spectra of the transmitted beam from a thin-film sample. X-ray microanalysis also has a limited spectral sensitivity that is associated with the inherent limitations of the Si(Li) solid-state detector and the background of ‘white’ X-radiation that partially obscures the characteristic signal. Errors in quantitative X-ray microanalysis are typically of the order of 1 % of the elemental composition, and the absolute limit of detection for any constituent element is usually no better than 0.2 %. Characteristic X-ray microanalysis may employ the K-, L- or M-spectra excited in the sample, depending on the atomic number of the constituents and the electron energy in the incident beam probe. In general, the incident beam energy is selected to optimize the spatial resolution with respect to the counting statistics. The former is best at lower beam energies, while the latter generally improve at higher beam energies. Two methods are commonly used to collect the characteristic X-ray spectra, namely wavelength dispersion and energy dispersion. In wavelength–dispersive spectroscopy (WDS) a proportion of the X-rays emitted from the sample is collected by a curved crystal and photons that have a wavelength which fulfills the Bragg condition for diffraction by the crystal are focused onto a detector (a gas proportional counter). By rotating both the diffracting crystal and the detector to change the Bragg angle, it is possible to scan across a range of wavelengths and record the excited X-ray intensity as a function of wavelength. In energy-dispersive spectroscopy (EDS) a solid-state, lithium-drifted silicon, Si(Li), detector is used to record the X-ray signal. The detector absorbs the photon energy and gives rise to current pulses whose intensity is proportional to the absorbed photon energy. The electrical pulses are first digitized and then counted in a multichannel analyser to develop a histogram of the number of pulses as a function of the energy of the collected photons. Although wavelength dispersion has better spectral resolution, energy-dispersive systems are able to record all the photons admitted to the detector simultaneously. However, the pulse count rate in EDS is limited by the response of the detector crystal, and this sets an upper limit to the rate of data collection. Although EDS is the most commonly available detection system, both systems are important, depending on the application requirements. Both EDS and WDS systems have limited detection efficiency for the long-wavelength radiation that is characteristic of low atomic number, light elements. Even so, it is possible to identify all constituent elements in the sample, including boron (Z ¼ 5) or even lithium (Z ¼ 3), and it is the problems of sample surface contamination, rather than limitations of the spectrometer, that restrict the experimental accuracy for light element microanalysis. To convert the recorded characteristic intensities into a quantitative analysis of the chemical composition, it is first necessary to subtract the background radiation count from the integrated characteristic intensity peaks. Reliable methods for optimizing background subtraction have been developed. Following background subtraction, the spectrum must be corrected for spurious peaks, most notably ‘escape’ peaks that are associated with fluorescent excitation of SiKa photons by the radiation incident on the detector. In many cases it is not possible to analyse for all elements present. Many software programs allow for additional chemical information, such as the assumption of chemical stoichiometry, to be introduced into the quantitative analysis software protocol. The size of the volume element from which an electron beam of a given energy will generate characteristic X-rays in the sample depends on the average atomic number or density, while the number of high energy electrons in the incident beam that are available for X-ray excitation will depend on the proportion that are lost to backscattering of the primary beam. The atomic number or Z

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correction, and the backscatter correction are combined in all software correction programs. A high Z element in a region of low average Z will have a higher probability of exciting characteristic X-rays than would be expected from the chemical composition, while the reverse is true for a low Z element in a region of high average Z. The X-ray photons detected by the spectrometer are generated beneath the sample surface, and these photons may undergo inelastic absorption before they escape from the sample. If microanalysis is to be made quantitative, then these absorption processes also have to be corrected for, based on our knowledge of the X-ray absorption coefficients of all the elements present for the characteristic radiation generated by each element individually. The characteristic line intensities are determined from the full width at half-maximum (FWHM) area of each peak after subtracting the background correction. All the corrections to the measured relative X-ray intensities have to be applied iteratively, assuming initially that the composition of the sample is given approximately by the measured relative intensities of the background-corrected, characteristic lines of the X-ray spectra. The initial corrections for atomic number, backscattering and X-ray absorption are all estimated based on this assumption. The corrected values for the relative concentrations are now used to re-evaluate each of the various correction factors, and this second iteration then serves to provide a more accurate estimate of the composition. In general, several iterations are required to yield a corrected composition whose value no longer converges. The extent of the microanalytical errors can be estimated by summing the calculated concentrations of each element and determining the deviation of the total from 100 %. In many software programs the calculated compositions are routinely normalized so that they always sum to 100 %, and these normalized values are presented as a ‘best estimate’ of the composition. This practice is acceptable, providing the microscopist recognizes that a useful checkon the accuracy of the correction procedure has been lost. For analyses performed on large, homogeneous particles, well away from phase boundaries, there is usually no problem, but the morphological information in a secondary electron image only reveals the surface and near-surface structure, whereas the X-ray data are largely sub-surface, and may contain data from phase regions that are not visible in the secondary or backscattered electron image. One further intensity correction is important for quantitative microanalysis, that due to fluorescence. If X-rays that have been excited by the incident electron beam are absorbed before exiting the sample, then they must give rise to other, additional excitation processes. The most important of these is fluorescent excitation of longer wavelength, characteristic radiation from other constituents that are present in the sample. While it is in principle possible to correct for fluorescence whenever necessary, it is not always possible to identify the origin of the fluorescent radiation. The electron excitation of X-rays occurs in a limited, sub-surface volume beneath the electron probe, but, once generated, the primary X-rays then have a mean-free path in the sample that is typically 10 mm or more. It follows that any subsequent fluorescent excitation can originate from a volume of the sample that is some three orders of magnitude larger than the source of the original, primary X-ray signal. While the most important application of X-ray microanalysis is to provide chemical information in SEM, it is also possible to detect characteristic X-rays that are generated in the thin-film specimens examined by TEM. In this case, the signal is very much weaker, both because of the small volume of material available for excitation in the thin film and

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because the probability of inelastic scattering of the electrons is much reduced at the very much higher beam energies used in TEM. Fluorescence effects are negligible in thin films, while absorption effects are much reduced and are proportional to the film thickness. The film thickness also limits spreading of the incident beam passing through the sample, so that the spatial resolution for X-ray microanalysis in TEM is greatly improved when compared with SEM microanalysis, and is limited primarily by the much poorer counting statistics. A field emission gun greatly improves the intensity of the beam probe and EDS microanalysis facilities are now often available on transmission electron microscopes. Much improved counting statistics for microanalysis in TEM can be obtained by recording the energy loss spectra. In electron energy loss spectroscopy (EELS) the light elements give the most readily detected signal, since the characteristic absorption edges of the light element, low atomic number constituents correspond to low energy losses. In the energy loss spectrum, the first peak, the zero-loss peak, contains information on the chromatic spread of the source, while plasmon peaks, adjacent to the zero-loss peak in the spectrum, contain information on the thickness of the sample. Despite the rapid, exponential decay of the energy loss signal with increasing energy loss, the characteristic absorption edges can also be detected in the high energy loss tail of the spectra, and many L-, M- and N-edges can be identified with specific high atomic number constituents in the sample. In addition to the characteristic edges, inelastic processes responsible for the energy loss spectra include photon excitations at the thermal excitation level of the zero-loss peak and electron transitions within the atom (energy losses of 1–150 eV). The plasmon excitations are associated with resonance in the conduction band of a metallic conductor (energy losses of 5–50 eV), but broad plasmon peaks are also found in nonconductors. There has been a steady improvement in the performance of the ‘hardware’ and the computer ‘software’ for microanalytical systems available for data processing and quantitative analysis in electron microscopy. This has been especially true for energy loss spectroscopy. The energy resolution in EELS is now an order of magnitude better than can be achieved using characteristic X-ray microanalysis, while the spatial resolution is now of the order of 2 nm. Nevertheless, quantitative microanalysis using EELS involves large errors, mainly in subtracting background corrections from the rapidly decaying tail of the energy loss spectrum. The remarkable energy resolution for EELS has revealed considerable fine structure in the absorption edges. This energy loss near-edge structure (ELNES) is associated with the chemical state of the atom, that is, the nature of the chemical bonding and the local, atomic coordination and symmetry in the solid. The extended energy loss fine structure (EXELFS) observed at energies above the edge is associated with local composition changes in the successive atomic shells surrounding atoms of the element responsible for the absorption edge. The EXELFS oscillations reflect changes in local atomic order. Microanalysis at this level is no longer a simple determination of the local chemical composition, but is beginning to answer questions on the chemical state of the atoms, their local packing symmetry and the degree of atomic order. Most recently, reliable energy-filtered transmission electron microscopy (EFTEM) has become commercially available. At the simplest level, all electrons which have lost energy can be excluded from the image-forming region in the electron microscope column. In this case, either diffraction patterns or bright-field and dark-field images are formed exclusively

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by elastically-scattered electrons in the zero-loss peak, providing a significant improvement in image contrast and revealing minor diffraction reflections. Even more remarkable are the images formed from the energy loss spectra of specific, characteristic absorption edges. The spatial resolution achieved for qualitative detection by EFTEM of individual elements is better than 2 nm, providing the electron microscopist with both morphological and chemical information at the level of a few hundred atoms.

Bibliography 1. V.D. Scott, G. Love and S.J.B. Reed, Quantitative Electron-Probe Microanalysis, Ellis Horwood, London, 1995. 2. D.B Williams and C.B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science, Plenum Press, London, 1996. 3. J.J. Hren, J.I. Goldstein and D.C. Joy (eds), Introduction to Analytical Electron Microscopy, Plenum Press, London, 1979. 4. R.F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum Press, London, 1986.

Worked Examples We now demonstrate some of the microanalytical techniques we have discussed by applying EDS and EELS to samples of polycrystalline alumina and a 1040 (0.4 % C) constructional steel. We start by using EDS in the scanning electron microscope. Our first specimen is polycrystalline alumina. Alumina is usually sintered (densified) with small quantities of additives (dopants) to prevent excessive grain growth and improve the sintering rate. Small quantities of impurities are also common in alumina (Ca, Fe or Si for example), and these may increase the rate of grain growth during sintering. It is important to be able to detect both impurities and dopants when they are present. Figure 6.28 shows a SEM micrograph from a polished and thermally etched alumina specimen, together with an EDS spectrum acquired from the entire region shown in the micrograph. The specimen was prepared from pure alumina, and intentionally doped with magnesium, together with a small amount of silicon and calcium. Automatic energy calibration and peak identification was performed by a computer program, and only the principle aluminium and oxygen peaks were detected. A thin-window EDS detector would have allowed detection of light elements, such as oxygen. The absence of Mg, Si, and Ca peaks in the EDS spectrum taken from the region imaged in Figure 6.28 was expected, since the total concentration of these dopants and impurities in this rather large sample volume is below the detection limit. However, an EDS spectrum, taken with the probe positioned at a single point located at a grain boundary, clearly shows the presence of both Mg and Si, although the amount of Ca at this grain boundary is still below the detection limit (Figure 6.29). It is clear from this result that both magnesium and silicon have collected at the alumina grain boundaries. We do not say that they have segregated, since we do not yet know the solubility limit of these elements at the sintering temperature. We cannot detect any of

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Figure 6.28 (a) Secondary electron image from thermally etched alumina. (b) EDS spectrum from the region shown in (a). Only Al and O peaks are recorded, together with carbon due to surface contamination.

the additives within the grains because the solubility limit of these cations is well below the detection limit of EDS. However, the presence of Si and Mg at the grain boundaries is an important finding, which implies that the alumina grains are saturated with these impurities. Figure 6.30 shows a TEM micrograph of a region taken from the same alumina specimen. Grain boundaries are indicated by arrows. EDS in the TEM can also be used to determine if silicon and calcium are present at specific boundaries or triple junctions. Figure 6.31 shows an EDS spectrum taken from the glass-containing triple junction identified in Figure 6.30. Significant peak intensities from both Si and Ca are detected, as well as from Mg. The

Microanalysis in Electron Microscopy 377 Al

O

Counts 1500

1000

500 Si Mg

0 1

2

3

4

Energy (keV)

Figure 6.29 EDS point spectrum recorded from a grain boundary in the alumina specimen. Al and O peaks are of course found, but also Mg and Si peaks, since these elements have collected at the boundary. In this particular case Ca, which was present in the specimen, is either below the detection limit or absent from this specific grain boundary.

Figure 6.30 TEM micrograph of the alumina shown in Figure 6.28(a). An amorphous phase (labelled glass) was detected at the triple junction.

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Figure 6.31 EDS spectrum taken from the triple junction shown in Figure 6.30. Mg, Si, and Ca are all detected. The Al signal is probably mostly from the neighbouring alumina grains, while the Ar signal is an artifact introduced during ion milling. Oxygen could not be detected with the EDS system used to collect this spectrum.

aluminium signal is at least in part from one or other of the alumina grains adjacent to the triple junction, while the argon signal is from argon ions that have been trapped in the thin TEM specimen during ion milling. We can also use parallel electron energy loss spectroscopy (PEELS) to microanalyse our alumina TEM specimen. Figure 6.32 shows portions of the PEELS spectrum acquired from an alumina grain in the energy loss region of the OK-edge (532 eV) and the Al L2,3-edge (73 eV). Quantitative analysis of the PEELS spectrum from the alumina grain confirms the stoichiometric concentration of Al2O3, and we can compare the ELNES of the Al edge from alumina to that from metallic aluminium (Figure 6.33). The observed differences in the two ELNES spectra reflect the differences in the nature of the chemical bonding of aluminium atoms in the ionic solid and the metal. We noted earlier that we could not specify whether magnesium or other dopants and impurities had segregated to the grain boundaries, or were enriched at the boundaries. The distinction between these two conditions depends on the equilibrium concentration of each additive in the bulk grains, and the value of the solubility limit for each species. Below the solubility limit, the element may segregate to the grain boundary if this will lower the grain boundary energy. If the same element is present at a metastable concentration level that exceeds the solubility limit, then the element may diffuse to enrich the grain boundaries in

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Figure 6.32 (a) Al L2,3 -edge (73 eV) and (b) the O K-edge (532 eV) in alumina.

order to lower the bulk concentration and approach the equilibrium solubility limit. Measuring bulk solubility limits at low concentrations is always difficult, and especially challenging for dopants and impurities in ceramics, when the solubility limit is often expected to be very low. EDS does not have a low enough detection limit for such a task, and so we will use WDS to measure the solubility limit for magnesium in Al2O3. Our sample is a polycrystalline Al2O3 that has been sintered for 24 h at 1600  C, after doping with magnesium at a level (5 atom %), well above the solubility limit. After sintering at 1600  C, the sample was taken directly from the furnace and quenched in water.

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Figure 6.33 Al L2,3 -edge in alumina and in aluminium metal. The position of the peak and the peak shape are significantly different in the two materials, reflecting the very different atomic bonding of the aluminium.

X-ray diffraction confirmed that two phases were present in the sample (Al2O3 and MgAl2O4). Hence, we are in the two-phase region of the Al2O3–MgO pseudobinary phase diagram and the magnesium concentration in the alumina grains should be at the solubility limit for 1600  C. The X-ray diffraction results are confirmed by SEM examination, which shows the MgAl2O4 phase is present in the form of platelets (Figure 6.34). For this experiment we use a WDS mounted on the same scanning electron microscope as our EDS system, and we must first maximize the WDS count rate with respect to the SEM operation conditions. We increase the beam current to a maximum and measure this beam current using a Faraday cup located on the sample stage. We now determine the WDS intensity as a function of working distance and magnification for a standard sample that is located on the same stage as our alumina specimen. Figure 6.35 shows results, recorded at 15 kVand using a thallium hydrogen phthalate (TAP) crystal in WDS. The maximum count rate corresponds to a working distance of 11.5 mm and a magnification greater than ·5000. The explanation for this result is shown schematically in Figure 6.36, which shows that the geometric configuration of the sample and detection assembly requires a minimum working distance and a minimum magnification if we are to maximize the collection angle for the wavelength-dispersive spectrometer. Once we have optimized the geometry of the system, we need to determine the intensity of the background, white radiation adjacent to the MgKa peak. This is demonstrated in Figure 6.37, which was recorded under the same working conditions for SEM that were used

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Figure 6.34 (a) Backscattered electron and (b) secondary electron SEM micrographs of the quenched Al2O3 sample. The MgAl2O4 phase is present as thin platelets.

when examining a pure sapphire crystal standard. This measured background can now be used to subtract the background from the WDS signal recorded from the MgO-saturated Al2O3. At 15 kV the depth of penetration of the incident electron beam into alumina is approximately 1.6 mm. From over 300 independent measurements giving a 95 % confidence level, the magnesium detection limit was found to be 4.6 ppm. This is really quite remarkable when compared with the limit of what could be done using EDS. The average concentration of magnesium in the saturated Al2O3 grains was found to be 132  11 ppm. Assuming that the quenching rate was fast enough, this then represents the solubility limit of magnesium in Al2O3 at 1600  C. We now examine a 1040 steel (0.4 % C) sample by EDS in the scanning electron microscope. Quantitative EDS analysis of the carbon content for any steel is almost impossible, since the carbon content is at the limit of detection for light elements and is often confused with carbon contamination in the system. Figure 6.38 shows an EDS spectrum taken from a large area of the specimen surface. A carbon peak is visible that would suggest a carbon content much higher than expected for 1040 steel. This is due to carbon

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Figure 6.35 WDS characteristic peak intensity in the scanning electron microscopes as a function of (a) the working distance for the sample and (b) the magnification of the image.

contamination at the surface from the breakdown of volatile hydrocarbons present in the vacuum system of the electron microscope. This signal has nothing to do with the carbon content of the alloy. However, Fe3C platelets are present in the specimen and this compound has a carbon content well above the detection limit for EDS. The carbide platelets are clearly seen in the pearlitic regions of the microstructure (Figure 6.39). The presence of the higher carbon content in the pearlite can be confirmed from an EDS point spectrum analysis taken with the probe beam located on the pearlite (Figure 6.40), which should be compared with Figure 6.38. In addition to pearlite, small dark areas are visible in the a-Fe grains and in the pearlite microstructure. Some of these dark regions may be Fe3C particles that have precipitated below the eutectoid temperature (723  C) during heat treatment. This can be checked by

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Figure 6.36 Schematic drawing of the SEM column geometry demonstrating how, at small working distances and low magnifications the backscattered electron (BSE) detector may ‘shadow’ the WDS detector, reducing the intensity of the signal that is acquired.

taking an EDS line-scan across some of the dark regions (Figure 6.41). In Figure 6.41 an X-ray line-scan for the carbon peak crosses two dark regions, one located within a pearlitic eutectoid region, and the second at the boundary between the pearlite microstructure and an a-Fe grain. The dark region at the interface between the pearlite and a-Fe has a significantly higher carbon content, and we conclude that this is a carbide particle. The dark region within

Figure 6.37 WDS background intensity for the Mg Ka peak region (recorded from a pure sapphire standard crystal).

Figure 6.38 EDS spectrum taken from an a-Fe grain in a 1040 steel. The carbon signal is far too high to correspond to the 0.4 wt % in the steel and certainly not to residual carbon dissolved in the a-Fe. This signal is in fact due to surface contamination.

Figure 6.39 Secondary electron SEM micrograph from an etched 1040 steel showing a-Fe grains and pearlitic regions of the eutectoid containing both a-Fe and Fe3C.

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Figure 6.40 EDS spectrum taken from a pearlite region in the microstructure of Figure 6.39. Note the increase in the carbon signal relative to the Fe signal when compared with Figure 6.38.

Figure 6.41 EDS line-scan from the carbon signal and a secondary electron SEM micrograph of the interface between pearlite and a-Fe.

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the pearlite does not show any higher an EDS carbon count than the background, and is most likely a surface pore. The EDS line-scan also shows the presence of an additional carbide particle (on the left) that was not immediately evident from the micrograph. The presence of Fe3C can therefore be detected quite easily from EDS line-scans, but a comparison of the EDS results with the secondary electron image from the same area is necessary to interpret the microstructure with any confidence. Quantitative analysis of the carbide phase would be much more difficult, since the carbon signal is strongly affected by the surface contamination.

Problems 6.1. Why is it important to know the chemical composition associated with microstructural features? Give three examples. 6.2. Why would you expect a fracture surface to be a ‘difficult’ sample for microanalysis in the scanning electron microscope? What purpose might be served by comparing microanalysis from a fracture surface with that from a polished and etched specimen of the same material? 6.3. Why does the best electron beam energy for microanalysis depend on the density of the sample? What are the disadvantages of working with too high or too low a beam energy? 6.4. What factors determine the width of the detection window for a characteristic Xray excitation line selected for quantitative analysis? How would you optimize the counting statistics to obtain the maximum signal-to-noise ratio for the selected line? 6.5. Distinguish between sensitivity and accuracy in microanalysis. 6.6. Discuss the statistical limitations on spatial resolution in microanalysis. 6.7. There is little point in attempting to analyse regions much less than 1 mm in diameter in the scanning electron microscope. Why? 6.8. For quantitative analysis, the characteristic X-ray spectrum excited by an electron probe should be corrected for background white radiation, atomic number effects, backscatter losses, X-ray absorption and X-ray fluorescence. In what order should these corrections be made? 6.9. An aluminium-copper alloy contains 3.5 % of copper. Would you expect the intensity of the Cu Ka radiation to be increased or decreased by the atomic number effect and the backscatter losses? Why? 6.10. A series of iron–nickel alloys are to be microanalysed. Discuss the relative importance of the absorption and fluorescent corrections for iron and nickel as a function of the alloy composition.

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6.11. A polished cross-section from a brazed joint is to be microanalysed. Would you recommend that the line of intersection of the joint with the polished surface be aligned parallel or normal to the X-ray collection system? Assume that the composition of the components being joined and that of the braze may contain copper, silver and zinc. Check the relative characteristic peak positions and the absorption edges for these constituents. 6.12. Why is electron energy loss spectroscopy now the dominant technique for thin-film microanalysis in the transmission electron microscope? 6.13. How would you expect the limit of detection in energy loss spectroscopy to vary with the atomic number of the constituent? 6.14. What are the advantages of parallel electron energy loss spectroscopy as an analytical tool in the transmission electron microscope? 6.15. Compare the energy resolution of EELS and WDS systems. 6.16. A joint between Al–4 wt % Cu and Cu–10 wt% Sn was checked by EDS in SEM using Ka characteristic lines and Figure 6.42 shows the distribution of EDS intensity as a function of distance from the interface. (a) Explain the expected difference between the relative measured intensity and the real chemical distribution at the interface. (b) Estimate the spatial resolution of the EDS line-scan based on the measured intensity values. Explain your assumptions. (c) What was the approximate accelerating voltage used to acquire the line-scan? Justify your estimate!

Figure 6.42

EDS line-scan of a joint between a Al–4 wt % Cu alloy and a Cu–10 wt % Sn alloy.

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Microstructural Characterization of Materials

Figure 6.43 EELS spectrum from a thin-film Al2O3 TEM sample.

6.17. An EELS analysis of a thin-film Al2O3 sample gave the low loss spectrum shown in Figure 6.43. Estimate the sample thickness. 6.18. The cross-section of a silicon sample is sketched in Figure 6.44. Certain regions of the sample are coated with a 500 nm thick film of titanium. The sample was characterized by a scanning electron microscope that was equipped with secondary electron (SE), backscattered electron (BSE), and EDS detectors that were located at take-off angles of 45, 70, and 30 , respectively. The accelerating voltage was 20 kV, and the incident electron beam was perpendicular to the sample surface. (a) Discuss the X-ray emission from the silicon beneath the titanium coating as a function of position on the sample. (b) Estimate the percentage of Si Ka photons that can escape from the sample in the direction of the EDS detector.

Figure 6.44 Schematic drawing of a Si wafer partially coated with Ti.

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Figure 6.45 Backscattered electron (BSE) micrograph of AlN, containing a secondary phase (bright contrast).

(c) Sketch the expected, qualitative intensity distributions for all three detectors for a line-scan across the surface of the specimen. Explain the relative changes in the intensities that you predict as a function of the beam position. To estimate the energy of the electrons that penetrate a thickness H of the titanium use the relationship: E2H ¼ E20 H·2:387 · 105 ðV2 Þ, where H is in nm and E is in V. Given:

Material Atomic Atomic mass Density Mass absorption Mass absorption Characteristic number (g mol1) (g cm3) coefficient for coefficient for X-ray (Ka) Ti-Ka (cm2 g1) Si Ka (cm2 g1) energy (eV) Ti Si

22 14

47.9 28

4.54 2.33

108 130

1458 347

4510 1740

6.19. During a study of an AlN sample, it was found, from secondary electron micrographs, that the sample contained 10 vol% of a second phase. Qualitative SEMEDS analysis showed that this phase contained yttrium and, possibly, oxygen and aluminium (Figure 6.45). What techniques would you suggest to determine

Figure 6.46 Schematic drawing of copper-filled vias in a silicon wafer.

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quantitatively both the structure and chemistry of the second phase? Include suggested operating parameters for the methods you select. 6.20. In the microelectronics industry, silicon-based devices must be characterized during processing. Figure 6.46 shows a schematic drawing of vertical contact lines (vias) filled with copper in a silicon wafer. (a) Which SEM signals would you select to characterize the shape and size of the vias, both before and after filling them with copper? Explain your choice. (b) Using a 20 kV incident electron beam in SEM, what spatial resolution would you expect for EDS of the copper vias? Explain quantitatively. (c) A sample containing copper-filled vias is coated with 100 nm of gold. Should an EDS signal be detectable from the silicon and copper? If so, explain the influence of the gold film on the intensity of the measured signal. Given:

Si Cu Au

Ka (keV)

Atomic number

Atomic mass (g mol1)

Density (g cm3)

1.740 8.046 9.712 (La)

14 29 79

28.0855 63.55 196.97

2.33 8.96 19.3

7 Scanning Probe Microscopy and Related Techniques So far, we have limited our discussion to microstructural probes that are based on visible light (the optical microscope), X-ray diffraction, or high energy electrons. We have discussed signals that are generated by both the elastic and the inelastic interaction of these probes with a carefully prepared specimen and we have established that microstructural information, characteristic of the sample, can be collected on the microstructural morphology, the crystal structure and the chemical composition of the phases that are present. Moreover, we have demonstrated that this information can be resolved over a very wide range of dimensions, from the everyday scale of visual observation, down to just a few atomic diameters, or even to the interplanar spacings present in the grains of individual crystals. In this chapter we take the discussion a stage further and describe what can be achieved when the probe of electromagnetic radiation or high energy electrons is replaced by a sharp, needle-shaped, solid probe that is brought into close proximity or into contact with the surface of the sample we wish to study. We shall discover that such a probe can provide us with additional microstructural information. This new information is beyond the range of the techniques we have discussed so far, and reveals details of both the surface structure and the surface properties of engineering materials at resolutions that, under suitable conditions, can image individual atoms. The information that is now available from scanning probe microscopy and some related techniques has proved to be of importance for applications that range from engineering problems in lubrication and adhesion, to softtissue, biological, interactions at membranes and across cell boundaries, as well as to electro-optical devices and sub-micrometre electronics.

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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Microstructural Characterization of Materials

7.1

Surface Forces and Surface Morphology

An observant reader will already have realized that, in both optical and transmission electron microscopy, the image plane of the source and the image plane of the specimen are inversely related, since the diffraction pattern from the specimen in reciprocal space is focused in an image plane of the source, while the image from the specimen, focused in real space, is convoluted by the angular distribution of the probe radiation that is emitted from the source. Over 40 years ago scientists working on field emission, at the National Bureau of Standards, outside Washington (now the National Institute of Standards and Technology NIST), placed a tungsten field emission tip, a ‘source’, in close proximity to a sample surface and monitored the changes in the field emission current that were associated with changes in surface topology, immediately beneath the tip field emitter, as the sample surface was scanned perpendicular to the emitter. At that time it proved quite impossible to maintain the required dimensional stability, and it was only 10 years later, with the development of reliable piezoelectric drives, that the thermal and mechanical stability necessary to ensure reproducible results was finally achieved. Today, a wide range of experimental methods are available for studying either the nanometre-scale contact forces between two solid objects, or the electrical properties of individual surface contacts, or even the atomic structure and surface chemistry of small surface areas. We have placed all these diverse techniques under a single heading, scanning probe microscopy, even though only two of the instruments, the atomic force microscope and the scanning tunnelling microscope are in fact ‘scanning probe’ instruments. We start with a brief discussion of surface-force measurements, and conclude the chapter with the remarkable results now being reported for atom probe tomography, a technique that is able to dissect suitable field ion microscope samples and reveal the three-dimensional chemical distribution of the individual atoms, identified by their atomic mass. 7.1.1

Surface Forces and Their Origin

When two solid surfaces approach one another, the interaction between them includes both attractive and repulsive components. These forces may be either long-range or short-range, and the interaction between the surfaces is strongly affected by the presence of surface adsorbates or by a gaseous or liquid environment in the space separating the solid surfaces. Before considering the multi-atomic interactions occurring when two surfaces are brought together, we first describe a text book, two-atom model for the individual atomic interactions. If the long-range forces between the two atoms considered are attractive, while the short-range forces are repulsive, then we can develop a simple, two-body model for the inter-atomic bonding of the atoms. This model can predict the qualitative attractive and repulsive forces between atoms from the known physical and chemical properties: the equilibrium, interatomic separation in the solid, the heat of formation and tensile modulus of the material, and its coefficient of thermal expansion. R R Figure 7.1 illustrates this simple atomic force model F(R) and its integration VðRÞ ¼ 1 FðRÞdR to give the potential energy of this two-body system as a function of the interatomic spacing E(R). The position of the minimum in the potential energy and the depth of the potential energy well are set equal to the equilibrium interatomic spacing and the atomic binding energy, respectively, while the

Scanning Probe Microscopy and Related Techniques

393

Force F

Attraction

Attractive force F A

Repulsion

Interatomic separation R Repulsive force F R

R0

Net force F N

Repulsion

Interatomic separation R Attraction

Potential energy E

Repulsive energy E R

Net energy E N E0

Attractive energy E A

Figure 7.1 Interatomic forces and interatomic potentials. The long-range attractive force and the short-range repulsive force balance at the equilibrium inter-atomic separation and the sum of the two forces can be integrated over distance as the atoms approach to give the atomic potential. The minimum of the atomic potential curve is the ground state of the system and the potential curve itself can be related to the physical properties of the corresponding solid (see text).

curvature at the bottom of the potential well and the asymmetry of this curvature are monotonic functions of the elastic modulus of the solid and its thermal expansion coefficient. In applying similar modelling concepts to two solid surfaces that are brought into close proximity, we will have to account for the multi-body interactions of all the surface atoms

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AB

AA

AA

CC AB

Distance Figure 7.2 Possible interactions between atoms on two contacting surfaces (AA), between surface atoms and atoms in a gaseous or liquid environment (AB), or between atoms in the environment (CC). Reprinted with permission from R.G. Horn, Surface Forces and Their Action in Ceramic Materials, Journal of The American Ceramic Society, 73(5), 1117–1135, 1990. Copyright (1990), with permission from Blackwell Publishing Limited.

from both surfaces. We will also need to consider the effects of surface roughness or local curvature, and, especially, the presence of a liquid or gaseous environment. As shown schematically in Figure 7.2, this leads to a three-phase model that includes the two solids and the inter-phase, environmental region. Again, both long-range and short-range forces must be considered, and it is now convenient to distinguish three interaction zones in the potential energy curve (Figure 7.3). At larger separations, only the long-range forces are experienced. These are usually attractive, and we refer to these larger separations as the non-contact region, defined as extending beyond the inflection point on our schematic potential–distance graph. This inflection point corresponds to a maximum in the net attractive force. At distances less than the separation corresponding to a potential energy equal to zero, we can define a contact region. In this regime the attractive forces are negligible and the short-range and repulsive forces dominate the interaction between the two bodies. In between these two regions, in the zone that includes the equilibrium spacing for the two surfaces and either side of the minimum in potential energy, we can

Repulsive force

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Intermittent contact region

Attractive force

Distance

Contact region

Non-contact region

Figure 7.3 Three regions can be distinguished on the potential energy curve for the interaction of two solid surfaces: a non-contact region dominated by attractive forces; a contact region dominated by the repulsive force; an intermediate, semi-contact region that includes the equilibrium separation and in which the attractive and repulsive forces are of similar magnitude.

define a semi-contact region for which the attractive and repulsive forces have similar magnitude. As we shall see, the interactions in these three regions can be separated in the atomic force microscope, and this provides unique information on the nature and distribution of the various surface forces. The strongest long-range forces between two solid surfaces are the Coulombic electrostatic forces, which may be either repulsive or attractive, depending on whether the total charge carried by the two surfaces is of the same or of opposite sign. At shorter distances, polarization, or van der Waals forces are experienced. These polarization forces are classified under three headings. The strongest forces are associated with permanent molecular dipole moments that create local electric fields and lead to ordered packing of an assembly of the surface dipoles. The electric field of a molecular dipole may itself polarize and attract neighbouring atoms, in proportion to the strength of the dipole and the polarizability of the interacting atoms. This second form of polarization force is known as the Debye interaction. Finally, random fluctuations in the electrical fields of any polarizable atom will also induce localized, attractive interactions with neighbouring polarizable atoms and generate forces that are termed London, or dispersion forces. van der Waals forces all decay rather rapidly (usually as d
7). The parameters that determine the strength of the interaction are the values of the dipole moment of the atomic or molecular entities at the surface and their polarizability, together with their separation. Asymmetric molecular adsorbates, such as carbon dioxide or water vapour, have large dipole moments that lead to strong van der Waals adsorption, while symmetrical molecular assemblies, such as methane, can have appreciable polarizability but no dipole moment.

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Polar groups that contain hydrogen, especially OH
and NH2
, establish hydrogen bonds that are strongly attractive at very short range, and are typical of the polar bonds that are critical for the existence of life on earth. The strong, short-range, repulsive forces between atoms are associated with the overlap of the atomic inner electron shells and these dominate the interaction as the solid surfaces are forced together. The repulsive forces may also dominate when soluble additives in a liquid medium, such as polymer molecules, prevent the solid surfaces from coming into direct contact, an effect termed steric hindrance. Charged surfaces in a nonpolar, liquid or gas environment of low dielectric constant, lead to strong, but rather unpredictable, electrostatic forces. The presence of a polar environment, with a high dielectric constant, leads to a layer of counter-charge adjacent to the charged solid surface of opposite sign. This is termed the Debye or double layer. Any surface charge, together with the diffuse layer of counter-charge, can be of immense practical importance and is readily investigated by techniques for surface force measurement. The most successful theory that has been developed to predict the surface forces responsible for the stabilization of colloid dispersions and the wetting of solids, was established some 50 years ago by Derjaguin and Landau in Russia, and, independently, by Verwey and Overbeek in Holland. The theory, usually referred to as the DLVO theory, has been modified over time in order to explain the factors that determine the forces between solid surfaces in the presence of surface segregation, adsorbate layers, or liquid solutions. DLVO theory is now used to develop chemical formulations for double layers that are used to inhibit particle contact by establishing metastable minima at nanometre particle separations. These adsorbates promote wettability and dispersion stability, as shown in Figure 7.4. 7.1.2

Surface Force Measurements

To fully understand surface forces we need to measure them under controlled conditions. The most successful approach to achieving this was developed by Jacob Israelachvili and his coworkers some 30 years ago. These workers constructed a sensitive force–balance instrument that was based on two curved, cylindrical mica surfaces mounted at right angles and brought into close proximity (Figure 7.5). The cylinders can be coated, for example by vacuum vapour deposition, in order to modify the surface structure, conductivity and composition of either or both substrates. The effective area of contact between the crossed cylinders used in this geometry is a circle. With the help of Israelachvili’s equipment, the force normal to the two surfaces can be measured to an accuracy of about 0.1 mN m
1, while the spacing between the surfaces in the contact region can be determined to an accuracy of about 0.1 nm. One of the most remarkable findings of this work has been the presence of surface force oscillations. These may be observed as a function of the surface separation. If the surfaces are separated by a polysaccharide or polyalcohol lubricant film, measurable force oscillations can be detected out as far as 10 nm (Figure 7.6). The oscillations are associated with the presence of steric hindrance and molecular forces which favour a complete, integral number of molecular layers of the boundary lubricant that is dissolved in the liquid medium between the two surfaces. A wide spectrum of micromechanical surface phenomena have been observed, dependent on the chemistry and surface crystallography of the substrates, and the composition of the liquid or gaseous medium.

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(a)

Energy

Double layer

Total

0 van der Waals

Separation distance (b)

Energy

Double layer

Total

0 van der Waals Secondary minimum

Separation distance Figure 7.4 A suitable double layer at the surface between two solids in a liquid environment can result in metastable minima in the surface energy of the system and determine the wettability of solid surfaces or the stability of particulate dispersions. (a) A weakly polar liquid generates a dispersed double layer. (b) A strongly polar liquid can result in a thinner double layer and a secondary energy minimum. Redrawn with permission from R.G. Horn, Surface Forces and Their Action in Ceramic Materials, Journal of The American Ceramic Society, 73(5), 1117–1135, 1990. Copyright (1990), with permission from Blackwell Publishing Limited.

7.1.3

Surface Morphology: Atomic and Lattice Resolution

We have established that surface forces normal to the surface can be measured extremely accurately, as a function of the separation of two solid surfaces, and that sub-nanometre vertical resolution has been achieved. We now need to consider the lateral resolution that might be possible when using a solid probe to explore the morphology of a solid surface and its chemical or physical properties. A flake of cleaved mica is an atomically smooth and almost featureless substrate, despite the presence of physically adsorbed moisture from the environment. The few, isolated

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Microstructural Characterization of Materials

Figure 7.5 Basic geometry for monitoring contact forces. Two curved mica plates are brought into close proximity, and the normal force between the plates is measured as a function of their separation.

cleavage steps on such a surface rarely interfere with surface force measurements (although dust particles can interfere a great deal). No detectable structure is normally visible on such a substrate. However, most other surfaces do contain a variety of morphological features that have varying degrees of structural order and are of varying practical importance. Surface roughness is seldom random in either amplitude or wavelength. Contributions to surface roughness may come from polishing scratches and oriented grinding or machining ridges, or perhaps from corrosive pitting or chemical etching, which often depends on the 3

2

F (mNm–1)

1

0

-1

Structural force due to molecular packing Attractive van der Waalsforce

-2

-3 0

2

4

6

8

10

D (nm)

Figure 7.6 Schematic illustration of the balance between attractive van der Waals forces and structural forces associated with steric hindrance. As the surfaces approach each other, the maxima and minima of the structural force define the possible equilibrium thicknesses for the multilayer lubricant film between the two solid surfaces. For small values of the lubricant film thickness, the attractive van der Waals force will be compensated by the repulsive structural or steric force. The seven smallest distances at which the attractive and repulsive forces are balanced are denoted by the black circles.

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crystal structure of the surface. The equilibrium surface of a polycrystalline solid may contain facetted crystal surfaces or grooved grain boundaries. Second phases may also be present in the material. Such second phases may either exist in the microstructure of the bulk material, or they may have been deposited on the surface as a coating, or result from a corrosion reaction. A second phase on the surface may be present as a continuous thin film, or as isolated, discontinuous or interconnected islands. If the scanning probe is the tip of a solid needle whose geometry can be approximated by a polished cone or pyramid, then it will have an effective tip radius that is usually in the range of 20 – 200 nm, much greater than the interatomic spacings on any solid sample surface. The value of the probe tip radius can usually be determined unambiguously by examining the probe in a scanning electron microscope. It is generally assumed that the effective contact area which is established at the sample surface as the probe approaches the specimen will have an area of at least 100 nm2. The probe tip radius directly limits the resolution that may be realized in a scanning probe image. One reason for this is illustrated in Figure 7.7. As the probe is scanned over the surface, the number of atoms beneath the probe that contributes to the measured signal will vary with the periodicity of the interatomic spacing on the surface. The relative amplitude of these variations in the force oscillations will decrease as the tip radius increases, but the wavelength of the oscillations will remain that of the atomic structure. For the large tip radii, these fluctuations will be undetectable, but for tip radii of

Figure 7.7 A nanoscale, solid probe, scanned over a solid surface, will experience an oscillating force that has the periodicity of the atomic spacing of the scanned surface, even though the probe tip radius may be an order of magnitude larger than the atomic spacing of the sample.

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Microstructural Characterization of Materials

Figure 7.8 The periodicity of the carbon ring structure on the basal plane of graphite can be revealed by the atomic force microscope, even though the probe tip has a much larger radius of curvature than the carbon ring repeat distance. Is this ‘true’ spatial resolution (see text)? (Reproduced from NanoTech America).

the order of 10 nm it has sometimes proved possible to resolve characteristic inter-atomic lattice spacings, such as the graphite ring structure on the basal plane (Figure 7.8) even though the measured force is due to a many-body interaction that is on a significantly larger scale. It has been argued that this is not ‘true’ spatial resolution and that only electrical measurements, which are dependent on the very sensitive response of field emission tunnelling to the interatomic separation, are able to resolve the atomic structure of the surface (Section 7.2). Images such as Figure 7.8 are certainly a good approximation to resolution of the atomic packing in the plane of the sample surface.

7.2

Scanning Probe Microscopes

In all scanning instruments, during the data collection process, the data are collected point by point (that is, pixel by pixel) and not integrated over time for the field of view seen in the image. Sequential data collection in scanning electron or scanning probe microscopy is therefore quite different from that used in optical or transmission electron microscopes, where data from all image points (pixels) are collected in parallel. In scanning probe microscopes, the rate of data collection is very slow and this limitation is exacerbated by mechanical and thermal drift of the sample and the imaging system, so that reliable data collected for an individual image comes from only a limited number of pixels. Images are

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often restricted to 256 · 256 pixels and it is unusual for the data set for each image to exceed more than 105 pixels when using a solid probe to scan the specimen surface. We first provide a historical chronology of the different scanning probe and other signals to be discussed in this chapter and illustrate this chronology with sketches that show the development over time of the concept of a sharp, solid probe (Figure 7.9). 1936 Field emission of electrons was first demonstrated: a negative potential was applied in a high vacuum to the sharp tip of a thermally smoothed, refractory metal wire. The needle-shaped sample was rigidly mounted by spot-welding to a heating filament that was attached to two electrodes [Figure 7.9(a)]. The field emission current i was shown to obey the Fowler-Nordheim equation: ! 1 3 ð
=wÞ2 2 w2 F exp
B ð7:1Þ i¼A ð
þwÞ F where m is the Fermi energy of the solid, f is the work function of the surface, F is the electric field strength acting at the surface, and A and B are constants. The projection image of the field emission current distribution over the hemispherical tip on a phosphor screen clearly showed the dependence of the work function on the crystal structure of the tip surface. 1956 Atomic resolution was first demonstrated in the field-ion microscope. Imaging was achieved by field ionization of gas ions at the surface of a cryogenically cooled and positively-charged sharp, refractory metal tip [Figure 7.9(b)]. The ions were accelerated to form a projection image of the crystalline tip surface in which the surface atoms at the edges of the regularly packed atom planes were clearly resolved. By increasing the electric field strength at the tip surface, it was possible to ionize the atoms at the edges of the crystal planes and remove them from the surface by field evaporation, to form symmetric, atomically smooth tip surfaces. The rigid support electrodes for the sharp tip were cooled to cryogenic temperatures. The field ion microscope was the first instrument that was able to achieve atomic resolution. 1967 In the atom probe, field-evaporated ions from the surface of a field-ion microscope specimen were chemically analysed by time-of-flight mass spectrometry to give mass-resolved chemical analysis of the sample at sub-nanometre resolutions [Figure 7.9(c)]. The metal tip was aligned to bring a selected region over a hole in the fluorescent image screen, so that atoms that were field-evaporated from the selected region could pass into the mass spectrometer. Over the following 40 years, areal detection of the field-evaporated ions over a wider angle became possible, leading to massive data sets of statistically significant, mass-resolved chemical information that possess sub-nanometre resolution and were derived from millions of atoms fieldevaporated from the sample needle. 1982 The feasibility of the scanning tunnelling microscope was demonstrated soon after piezoelectric control of vertical and lateral displacements was shown to be capable of the mechanical and thermal stability needed for the atomic resolution of a scanning probe instrument [Figure 7.9(d)]. The scanning tunnelling microscope was able to probe the electronic structure of the surface region down to atomic resolution. The thermodynamically stable restructuring of the packing of surface atoms and the

402

Microstructural Characterization of Materials Tip sample

Tip sample

R~1 nm

R~0.1 µm

ε

T=20–80 K F ~ 0.5 Vnm–1

V

F ~ 0.05Vnm–1 ε Screen

ε

V

+

Screen

+

+

(a)

(b)

Tip sample

Tip probe R~0.1 µm

V R~1 nm Screen

+

+

V

Time of flight spectrometer

~1 nm Sample

I

+

Multichannel detector

(c)

(d)

Tip probe

Cantilever force meter

R~1 nm

(e) Scan ~1 nm Sample

Figure 7.9 Fifty years of tip assemblies, 1936–1986 (see text). (a) Field emission tip assembly. (b) a Field-ion microscope tip. (c) Atom probe tip assembly and time-of-flight mass spectrometer. (d) Scanning tunnelling probe assembly. (e) Atomic force probe scanning assembly and cantilever force monitor.

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existence of several crystal structures that were unique to the surface was discovered. The sharp tip was rigidly mounted on a support frame that was attached to a mechanically stiff, cylindrically symmetric, piezoelectric actuator that allowed for sub-nanometre control of the separation of the probe tip from the specimen surface and the lateral positioning of the probe. 1986 Atomic force microscopy was developed by mounting the solid, needle-like probe at the end of a flexible cantilever and monitoring the displacement of the cantilever due to the surface contact forces [Figure 7.9(e)]. The flexible cantilever probe assembly was combined with a rigid, piezoelectric x-y-z actuator module and the movement of the cantilever was monitored by a split-beam laser detection system. The atomic force microscope, unlike its predecessors, no longer required a vacuum, but could be operated under atmospheric conditions or in a controlled gaseous or liquid environment. Resonant vibration of the cantilever allowed the tip to probe a wide range of electrical, magnetic and mechanical properties of the surface at lateral resolutions in the nanometre range. By controlling the separation of the probe at fixed distances from the sample surface it proved possible to operate the atomic force microscope in either contact, semi-contact or non-contact modes, revealing details of the dominant surface forces for a wide range of environmental conditions and engineering materials, including soft, compliant materials such as adhesives and biological tissues. 7.2.1

Atomic Force Microscopy

The basic components of a scanning probe system for either scanning tunnelling or atomic force microscopy are shown in Figure 7.10. The table-top instrument is mounted on a rigid base of high damping capacity, in order to ensure freedom from extraneous mechanical vibrations. The chamber is fully enclosed, both to provide thermal stability and to allow for environmental control. A high vacuum is not necessary for atomic force microscopy. The Computer Microscope

X,Y

Z

Probe

Digital Signal Processor

Scanner

Controller

Electronics Interface

Detector Sample

Graphic Display

Anti-Vibration Table

Figure 7.10 Basic components of a scanning probe microscope for scanning tunnelling or atomic force microscopy are designed to combine sub-nanometre mechanical and thermal stability with sub-nanometre position control and rapid response times (see text).

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Microstructural Characterization of Materials

probe assembly includes both the probe cantilever mount and the sub-nanometre precision, piezoelectric position control system for the probe tip. The sample stage is adjustable to allow for optical alignment of the sample with respect to the probe and to permit coarse adjustment of the initial probe–sample separation. In atomic force microscopy, the splitbeam laser system used to monitor the mechanical deflection, and the amplitude and frequency of vibration of the probe cantilever are both within the microscope chamber (Figure 7.11) but with external control of the mechanical detection system alignment. The scanning of the probe and the probe–sample separation are under analogue control. All the parameters that are transmitted by the graphics interfaces to the display and control monitors are digitized. The heart of the atomic force microscope is the cylindrical, piezoelectric position control system. Movement in the z-direction is achieved by applying a voltage across the piezoelectric tube wall that contracts or expands the axial length of the tube. Movement in the x–y plane is achieved by applying voltages of opposite sign across diagonal segments of the tube, causing it to deflect by bending in either the x or y directions. Two other components are no less important to the successful functioning of the atomic force microscope: These are the design of the cantilever for the probe tip, and the detection system for the tip displacement.

D

Laser

et ec to r

Primary lens

Mirror Mirror Piezo tube scanner Tracking lens

Cantilever holder Sample

Figure 7.11 In the atomic force microscope, the controls for laser alignment and coarse adjustment of the sample position are external to the environmental chamber, but the photodiode optical assembly is inside the chamber.

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V-shaped cantilevers [Figure 7.12(a)] are flexible about the points of support of the V, but possess high torsional rigidity. They are intended for scanning the sample along the axis of the cantilever. Alternative, simple beam cantilevers [Figure 7.12(b)] are much less flexible than the V-cantilever perpendicular to the beam axis, but they are sensitive to torsional displacements about an axis parallel to the cantilever beam. They can therefore be scanned either parallel to the cantilever beam, as in conventional scanning, or perpendicular to the cantilever, when they provide information on the local frictional forces that act parallel, rather than normal, to the plane of the sample surface. Many different tip materials have been used for atomic force microscopy, including diamond, tungsten and tungsten carbide, but silicon nitride is often preferred, since this material possesses good chemical and physical resistance to tip damage. The high elastic rigidity of silicon nitride reduces hysteresis losses and mechanical phase delays in the signal generated, effects that would be associated with the tip rather than the sample. Silicon nitride is therefore the tip material of choice for contact mode imaging (see below), but the tip radii are typically only of the order of 20 – 60 nm, which places atomic resolution beyond the reach of silicon nitride tips. One alternative, silicon tips, can be integrated into a single-crystal, silicon cantilever. These are now often prepared by focused ion beam milling (Section 5.8). They provide higher resolution, since the nominal tip radii are typically less than 10 nm. Silicon tips are therefore preferred for applications in the semi-contact region, usually explored by using a vibrating cantilever (the tapping mode of operation, see below). An additional ‘enabling technology’ for atom force microscopy is the detection system for determining the tip displacement as it is scanned over the sample surface. This is usually based on a split photodiode detector operating with a solid state laser source (Figure 7.13). As the cantilever is deflected, it reflects a laser beam which then scans across a small gap between the two halves of the photodiode, so that the signal generated depends on the proportion of the light falling on each half of the photodiode. The cantilever is gold-coated to maximize reflectivity and the detector is tuned by adjusting the tilt of a mirror. The length of the optical arm controls the maximum deflection amplitude that can be measured for each setting. If a piezoelectric drive is used to vibrate the cantilever close to its resonant

Figure 7.12 Cantilever assemblies for the atomic force microscope. (a) A thin, V-shaped design provides maximum bending sensitivity. (b) A simple beam that permits both flexure and torque of the cantilever.

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Microstructural Characterization of Materials

Figure 7.13 Movement of the cantilever probe support is monitored by a laser beam that is reflected onto a split photodiode and generates a signal that is proportional to the deflection of the cantilever.

frequency, then the alternating signal from the split photodiode can be compared with the drive signal in order to measure phase shifts in the signal that are related to nanostructural features on the sample surface. Finally, it is also possible to apply a bias voltage to the probe and then monitor the electric current between the probe and the sample. This is, of course, the basis of the scanning tunnelling microscope, but a bias voltage may also provide some additional flexibility in the operation of an atomic force microscope. 7.2.1.1 Data Collection and Interpretation. The principle of operation of the atomic force microscope may be simple, but it should not be forgotten that the instrument is monitoring the surface topology and behaviour of materials in a spatial regime that may extend down to atomic separations. Not surprisingly, these measurements are sensitive to many artifacts, some of which are still poorly understood. For example, piezoelectric scanners show non linear hysteresis and their response may change with time, especially in a new instrument (a process termed ‘ageing’). This requires frequent calibration of the system. Nonlinearity of the piezoelectric response also leads to distortion of the image, unless this is specifically corrected. Time-dependent drift of the atomic force microscope system is very common, leading to further image distortions that must be recognized. To ensure that the deflection of the cantilever is accurately measured, the electronic detection system must be tuned, both to minimize noise and to ensure a fast electrical response of the system in order to follow the cantilever movement. The plane of the sample surface should also be as near as possible coplanar with the x–y scan of the probe tip. Expecting artifacts to be present is a good defence against being misled into interpreting any imaging defect as a feature of the surface topology. 7.2.1.2 Modes of Operation in Atomic Force Microscopy. The basic structural features of the atomic force microscope and the scanning tunnelling microscope are compared in Figure 7.14. In the scanning tunnelling microscope, it is the current–voltage characteristics i (V) that are measured as a function of the probe separation z at a site in the x–y plane of the

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Figure 7.14 (a) In scanning tunnelling microscopy the current–voltage characteristics i(V) are measured as a function of the probe separation z at a chosen site in the x–y plane of the sample surface. (b) In atomic force microscopy the displacement of the probe tip, mounted on the end of a flexible cantilever beam, is measured. (c) Vibrating the cantilever beam of an atomic force microscope close to its resonant frequency allows the amplitude, frequency and phase of the oscillations to be monitored as additional signals that provide spatially resolved data on the properties of a sample surface.

sample surface [Figure 7.14(a)]. In the atomic force microscope, it is the displacement of the probe tip at the end of the cantilever that is measured [Figure 7.14(b)]. Vibrating the cantilever probe by a tuned piezoelectric drive close to its resonant frequency, allows the amplitude, frequency and phase of the oscillations to be collected as additional signals that provide spatially resolved data on the properties of the sample surface [Figure 7.14(c)]. A simple, straightforward application of the atomic force microscope is to monitor surface topology in the contact mode. The probe is lowered onto the sample until a surface repulsion is detected as a positive deflection of the cantilever. Two options are then possible: either the changes in the repulsive force may be measured by the deflection of the cantilever,

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Microstructural Characterization of Materials

while keeping the height of the piezoelectric assembly fixed, or a constant repulsive force may be set as a fixed elastic deflection of the cantilever, while the changes in height that are needed to maintain this constant force are monitored. Under atmospheric conditions, films of moisture are often present on the surface and can result in a capillary attraction that is not present, either in vacuum or when the chamber is filled with dry nitrogen. Electrostatic charging of a nonconducting surface may also result in unpredictable attractive or repulsive electrostatic forces between the probe tip and the sample. Most of these effects can be neutralized when the scanning probe is operated in contact mode, providing the deflection of the cantilever is positive. To ensure maximum deflection sensitivity, a thin, V-shaped cantilever is used, with a silicon nitride tip to minimize wear damage to the tip. Tip wear is often a problem in contact mode and wear debris can collect on the probe tip. For soft, compliant samples it is important to work at a constant, pre-set cantilever deflection in order to limit damage to the sample. Resonant excitation of the cantilever has proved to be most useful in the semi-contact regime, for which the attractive and repulsive surface forces have similar magnitudes (Figure 7.3). This mode of operation is frequently referred to as the tapping mode, since to a first approximation the probe tip ‘taps’ the sample at its point of closest approach to the sample surface. A ‘spring-and-dashpot’ model for the viscoelastic, ‘tapping-mode’ behaviour of the probe–surface interaction is useful (Figure 7.15). In the semi-contact regime the force on the probe tip due to the presence of the sample changes sign during the deflection cycle, as the tip moves past the minimum potential energy position. This is shown schematically in Figure 7.3. The resultant ‘damping’ alters both the amplitude of the oscillations and their phase with respect to that of the drive crystal. The ‘phase’ image in tapping mode can be extremely sensitive to the elastic properties of the sample and is capable of revealing details of the nanostructure with excellent spatial resolution (Figure 7.16). As the probe approaches a sample surface, the initial deflection of the cantilever is normally attractive. Pre-setting the deflection of the cantilever to a negative value will

Figure 7.15 ‘Spring-and-dashpot’ model for simulating damped oscillations of the cantilever in semi-contact with the sample and interpreting changes in the amplitude or phase of the oscillations.

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Figure 7.16 A tapping-mode atomic force image of partially crystalline poly(ethylene oxide) uses the phase shift across crystallite boundaries to reveal the structure. (Reproduced from NanoTech America).

collect subsequent data on this attractive force in the non-contact mode, essentially without any repulsive force interaction. The major effect observed is an increase in the amplitude of the vibrating probe oscillations that is detected by the split photodiode and can then be translated into a direct measurement of the changes in the attractive forces as the tip is scanned across the surface. As noted previously, torque displacements can also be monitored at high resolution by using a focused ion beam machined, silicon tip integrated into a silicon beam. It has proved possible to image the basal plane of graphite using this technique (Figure 7.17). At this scale, we should avoid interpreting the results in terms of atomic ‘surface topology’, as though the atoms were packed like billiard balls, and remember that we are really looking at electrostatic and electrodynamic interactions between many neighbouring atoms. By applying a bias voltage between the sample and the cantilever probe tip, and then monitoring the tip current, it has also proved possible to operate the atomic force microscope as a scanning tunnelling microscope. Image features are then associated with the electrical properties of the surface. These electrical signals are interpreted in terms of the spreading resistance of the contact, the contact potential or the local capacitance across the gap between the probe tip and the sample surface. Finally, magnetic domain structures can also be imaged by atomic force microscopy. The probe–surface separation is then increased beyond the range of the attractive van der Waals forces, leaving only the magnetic forces between a suitable tip and the ferromagnetic sample. Clearly the resolution when imaging magnetic domains will be limited to the separation distance between the probe and the sample, typically several nanometres.

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Microstructural Characterization of Materials

Figure 7.17 Graphite basal plane imaged in tapping mode at the limit of resolution of the atomic force microscope. (Reproduced from NanoTech America).

7.2.2

Scanning Tunnelling Microscopy

In scanning tunnelling microscopy the probe tip is scanned over the surface of the sample in vacuum at a controlled probe–sample separation and the current–voltage characteristics determined for each pixel point. The interpretation of the results is based primarily on the Fowler–Nordheim equation [Equation (7.1)]. It is possible to probe the local density of electron states at the sample surface quite accurately and, under suitable circumstances, the sensitivity of the tunnelling current to the probe–surface separation allows individual surface atoms to be resolved. Unlike atomic force microscopy, scanning tunnelling microscopy is limited to materials that are electrically conducting, or at least semiconducting, and the microscope chamber has to be maintained under ultra high vacuum to prevent adsorption and surface contamination from the gas phase. As in atomic force microscopy, the mechanical and thermal stability of the scanning probe system often dominates the performance, and drift of the probe tip does occur, so that measurements cannot be made over extended periods of time. This limits the number of pixels from which data can be usefully accumulated for any given image field. As noted previously, the total number of pixels in a scanning probe image is always at least an order of magnitude less than can be realized in a secondary electron image acquired in the scanning electron microscope. Although the inherent resolution is better, the modes of operation of the scanning tunnelling microscope are more restricted than those of the atomic force microscope, since it is primarily the variations in work function and surface density of states that are being detected, rather than the surface topography of the sample. It is important to recognize that

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the work function is a surface property and not a bulk property of the solid. In particular, the work function is not the energy that is required to extract an electron from the top of the conduction band and take it to infinity, which would be a bulk property, but rather the energy needed to take a conduction electron to just beyond the solid surface. The work function is therefore sensitive to the atomic packing of the surface from which it was extracted and can be dramatically affected by crystal structure, surface contamination, adsorbates and the presence of segregants. Since, according to the Fowler–Nordheim [Equation (7.1)], the current passing through the tip depends exponentially both on the work function and on the effective distance between the tip and the sample, variations in these two variables are superimposed as the tip traverses the surface. In general, as in the atomic force microscope, more than one mode of operation is possible. In the first instance, the scanner assembly is maintained at a constant height (constant z in the microscope) as it is scanned in the x–y plane. The changes in the tunnelling current are then monitored and interpreted as either changes in probe–sample separation or work function. Alternatively, a given tunnelling current can be pre-selected, and the height of the scanning head is then continuously adjusted, in a feedback loop, to keep the electron field emission current constant as the tip is scanned over the sample surface. The changes in height are now interpreted as changes in sample topography. A third option is to scan the tip bias voltage at each x–y pixel location so as to obtain a local value for the voltage dependence of the tunnelling current i(V) for both a given tip–sample separation Dz and x–y location. The slope of the i(V) curve taken at a known location x, y, Dz, can now be interpreted in terms of the local density of states. Since the local density of states depends strongly on the atomic structure, this mode of operation frequently results in excellent atomic resolution of the surface structure and atomic packing. 7.2.2.1 Resolving Surface Morphology: Restructuring. Some examples of atomic resolution are shown in Figure 7.18. These include the {1 0 0} and {1 1 1} surfaces of silicon, the basal plane in graphite and an example from a gallium arsenide detector crystal. In all cases, it is the electronic nanostructure associated with the surface packing of the atoms, the atomic morphology, that has been resolved. The contrast in the images also depends on the bias voltage applied between the probe tip and the sample. On this atomic, sub-nanometre scale, it makes little sense to talk of ‘surface topology’, since the surfaces are often atomically smooth. In such cases, the probe is ‘responding’ to the local modulations in the electronic structure of the array of surface atoms that are present. One of the more unexpected discoveries revealed by the scanning tunnelling microscope has been the extent to which the atomic separations in the surface layers of a crystal deviate from their bulk values. In several cases the new atomic packing results in surface arrays that do not even have the same symmetry as the bulk lattice. We have already described the reconstruction of the {1 1 1} planes of silicon into a 7 · 7 two-dimensional unit cell [Section 1.1.2.4 and Figure 7.18(b)], but this is by no means a unique case. Figure 7.19 shows a rather complex reconstruction in which the hexagonal symmetry of the (0 0 0 1) plane of the InGaAs parent lattice has been lost and replaced by a 2 · 4 rectangular unit cell. It should be no surprise that both the chemical and the microelectronic industries have a major interest in these phenomena, since they have a direct bearing on both catalytic activity and electronic surface states.

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Figure 7.18 Examples of STM micrographs showing (a) the {1 0 0} surface of silicon at low magnification, (b) the reconstructed 7 · 7 (see text) {1 1 1} surface of silicon,(c) the basal plane in pyrolitic graphite, and (d) the {1 1 0} surface of gallium arsenide with zinc acceptors (triangular features). (Reproduced from NanoTech America). (See colour plate section)

Surface relaxation and complete restructuring are not the only options for changes in atomic packing at a surface. Some surfaces can become ‘rumpled’ on the atomic scale, if this will reduce the surface energy through a less symmetrical configuration. It has also been established that the lattice parameters of colloidal nanoparticles may differ substantially from those of bulk materials that have the same crystal structure and composition. The assumption that the atomic packing and morphology at a free surface is derived by sectioning parallel to a defined crystallographic plane and then discarding one half of the crystal is an inaccurate and even misleading approximation. With the scanning tunnelling microscope, the equilibrium surface morphology of metallic conductors and semiconductors can now be explored in atomic detail. 7.2.2.2 Electron Energy Levels: Scanning Tunnelling Microscopy Spectroscopy. Although the scanning tunnelling microscope can be used to probe the i(V) characteristic as a function of the probe separation Dz at any point on the surface, the limited, long-term, thermal and mechanical stability make it difficult to obtain such spectroscopic data from

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Figure 7.19 A 2 · 4 restructuring of the (0 0 0 1) surface on InGaAs. Reproduced from P. Vogt, K. Lu¨dge, M. Zorn, M. Pristovsek, W. Braun, W. Richter and N. Esser, Atomic structure and composition of the (2 · 4) Reconstruction of InGaP(0 0 1), Journal of Vacuum Science and Technology B, 18(4), 2210–2214, 2000.

more than a restricted number of pixel points. Nevertheless, it has proved possible to probe the structures associated with selected microelectronic components, such as a p-n junction, or quantum dots. A spectroscopic data set that has been collected from a predetermined pixel location and at a given probe separation, can be analysed in terms of the gradient of the i(V) characteristic, di/dV, in order to extract information on the local density of surface states. The bias voltage applied to the probe can be either positive or negative, while the frequency response of the current signal may also provide information well into the megahertz region, offering the possibility of determining local dielectric loss factors and other data that are related to polarization forces (the dipole moments and polarizabilities of the surface atoms and molecular assemblies). This subject is well beyond the objectives we have set for the present text, but is nevertheless an exciting area of research for solid-state device technology and condensed-matter physics.

7.3

Field-Ion Microscopy and Atom Probe Tomography

The evolution of the simple field-ion microscope (Figure 1.10) into the atom probe (Section 7.2) occurred some 40 years ago, but the initial results had limited impact on materials characterization for two very good reasons: first, the data sets that were collected contained counts from, at most, a few thousand atoms. This was insufficient to provide convincing statistical evidence that they represented the ‘real’, nanometric scale chemical compositions for the constituent elements of a sample. Secondly, not only was it necessary for the minute, sub-micrometre tip of the sample needle to be a metallic conductor, but it

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also had to have sufficient mechanical stability to withstand the gigantic stress exerted by the applied electric field during field-evaporation of the surface atoms from the specimen tip. The introduction of time-of-flight mass spectrometry, with sub-nanosecond resolution, provided the technology that made the atom probe possible. Instead of relying on a magnetic field to analyse the mass spectrum of the beam of evaporated ions, the time-offlight mass spectrometer measured the time required for an ion, evaporated from the specimen tip surface by a voltage pulse, to reach a detector placed at the end of a 1 m long flight tube. This time, typically measured in nanoseconds, was directly proportional to the ratio of the electric charge carried by the ion ne, and its isotopic mass m, where n is the charge on the ion and e is the electronic charge. It therefore follows that the pure elements give mass spectra that contain several spectral lines, both because the different isotopes of a single element can be well-separated by the time-of-flight mass spectrometer, and because the field evaporation process may generate ions of more than one charge from a single atomic species. A complex but well-resolved mass spectrum is shown in Figure 7.20. In many cases, molecular ions are also collected and are often associated with adsorbed residual gases, such as carbon dioxide and water vapour, that have been trapped at the sample surface. Improvements in electronic detection have enabled the mass spectrometer flight tube to be shortened, while the development of channel-plate charge multipliers and charge coupled device (CCD) collectors has made it feasible to increase the collection angle from the tip and obtain a ‘field evaporation image’ by summing the counts from given ne/m elemental spectral peaks over time. The data are accumulated for each pixel of the CCD collector and for each field evaporation excitation pulse. Finally, there is now no need to apply a voltage pulse to the tip in order to excite field evaporation. Instead, an applied electric field can be used to reduce the energy barrier for field evaporation of an atom to any predetermined level and a pulsed laser then used to trigger the release of ions from the tip surface. This has resulted in a significant improvement in the time resolution of the mass spectrometer. Successive layers of atoms can be field evaporated and the charge-to-mass ratio of each captured ion can be recorded as a function of both the pixel location on the CCD detector (the x–y plane of the field-ion image at a given time) and the pulse number in the sequence of laser or voltage excitation pulses (the z-coordinate of the sample tip). Atom probe tomography can provide three-dimensional atom-by-atom chemical analysis at the nanometre level. 7.3.1

Identifying Atoms by Field Evaporation

The time-of-flight mass spectrometer has a time-scale that depends on the length of the flight tube and the accelerating voltage, but is typically of the order 100 ns. It follows that a time resolution of better than 1 ns is essential for good peak separation and that gigahertz electronics is necessary to process the results. The capacitance of the sample tip and extraction electrode is a significant factor in determining the mass resolution, as is the response of the channel-plate/CCD assembly. Assuming that the mass peaks are sufficiently sharp for the different values of ne/m to be separated unambiguously, we still need to know both the charge on the ion and its isotopic mass before we can identify the chemical species. The charge on the ion field-evaporated

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from a given sample tip is determined by the electric field required to ionize that particular chemical species. Intuitively, we might expect the ionic charge to be equal to a known chemical valency of the element, but this is incorrect. All field-evaporated species are positively charged, since a positive potential is applied to the sample tip, and the energy required to field-evaporate an ion Mnþ is approximately equal to the sum of the ionization

Figure 7.20 Local electrode atom probe time-of-flight mass spectrum of an Al–0.1Zr–0.1Ti (atom%) alloy containing Al3(Zr1-xTix) (L12 structure) precipitates. (a) Mass spectrum for the entire analysis, representing 7.67 · 105 identified atoms, of which 5.21 · 103 atoms (0.679  0.009 atom%) are identified as Zr and 2.16 · 103 atoms (0.282  0.006 atom%) are Ti. (b) Mass spectrum indicating the core composition of the Al3(Zr1-xTix) precipitate. This spectrum represents 8.88 · 103 atoms, of which 1508 atoms (17.00  0.40 atom.%) are Zr and 335 atoms (3.77  0.20 atom%) are Ti. (c) Mass spectrum indicating the composition of the matrix (a-Al solid solution) surrounding the Al3(Zr1-xTix) (L12 structure) precipitate. This mass spectrum represents 2.09 · 105 identified atoms, of which 224 atoms (0.107  0.007 atom.%) are Ti. There is no evidence of Zr (<0.010 atom%), indicating that all detectable Zr atoms have partitioned to the Al3(Zr1-xTix) precipitate. These data are impressive since they represent solute analysis in very dilute (0.1 atom.%) alloys. The isotopes of Ti and Zr are clearly resolved, and the Zr atoms are observed in two charge states (2þ and 3þ). Courtesy of David Seidman.

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potentials Sn I n minus nf, the work function of the sample surface in eV multiplied by the charge on the evaporated ion. The minimum field strength required to evaporate an ion of given charge depends on the relative values of the ionization potentials that are needed to excite an atom to a given charge state. In most cases, the observed field-evaporated ions are doubly charged, but singly charged ions are frequently found, while some species that are only ionized at very high field strengths may be triply charged. Although doubly charged ions are expected to dominate, it is quite possible for any given element to appear in the mass spectra with more than one charge, as seen in Figure 7.20. Excellent mass resolution is available in these spectra and isotope peaks from a given chemical species can be identified from their relative abundance in the earth’s crust. The isotopic abundance of the chemical elements in the earth’s crust is well-documented and may be used to resolve ambiguities in the mass spectrum that may arise when two ions have similar charge-to-mass ratios. This can be quite common when molecular ionic species are present. 7.3.2

The Atom Probe and Atom Probe Tomography

The design of one modern atom probe is shown in Figure 7.21. Several samples can be loaded into the ultra-high vacuum chamber through an airlock at the same time. The samples are then moved into position, one at a time, over the field-evaporation electrode. The sample is first aligned optically, before forming a field-ion microscope image using an appropriate image gas (usually helium for refractory metals and alloys, or neon in the case Single Atom Detector Computer Controlled Timing System

DC High Voltage

Time-Of-Flight Mass Spectrometer

Energy Compensating Lens

HV Pulse Pulsed Laser

Airlock

Channel Plate

Preparation Chamber

Specimen

3DAP Detector

Figure 7.21 Atom probe tomography is made possible by combining the high spectral resolution of tune-of-flight mass spectrometry with the atomic resolution of the field-ion microscope and pulsed field evaporation. The sub-nanosecond time resolution of the recording system is capable of storing data for millions of ions and identifying both their chemical species and their relative atomic locations in the sample.

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of steels). An image intensifier simplifies both viewing and recording. At this stage, the tip is shaped by pulsed field evaporation to remove absorbed debris and leave an atomically smooth, symmetric, tip surface. Final alignment of the sample ensures that a nanostructural area of interest is centred on the axis of the mass spectrometer collection aperture. Typically, a crystallographic direction of interest will be chosen to lie on the axis of the spectrometer. Data collection can now begin. Voltage or laser pulses are used to field-evaporate the atoms at the edges of the closepacked planes of the structure. The data from each evaporation pulse are digitally binned, according to the charge-to-mass ratio of the ion ne/m (defined by the time of arrival of the ion at the detector) and the x–y pixel coordinates for the ion on the CCD detector. The zcoordinate is registered as a pulse time. The system records the data from each separate pulse and the time register is then incremented by one step before a further pulse is used to generate the next ion shower to be detected at the CCD. The complete data set for each ion species is recorded in three-dimensions but all three x, y and z dimensions have to be calibrated and this is not trivial. Calibration usually assumes known values for the lattice parameters and uses field-ion images of the lattice planes to determine an x–y scale for the CCD plate coordinates. Since the projection is never a simple point projection (the tip is not a simple hemisphere), the effective magnification varies over the CCD plate and this information must be incorporated into the calibration software. In addition, as field evaporation proceeds, the effective tip radius of the sample increases, so that the local x–y magnification has to be continuously recalibrated. Often the tip is approximately conical and the changes in magnification are ‘well-behaved’, but this is not always the case, especially if polyphase samples that contain coarse particles are being analysed. Calibration of the z-axis scale is determined from the number of pulses required to remove the layers of a single crystal plane with known d-spacing. Typically, less than 100 atoms are removed by each evaporation pulse, and up to 100 pulses may be required to remove a single lattice plane of atoms. The ion detection efficiency of the time-of-flight mass spectrometer and CCD collection plate is remarkably high (at least 50 % and sometimes better than 80 %). The collection efficiency depends mainly on the sample dimensions and the collection system. Although the collection efficiency depends very little on sample composition, this is not quite true for large, second-phase particles, since such particles can affect the surface topography appreciably during field evaporation, especially when they lie off the axis of the sample, as is usually the case. Figure 7.22 shows a few striking examples of atom probe tomographs from some engineering alloys. When compared with every other available technique, the quality of the sub-nanometre chemical information is quite extraordinary. In each case the recorded counts are due to individual ions. Although not every ion can be detected, most are, and the atom probe is now providing atomic-scale chemical information on the earliest stages of precipitation and segregation in key engineering alloys.

Summary In previous chapters, microstructural characterization has been discussed in terms of the analysis of a signal that was generated when a beam of electromagnetic radiation or energetic electrons interacted with a prepared specimen. The discussion of these

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Figure 7.22 Some examples of the applications of atom probe tomography in alloy development. (a) Variations in concentration in a copper (red) and silver (green) nano-

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electromagnetic and energetic electron probes is now expanded by considering the interaction of a solid probe, in the form of a sharp point or needle. The location and separation of the probe from a solid specimen surface are controlled with sub-nanometre accuracy. Some ‘microscopes’ have been developed, in which the sample is itself the source of the signal, and no separate probe is present. The first such instrument to yield atomic resolution was the field-ion microscope. In this instrument, the specimen is the tip of a sharp needle. Such sharp needles are also the electron sources used in field emission guns for highresolution scanning and transmission electron microscopy, and the same ‘needle’ geometry is used for the solid probe in scanning tunnelling and atomic force microscopy. In the atomic force microscope the sharp probe tip is mounted on a cantilever beam whose deflection can be monitored while the probe is scanned over the surface. The cantilever can also be vibrated during scanning, at approximately its natural frequency, and the changes in either the deflection amplitude of the cantilever, or the phase shift of the tip signal with respect to the driver can be monitored. In order to monitor changes in the attractive surface force between the probe and the sample, the probe tip can be scanned in the so-called, non-contact regime, far enough from the surface to ensure that attractive forces dominate the interaction. Alternatively, it is possible to monitor the contact regime, with the tip probe close to the surface, where the interaction is dominated by repulsive forces. The probe can also be scanned close to the equilibrium, zero-force separation, in the semi-contact regime. In this semi-contact regime, vibrating the probe leads to a tapping mode of operation in which the probe moves in and out of contact with the surface. The ‘tapping’ signal is very sensitive to changes in amplitude and phase, yielding high resolution images of the surface atomic features and surface properties. The tapping mode can provide information derived from changes in the amplitude, frequency or phase of the cantilever vibrations with respect to the phase of the driver. Although the information is, for the most part, qualitative, the nanoscale image contrast can be linked directly to differences in surface elasticity, for different surface phases, or crystallographically determined elastic anisotropy of a single phase polycrystal. This information is not available in any other instrument. The scanning tunnelling microscope employs a bias voltage between the scanning probe tip and the sample surface, in order to monitor the voltage dependence of the tunnelling current as a function of the probe–surface separation. The current–voltage characteristic can provide information on the local density of states, while the scanned tunnelling image is

3 composite. [Reprinted from F. Wu, D. Isheim, P. Bellon and D.N. Seidman, Nanocomposites Stabilized by Elevated-Temperature Ball Milling of Ag50Cu50 Powders: An Atom Probe Tomographic Study, Acta Materialia, 54(10), 2605–2613, 2006]. (b) Segregation of silicon to a grain boundary in iron [Reprinted from B.W. Krakauer and D.N. Seidman, Subnanometer Scale Study of Segregation at Grain Boundaries in an Fe(Si) Alloy, Acta Materialia, 46(17), 6145–6161, 1998]. (c) Evolution of a Second Phase in an Ni-Al-Cr Alloy [Reprinted from C.K. Sudbrack, K.E. Yoon, R.D. Noebe and D.N. Seidman, Temporal Evolution of the Nanostructure and Phase Compositions in a Model Ni–Al–Cr Alloy, Acta Materialia, 54, 12, 3199–3210, 2006]. Copyright (2006), with permission from Elsevier. (See colour plate section)

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capable of resolving the packing of surface atoms and atomic spacing. The scanning tunnelling microscope has revealed that many crystal surfaces undergo previously unsuspected rearrangement and restructuring, in order to minimize the surface energy of the solid. The development of the atom probe as an extension of the field-ion microscope, together with wide-angle, time-of-flight mass spectrometry has made atom probe tomography a practical reality. Individual atoms from the smooth tip of a sharp specimen needle are fieldevaporated as ions by a pulsed electric field or pulsed laser. The chemical species of the ions can be identified, from their charge-to-mass ratio in a time-of-flight mass spectrometer, and then counted as a function of their location on the tip surface (x–y coordinates) and the number of pulses in the field-evaporation sequence (the z-coordinate). Data sets consisting of millions of ions have now provided three-dimensional chemical nano-analysis of precursor precipitate clusters, grain and phase boundary segregation, and other features of the nanostructure of engineering materials. The sample materials that can be examined by atom probe tomography are limited to those metals and semiconductors that are capable of withstanding the mechanical stresses generated by the electric fields( MVcm
1) needed for controlled field evaporation.

Bibliography 1 J. Israelachvili, Intermolecular and Surface Forces, 2nd edn, Academic Press, New York, 1991. 2 R. Wiesendanger, Scanning Probe Microscopy and Spectroscopy, Cambridge University Press, Cambridge, 1994. 3 M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Kluwer/Plenum, New York, 2000. 4 V.M. Mironov, Fundamentals of Scanning Probe Microscopy, Russian Academy of Sciences, Nizhniy Novgorod, 2004, 5 B. Bhushan, H. Fuchs and S., Hosaka, Applied Scanning Probe Methods, Springer, New York, 2004.

Problems 7.1. In the absence of a specimen in the transmission electron microscope, what information on the microstructure of the electron source ought to be present in the diffraction image plane? 7.2. In scanning probe microscopes, the probe and the sample form a single system. What physical and mechanical properties of the probe are necessary to ensure that the properties and structure of the probe do not affect the results? 7.3. What is meant by the term Hamaker coefficient? Under what circumstances might you expect the Hamaker coefficient to be negative? 7.4. When two hydrophobic surfaces that are separated by a thick film of water are brought together, a puzzling attractive force can sometimes be measured at

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distances that greatly exceed those associated with polarization forces. Can you suggest a possible explanation? 7.5. Discuss how the charge on the counter-ions can affect the structure of a double layer formed at the surface of a solid that is immersed in a polar solution. 7.6. Explain the term steric hindrance. Why should a nonpolar, polymer molecule prevent dispersed particles from flocculating (aggregating together in the dispersion)? 7.7. Explain the origin of multiple minima in a surface force–distance curve measured in the presence of a solution of a long chain alcohol or a polysaccharide. 7.8. Distinguish between the lattice resolution of atomic planes observed by high resolution electron microscopy, and atomic resolution observed on the surface of a semiconductor by scanning tunnelling microscopy. 7.9. What are we ‘seeing’ when we observe the 7 · 7, restructured {1 1 1} surface of silicon in the scanning tunnelling microscope? 7.10. Why does field emission tunnelling have a better potential for atomic resolution than that available by monitoring the surface forces experienced at the tip of an atomic force microscope? (Hint: compare the Fowler–Nordheim field emission equation to the DLVO surface force–distance model.) 7.11. What might you expect to find in the surface composition of an electroplated gold layer examined by atomic force microscopy in air? One word answers are not acceptable! 7.12. What damage mechanisms account for the rather limited working life of a silicon nitride atomic force microscope cantilever tip? 7.13. An atomic force microscope operated in tapping mode can be used to monitor the elastic response of a soft tissue sample surface. How could this information be quantitatively related to the elastic constants of the material? 7.14. Scanning tunnelling spectroscopy is a powerful tool for probing the density of states of a semi conductor and exploring the structure of a p-n junction and other features of solid-state device technology. What are the limitations of this technique when applied to semiconductor surfaces? 7.15. Atom probe tomography has produced some remarkable results and commercial instruments are available. Nevertheless, relatively few papers have been published and the development process has taken over 40 years. Why?

8 Chemical Analysis of Surface Composition Chemical analysis of the composition is an important element in the characterization of an engineering material at all stages of the materials cycle (extraction, manufacture, service and, ultimately, disposal or recycling). Nevertheless analytical chemistry is not a topic for this text, which has been restricted to microstructural aspects of the characterization process. In Chapter 6 we explored microanalytical methods of chemical analysis that can be combined with electron microscopy, either for bulk samples, using the scanning electron microscope, or for thin-film specimens in the transmission electron microscope. Chapter 6 covered X-ray microanalysis by energy-dispersive or wavelength-dispersive spectroscopy, as well as electron energyloss spectroscopy (EEELS). In the present chapter we extend our treatment of microanalysis to three further analytical methods that, in this case, focus on the chemistry of the surface layers of an engineering component. These additional tools have major applications in surface engineering, especially for the study of solid-state devices. These applications include opto-electronic and superconducting materials, thin-film devices, and both radiation and chemical detectors. Before beginning, we should first list some of the many available methods of chemical analysis that will not be covered in this chapter: .

.

.

Atomic absorption spectrometry depends on the detection of characteristic absorption spectra for atomic species. A beam of individual atoms is commonly generated by laser vaporization of a sample and the absorption spectra for the constituent species are recorded using a white light source. Optical emission spectroscopy is the inverse of atomic absorption. It measures the intensities of the characteristic emission lines of the different atomic species. In this method, a spark source is usually used to generate the signal. Infrared spectroscopy is based on absorption spectra that are detected in the infrared range when the sample is placed in the path of a suitable infrared radiation source. The infrared absorption spectra can be used to extract information on the chemical bonding in the system.

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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.

.

.

.

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Raman spectroscopy employs inelastic photon scattering, usually after laser excitation. Photon absorption at the wavelength of the excitation radiation is accompanied by photon emission at longer wavelengths. These Raman emission lines reflect the decay of excitation states that are primarily associated with variations in the chemical bonding of the sample material. Electron spin resonance is observed in paramagnetic materials subjected to microwave frequencies. The observed resonances correspond to specific electronic states in the material, for example those associated with specific chemical valences. Nuclear magnetic resonance (NMR) occurs at radio frequencies and corresponds to resonance of the magnetic moment associated with an atomic nucleus for some isotopes of a given chemical species. Imaging by NMR has had a major impact on noninvasive, diagnostic medicine by medical imaging. In particular, it has vastly extended both the resolution and the sensitivity of X-ray radiography and ultrasound medical imaging methods. Fluorescence spectroscopy makes use of characteristic X-ray excitation generated by exposing the sample to a beam of high-energy, ‘white’ X-rays. In contrast to the spectrum of characteristic X-rays generated by an electron beam, there is no excitation of background Bremmstrahlung radiation, and the characteristic lines from minor trace elements can often be detected. Rutherford backscattering employs the simple Newtonian laws of momentum transfer that apply when a beam of MeV energy ions (usually helium) is backscattered after colliding with individual atoms in a solid specimen target. The energy and angular distributions of the backscattered ions reflects their momentum distribution and can be calibrated with excellent depth resolution.

Many other techniques of bulk chemical analysis exist. With a few exceptions, notably high resolution scanning secondary ion mass spectrometry (Section 8.3), microbeam excitation of Raman spectra and infrared absorption, none of the above techniques has the microanalytical capability associated with sub-millimetre spatial resolution. Table 8.1 summarizes the different excitation probes that can be used to excite a solid sample and the corresponding composition-sensitive signals for all the analytical techniques that have been outlined above, as well as those that will be discussed in this chapter.

8.1

X-Ray Photoelectron Spectroscopy

X-ray photoelectron spectroscopy (XPS) is also known as electron spectroscopy for chemical analysis (ESCA).1 This technique is used to probe most of the energy levels in the atom that are revealed by electron energy loss spectroscopy (Section 6.2), but without any spatial resolution. XPS finds its place in this chapter because of its depth sensitivity, which is on the nanometre scale. The photons in a beam of monochromatic, characteristic X-rays that is incident on a solid target are absorbed by the atoms in the sample and give rise to secondary electrons that are ejected from the target with a kinetic energy Es, equal to the difference between the energy of the incident photon hn and the energy that was required to displace the secondary electron from the target atom. This excitation energy is the sum of the binding energy Eb that is 1

The authors prefer XPS, since this name relates directly to the physics of the technique.

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Table 8.1 The probes and corresponding signals for common methods of analytical chemistry.a Name of method

Probe used

Signal detected

Optical spectroscopy

Visible light

Infrared spectroscopy Raman spectroscopy Electron spin resonance Nuclear magnetic resonance X-ray fluorescence X-ray photoelectron spectroscopy* Auger electron spectroscopy*

Thermal excitation Laser excitation Microwave excitation Radiowave excitation ‘White’ X-rays X-rays

Absorption and emission in the visible range Infrared radiation Infrared emission Chemical bonding states Selected isotope resonance Characteristic X-rays Secondary electron emission Secondary electrons from Auger transitions Characteristic X-rays Absorption edges

X-ray microanalysis* Electron energy loss spectroscopy* Secondary-ion mass spectroscopy* Rutherford backscattering a

Usually energetic electrons Energetic electrons Energetic electrons keV inert ion beam MeV (helium) ion beam

Sputtered target atoms and ions Angular dependence of backscatter intensities

An asterisk indicates the method is treated in this text.

required to raise the excited electron to the Fermi level and the orientation-dependent workfunction f that is required to bring the electron from the Fermi level into the vacuum just outside the surface: ES ¼ hn
ðEb þfÞ

ð8:1Þ

On rearranging these terms, the binding energy of the photoelectron is given by: Eb ¼ hn
ðES þfÞ

ð8:2Þ

The electron binding energy in the atom increases as the energy of the emitted photoelectron decreases. A schematic energy diagram for the secondary emission of a 2p photoelectron from copper is shown in Figure 8.1, while Figure 8.2 shows the complete copper photoemission spectrum. In addition to the lines corresponding to the emission of electrons from energy levels in the inner electron shells, the emission spectrum also includes Auger electrons that are associated with energy transitions within an atom after excitation by an incident X-ray photon. The Auger peaks can be separated from the photoelectron spectrum by recording two different spectra using two characteristic, monochromatic X-ray wavelengths, that is, two different photon excitation energies, for the photoelectron excitation process. The Auger peaks will always appear in the same position, but the photoemission lines will all be shifted by the energy difference between the energies of the two incident X-ray beams used to excite the signal. The electron binding energies that are of interest for a given sample may exceed 1 keV. The relation between the energy of an X-ray photon and its wavelength is l ¼ hc/E ¼ 1.24/V nm, where c is the speed of light, E is the energy of the photon and V is in keV. It follows that that suitable X-ray excitation wavelengths for photoelectron spectroscopy are of the order of 0.1–1 nm.

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φ 4s 3d

EVAC EF

3p

hV 3s

2p

3/2 1/2

2s

1s

Figure 8.1 A 2p photoelectron is emitted from a copper atom as the result of the excitation of the atom by absorption of an X-ray photon.

8.1.1

Depth Discrimination

The background signal in the photoelectron spectrum, for example in Figure 8.2, arises from multiply scattered, secondary electrons that are generated in the deeper, inner layers of the sample. If secondary electrons are inelastically scattered before they can escape from the surface, then they can only contribute to the background signal, rather than to the characteristic peaks in the spectrum. It follows that the observed peaks are due to photoelectrons that are generated in the immediate surface layers, at a depth which is less than the mean free path of the secondary electrons excited in the material. By comparison, the higher background levels on the high binding energy side of each peak, to the left of the peaks in Figure 8.2, are due to inelastically scattered electrons that originate in the deeper layers of the target and correspond to a lower energy for the collected signal. In practice, both elastic and inelastic scattering will occur, and the depth dependence of the intensity for a specific peak leads to the definition of an attenuation length: a measure of the loss of detected signal intensity, normal to the sample surface, as a function of the excitation depth (Figure 8.3). The broad minimum for photoelectron energies between 10 eV and 500 eV corresponds to emission depths of between 2 and 5 atomic layers and demonstrates the power of the technique to analyse surface composition and chemical binding. XPS is extremely sensitive to chemical changes occurring at the surface, and has a depth resolution of just one or two atomic layers. However, as noted previously, no lateral resolution is available so there is no imaging capability.

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Intensity (a.u.)

Cu Al Kα XPS 2p3/2 3p 2p1/2

3s LMM Auger

valence

2s

x10

1000

500

0

Binding energy (eV) Figure 8.2 Photoelectron spectrum from pure copper includes both the photoelectrons and Auger electrons that are associated with energy transitions occurring within the atom after excitation (Section 8.2). Reprinted from G.C. Smith, Analysis of nanometer-sized precipitates using atom probe techniques, Material Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

Au

Mean free path (A)

100

Al Au

50

Au Ag

Ag Au

Ag Au

Ag

10

5

Ni Be{ Be Ag P Be Fe W Ag Ag Mo Be

Ag Au Be{ Ag Ag C Mo

Au C C Be

Mo W

3 2

5

10

50

100

500

1000

2000

Electron energy (eV) Figure 8.3 Attenuation length for electrons detected normal to the surface for a variety of different solids as a function of their energy. Note the broad minimum in the range 10–500 eV that corresponds to just a few atomic layers. (Courtesy of L.C. Feldman and J.W. Mayer, Fundamentals of Surface and Thin Film Analysis, Elsevier Science Publishing Co.).

428

8.1.2

Microstructural Characterization of Materials

Chemical Binding States

The binding energies of the electrons in the outermost shell of an atom are sensitive to the chemical state of the atom; the strength and nature of the chemical bonds. Photoelectron emission from the same atomic species, but in a different coordination or binding state, will generate multiple peaks in the photoelectron signal. Figure 8.4 is a particularly elegant example in which the 1s binding energies for carbon in an organic molecule are easily distinguished by peak splitting. These energy differences are of the order of a few eV for the different carbon binding states. Similar effects have been observed for the oxidation states of metals. Figure 8.5 shows the different 2p states in a titanium atom. The higher binding energy in the oxide results in a large peak shift of 4–5 eV.

F

F

O

C

C

O

H

C

C

H

H

H

Intensity (a.u.)

F

H

0 295

290

285

280

Binding energy (eV) Figure 8.4 Binding energy of a 1 s electron in carbon is a clear indication of its chemical binding state, with well-resolved peaks for each of the four carbon atoms in an ethyltrifluoroacetate molecule. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

Chemical Analysis of Surface Composition

Intensity (a.u.)

2p3/2

TiO2

429

2p3/2

Ti Metal

2p1/2 2p1/2

470

460

450

Binding energy (eV) Figure 8.5 Oxidation states on metal surfaces are resolvable by XPS. The growth of TiO2 on titanium metal is accompanied by 2p peak shifts of 4–5 eV. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

8.1.3

Instrumental Requirements

Since the XPS signal comes from the first few atomic layers of the sample, it is important to ensure that the sample surface remains uncontaminated throughout the analysis. An ultrahigh vacuum chamber is required. Surface layers of contamination are usually removed by argon-ion sputtering with ion beam energies of a few kV. The arrival rate at the sample surface of gaseous species from the environment depends on their molecular weight, the temperature of the gas and the gas pressure. For air at a pressure of 10
6 Torr at room temperature, a complete monolayer of the gas molecules will arrive at the surface (but not necessarily stick) in just 1 s. The time required for monolayer coverage of the sample by adsorption depends linearly on the pressure, so we should conclude that pressures below 10
8 Torr are essential in an XPS chamber if reasonable time is to be available to collect the experimental data before environmental contamination of the clean surface invalidates the results. There is no reason why any system for electron spectroscopy should be limited to only one type of excitation probe, and it is possible to purchase commercial ultra-high vacuum chambers that combine an analyser system containing several probe sources, specimen stages and both source- and specimen-exchange carousels. Figure 8.6 illustrates schematically some of these options. The XPS specimen is cleaned by sputtering under an incident ion beam which is usually argon. The signal is generated by either a monochromatic X-ray or a high energy electron beam. An electron beam can be scanned over the surface in a fine-focus raster and a secondary electron image of the surface can be used to position the sample. The photoelectron signal is usually analysed electrostatically, in a focusing analyser having an energy resolution that is better than the inherent peak width (typically 1 eV). The emitted electrons are focused onto a defining input slit and the electrostatic field in the

430

Microstructural Characterization of Materials Concentric Hemispherical Analyser

Input Slit

Output Slit

Detector Ion Gun Lens Electron Gun

Scintillator/ Photomultiplier

X-Ray Source

Sample

Figure 8.6 Schematic layout for an ultra-high vacuum electron spectrometer that may be tailored to incorporate various surface probes and specimen stages, providing facilities for both Auger analysis and X-ray photon spectroscopy. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

analyser then focuses the electrons of a given energy onto an output slit, where the electrons are detected and the signal subsequently amplified. Scanning the analyser voltage then scans the focus of the spectrometer for the full range of the different electron energies. It is the width of the output slit that determines the energy resolution of the spectrometer. The provision of a specimen carousel, capable of accommodating a number of different samples inside the ultra-high vacuum of the spectrometer, dramatically reduces the average time required to change specimens. High vacuum airlocks, that incorporate pre-bake facilities for degassing the samples prior to admitting them into the spectroscope chamber, further reduce sample exchange times, usually to no more than a few hours, despite the stringent ultra-high vacuum requirements. Several degrees of rotational and positional freedom are also provided with the specimen stage. These allow tilting of the specimen with respect to both the monochromatic X-ray or high energy electron beam and the spectrometer assembly. This is important, since both the take-off angle of the signal from the surface and the incident beam angle can affect the recorded spectrum, either as a result of crystalline anisotropy of the specimen or, more often, because the increased secondary electron path length at shallow take-off angles significantly reduces the emission depth, changing the thickness of the sampled surface nanolayer.

Chemical Analysis of Surface Composition

8.1.4

431

Applications

The information available from XPS helps to fill an important gap in our ability to characterize surface chemistry. The reaction of solids with the environment takes place at surfaces and interfaces, and the chemical sensitivity of XPS analysis make it a preferred choice for studies of gas phase absorption and catalysis that involve partial coverage of less than a monolayer on the sample surface. The sensitivity of the technique is often sufficient to detect the energy differences that are associated with surface atoms of the same species that differ only in their coordination numbers; for example, atoms sited on low index surfaces, close-packed ledge atoms or atoms at kink sites. With the rapid development of electronic device technology, especially thin-film detectors and electro-optical systems, XPS is proving a powerful method for quantifying chemical changes that occur at the surface. What XPS cannot do is to analyse the surface composition of complex, multiphase samples or to provide useful, lateral, surface resolution. To meet these requirements, Auger electron spectroscopy (AES) is usually more suitable.

8.2

Auger Electron Spectroscopy

In terms of materials surface characterization, AES is probably the most useful of the three techniques that we concentrate on in this chapter. Field emission sources for the electron excitation beam in Auger systems have now been introduced. This has significantly increased the signal intensity and dramatically improved the spatial resolution for Auger mapping of an elemental distribution. The combination of Auger mapping and controlled sputtering of the sample surface makes it possible to reconstruct a surface morphology and chemistry in depth. Nevertheless, there still remain problems associated with the roughened topology of a sputtered surface and blurring of composition gradients due to focused atomic collision sequences during bombardment by the sputtering ions. Even so, sub-micrometre lateral resolution for Auger mapping is now routinely achieved. A typical energy scheme for Auger nonradiative emission, that is, electron emission with no accompanying photon emission, is shown in Figure 8.7. Excitation of the emitting atom may be by either an incident energetic electron beam or by photon excitation (hence the Auger peaks in the XPS spectra), but electron excitation is generally preferred. By scanning the electron beam over the sample surface, a secondary electron image can be recorded, as in the scanning electron microscope. A region for Auger analysis is then selected. The incident electron beam may be focused and scanned across the surface with sufficient lateral resolution of the Auger information for Auger imaging and mapping, providing the signal intensity is sufficient. This is always the case if the source of the high energy electron probe is a field emission gun. The characteristic energy of the Auger electrons is determined by three energies that are a function of the excitation levels in the atom: the energy of the excited state, for example EL3 in the case illustrated in Figure 8.7, the energy released by the M-electron that subsequently fills the vacant L-hole, EM 1 , and the energy that is absorbed in allowing the Auger electron to escape to the vacuum from its original M-level, EM 2;3 with an energy A EL 3 M 1 M 2 ;3 . It follows that (Figure 8.7), A EL 3 M 1 M 2;3 ¼ EL 3
EM 1
EM 2;3 .

432

Microstructural Characterization of Materials

AEL

= EL - EM - E*M 3 1 2,3

3M1M2,3

EVAC

VEF M2,3

M1

L3 L2

L1

K

Figure 8.7 An Auger electron originating from the M-level is emitted from copper as a result of an atom that has been excited to the L-state decaying to the M-state. The energy of the Auger electron is determined by the energy of the excited state, the energy released by the M-electron that fills the vacant L-hole, and the energy absorbed as the Auger electron escapes to the vacuum from its original M-level.

This is an introductory text and we have simplified the potential complexity of atomic absorption and emission spectra. In any case, most physical methods of chemical microanalysis seldom resolve the fine-energy structure of either the absorption or emission processes. However, this is not the case in Auger spectroscopy, for which the well-resolved fine structure of the Auger peaks provides an important tool for analysing the chemical bonding through the quantitative interpretation of adjacent Auger peaks. The energies of the K-, L- and M-states are always well-separated. Not all quantum excitations may be observed, but all the possible transitions, and practically all their relative transition probabilities, are now both well-known and well-understood. There is therefore little ambiguity in data interpretation, providing that the operator remains aware of the importance of meticulous calibration and the need to refer to the professional literature. Auger excitation is particularly complex, since three separate energy states contribute to the energy of the Auger electron. This results in far more Auger energy signals for a given material than is possible, either in an XPS photoelectron spectrum or when analysing the characteristic X-ray signals in X-ray microanalysis. Auger electron excitation is actually in competition with characteristic X-ray excitation, and the energy absorption edges that are used in EELS analysis with an electron beam ‘probe’ are the adsorption edges that are

Chemical Analysis of Surface Composition

433

associated with both Auger excitation and the same characteristic X-ray excitation. As we have already noted, the low atomic number constituents in a sample give weak characteristic X-ray signals, while their corresponding absorption edges are strongly defined. It follows that the Auger electron emission signal is strongest for these same low atomic number elements, so that the greatest strength of Auger spectroscopy is its excellent sensitivity to changes in the chemical concentration and chemical binding of these low atomic number constituents. 8.2.1

Spatial Resolution and Depth Discrimination

The energy range of Auger electrons is the same as the range of energies observed for photoelectrons, so that the Auger electrons can only escape from the same limited depth below the sample surface: typically two to five atomic layers. However, in Auger excitation by electrons, it is possible to focus the incident electron beam to a fine probe and limit the region for Auger analysis to the small surface area immediately beneath the probe. As in the scanning electron microscope, the total electron current in the probe depends sensitively on the probe diameter and falls rapidly as the size of the probe is reduced. As in scanning electron microscopy, the ultimate limit on probe size for a given source is set by the electro-optical parameters: electron wavelength (the beam energy) and lens aberrations (the performance of the electromagnetic lenses). A field emission electron source increases the available probe current by some two orders of magnitude. While the probe size depends on the electro-optical parameters of the instrument, the virtual source of secondary electrons (which includes Auger excitation) depends on both the probe size and the volume from which back scattered electrons are emitted, since the backscattered electrons near the surface also generate secondary electrons. Because the Auger signal is derived from the first few atomic layers at the surface, it is strongly dependent on the vacuum conditions in the instrument. The rate of contamination under the electron beam is a very important factor that may limit performance in AES, far more so than it was for any of the previous methods of microstructural characterization that we have discussed. This includes highresolution lattice imaging in transmission electron microscopy, although the thin-film transmission electron microscope specimens are also very contamination-susceptible. As in XPS, ultra-high vacuum chambers are essential, with residual pressures significantly better than 10
9 Torr. To achieve these conditions periodic bake-out of the assembly is required in order to degas both the specimens and the surrounding surfaces. The optimum electron probe size is a compromise between the limitations on mechanical tolerances imposed by a bakeable, ultra-high vacuum system, the statistical limitations of signal detection that are determined by the probability of observing a given Auger electron transition, and the electron optics of probe formation. With the exception of the ultra-high vacuum condition, these requirements are all similar to those that we have discussed before with respect to both EELS and thin-film X-ray microanalysis. In the case of AES systems, electron beam energies of several keV are generally used, with probe diameters of the order of 50–100 nm. Spatial resolutions for Auger mapping of as little as 10 nm have been reported for high intensity LaB6 or field emission electron sources with high performance electron optics, but a more realistic resolution for most sample conditions would be of the order of 0.1 mm due to Auger electrons generated by back scattered electrons. As in scanning electron microscopy, the total secondary electron signal that contains the Auger signal exceeds the incident beam current by at least two orders of magnitude, so that

434

Microstructural Characterization of Materials

there is no problem in recording a secondary electron scanning image of the surface at a spatial resolution equivalent to the spatial resolution of Auger analysis. 8.2.2

Recording and Presentation of Spectra

The accepted method for presenting Auger data graphically is as the differentiated signal rather than as the raw data in a collected energy spectrum. In Figure 8.8 we compare these two graphical methods of presentation. The differentiated presentation effectively removes the background signal and each Auger peak is defined by twin maxima and minima in the differentiated signal. These correspond to the maximum slopes of the leading and trailing edges of the Auger peak. In the differentiated signal the position of the Auger peak is defined by the zero point between the twin peaks of the differentiated signal, while the distance between the maxima and minima defines the peak width. These two parameters, peak position and peak width, are usually quite sufficient to identify the atomic transition responsible for the Auger signal, and hence the corresponding chemical species. This is often achieved by a simple visual comparison with an atlas of published spectra.

Intensity (a.u.)

Direct

Differentiated

0

500

1000

Kinetic energy (eV) Figure 8.8 Auger electron spectrum from copper can be presented as either the energy dependence of the collected signal intensity (the direct signal) or as the differential of this direct signal. The differential form is that usually published in the literature. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

Chemical Analysis of Surface Composition

435

Nevertheless, quantification of the Auger signal, as opposed to simple element identification, requires the same careful attention to the appropriate correction procedures as for X-ray microanalysis. In that case, we saw that data collection of the complete characteristic X-ray signal in the region of a peak was necessary in order to derive the total, integrated characteristic X-ray intensities (Section 6.2.4). 8.2.3

Identification of Chemical Binding States

Since the Auger electrons, like the photoelectrons, originate from the outer shells of the atoms, they are sensitive to the chemical binding states of the atom. In general, two effects are possible: either a simple energy shift of a main Auger peak that directly reflects a change in the binding energy of the atom, or a change in the energy loss structure on the low energy side of an Auger peak. An example is given in Figure 8.9 for an aluminium Auger peak originating either from aluminium metal or from the stable stoichiometric aluminium oxide. This low energy loss structure, observed in Figure 8.9, is analogous to the energy loss near-edge structure (ELNES) (Section 6.2.4), and can provide a wealth of additional chemical information that is only beginning to be explored. One difficulty in Auger fine structure analysis is to separate crystallographic effects from chemical effects. For example, the channelling of high energy electrons in the incident beam down a low-index direction may affect both the intensity and the structure of the Auger peaks, independent of any chemical effects, and lead to an orientation dependence of the recorded spectra.

Figure 8.9 Differentiated Auger peak for metallic aluminium compared with that for aluminium oxide, showing an 18 eV shift in the peak position and extensive differences in the energy loss structure. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

436

8.2.4

Microstructural Characterization of Materials

Quantitative Auger Analysis

As in the case of quantitative X-ray microanalysis, the role of quantitative analysis in both XPS and AES is to derive data on the chemical composition from a comparison of the measured intensities of specific spectral lines. The spectrum recorded from an unknown specimen is compared with that from a known standard obtained under the same experimental conditions. For a simple binary system, and assuming a linear dependence of composition on the measured intensity IA or IB, this implies an atom fraction of A, XA, that is given to a first approximation by: I A =I 0A  XA ¼
I A =I 0A þI B =I 0B

ð8:3Þ

The superscript 0 refers to pure standards of A and B. In practice, the standard intensity values are now available in the literature as calibrated sensitivity factors S and the atom fraction of A for a multi-component system is approximated by: I A =SA X A ¼ Pn i¼1 ðI i =Si Þ

ð8:4Þ

This equation may be satisfactory as a first approximation, but it omits corrections that arise from differences in emission probability. The emission probabilities depend on composition and are due, among other things, to changes with composition of the backscatter coefficient for the incident electron beam (compare the corrections for quantitative X-ray microanalysis, Section 6.1.3.1). For the simple binary alloy case, a matrix factor FAB can be included in the correction equation: X A ¼ F AB :

I A =SA I B =SB

ð8:5Þ

where FAB is assumed to be given by: FA AB ðX A !0Þ

   1þRA ðEA Þ aB 3=2 ¼ 1þRB ðEA Þ aA

ð8:6Þ

or 

FA AB ðX A !1Þ

1þRA ðEB Þ ¼ 1þRB ðEB Þ



aB aA

3=2 ð8:7Þ

where aA and aB are the atomic diameters of the two constituents and RA and RB are their backscatter coefficients. As in quantitative X-ray microanalysis, the problem is to devise an iteration procedure for a multi-component system. In the multi-component case, the matrix factors are estimated from the initial, measured intensity ratios, by assuming that the intensity ratio gives an approximate, zero-order composition ratio. The calculation can then be reiterated. As in the case of X-ray microanalysis, the concentration data reported in the literature very often omit details of their correction procedures. This is unfortunate, since the reader is then unable to judge the experimental significance and accuracy of the published data.

Chemical Analysis of Surface Composition

8.2.5

437

Depth Profiling

The excellentdepth discriminationinbothXPS and AES, that isthe resolutionperpendicular to the surface, makes them ideal tools for the investigation of thin-film devices, multilayered structures and distributed interfaces. This includes the analysis of microstructural features that are only revealed after sputtering, and not necessarily associated with the original free surface. Controlled sputtering of the surface by a chemically inert beam of energetic incident ions is the preferred method for obtaining a concentration profile. Sputtering rates depend both on the relative mass of the incident ion with respect to the sputtered species, and on the incident angle at which the ion beam impinges on the sample, as well as on the ion energy and the ion beam intensity. A Newtonian, billiard-ball model has been quite successful in predicting the main features of the sputtering process. In such a Newtonian model, the maximum energy Emax that can be transferred to the sputtered atom at the surface by an incident ion with kinetic energy E0 is given by: Emax 4m1 m2 ¼ E0 ðm1 þm2 Þ2

ð8:8Þ

where m1 and m2 are the atomic masses of the struck atom and the incident ion. The primary assumption made here is that the struck atoms react independently of their neighbours, so that the reaction time for the collision with the sputtered atom is small compared with the reaction time of its neighbouring atoms. This will certainly be true for high values of E0. For the special case m1 ¼ m2, then Emax ¼ E0. The larger the mismatch in the atomic masses of the incident and the sputtered particle, the smaller the value of Emax. The choice of argon as the preferred sputtering ion is dictated by its chemical inertness and its atomic mass, not too different from that of the major constituents in many engineering materials. The maximum value of energy transferred corresponds to a direct, knock-on collision in the forward direction. The sputtering process is complicated by multiple collisions in which a primary knock-on event ejects a neighbouring atom. Careful calibration of the sputtering rate is important for depth profiling by controlled sputtering. Multilayer calibration samples with known layer thickness and composition can be prepared by several standard methods, for example by sputter deposition. A low-angle, taper section through such a multilayer will show a series of interlayer interfaces that migrate steadily across the field of view as sputtering proceeds. This provides a reliable method for calibrating both the sputtering rate and the thickness removed. In analysing the depth profile data, it is necessary to allow for sputtering ion damage. The incident ions not only sputter away the surface atoms, but also inject point defects into the surface layers. The depth of the damage depends rather sensitively on the angle of incidence of the sputtering ions, as well as on the nature of the point defects that are created. Focused collision sequences due to a primary collision, can result in the injection of defects to some distance below the position of the primary event. Low angles of incidence for the sputtering beam reduce the depth of damage, but do not affect radiation-induced diffusion of constituents that is associated with the high point defect concentrations generated during ion bombardment. In particular, sharp concentration gradients in multilayer samples are often blurred during sputtering by inter-diffusion of the chemical constituents. In depth profiling, Auger spectra are usually recorded at sputtering intervals that correspond to the removal of less than an atomic layer. Auger spectroscopy has also proved exceptionally successful in the study of grain boundary embrittlement associated with

438

Microstructural Characterization of Materials

Figure 8.10 Auger spectrum from a brittle grain boundary failure in a nickel alloy. Carbon, boron and phosphorous are all enhanced at the boundary, indicating boundary segregation of these elements. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

impurities in steels and other alloys. The impurity segregant responsible for embrittlement is often localized within a few atomic layers of the grain boundary surface. For Auger analysis, the embrittled material can be fractured in a jig, mounted under ultra-high vacuum, and the sample is then transferred directly to the Auger analysis chamber without breaking the vacuum. Emission of occluded gas trapped in the sample occurs at the moment of fracture and makes it undesirable to fracture the sample within the Auger chamber itself. Even if an alloy is not embrittled by a segregant that is of interest, it is still possible to study grain boundary segregation phenomena in considerable detail. The sample is first loaded cathodically with hydrogen, to induce hydrogen embrittlement, and then fractured in the vacuum chamber. An example of results obtained from a nickel alloy is shown in Figure 8.10. The differentiated Auger peaks for phosphorus, boron and carbon are clearly visible, but decrease rapidly as successive layers of atoms adjacent to the fracture surface are sputtered away. Accurate quantitative data on the depth distribution of segregation for these elements is obtained, even in the absence of an Auger image of their spatial distribution. 8.2.6

Auger Imaging

As in the case of X-ray mapping (Section 5.3.1), the primary factor limiting acquisition of a resolved Auger image is the low signal intensity and the poor counting statistics, especially in the absence of a field emission source for the incident electron beam. The same relations noted for X-ray mapping apply, linking the number of counts needed per pixel point and the number of pixel points required to develop a useful image of the microstructural features. The human factor often plays a determining role, in addition to the mechanical and

Chemical Analysis of Surface Composition

439

electrical stability of the system, and 100 s is close to the limit of data acquisition times that a less dedicated user is prepared to accept for routine applications. As in the case of characteristic X-ray microanalysis, there are some further possibilities for improving the spatial ‘resolution’ in AES, especially in deriving a spectrum from a localized area. As in X-ray microanalysis, thin films can provide information from the region beneath a focused probe that is less than 20 nm in diameter, simply because, in the thin film, there is very little spreading and backscattering of the incident beam by inelastic scattering. In Auger spectroscopy, the signal is coming from the first few atomic layers of the sample, but it may be generated either by primary inelastic collisions of the incident beam, or by secondary collisions of backscattered electrons. For incident beam energies of the order of 10–20 kV, the area of emission for backscattered electrons reaches of the order of 50–100 nm in diameter, while the backscatter coefficient R increases rapidly with atomic number (Section 6.1.3.1). It follows that, while most of the Auger signal from a low atomic number matrix will be localized within the diameter of the focused incident beam probe, that from a high atomic number material will be distributed over the diameter of the backscattered electron distribution. This is illustrated schematically in Figure 8.11. Increasing average atomic number

Increasing incident beam energy

Primary Auger Excitation

Excitation by backscattered electrons

Sample

Sample

Sample

Sample

Figure 8.11 Schematic drawing showing the influence of atomic number and incident beam energy on primary Auger excitation and Auger excitation by backscattered electrons.

440

Microstructural Characterization of Materials

Figure 8.12 Auger images of aluminium conduction lines on silicon. (Courtesy of J.B. Wachtman, Characterization of Materials, Butterworth-Heinemann).

Providing the incident beam can be focused to a fine, nanometre-scale probe and the source intensity improved by using a field emission gun, then good resolution should be easier to obtain in an Auger signal than from a characteristic X-ray map, and very much easier in the case of a low atomic number matrix. If we set the upper limit of atomic number at Z ¼ 15, then this criterion will include silicon microelectronics technology, the aluminium aircraft alloys and all polymers! An example of a well-resolved Auger image is shown in Figure 8.12. Some care is required to avoid errors of image interpretation that are associated with the topography of the sample surface. In Figure 8.13, the edge of a surface step may shadow the signal from the matrix collected by the detector, while electrons penetrating the edge of a feature on the surface may generate a spurious matrix signal in a nearby zone. These topological effects are analogous to those discussed earlier for topological contrast in the scanning electron microscope (Sections 5.3.3.1 and 5.3.4.2).

8.3

Secondary-Ion Mass Spectrometry

We have seen how a beam of energetic ions incident on a surface will sputter the surface layers, displacing surface atoms into the vacuum, and how ion-beam sputtering (ion milling) can be used to prepare thin films for transmission electron microscopy (Section 4.2.1.3). This same sputtering process is the basis for the focused ion beam (FIB, Section 5.4.4). Ionbeam sputtering is also used in ultra-high vacuum Auger spectroscopy, both to clean the specimen surface and to remove successive atomic layers sequentially for depth profiling (Section 8.2.5). The next logical step is to analyse the sputtered ion signal, and this is precisely the function of the ultra-high vacuum secondary-ion mass spectrometer. In ion milling, for the preparation of thin-film electron microscope samples, the sputtering rate may be several micrometres per hour, corresponding to at least 50 atom layers per minute. This is more than an order of magnitude too fast for either sputter cleaning of Auger specimens or for depth profiling in Auger analysis. These atomic-scale

Chemical Analysis of Surface Composition

441

Figure 8.13 Schematic influence of ‘shadowing’, beam-penetration and backscattering on an Auger line-scan from silicon that has been partially covered by an aluminium conduction line. Reprinted from G.C. Smith, Analysis of nanometersized precipitates using atom probe techniques, Materials Characterization, 25, 1. Copyright (1990), with permission from Elsevier.

sputtering processes require a controlled sputtering rate of no more than a few layers per minute. The sputtering rates of interest for secondary-ion mass spectrometry, (SIMS) analysis are even lower, reflecting the extreme sensitivity of the SIMS ion detection system, which is capable of measuring atomic concentrations in the parts per billion (10
9) range. The sputtering ions in a SIMS system are usually positively charged, inert Arþ gas ions, but for some purposes a chemically active sputtering ion is desirable: either Csþ or Ga2þ to promote sputtering of electronegative elements, and O2þ to promote sputtering of electropositive constituents. The energies of the sputtering ions are typically 1–30 kV. The initial translational energy distribution of the ions sputtered from the sample surface depends on their complexity and can extend from a few eV up to as high as 100 eV (Figure 8.14). To ensure maximum conversion of neutral sputtered atoms to ions, the region of the sample surface may be bathed in a flux of low-energy electrons trapped in a cylindrically symmetric magnetic field. The sputtering yield in SIMS depends on a wide range of factors. These include the ionization potentials (positive ions) or electron affinities (negative ions), the mass of the impinging ion, the chemical activity of the target species, the incident angle of the beam on the surface, the take-off angle for the sputtered ions, and, not least, the composition of the target. The yields vary over many orders of magnitude and trace elements in the sample often have a major influence on the yields from other constituents. As would be expected, and as a general rule, the yield for positive ions decreases as the ionization potential increases, while the yield for negative ions increases with the electron affinity.

Microstructural Characterization of Materials

M1+

Relative intensity

442

M2+

M3+

5

10

15

20

Energy (V) Figure 8.14 The translational energy of the sputtered species in SIMS may range up to 100 eV. Molecular sputtered ions (M2, M3) have lower translational energies because energy is transferred to internal, molecular vibrational modes.

The rather sensitive dependence of the yield for a particular species on the composition of the target makes quantitative analysis completely dependent on accurate calibration standards. For example, the yield of copper ions from an aluminium alloy containing 2 % of copper actually exceeds the copper ion yield from a pure copper target that has been exposed to the same sputtering and ion collection conditions. The wide variations in the ion yield from pure targets, seen as a function of their atomic number, is related to the cyclic variations in ionization energy across the periodic table of the elements (Figure 8.15). In addition to the sensitivity of the yield for a particular species to the target composition, the impinging ion species can also have a dramatic effect. In comparing oxygen ion sputtering with caesium ion sputtering, differences in the yield of four orders of magnitude have been measured. This has been explained by the effect of the electron affinity of implanted oxygen in inhibiting ionization of the target, as opposed to the effect of implanted caesium in reducing the work function at the surface and therefore enhancing ionization. 8.3.1

Sensitivity and Resolution

The ions sputtered from the surface of the target specimen have low kinetic energies and must be accelerated into the spectrometer, where an electrostatic analyser is used to limit the energy spread of the collected ion signal before it is admitted to the mass analyser (spectrometer). Three types of mass analyser have been used. Quadrupole analysers have good mass resolution but are limited to the lighter ions [less than 103 amu (atomic mass units)]. The sensitivity is correspondingly low. Magnetic sector analysers have rather poor mass resolution but can analyse the full range of particle masses and are far more sensitive. Both of these analysers are sequential instruments, in which the

Chemical Analysis of Surface Composition 24

Calculation:

23 22

443

and

Experiment:

21 20

I/Z (eV)

19 18 17 16 15 14 13 12 11 10 9 8 0

10

20

30

40

50

60

70

80

90

100

Atomic number Z Figure 8.15 Measured variations in the ion yield from pure targets as a function of the atomic number. (Courtesy of L.C. Feldman and J.W. Mayer, Fundamentals of Surface and Thin Film Analysis, Elsevier Science Publishing Co.).

signal must be scanned across a detector slit at the entrance to the mass analyser, in order to record each mass peak separately. For this reason, they have sometimes been replaced by time-of-flight (ToF) mass spectrometers that were described previously for atom probe tomography (Section 7.3). The ToF spectrometer records the time taken for a sputtered ion of given kinetic energy to reach the detector from the target. The mass resolution can be as good as a quadrupole analyser and there is no limitation on the detectable mass range. The sensitivity is excellent, and most particles entering the ToF spectrometer can be detected. Moreover the system operates in parallel, and not sequentially, so that ions of all masses that are generated by a short pulse of incident ions can be detected simultaneously. Indeed, the major disadvantage is this need to pulse the signal entering the spectrometer flight tube. Pulsing the incident ion beam, by pulsing a magnetic field that deflects the incident ion beam off-axis, is just one option. Electrostatic deflection of the sputtered ions outside the detector entrance slit is a viable alternative. Since the flight times in the spectrometer require electronics with subnanosecond resolution, care must be taken to ensure that parasitic capacitance effects are minimized. Electrostatic pulsing of the sputtered ions may therefore be preferred, even though this means that a large fraction of the sputtered ion signal is lost. A second disadvantage of the ToF spectrometer is fundamental to the technique: since all the sputtered ions are accelerated through the same voltage, ions of higher charge are recorded at shorter times This means that the spectrum is recorded as a function of the

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Figure 8.16 ToF spectra from a silicon wafer examined before (a) and after (b) an organic cleaning treatment. (Reproduced by permission of R.D. Cormia, Advanced Materials & Processes, 12, 18 (1992) ASM International).

mass-to-charge ratio rather than the mass itself. Inevitably there are occasional ‘coincidences’ in the spectra and hence some ambiguities of interpretation. However, for elements with more than one isotope, most of these ambiguities can be resolved, since the ToF spectrometer is quite capable of resolving to better than 0.1 amu, and knowledge of the isotopic abundance for the elements is usually sufficient to remove the ambiguity. An example of a ToF spectrum is given in Figure 8.16, in which a silicon wafer was examined before and after an organic cleaning treatment. The peak at m/ne ¼ 18 in the spectrum from the wafer before cleaning is probably associated with H2Oþ ions. Under suitable conditions, the sensitivity of the SIMS technique enables surface and sub-surface impurities to be detected at the ppb (parts per billion) level. Accurately calibrated data may be interpreted quantitatively to between 10 ppb and 100 ppb. This is far better than can be achieved using the alternative techniques of microanalysis that we have discussed. 8.3.2

Calibration and Quantitative Analysis

As noted previously, the only satisfactory method for converting the relative intensities of the mass peaks in SIMS into mass concentrations is by recording data from known calibration samples. This can be done accurately by injecting high-energy ions of the

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impurity of interest into a pure sample of the matrix. The sub-surface distribution of the implanted impurity can be calculated from knowledge of the ionic mass, the incident energy and the matrix density. This is done using Monte-Carlo computer simulations. These can be compared directly with changes in the SIMS signal for the impurity while the surface of the ion-implanted standard is eroded by sputtering. Such carefully prepared calibration curves should be satisfactory over several orders of magnitude in concentration, with a limit of detection of about 100 ppb. Quantitative SIMS analysis has proved successful despite the serious calibration problems, and is usually based on the use of a calibrated parameter termed the relative sensitivity factor (RSF): IR IE ¼ RSFE · CR CE

ð8:9Þ

E and R refer to the element to be analysed and a reference element, respectively, while I and C are the measured secondary-ion intensities and the true atomic concentrations of the two species concerned. It is usual to choose the major component of the matrix as the reference element and, for trace element analysis, it is generally assumed that the matrix composition remains constant. Replacing the suffix R by the suffix M (for matrix) and rearranging the above equation we can write: C E ¼ C M RSFE ·

IE IM

ð8:10Þ

For trace element analysis, CM is constant, and the ratio of the measured ion intensities can be simply multiplied by a value of RSF taken from tabulated literature data. This treatment is also satisfactory for trace element analysis of semiconductors, based on stoichiometric compounds such as GaAs or InSb, taking either of the major constituents as the reference element. Quantitative analysis of trace impurity, dopant or alloy concentrations up to a maximum of 1 or 2 %, can be satisfactorily treated by the use of calibrated standards, but this is not possible for higher concentrations, since these are especially susceptible to changes in ion yield as a function of concentration. However, it is precisely in this range of composition that other techniques of surface analysis, especially Auger spectroscopy (Section 8.2), are capable of providing data that is both accurate and unambiguous. 8.3.3

SIMS Imaging

The ion beam impinging on the target can be focused to a probe that has a diameter of as little as 0.1 mm and this probe may be rastered across the target surface, as in the focused ion beam (FIB) microscope. The sputtered ions of the different target species can then be accelerated and detected to form an image of the target in which only ions of a specific mass (or mass-tocharge ratio) will be present. In quadrupole and magnetic spectrometer systems, once the signal has been electrostatically filtered, an aperture behind the mass analyser, placed in the back-focal plane of a collecting ‘lens’ can be used to select those ions having the required mass. The available spatial resolution is poor, and SIMS imaging has limited applications. As in scanning electron microscopy, the final image that maps the distribution of the elements is viewed on a monitor with the same time-base as the scanning coils of the ion beam probe that

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is focused on the target sample surface. Although the lateral resolution is very limited (certainly no better than 1 mm), the sensitivity is high, while successive scans can reveal changes in the sample with a depth resolution of better than one monolayer. As in Auger spectroscopy, depth profiling is often used to characterize layered structures, but blurring of the composition profile by radiation damage effects is a major problem. A more rewarding technique is to compromise between the ‘point’ analysis of a static SIMS analysis and the poor spatial resolution of a dynamic SIMS image by using a line scan. As in X-ray microanalysis (Section 6.3.2.1), line analysis gives far better counting statistics and may provide a satisfactory insight into the relationship between the microchemistry and the microstructural morphology, especially when combined with an alternative method of imaging the sample, such as the FIB. The range of concentrations that may be detected by SIMS covers some four orders of magnitude, while the depth resolution is better than 1 nm. No competitive technique presently available can achieve this same combination of depth resolution and sensitivity. Finally, the development of high resolution SIMS, based on FIB technology, is still in its infancy. A Gaþ ion beam, extracted from a liquid Ga source, provides an exceptionally fine ion-probe that could, in principle, have a far better analytical image resolution than is currently available in SIMS, perhaps as low as 50 nm.

Summary The chemical composition is the third class of information, following the crystal structure and microstructural morphology, that is required to complete the microstructural characterization of a material. However, the sensitivity of many material properties to the surface condition of the sample has led to a more specific requirement, for an assessment of the surface chemistry (the composition of the first few atomic layers of the sample). Examples of the importance of surface chemistry range from the segregation of impurities on a brittle fracture surface, to the composition of catalytically active layers on an inert substrate, and the chemistry of thin-film, semiconductor components or electrooptical detector devices. The chemical analysis of such surfaces is generally beyond the reach of the microanalytical methods that we have discussed earlier in Chapter 6. For example, a high-energy electron probe generates a characteristic X-ray signal that comes from a sample thickness of the order of 1 mm or even more, while the electron energy loss spectrum detected in a transmission electron microscope requires a sample thickness of the order of the extinction distance (typically about 10 nm). Of the wide range of physical phenomena that are used to derive chemical information from a sample, only three result in a signal which is sufficiently localized at the surface to ensure that only the surface layers of atoms contribute to the signal. These three methods are X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES) and secondary-ion mass spectrometry (SIMS). In XPS a secondary photoelectron signal is excited by an incident X-ray beam. Since low energy, secondary electrons can only escape from the sample if they are created in the first few surface layers of atoms, only these first atomic layers are sampled by the detected signal. The energy of the photoelectrons is sensitive to the local work function and the binding energy of the photoelectron. As a result, the technique is able to distinguish

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different chemical binding states, for example those associated with the different valency states of a polyvalent cation. Although XPS has no useful lateral resolution, the sensitivity of the technique to the chemistry of a single surface layer of atoms and the ability to detect changes associated with the early stages of adsorption, corresponding to just partial coverage of the surface by an adsorbate, combine to make this a very useful tool for materials characterization in surface science. AES differs from XPS in that the relaxation process for an atom in an excited energy state occurs by the emission of an electron whose kinetic energy is determined by the differences in energy for specific energy states of the excited atom. The energy of the Auger electron is therefore independent of the electron beam or X-ray photon energy used to excite the emission, unlike the energy of the photoelectrons emitted in XPS. Although the range of the Auger electrons escaping from the sample is similar to that of photoelectrons (and both originate from a depth of two to five atoms beneath the surface), the lateral resolution achievable in AES is limited only by the counting statistics for the Auger electron signal. As in microanalysis using characteristic X-radiation, the cross-section for Auger excitation is small, so that large electron beam currents are needed to excite an Auger signal and provide adequate counting statistics. A conventional electron source requires a minimum electron beam diameter approaching 1 mm if adequate Auger counting statistics are to be recorded. Far better spatial resolution for Auger analysis, perhaps as low as 10 nm, is achievable using a field emission source for the electron beam. Auger spectra are conventionally presented as a differentiated intensity signal, in which the peak position is determined by the crossover of the differentiated signal from a given Auger peak. The shape of the differentiated peak is sensitive to the chemical state of the excited atom, so that the Auger spectrum can be an important diagnostic tool for analysing the nature of the chemical bonding at the surface of the sample. Quantitative Auger analysis is possible, based on a comparison of measured, integrated intensities relative to known standards. In general, quantitative Auger analysis is not very accurate, partly because of the difficulties of calibration but also because the collected signal is usually from a comparatively large, thin area, so that the composition may well vary within the area analysed. In practice, AES is often combined with ion sputtering of the surface in order to remove successive layers of atoms, so that the Auger spectrum is then recorded as a function of the sputtered depth. The application of such depth profiling to research and development for thin-film devices is well-established. Quantitative depth profiling is not easy, primarily because it is difficult to ensure a uniform sputtering rate over the area to be analysed. Depth profiling is a standard procedure in the development of electronic and electro-optical devices, and has also proved extremely useful for the study of mechanical embrittlement associated with trace impurity segregation in structural materials. A high-intensity, field emission electron source in an Auger system can improve the spatial resolution to of the order of 100 nm or better. It is then possible to record an Auger image by scanning the incident probe beam over the surface and collecting the Auger electrons characteristic of any specific element of interest. Statistically significant images require large numbers of Auger electrons, usually between 100 and 1000 per pixel, and it follows that the signal intensity must be adequate if Auger imaging is to be justified. SIMS is far more mass-sensitive than either XPS or AES, and it is usually possible to detect trace impurities or dopants present at concentrations of the order of 10 ppb (parts per billion, 10
9). However, the ions detected are those sputtered from the surface, so

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that this technique is destructive, and removes successive atomic layers during the analysis. The SIMS technique is difficult to calibrate because the yield of a particular mass species depends sensitively on the concentrations of other elements present in the sample. In addition, since it is either the mass or the mass per unit charge that is determined, and most elements have more than one isotope and, often, more than one valency, a knowledge of the isotopic abundance and ionic charge is usually necessary to eliminate ambiguities. It is difficult to localize the signal to better than 1 mm, so that, although it is possible to collect secondary-ion images, in SIMS this is usually at the cost of losing the depth resolution, and always with rather poor lateral resolution. Even so, high resolution SIMS has been developed using highly focused ion beams scanned across the sample in a raster. Images with a spatial resolution of approximately 50 nm have been reported, although not for commercial instrumentation. It is important to recognize the major differences between the three surface analysis techniques discussed in this chapter and the methods of microanalysis discussed in Chapter 6. Neither X-ray microanalysis nor EELS are capable of providing analytical information that is localized to the immediate vicinity of a solid surface (the first few atomic layers). Although the methods of microanalysis discussed in Chapter 6 are influenced by the presence of thin surface films and contamination (which are therefore to be avoided), the primary objective of microanalysis is to determine bulk concentration on the microscale. The methods of surface analysis, introduced in this chapter, are tailored to determining chemical composition and composition changes that are present at, and immediately adjacent to, the surface itself.

Bibliography 1. J.M. Chabala, K.K. Soni, J. Li, K.L. Gavrilov, and R. Levi-Setti, High Resolution Chemical Imaging with Scanning Ion Probe SIMS. Int. J. Mass Spectrom. Ion Processes, 143, 191–212, 1995. 2. L.C. Feldman, and J.W. Mayer, Fundamentals of Surface and Thin Film Analysis, Elsevier Science Publishing Co., Inc, London, 1986. 3. J.B. Wachtman, Characterization of Materials, Butterworth-Heinemann, London, 1993. 4. J.M. Walls, and R.S. Smith, (eds), Surface Science Techniques, Elsevier Science Ltd, Oxford, 1994. 5. A. Benninghoven, E.G. Rudenauer, and H.W. Werner, Secondary Ion Mass Spectroscopy, John Wiley & Sons, Ltd, New York, 1987.

Worked Examples In order to demonstrate some of the techniques discussed in this chapter, we will focus on the characterization of samples of aluminium on a Ti/TiN/SiO2/Si stack prepared by chemical vapour deposition (CVD). We begin with Auger spectroscopy, which can be used to examine the surface of the aluminium, as well as the relative thickness of the different layers. Figure 8.17 shows an uncalibrated Auger electron intensity sputter profile, taken through the aluminium layer, the TiN film and terminating in the titanium substrate. The spectrum was taken from a specimen after deposition of a very thin, discontinuous aluminium film

Chemical Analysis of Surface Composition

Auger signal (peak-to-peak)

6

449

N+Ti

4

Ti N

2

C 0 2

4

6

8

10

12

14

16

Sputter time (min) Figure 8.17 Auger signal sputter profile from a multilayer specimen of aluminium deposited on a TiN layer on a titanium substrate. The spectrum was taken after deposition of a very thin, discontinuous aluminium film (that is, isolated islands of aluminium).

Atomic concentration (%)

(islands of aluminium on the TiN). A relatively strong oxygen signal is detected for short sputtering times (less than 2 min). This is due to the native oxide that had formed on the aluminium when it was exposed to air. In addition, a surface layer of carbon is evident, again a contaminant associated with exposure to air. Both the oxygen and the carbon signal drop to background levels after the initial sputtering, although the oxygen signal persists to a deeper level, presumably because of oxygen adsorption on TiN regions which were not covered by aluminium.

40

Ti N

20

C 0 2

4

6

8

10

12

14

16

Sputter time (min) Figure 8.18 The same Auger signal profile as Figure 8.17, but now calibrated for atomic concentration.

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Microstructural Characterization of Materials Ti 2

Ti 1

L3M23V L3M23M23

Ti Ti

L

Auger signal

I 420

L3VV

N (in Si3N4)

L

I 385

KL23L23

300

350

400

450

300

350

Energy (eV)

400

450

Energy (eV)

Figure 8.19 Standard Auger signals from the various atomic species present in the multilayer sample.

Signals from titanium and nitrogen are also evident, but its impossible to differentiate between TiN and titanium, since the Auger peaks overlap and the TiN and titanium films are very thin. Figure 8.18 is a calibrated Auger profile of the same specimen. An increase in the titanium and nitrogen signals with depth is clearly seen while the aluminium signal decreases. We can also use Auger spectroscopy to differentiate between titanium metal and titanium cations in TiN. Figure 8.19 shows standard Auger signals from the various atomic species in 100

Atomic concentration (%)

90 80 70 60 50

Ti

40 30 20

N

C

10

C

0 0

4

8

12

16

20

24

28

32

36

40

Sputtering time (min) Figure 8.20 Auger signal sputter profile from a similar specimen to Figure 8.17, but with a much thicker, continuous, aluminium film.

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451

Figure 8.21 TEM cross-section micrograph of the specimen from which the Auger sputter profiles in Figure 8.20 were obtained. This micrograph can be used to calibrate the sputtering rate.

our sample. A clear difference is evident in both the peak position and the peak shape for the Ti metal, and the TiN and TiOx ceramics. Figure 8.20 shows an Auger sputter profile from the same type of specimen, but this time with a much thicker, continuous aluminium film. A definite drop in the oxygen signal is visible after sputtering through the initial surface layers. (Note that the increase in the oxygen signal after about 24 min of sputtering corresponds to an SiO2 layer beneath the Al/ TiN/Ti sandwich.) In addition to verifying the purity and concentration depth profile of our films, we can also determine the thickness of the individual layers, by calibrating the sputtering rate. This is relatively easy, since we already have transmission electron microscopy (TEM) results for these materials (Chapter 4), so we need only compare Figure 8.20 with a cross-section TEM micrograph of the same specimen (Figure 8.21).

10 9

N(E)/E

8 7 Al

6 Al2O3

5 4 3 2 1 0

88

86

84

82

80

78

76

74

72

70

68

Binding energy (eV) Figure 8.22 XPS spectrum from the surface of the aluminium specimen before sputtering, showing the presence of Al2O3 in addition to metallic aluminium.

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Microstructural Characterization of Materials 10 O1s

9 8

6 5

SiLMM

0 1400

1200

1000

800

600

400

200

Al2p

O2s, Ar3s

1

Si2p

2

Si2s

OKLL OKLL

3

Al2s

C1s

4

Ar2p3

N(E)/E

7

0

Binding energy (eV) Figure 8.23 XPS spectrum taken over a larger energy range than that in Figure 8.22, showing the presence of carbon contamination on the surface of the sample.

Finally we can confirm the oxidation of the multilayer specimen surface using XPS. Figure 8.22 is an XPS spectrum taken before sputtering. A strong signal from Al2O3 (74– 75 eV) is clearly evident, in addition to the metallic Al 2p peak (72 eV). There is also clear XPS evidence for surface contamination by carbon, in an XPS spectrum that was taken over a wider energy range, but at a slightly lower energy resolution (Figure 8.23). The carbon 1s XPS peak (285 eV) confirms the previous Auger spectroscopy findings. After sputtering for 5 min, the superficial Al2O3 layer is completely removed (Figure 8.24). 10 Al2p

9

N(E)/E

8 7 Al 6 5 4 3 2 1 0 88

86

84

82

80

78

76

74

72

70

68

Binding energy (eV) Figure 8.24 XPS spectrum After sputtering for 5 min. The Al2O3 signal is no longer visible.

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Problems 8.1. How ‘thick’ is the ‘free surface’ of a solid? Justify your answer! Is there a distinction between the physical thickness, reflecting the electronic structure, and the chemical thickness, reflecting composition variations? 8.2. Give three examples for which you would expect that surface analysis of a solid sample would be more significant than microanalysis of a polished section. 8.3. Analytical information can be obtained from a wide range of signals. What properties of photoelectrons, Auger electrons and secondary ions make them the preferred signals for surface analysis? 8.4. Compare the limits of detection for chemical analysis using photoelectrons, Auger electrons and secondary ions. What factors restrict our confidence in a quantitative analysis of the composition for each method? 8.5. An XPS signal contains information on the chemical binding state of the elements detected. Suggest how this type of information might be useful in the study of a metallurgical failure. Distinguish between a purely mechanical failure and one associated with environmental attack. 8.6. Auger spectroscopy is often used for depth profiling in thin-film devices. Discuss the sources of error involved in plotting concentrations determined by Auger spectroscopy as a function of sputtered depth. (Consider especially the topography of the sputtered surface, ‘knock-on’ radiation damage, and the quantitative calibration of the Auger signal.) 8.7. Argon (atomic weight 40) is usually used to sputter-clean surfaces for Auger spectroscopy and to remove successive atomic layers in depth profiling. Compare the expected sputtering efficiency for aluminium, iron and tungsten (atomic weights 27, 56 and 184) by energetic argon ions and suggest some possible strategies that might be used to improve the reliability of depth profiling. 8.8. Secondary-ion mass spectrometry frequently gives a larger intensity signal for sputtered copper ions in an age-hardened copper alloy than from a pure copper, calibration standard. Suggest some possible physical reasons for this observation. 8.9. What are the advantages and disadvantages of a time-of-flight mass spectrometer, as opposed to a quadrupole or magnetic sector analyser, for secondary ions? 8.10. The spatial resolution in conventional secondary-ion mass spectrometry (SIMS) is limited to about 10 mm, but in principle this could be improved by taking a line-scan rather than a TV-raster image. Assuming that the only limitation is in the counting statistics, estimate the expected resolution of a SIMS line-scan. 8.11. Multilayer films are often sectioned at an inclined angle to the surface prior to Auger analysis. What are the experimental advantages of such a wedge-shaped specimen geometry?

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8.12. To a first approximation, an Auger electron intensity peak can be described by a Gaussian intensity distribution about an energy E0 according to:  2 N0
e ð8:11Þ exp NðeÞ ¼ 1=2 2 2s2 ð2ps Þ where e ¼ E-E0.

X

Figure 8.25 substrate.

Schematic drawing of alternative methods for Auger analysis of a thin film on a

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(a) Find an expression for the total number of electrons counted in an Auger peak. (b) Derive an expression for the peak-to-peak height in a dN/de plot. (c) Sketch a graph of both dN/de and N(e) on the same axis. (d) Assuming that the surface concentration of a specific element is proportional to the number of Auger electrons having an energy E0, when might it be appropriate to use the differentiated signal, peak-to-peak height to estimate a surface concentration? 8.13. A thin film of thickness x was deposited on a single crystal, silicon substrate. Figure 8.25 shows schematically three ways in which the film could be characterized by Auger spectroscopy, using an incident electron beam 200 nm in diameter: auger depth profiling using Ar ion sputtering perpendicular to the surface; cleavage of the wafer followed by an Auger line-scan of the cross-section (without sputtering); and mechanical polishing at a wedge angle of 10 followed by an Auger linescan analysis (without sputtering). (a) What do you think is the minimum film thickness that could be measured by each of these three methods? (b) List the advantages and disadvantages for each of the three methods. (c) Suggest an experimental methodology that would optimize the accuracy of an investigation of this sample by auger electron spectroscopy. 8.14. A thin film of element A was deposited on a substrate of element B. Auger analysis was conducted without sputtering and the Auger signal intensity from element B was measured. The mean free path of these Auger electrons was lB. For an uncoated substrate, the Auger electron intensity for this peak was I0. (a) Assuming that the layer of A has a uniform thickness t, what do you expect the intensity of the signal to be as a function of the thickness of A? Sketch the expected graph with a thickness scale. (b) Assuming that the element A formed discontinuous islands on the substrate during the deposition process, sketch a graph showing the Auger signal from B as a function of coverage by A. (c) Assume you have received a substrate of element B, upon which was deposited a thin film of A, but that you do not know if the morphology of the film of A is continuous or discontinuous. How could you determine this by Auger electron spectroscopy?

9 Quantitative and Tomographic Analysis of Microstructure In Chapters 6 and 8 we outlined some of the methods available to convert a variety of spectroscopic data into quantitative estimates of chemical composition, while in Chapter 7 we saw that atom probe tomography was capable of combining both three-dimensional chemical and morphological information with a resolution on the sub-nanometre scale. In Chapter 1 we also noted that parameters derived from microstructural observations, such as grain size, particle size and the volume fraction of a second phase, were directly related to structure-sensitive engineering properties. However, we have not yet attempted to quantify the three-dimensional microstructural parameters that can be extracted from routine, twodimensional projection-image data, nor have we described the methods that are available for us to estimate numerical values for the three-dimensional parameters that can be derived from measurements taken from two-dimensional images. In part, the difference in the treatment of chemical and morphological information is a consequence of the way in which we usually regard information associated with microstructural chemistry and microstructural morphology. An image of the microstructure is, for many purposes, seen as an end in itself (and ‘worth a thousand words’, as Confucius is said to have expressed it). The image is often discussed as though it were itself the object – pearlite, a eutectic or dendrites, to give but a few examples of terms used to describe both image features and microstructural constituents. This is not the case with spectroscopic data, and it is only in the event that the spectrum is used to identify the simple presence of an element (provide a ‘fingerprint’ of the element), irrespective of its concentration in the matrix, are we likely to be satisfied with the mere identification of a characteristic intensity peak in the spectrum. Crystallographic data are also frequently used to identify the presence of a phase, irrespective of its volume fraction in the sample (the phase concentration) or its distribution in space (the phase morphology). Any relation between the crystal orientations and the geometrical axes of an engineering component, corresponding to the existence of preferred orientation, is then seen as a topic requiring special investigation. The X-ray determination of particle size or grain size for powders or crystallites, and the presence of microstresses,

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

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Microstructural Characterization of Materials

which can both be determined from measurements of X-ray peak width and peak shift, are also techniques of quantification that, while available when required, are generally seen as subsidiary to the main ‘business’ of diffraction studies, generally considered to be phase identification and structure determination. In this chapter we concentrate on the quantification of microstructural morphology and examine the options available for three-dimensional reconstruction of information derived from serial sections. A general term often used for the computer-generated imaging of three-dimensional data sets is tomography, although this is a partial misnomer, since it is derived from the Greek and means a drawing (graphos) of a section (tomo). In modern usage it refers to the two-dimensional presentation of the three-dimensional data using a computer program that allows the projection to be rotated about any axis, usually in defined steps of the rotation angle, or to be viewed as a selected two-dimentional section taken through the three-dimentional data set. The earliest work in this area was for the study of anatomy (the morphology of human and animal physiology) and histology (the morphology of biological tissues or isolated, single cells). The study of the spatial relationships between objects is termed stereology, and the basic stereological concepts have been known and applied for over a century, primarily in the very diverse fields of medicine (that is, anatomy and its microstructural counterpart, histology) and geology (again, both on the macroscale, as the structure of the earth’s crust, and on the microscale associated with mineralogy). The availability of ever-increasing computational speed and memory has led to the development of a wide range of software programs that are devoted to one or other of the two key issues for the quantitative interpretation of image data, namely image data processing and image data analysis. Image data processing is concerned with ‘correcting’ the raw, digitized data by removing random background, enhancing or reducing contrast, and averaging the measured data over neighbouring pixels (data smoothing). Image data analysis is concerned with extracting quantitative measurements of selected parameters from the image and combining data sets, for example, those from successive serial sections. Image data analysis is the principle topic of this chapter. We assume that the recorded image has already been ‘processed’ to maximize the information content and minimize background noise and artifacts, and we focus our discussion solely on image data analysis.

9.1

Basic Stereological Concepts

The quantitative analysis of two-dimensional image data is very much a ‘one step back, two steps forward’ process. A section is first prepared from a three-dimensional object and imaged in two dimensions (‘one step back’). These image data (possibly recorded as a function of time) are then analysed. The expectation is that the image analysis will result in a quantitative estimate of microstructural parameters that are relevant to the three-dimensional object, that is, the ‘two steps forward’. In some cases, two-dimensional image analysis of a microstructural parameter can be unambiguous, and we refer to such parameters as being ‘accessible’ from the two-dimensional image, although with varying degrees of accuracy. In other cases we can only estimate the three-dimensional microstructural parameters from two-dimensional data on the basis of an assumed model for the microstructural features. Such microstructural parameters are then said to be ‘inaccessible’.

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Before we discuss these complications in any detail, we first need to examine some of the microstructural factors that may affect stereological analysis. 9.1.1

Isotropy and Anisotropy

Microstructural anisotropy can take two forms, since the term anisotropy is applied to both anisotropy of the morphology of visible microstructural features and anisotropy of the crystallographic orientation of the crystalline phases that are present in the microstructure. In both cases, the reference coordinates are the geometry of the engineering component that is being studied. If the microstructure is isotropic, with respect to both microstructural morphology and crystallographic orientation, then both the diffraction spectra and the microstructural image will be independent of the plane of the sample section that has been taken through the component or of any selected direction within the section plane. Crystallographic anisotropy is best termed preferred orientation, in order to avoid confusion between morphological anisotropy and crystalline anisotropy. Crystalline anisotropy is associated with the orientation dependence of the physical properties of the crystal lattice, for example the variation of the elastic tensile modulus measured along the different crystal directions. Preferred orientation is usually determined from an analysis of the orientation dependence of the diffraction peak intensities that have been measured in different spatial directions within the coordinate system of the sample (Section 2.4). Morphological anisotropy implies that one or more microstructural parameters depend on the orientation of the direction or plane with respect to which that parameter has been measured. An obvious example is the elongation of the grains in a metal bar due to plastic elongation of the specimen. In this case, the microstructural change in the ratio of the grain length to grain width (the aspect ratio) will depend both on the total elongation of the bar and the plane of the selected microstructural section. If the plane of the sample is defined by the direction of tensile elongation and its normal, then the aspect ratio observed for the grains seen in the plane of the section will be a maximum, while the distribution of this aspect ratio (maximum and minimum values of the aspect ratio in this longitudinal section) will be related to local, crystallographically determined, variations in ductility and stereological constraints that are imposed by neighbouring grains in the material. The grain aspect ratio in a given section of a rolled metal sheet (Figure 9.1) will depend on the mechanical processing history. If the sheet was rolled as a continuous strip, in a series of individual passes, then the grains will all be elongated along the direction of rolling, but if the reduction in thickness is the result of rolling by equal amounts in two directions at right angles (cross-rolling), then the grains will be flattened in the plane of the sheet, rather than elongated. In general, at least two sections will be needed to characterize morphological anisotropy in a rolled sheet (taken perpendicular and parallel to the plane of the sheet), so that the sample sections contain all three of the principle directions in the product. These are usually termed the longitudinal, transverse and through-thickness directions. A similar example is given by the distribution of nonmetallic inclusions in a metal sheet, but in this case it is the distribution of the inclusions, which constitute a second phase in the system, that is anisotropic, and not only the shape of the inclusions (Figure 9.2). Most composite materials exploit mesostructural anisotropy in order to optimize their engineering properties and to minimize the weight or the dimensions of a structural component in an engineering system. Composite lay-ups of resin-bonded sheets of fibre can

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Figure 9.1 Anisotropic grain shapes in a rolled copper sheet revealed by etching a specimen that has been sectioned parallel to the rolling direction. (Courtesy of Metals Handbook, American Society for Metals).

be oriented with the fibres aligned at predetermined angles in a variety of geometries or architectures, in order to obtain the desired mechanical properties. These properties are usually the strength and stiffness of the composite component. In short-fibre reinforced composites the lengths of fibre are randomly distributed in a principal plane of the product,

Figure 9.2 The same copper sheet, imaged unetched but after annealing. Oxide inclusions are visible, anisotropically aligned along the original rolling direction. (Courtesy of Metals Handbook, American Society for Metals).

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but all the reinforcing fibres lie within that plane. The mechanical properties will exhibit inplane isotropy, but through-thickness anisotropy. Finally, we should note that any individual particle in the microstructure may exhibit anisotropy that is associated with either its shape or its crystal structure (or both), but the particles themselves may still be distributed randomly in the sample. If the morphology is unaffected by the plane of the section, then the sample is considered isotropic, irrespective of the shape of the individual grains or particles. 9.1.2

Homogeneity and Inhomogeneity

Processing technology is usually geared to ensuring the homogeneity of a material product, so that any random sample selected from an engineering component will have sensibly identical microstructure and properties. This is not always the case. For example, cast ingots of steel and other alloys may be hot-rolled in stages to produce a homogenous finished product (either sheet, rod or profiled bar), but in the early stages of this hot-working process, a distinction must be made between material that is derived from the top, the bottom or the middle of the original ingot. The heavier, nonmetallic inclusions in the metal will tend to be concentrated at the bottom of the ingot, while the higher impurity levels and excess alloy concentrations are usually to be found, partially segregated, in the last fraction of the liquid metal to solidify, together with the lighter inclusions, at the top of the original ingot. The end product may therefore be significantly inhomogeneous, with respect to both the inclusion content and, to some extent, the alloy composition. We should distinguish between several types of inhomogeneity. In the example of the cast ingot, we note that the inclusion count and the alloy composition are likely to vary independently. The inclusion count shows morphological inhomogeneity, while the alloy content shows chemical inhomogeneity. Crystallographic inhomogeneity may also be observed in the ingot, as an inhomogeneous distribution of the preferred orientation that reflects the solidification direction in different regions. As another example, consider a ductile metal bar that undergoes shear during extrusion (Figure 9.3). The amount of shear is a function of the distance from the axis of the bar, and the preferred orientation (the deformation texture) will therefore vary across the section,

Die

Extrusion Direction

Figure 9.3 Schematic representation of the distribution of shear deformation across an extruded rod.

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Microstructural Characterization of Materials

with the maximum shear component in the exterior layers. This crystallographic inhomogeneity may be retained after the metal has been annealed, and can show up as an annealing texture (the preferred orientation characteristic of the deformed metal after recrystallization). Other mechanical-forming operations have similar effects on the texture and its distribution in the final component. Forged components usually retain a marked macrostructural inhomogeneity that is on the scale of the component dimensions. Such inhomogeneity may be beneficial to the toughness and fatigue resistance of the component. Figure 9.4 is from a forged steel cam shaft that shows lines of macrostructural flow which inhibit fatigue crack propagation through the load-bearing cross-section. It follows that there are many instances where the microstructure of an engineering component cannot be adequately characterized from either a single sample, taken from the bulk, or a single section, taken through the sample. An extreme situation would be a solidstate semiconductor device that consists of a large number of micro-components densely arrayed on a single silicon chip. For such a case, focused ion beam milling could be used to prepare several cross-sections for transmission electron microscopy (Section 5.2.1), in which each section includes several of the interfaces between the active and the passive microelectronic components. Similar considerations may apply to coated or surface-treated engineering products, comprising surface layers that vary in both microstructure and

Figure 9.4 Macroscopic development of textured ‘grain’ during a forging operation. (Reproduced from J.L. McCall and P.M. French, Metallography in Failure Analysis, Copyright (1978), with kind permission of Springer Science and Business Media).

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composition through the layers. This would be true for electroplated components, chemical vapour deposited coatings, thin-film devices and ceramic-coated cutting tools. 9.1.3

Sampling and Sectioning

In microstructural characterization, the question most often posed is whether an observed microstructure is substantially the same as or significantly different from some other microstructure. Since microstructures are examined on a diminishing length scale as the magnification is increased, we need to distinguish between the microstructural variability within a given selected sample, and the variability observable between a set of different samples. Moreover, within any region of a selected sample, the different microstructural features will each have their own characteristic length scale. Such length scales may correspond to either a particle size or a particle separation, or to any other specific microstructural feature. For example, a eutectic structure (Figure 9.5) could be characterized by the size of the eutectic colonies, the separation of the colonies, the shape, spacing and dimensions of the lamellae within a colony, or the volume fraction of each phase in the eutectic. (Remember that three-phase ternary eutectics may also exist in engineering alloy systems!) We will limit our discussion to three length scales for microstructural ‘sampling’ that might affect the statistical significance of any attempt to quantify the data derived from a microstructural investigation: 1. First, the macroscale over which the samples were selected with respect to the geometry of an engineering component. On this scale, someone will have to decide if a single sample is sufficient, or whether sections should be taken from different regions of the component or in different orientations, in order to test for inhomogeneity and anisotropy. If several samples are to be investigated, for example, from a set of components that have received different heat treatments, then we will need to ask if it is sufficient to take a single section for each condition, or whether several, nominally ‘identical’ samples need

Figure 9.5 Eutectic colonies in a tin alloy are characterized by the size and shape of the colonies, the volume fraction of the phases present and the spacing of the lamellae within the colonies. (Courtesy of Metals Handbook, American Society for Metals).

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be taken in order to ensure the reproducibility of the results. As an example, a useful quality-control test might be to take several representative samples from each production batch of a component and test for uniformity of the grain size. The statistics of sampling will require data from a minimum of three samples if some estimate is to be made of both the mean and the variance of any measured parameter. This requirement corresponds to assuming a Student’s-test statistical distribution in which the total number of ‘degrees of freedom’ for a set of n data points is equal to n-2. 2. For any given microstructural cross-section, it will be important to check the variability of the microstructure across the sample section, that is the mesostructural variability, and the magnification selected for any recorded image will determine the diameter of the field of view in real space. The features of interest must be clearly resolved in the image, but, at the same time, a sufficient number of features must be present in the field of view in order to ensure adequate sample statistics for each selected image area. For small features that are widely separated, these conditions may be in conflict and require that several images be recorded at different magnifications. The statistics of the mesostructure will determine whether or not we can make a statement about the uniformity of the morphology within a sample with any confidence. In cases where inhomogeneity is an important issue, it is usually possible to choose an appropriate magnification and then select a regularly-spaced set for the imaged fields of view, in order to derive a statistically significant data set. For example, a surface coating could be sampled by recording a set of images on a section normal to the coating, but with the images at a shallow angle to the intersection of the plane of the coating with the section (Figure 9.6). A set of three or more such lines will improve statistical estimates for the depth dependence of the relevant parameters. 3. Finally, the distribution of the microstructural features within any given field of view will determine the statistical significance of estimates of the microstructural parameters that are based on this single image field. In general, estimates taken from other fields of view on the same section, or from other sections, will give different values for both the measured parameters and their variance. It is therefore important to distinguish between errors that are associated with the analysis of a single field of view and errors associated with a complete data set that comprises many fields of view taken from either the same or from different samples. We will return to this point later.

Coating

Substrate

Figure 9.6 One possible array of imaging sites (þ) for representative, high-magnification sampling of a coated specimen section.

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It is usually possible to select an effective ‘field of view’ that is larger than any individually imaged region, either by recording a ‘panorama’ of overlapping images from neighbouring regions or by scanning an image ‘strip’ along a selected direction in the plane of the section. Such procedures may be unavoidable if the features to be recorded are both small and widely separated, since it may not be possible to resolve a sufficient number of features in any single image to provide sufficient statistical data. Some balance has to be struck in selecting samples from the different length scales that we have chosen: the macrostructural, the mesostructural, the microstructural or, possibly, the nanostructural. Visual inspection (or common sense) may allow us to dispense with formal sampling on the macroscale, and this is perfectly legitimate providing we are aware of what we have done. It is much less legitimate to dispense with meso sampling. At the very least, the different regions of any chosen section should be scanned for visually significant differences in the microstructure. Ideally, steps should also be taken to ensure that the variance of a measured parameter, such as grain size, associated with mesostructural sampling matches that found for the field of view of any single microstructural sample. In other words, the statistical errors derived when comparing different regions should exceed significantly the errors determined for any one region. The availability of methods of microstructural characterization that extend down to the nanoscale has resulted in new challenges for the statistical analysis of image data. In the atomic force microscope, the number of pixel points in a given image data set is far less than we have grown used to in other instruments. The atomic force microscope field of view is limited and, as in the early days of electron microscopy, it is all too easy for the subjective microscopist to ‘select’ images as being ‘typical’ of a phenomenon to be demonstrated. One further point, it is good practice to select areas of the microstructure for quantitative analysis that lie within an imaged region, leaving a well-resolved, microstructural ‘border’ outside the area to be analysed. This is illustrated in Figure 9.7, and avoids selecting a region for analysis adjacent to an obvious defect in the sample (due to either processing or sampling). This procedure also enables the observer to assess the importance of possible

Figure 9.7 The region to be analysed quantitatively should lie within a well-resolved, microstructural ‘border’ (See text).

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edge effects that can arise, for example, at grain boundaries and grain boundary junctions. The microstructural ‘border’ should be selected to have an area approximately equal to p theffiffiffi area that will be quantitatively sampled, corresponding to a linear dimension that is 2 greater than the actual analysed area and a border width that is approximately 20 % of the analysed image dimensions. 9.1.4

Statistics and Probability

A number of statistical functions and procedures have proved useful in the quantitative analysis of microstructural data, but a detailed description of these functions is beyond the scope of this text. Nevertheless, a very brief account of some statistical tools is in order. In general, the measured values of any parameter will be distributed about a statistical mean or average value, and the width of this distribution can be defined and measured to describe the spread of the measured values in terms of a ‘statistical error’. Moreover, any statistical function that is derived from data related to frequency of observation can also be interpreted as a probability distribution. The probability distribution, or frequency function, determines the probability that a parameter (in this case, a microstructural parameter) will have any given value. If the statistical frequency function is found to be skewed (asymmetric), then this also may be described by an appropriate statistical parameter. Different statistical tests may be applied to the data collected, providing the appropriate mathematical conditions are met, and may be used to decide, for example, whether two different data sets come from the same population, or whether the values of two different parameters can be correlated with one another. We do need to clarify what we mean by probability since this concept has always presented problems. There is considerable overlap between ‘probability theory,’ the study of parameters that are related by a stochastic or, equivalently, nondeterministic function, and statistical analysis, that is the estimation of a parameter based on a frequency analysis performed by sampling from a well-defined population. In many probability-based processes, subsequent events are linked, so that a single event in a chain determines the probability that the next event will occur. Such sequences are termed Markov processes and may be observed in more than one dimension (chain-branching in polymerization is just one example). A further complication arises when a phenomenon can be interpreted as fractal in nature: the process is then fully deterministic, but not amenable to analytical analysis. Weather forecasting is one area in which fractal analysis has resulted in major improvements in forecast accuracy. Fractal analysis has been fruitfully applied to the propagation of fracture in brittle materials. The interested reader is encouraged to consult other texts. Although these topics are frequently of direct relevance to the quantitative analysis of microstructure, they fall well outside our present mandate. As a very good first approximation, all measurements of microstructural parameters are subject to statistical errors that are associated with just three separate factors: 1. The inherent variability of the parameter in the bulk material, for example, the grain size of a polycrystal. This means the errors associated with the statistical distribution of the parameter that we would like to determine. 2. The statistical errors that are associated with the methodology of sample selection for microstructural observation. In particular, errors associated with the number, position, orientation and size of the features that are available for analysis.

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3. Any errors that are associated with our methods of observation or data collection and specimen preparation, as well as the method of measurement we have selected. Under this heading we include the artifacts due to polishing and etching procedures used to prepare the section, the performance and characteristics of the microscope, the properties of the recording media or the values of the digitizing parameters (pixel density, dynamic range or channel width), as well as any intrinsic error associated with the definition of the microstructural parameters. It is not always easy to separate the inherent variability of a bulk material parameter from errors that are due to specimen preparation, or the methods of observation and data collection or recording. The essential requirement is to reduce these secondary effects to a level where they no longer affect the significance of the results. For example, if it is not possible to exclude these errors from the measurement of grain size, then clearly it is also impossible to accurately quantify variability in the grain size of the bulk material.

9.2

Accessible and Inaccessible Parameters

The distinction between accessible and inaccessible bulk microstructural parameters is often basic to the quantitative interpretation of morphological data collected from a twodimensional sample section, but is often ignored. There are a very limited number of accessible microstructural parameters characteristic of the bulk structure that can be derived directly and unambiguously from a two-dimensional section. The volume fraction of a second phase is one of these accessible parameters. However, the particle size, the volume per particle and the number of particles per unit volume cannot be derived from observations made on a planar section without making considerable assumptions about the particle shape. (For example, we could assume that all the particles have only positive curvature with no re-entrant angles.) Figure 9.8 makes clear why such simplifications are often necessary when modelling microstructure. A section through a ‘doughnut’ or any other feature with a partially concave

Figure 9.8 A feature having regions of both positive and negative curvature may intersect the section surface to appear as more than one area intercepted on the section.

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interface can appear in the plane of the section as two disconnected areas of second phase. Dendritic grains are an excellent example of a class of feature that has both positive and negative curvatures. The dimensions characteristic of dendritic growth: the number of grains and the spacing of the branches, for example, cannot be derived from a single metallographic section. Serial sectioning can provide information perpendicular to the plane of the sections. In serial sections, thin layers of known thickness are removed sequentially, using fiducial markers to ensure that image data are recorded from the same area at each stage. Serial sectioning techniques are standard practice in many histological studies of soft, biological tissues and cells, and have now been extended to many engineering applications for the three-dimensional microstructural characterization of materials (Section 9.5). Providing the features of interest in the microstructure have linear dimensions that exceed the thickness of the layers removed at each stage, then each feature observed on the nth section can be related to the cross-section of the same feature on the (n
1)th and (n þ 1)th sections to derive a three-dimensional reconstruction of the bulk microstructure. The ‘resolution’ perpendicular to the sample surface will, of course, depend on the separation of the serial sections. Mechanical polishing can be automated to an accuracy of better than 10 mm with no special precautions, while focused ion beam (FIB) milling can remove controlled thicknesses of about 10 nm. Clearly, the latter technique has major potential for electron microscope investigations and is now being used to scan ‘through’ thin-film devices in microelectronic and electro-optical systems. Microtomes (rigidly mounted knives that can cut thin slices from a sample) have been in use in the biological sciences for many years and are used to produce serial sections of histological samples. In the atomic force microscope, diamond-knife microtomes have been used for in situ serial sectioning. All the above techniques have disadvantages: mechanical damage, especially when microtome slices are prepared, and radiation damage generated during ion milling. 9.2.1

Accessible Parameters

Only a few bulk microstructural parameters can be accessed from a two-dimensional section without making serious morphological assumptions. Nevertheless, it is possible to define some parameters so that they become accessible from the two-dimensional image. Two examples are grain size and particle size, as we shall see below. Furthermore, a single assumption is all that is necessary to make the transition from ‘inaccessible’ to ‘accessible’ for a wide range of other parameters. This simple assumption is that all interfaces are convex, since no completely convex particle (a particle that possesses only positive curvature) can intersect a planar section discontinuously. Thus, for convex particles only, every area of a grain or second phase particle that is observed on a planar section represents a single particle or grain in the volume of the bulk material. Several other parameters that are characteristic of a planar section can be determined quantitatively, but these are not bulk parameters. An example would be the number of particles of a second phase per unit area of the section. Such two-dimensional, but ‘accessible’, parameters are of limited value, although they have their uses. For example, maximum allowed inclusion counts, taken parallel and perpendicular to a principle axis of

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the component, are often written into industrial acceptance criteria for a product, as a quality control, standard requirement. 9.2.1.1 Phase Volume Fraction. The volume fraction of a second phase fV is a bulk microstructural parameter that, in the case of a fully crystalline material, can usually be determined by quantitative X-ray diffraction analysis (Section 2.4). However, many materials contain noncrystalline phases that cannot be identified with any reliability by X-rays, and there may be other reasons why the microstructural determination of fV from a planar section is to be preferred to a diffraction analysis. A remarkable result of stereology (the study of spatial relationships) helps us to select a reliable method for measuring fV from a planar sample section: fV ¼

V A L P ¼ ¼ ¼ V 0 A0 L0 P0

ð9:1Þ

where V, A, L, and P are the total volume of the second phase, the area of the phase intersected on a random planar section, the line length traversing the phase for a randomly oriented line across the section, and the number of points falling within the phase for a random array of points superimposed on the same planar section, respectively. The subscript 0 refers to the total volume of the sample, the total area of the section, the total line length examined and the total number of points in the test array. It follows that the same result is to be expected for fV regardless of whether fV is measured by sampling a volume fraction, an areal fraction, a line fraction or a point fraction. We can devise an experiment that could confirm this result (Figure 9.9): First a unit volume of the sample is immersed in a suitable medium to dissolve the matrix and the particles could then be collected, to determine their volume, by weighing, providing their density is known [Figure 9.9(a)]. Secondly, a micrograph of a ‘random’ section of the sample could be recorded and digitized, and the total area of particles intersected per unit area of the section could be assessed from the total number of pixels associated with the sectioned particles [Figure 9.9(b)]. Thirdly a random array of test lines could be superimposed on the sample, and the total length of line falling within the particle sections could be determined and divided by the total length of the test line [Figure 9.9(c)]. Finally, a random array of pixels could be selected from the random test section and the proportion of pixels falling within the areas corresponding to the sectioned particles could be determined [Figure 9.9(d)]. Which of these methods is ‘best’? Dissolving away the matrix is only likely to be feasible in exceptional circumstances, and is likely to be both time-consuming and prone to experimental errors that will be difficult to gauge. Areal analysis is a natural choice for any digitized image, since all the available data on the sample section are included and it is only necessary to set intensity thresholds and determine the proportion of the pixels scanned whose intensity falls within the set intensity window. Of course, the pixel spacing has to be appreciably less than the size of the particle sections (and, preferably, less than the resolution in the observed image). The plane of the section is important and should be chosen to coincide with the principal axes of the component, while close attention must be paid to any statistically significant variations in the results that might be associated with microstructural anisotropy or microstructural inhomogeneity.

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Plane of Section (a) Dissolution of the matrix

Plane of Section (c) Random lines

(b) Areal analysis of a cross-section

Plane of Section (d) Random points

Figure 9.9 Possible methodologies for volume fraction analysis based on Equation (9.1) and using either (a) volumetric, (b) areal, (c) linear or (d) point analyses.

Linear analysis could also be very useful, especially if the particle spacing is very much greater than the particle size. The most efficient methodology for acquiring digitized data for linear analysis would then involve spacing the test lines at a distance comparable with the spacing of the features of interest (the particle sections), while controlling the pixel size along the lines to well below the particle section diameter, and preferably less than the microscope resolution. Such a strategy would allow a much larger number of particles to be sampled in the section area of the specimen, but without loss of particle size resolution. The disadvantage is that the orientation of the test lines is also a significant variable that must be controlled. In a linear analysis, data from a predetermined total number of line scans would be collected and the number of times a selected, threshold intensity was exceeded would be recorded. Normally, the scan lines are separated by a set interval and their orientation is determined by the sample orientation with respect to the raster. Anisotropy in the plane of the section can then be assessed by rotating the sample with respect to the scanning raster (Figure 9.10). Before the introduction of digitized imaging systems, point analysis was by far the most effective method of assessing volume fraction, since a grid of test points placed over an image could be used by an observer to accumulate numerical data rapidly. The method is not entirely obsolete, since maximizing digital data collection from a large number of widely separated, small features may still be best achieved by spacing the pixels in a regular array whose separation corresponds roughly to the spacing of features. This procedure reduces the amount of data that needs to be acquired for any given level of significance.

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z

Line scan

x

y Figure 9.10 An areal scan will detect anisotropy between sections taken in different orientations through an engineering component, while a line scan will detect anisotropy in the plane of a section.

If visual assessment by an observer is still required, perhaps as a rough check on the results obtained using a software package, then the perceptual limitations of the human brain need to be appreciated. Most observers can assess and record, without actually counting, the number of points in a nine-point grid (‘tictac toe’) (Figure 9.11) that lie within

X

X

X

X

X

X

X

X

X

Figure 9.11 A 3 · 3 grid of test points can be used to visually assess and record the number of points falling on the sectioned second phase particles. The grid image is projected on the microscope image at the appropriate magnification and the specimen is then displaced by fixed intervals in order to acquire a data set for the proportion of grid points that lie within the boundaries of a selected phase in the microstructure.

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the particles of a second phase. A regular array of test points, rather than a random array, actually reduces the statistical error, since the sampling array is then uniform. From Equation (9.1), the areal, lineal and point-scan methods for assessing the volume fraction of a second phase are all expected to yield the same result for a randomly positioned sampling probe, that is, a random planar section, random test lines or a random grid of test points. The results should also be equivalent if the microstructure is itself ‘random’, which may often be approximately true. However, observed differences between these three methods are themselves useful for assessing microstructural order, both isotropy and homogeneity, and, with care, this assessment can also be made quantitative. The guiding principle can be very simply stated: we require minimum error for minimum effort, and we will see below (Section 9.3) how this principle can be used to determine a strategy for quantitative microstructural analysis. In the meanwhile we should note that there is no point in collecting massive data sets unless some useful procedure for data interpretation has been devised. The three principle procedures of areal, linear and point analysis can often be usefully combined. For example, areal analysis data at a high magnification in the scanning electron microscope may be collected for a set of regularly spaced locations on the sample section. The average values of a chosen parameter at each location can then be compiled into a frequency distribution function that corresponds to a measure of homogeneity or isotropy across the complete data set. 9.2.1.2 Particle Size and Grain Size. Any bulk determination of both grain size or particle size should be based on an assessment of the area of interface present per unit volume of the bulk material, just as the measurement of volume fraction of a second phase was based on the assessment of the volume of the second phase particles per unit volume. In the case of particle size, it is the area of interphase boundary per unit volume of the second phase that we wish to know. Defined in this way, we obtain inverse particle size or inverse grain size, and therefore both these parameters have the dimensions of inverse length. These parameters are measures of the surface to volume ratio. In mathematical terms, for convex particles, both parameters estimate the total curvature of the interface, defined as dS/dV. The simplest example is that of a spherical particle of radius r, whose volume is 4/3pr3 and whose surface area is 4pr2. It follows that dS/dV ¼ (dS/dr)/(dV/dr) ¼ 8pr/4pr2 ¼ 2/r. That this is indeed the particle-size parameter most commonly of interest becomes clear if the driving force for reducing the internal energy of the system is considered. The total surface energy of the system is gS, where g is the surface energy per unit area, so that the driving force for a reduction in the internal energy is d(gS)/dV. If the surface energy per unit area is assumed constant, then this reduces to g(dS/dV). For the case of a stable soap bubble, the surface tension force is balanced by the difference in pressure DP between the inside and the outside of the bubble, leading to the well-known Laplace equation: DP ¼

2g r

ð9:2Þ

For the general case, the driving force for any reduction in total surface energy is given by the relationship d(gS)/dV. A reduction in g may result either from dopant or impurity segregation or, in the case of anisotropic systems, by faceting on low energy crystallographic planes, which reduces the driving force by reducing the average interface energy.

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Figure 9.12 Unit volume from a fully interconnected two-phase microstructure that has zero curvature. The principal radii at every point on the phase boundary are equal and opposite. In this sketch the volume fractions of the two phases are equal, but this is not a necessary condition for achieving zero curvature of the interface.

If g is assumed constant, then an increase in curvature of a single particle will decrease the stability of this particle, since small particles are less stable than large ones. In a polycrystalline sample grain growth occurs by reducing the ‘curvature’ through a reduction in total grain boundary area per unit volume, so that dS/dV is then negative. The grain size is the parameter that is easiest to visualize, and several alternative definitions of grain size have been given in the literature. For a material with morphological anisotropy, the measured data are often interpreted as an orientational dependence of either the mean intercept length, or the mean caliper diameter, with maximum and minimum values quoted for the plane of a selected section. We prefer to select an anisotropyindependent parameter for the grain size and therefore concentrate on a definition that is based on the inverse of the total curvature of the boundaries in the sample, dS/dV, even though this definition is valid only for convex particles. A network of second-phase particles that is interconnected in three dimensions may have a total curvature which is positive, negative, or even zero (Figure 9.12). Clearly, S/V remains a valid measure of the microstructural scale, with the dimension of inverse length, even though it is now no longer directly related to the total curvature. We start with a line element of length Dl on a section containing a test grid of parallel lines that are separated by a distance d (Figure 9.13). Averaging over all angles between the line segment and the normal to the grid, the probability of the line element intersecting the grid, p, is given by: p¼

2Dl pd

ð9:3Þ

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∆l

d Figure 9.13 A line element that is projected onto a test grid in the same plane of the sample section has a fixed probability of intersection.

For a total length of line L made up of randomly oriented segments, the average number of
will be given by: intersections N
¼ 2l N ð9:4Þ pd and if the total test area on the section is A, then the total length of test line will be A/d ¼ L, so that the average number of intersections per unit length of test line is linearly related to the total length of intercepted interface per unit area of the section:  
l 2 N ð9:5Þ ¼ L A p Now consider the intersection of an irregular particle having a volume Vand a surface area S with a set of parallel test planes of separation d (Figure 9.14). The average area intercepted in a section, A, is given by: A¼

V d

ð9:6Þ

while the average length of the intercepted boundary on the section is:
I ¼ pS 4d It follows that the surface-to-volume ratio is given by: !
S= ¼ 4 I V p A


ð9:7Þ

ð9:8Þ

Adapting the relationship for the length of boundary intercepted by unit area of the section, derived above, to the case where we are interested only in the portion of test grid falling

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d

Figure 9.14 Sectioning of an irregular particle by test planes having a fixed spacing d.

within the area of the second phase particles, then the average intercepted boundary length
and we obtain, for a given phase a. on the section is L,  
N S ð =V Þ a ¼ 2
ð9:9Þ L a So, as long as only the length of the test line falling within the particles sectioned by the test plane is counted, the surface-to-volume ratio for the a phase, is just twice the number of intercepts made by a unit length of test line with the traces of the a phase interfaces on the surface section. In the measurement of grain size, the grid of test lines covers the total area of the section, while each boundary trace is shared by two grains, so that each intercept is shared by the neighbouring grains, and the relationship for the surface-to-volume ratio of a polycrystalline material, defined by the inverse grain size S/V, becomes: S


=V ¼ N L

ð9:10Þ

In the literature, it is sometimes difficult to know exactly what measure of grain size or particle size has been used, since very often ‘correction’ factors are introduced in order to convert a measured mean intercept length into an estimated ‘grain size’. One such correction factor commonly used is 4/p (see above). If it is the driving force for grain growth or particle coarsening that is of interest, then this force is given by d(gS)/dV. As noted previously, the driving force may be reduced if g is reduced by segregation or surface faceting. The driving force is larger for smaller particle sizes equivalent to an increase in total curvature, providing the volume fraction of particles remains unchanged. In most cases, it is advisable to use the accessible bulk parameter, the surface-to-volume ratio, which has the dimensions of inverse particle or grain size, rather than any alternative definition of grain or particle size.

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Microstructural Characterization of Materials

One further case of major importance occurs when the second phase particles have a finite probability of touching one another, known as contiguity. This concept needs to be distinguished from continuity, which is a percolation condition corresponding to a continuous, interconnected path through the second phase in the sample. A good example would be the continuity condition for the onset of electrical conductivity when an electrically insulating matrix contains a conducting second phase. Two types of interface are now of interest: grain boundaries between two contacting particles of the same phase: interfaces that are shared between the contacting particles, and the interphase boundaries that separate the particles from the matrix. The relation for the surface-to-volume ratio must be modified and becomes:  
aa þ2N
ab N S ð9:11Þ ð =V Þ a ¼
a L For approximately spherical particles, a percolation threshold for continuity occurs at a volume fraction close to 0.3, and leads to three distinct morphologies for such a two-phase material. When the volume fraction is below 0.3, the minor phase is dispersed in the major phase, predominantly as isolated particles. For volume fractions between 0.3 and 0.7 the two phases form two interconnected, continuous networks that can be idealized by the zerocurvature model shown in Figure 9.12. Above 0.7 (a second percolation point) the phase roles are reversed: the ‘second’ phase now becomes the matrix while the original ‘matrix’ phase is predominantly present as isolated particles. Both stress corrosion cracking and comminution (grinding) processes are good examples of morphologies that exhibit similar, twin percolation thresholds. In stress corrosion cracking, isolated micro cracks join up to create a continuous ‘leakage’ path through the component (the first percolation threshold). Eventually, all grain contacts are lost across a stress–corrosion failure surface that corresponds to a second percolation point. Similarly, in comminution an applied pressure generates micro cracks in coarse aggregates that can join together at a percolation threshold to form a continuous crack network. A second percolation point is reached when intersecting cracks isolate the fragments of the solid (crushing the coarse aggregate). Viewed on a two-dimensional section, only a single percolation threshold can be observed, corresponding to the transition point at which the ‘minor’ phase becomes the continuous matrix. It follows that the wide, technologically important region of compositions in which the microstructure contains two interpenetrating, interconnected and continuous phases can only exist in three dimensions and cannot be accessed by a two-dimensional section. 9.2.2

Inaccessible Parameters

We have already touched on problems associated with morphological anisotropy, and have demonstrated how the problems may sometimes be bypassed by using definitions for grain and particle size that are independent of any anisotropy in the material, and are determined only by a surface-to-volume ratio. Nevertheless, some measure of morphological anisotropy is often required, and we will therefore discuss the limitations to the determination of inaccessible parameters from two-dimensional sample sections. A rather unsatisfactory example of an inaccessible parameter is dislocation density. In principle, dislocation density might be quantitatively evaluated from diffraction-contrast

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images in thin-film transmission electron microscopy (TEM), using one of two complementary methods. In the first, the number of intersections that the dislocations appearing in contrast make with the top and bottom surfaces of a foil sample with visible area A is estimated, while in the second the number of intersections made by the dislocation diffraction contrast images with a superimposed test grid is used. In the first instance the test area is actually 2A, since intersections that occur with both the top and bottom surfaces of the sample foil are counted, while in the second method the test area is Ld, where L is the total length of test line in the superimposed test grid and d is the foil thickness. Neither method is particularly useful, since estimates of dislocation density that are made by either method are liable to major errors for several distinct reasons: 1. There are serious problems associated with changes in the dislocation contrast as a function of the contrast parameter gb, where b is the Burgers vector of the dislocation and g is the reciprocal lattice vector of the diffracting planes (Section 4.3.2). 2. The thin-film specimens have a variable thickness and there are many problems associated with determining the thickness to any degree of accuracy. 3. Surface image forces experienced during sample preparation are expected to cause some dislocation rearrangement at the surface. The two possible test areas defined above are orthogonal to one another, and are therefore unlikely to give the same values for the measured dislocation density. 4. Diffraction contrast from dislocations is typically quite diffuse, and, even when the dislocations are imaged away from the Bragg condition, the widths are usually a few nanometres. It follows that the dislocation images frequently overlap and serious resolution errors are to be expected. 5. Dislocations normal to the plane of the surface cannot be counted on a superimposed test grid, since they appear as points, while dislocations parallel to the surface do not intersect the foil surfaces, but will intersect the test grid. It follows that, even if the dislocation density, best defined as dislocation line length per unit volume, could be estimated from the number of intercepts per unit area, then this estimate is liable to be strongly biased. Of course, many measurements of dislocation density have been published, based on thinfilm electron microscopy, but the results are error-prone. Assuming that good contrast and minimum contrast overlap are required, then reasonable estimates of dislocation density are certainly achievable for the early stages of work-hardening, corresponding to a dislocation density of perhaps 1014 m-2, but not for either annealed or heavily work-hardened materials. The origin and classification of errors in quantitative microstructural analysis is summarized later (Section 9.3). 9.2.2.1 Aspect Ratios. As we have repeatedly emphasized, the estimation of morphological anisotropy in three dimensions is not readily accessible. In particular, particle and grain shape are difficult to reduce to a single parameter. However, we have seen that an unambiguous determination of grain and particle size is accessible in terms of the surfaceto-volume ratio, equivalent, for convex shapes, to a curvature parameter that is not sensitive to morphological anisotropy. A common solution is to seek a single measure of shape, even

Microstructural Characterization of Materials

a=b
Probability of intersection

478

a

a=b
c

a

a a

Oblate ellipsoid (plate-like)

c Prolate ellipsoid (needle-like)

log(a0/b 0)

-

0

+

Log of the aspect ratio

log(a0/b 0)

Figure 9.15 Frequency distributions for the aspect ratio of planar sections through oblate (‘hamburger shaped’ or plate-like) and prolate (‘cigar shaped’ or needle-like) particles.

though it is recognized that any such shape factor involves some serious assumptions that may be difficult to justify. The shape factor usually chosen is the grain or particle aspect ratio, and the simplest assumption is that the particles can be treated as either oblate spheroids, that is lenticular particles, with an aspect ratio greater than one, or elongated ellipsoids, that is prolate spheroids or acicular particles, with an aspect ratio less than one. In both cases the particles are assumed to preserve cylindrical symmetry. Although the distribution of apparent particle shapes observed on a planar section will depend on the aspect ratio (Figure 9.15), for a single section of a particle there is no way of distinguishing between an elongated ellipsoid and an oblate spheroid (Figure 9.16). Prolate Ellipsoid

Oblate Ellipsoid a

a

Plane of Section c

Figure 9.16 An oblate ellipsoid (a plate-like particle) and a prolate ellipsoid (a needle-like particle) may give identical sections in the plane of a sample section.

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Crystallographic anisotropy is often associated with morphological anisotropy, and, in such cases the second-phase particles, are usually either plate-like or needle-like, rather than ellipsoidal, but once again the same problem exists. The surface section of any oblong features could represent either needles or plates, even though the section probability distributions (Figure 9.15) are quite different. As shown in Figure 9.15, plate-like particles and oblate spheroids are most likely to be sectioned as elongated, acicular shapes, while acicular needles and prolate spheroids are most likely to be sectioned as small particles of low aspect ratio. The ratios of the maximum to minimum caliper dimensions for particle cross-sections observed on a planar section from the sample (Figure 9.17) can be used to estimate the bulk aspect ratio, by assuming that the maximum value for the experimental aspect ratio on a given section coincides with the maximum value of this parameter in the bulk material. This will only be true if all particles have the same aspect ratio. Also, since the ‘most probable’ shape for the intersection of an acicular particle has a low aspect ratio, an underestimate of the true ratio is to be expected. 9.2.2.2 Size and Orientation Distributions. Average values of the accessible bulk microstructural parameters can usually be determined to of the order of a few per cent, providing sufficient data are collected from representative sample sections, but the determination of a size or orientation distribution for the microstructural features is much more difficult. There are two basic problems. In the first place, the volume fraction of the smallest particles may be low, but the detection errors associated with their small size will almost certainly lead to large counting errors. In the limit, when the smallest particle size reaches the resolution limit, these errors become indeterminate. At the other end of the size distribution scale, the contribution to the total volume fraction of a second phase that is made by the few large particles may be very significant, but the poor counting statistics associated with the small number of particles severely limits the accuracy of the data

Plane of Section

Dmin

Dmax

Figure 9.17 The maximum (Dmax)and minimum (Dmin) caliper dimensions of a sectioned particle can be measured, and the maximum value of the aspect ratio in the section then used to estimate the bulk aspect ratio or shape factor.

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Microstructural Characterization of Materials

collected. Thus, for small particles, detection errors restrict the accuracy, while the counting statistics for the largest particles, will be unfavourable for quantitative analysis. A third problem is associated with the methodology of the analysis of particle size data from a surface section that results in the cumulative propagation of errors from the measurement of the largest size fractions down to the smallest. We need a sectioning function for particles of a given size D, which will describe the probability Pd that the apparent particle size on a planar surface section will be less than d. Assuming spherical particles, we obtain the simplest possible sectioning function: ð1
Pd Þ2 ¼ 1
ðd =D Þ2

ð9:12Þ

We can now analyse the raw data for the number of particles observed on the surface section in each size group of the section (Figure 9.18), since we know that the largest areas of particle sections that are imaged can only be due to the largest particles. We therefore calculate, using the sectioning function, the contribution of this largest particle fraction to the smaller intercept sections. The statistical error for these large particles will be large, so we adjust the size intervals for the measured sizes to optimize the accuracy: d1/d2 ¼ d2/ large number of small particles. It is also d3 ¼ . . .dn/dnþ1, thus avoiding a disproportionately pffiffiffi common practice to choose d1/d2 ¼ 1/ 2, in order to halve the area of the selected particle sections as we go to successively smaller sizes. The errors in this analysis propagate through the distribution, and will be a maximum for the smallest size fraction, and may often result in unacceptable, negative values. Bearing in mind that the volume of a particle varies as the cube of the particle size, these errors are concentrated in the smallest sizes and may not prove serious.

Figure 9.18 Micrograph of a low carbon steel, containing a grain boundary (arrowed) that cannot be detected by a computer program for automatic grain size measurements. (Courtesy of Metals Handbook, American Society for Metals).

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Orientation distributions of microstructural features often have engineering significance, as, for example, in a forged bar or a directionally solidified ingot. Several methods have been described for analysing such oriented microstructures. It is important to distinguish between some of the different possibilities. The case of partial alignment in three dimensions, has been treated by applying the orientational dependence of an intercept analysis made with a parallel array of test lines. Some symmetrical cases, for example, partial orientation parallel to one symmetry axis of the component, but with a random distribution in the plane normal to this axis, can be easily analysed. The presence of two populations of grains or particles, one randomly oriented and the other partially oriented, may also be important. An example would be a partially recrystallized sample, containing both elongated, cold-worked grains and equiaxed, fully recrystallized grains. Composite materials, containing anisotropically distributed reinforcement, used to achieve highly directional mechanical properties, present their own problems. Such materials may include a variety of woven reinforcement in which the distribution of the fibres within the reinforcing weave is an important variable.

9.3

Optimizing Accuracy

The quantitative analysis of microstructural data used to be a boring, time-consuming and rather unrewarding exercise, but this is no longer the case. First of all, digitized image processing and image analysis have made data collection fast and efficient, and have all but eliminated photographic recording (though not the need for visual judgement!). Secondly, a wide range of reliable software packages are readily available to improve both the quality of the image and the accuracy of the quantitative analysis. Computer-based analysis has dramatically improved the counting statistics and reduced the statistical errors. The accuracy of quantitative microstructural analysis may depend on a wide range of factors, but it is convenient to start by asking three key questions whose answers usually determine the practical significance of quantitative data: 1. Is the sample representative of the object? We have already discussed the problem of sample selection (Section 3.3.1). In what follows we will assume that this first question has been answered positively, and that we need only consider the accuracy of our analysis of the selected sample, independent of the original bulk object. 2. Is the image representative of the sample? This is a question that we will try to answer before we discuss the possible errors of measurement. Our problem here is to identify artifacts that may appear in the image, then determine their cause and, finally, minimize their incidence in our data. 3. How can we optimize the quantitative analysis of the image? There is actually a two-part answer to this question. First, we need to know what accuracy is achievable. pffiffiffiffi In Poisson statistics the error in determining the mean of a distribution is equal to 1/ N , where N is the number of measurements (that is, the size of the data set). Other statistical functions also show a similar inverse dependence of the statistical error on the number of measurements that are made. Assuming that each measurement requires the same finite time, a significant reduction in the statistical errors of analysis can only be achieved by

482

Microstructural Characterization of Materials

either devoting more time to data collection, or to improving the rate of collection. Modern computing facilities have made it possible to collect and analyse enormous data sets that include large numbers of variables within the set. The second part of the answer brings us back to the need to sample the microstructure of the material on more than one length scale. On the macroscale, the microstructure of a series of cast components may depend on the flow rate of molten metal into the mould and its preheat temperature. We will then require sections taken from several different components if we wish to detect these macro effects. However, the microstructure of a casting may also vary across the plane of a section prepared for examination, and these mesoscale variations within a section can be detected by collecting several images from each section. However, when the microstructure of a single, individual image is analysed, it is the microscale which is being sampled, and we must recognize that no single image can provide information on mesoscale or macroscale variations in the cast components. In order to optimize the efficiency of quantitative microstructural analysis we must first decide whether it is necessary to sample on all scales, from the macro to the nano, or whether it is safe to assume that the microstructure is homogeneous on the macroscale, and then restrict our analysis to a single sample. In such a case, it will be important to ensure that the errors associated with our analysis of a single image (the microscale) are less than those involved in comparing a set of images from different regions of the sample (the mesoscale). At this point we should re-emphasize the importance of good sample preparation. Artifacts in optical micrographs are commonly associated with poor polishing and etching methods (Section 3.3) that may result in the following defects: 1. Scratches and the traces of scratches that are revealed by etching. Automated image analysis systems often count such features as the traces of ‘boundaries’. 2. Occluded particles of polishing media, embedded in a soft, ductile matrix. Such features are interpreted by automated image analysis systems as ‘inclusions’ or ‘second phase particles’. 3. Pull-out of grains from the matrix of the sample during final polishing. This can occur in brittle, polycrystalline materials and is often interpreted as residual ‘porosity’. 4. Rounding of edges at second-phase interfaces and pores. This effect is often associated with elastic mismatch of a second phase. The rounding increases the apparent areal fraction of the microstructural features. 5. Poor contrast at interfaces and failure to reveal some grain boundaries. This leads to an underestimate of the surface-to-volume ratio and an overestimate of the grain or particle size. 6. Over-etching of the microstructural features. This results in increased resolution and sectioning errors (see below). Figure 9.18 shows one example of an artifact. If the specimen is polished and etched using standard procedures, some grain boundaries may not be revealed, while individual grains may vary in contrast due to orientation-dependent chemical staining. Heat-treating the sample to decorate the grain boundaries with small precipitates has no effect on the grain size but may sometimes improve the uniformity of the etch response of the boundaries.

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Lightly polishing a sample after etching can then remove the chemical staining from individual grains, improving the contrast of the grain boundaries for automated quantitative analysis. A great deal depends on our awareness of the microstructural limitations of quantitative analysis, and of the partially conflicting requirements for accuracy and rapid data acquisition. Careful sample selection, good specimen preparation, the acquisition of a sufficiently large data set and reliable software algorithms are all important components of a successful quantitative microstructural analysis. We now consider in more detail the optimization of accuracy with respect to minimizing the effort needed to collect and analyse data. 9.3.1

Sample Size and Counting Time

In any statistical analysis, the larger the size of a data set the smaller the statistical error. Unfortunately, all methods of analysis involve more than one source of statistical error. For example, in determining the volume fraction of a second phase, both the number of the second-phase particles in the field of view of the image and the number of measurements of the particle size that are made for this image are important. At too high a magnification the resolution error may be smaller, because of a higher numerical aperture objective lens, but the field of view may only contain a very limited number of particles, leading to a large sampling error. Each measurement requires time, even in a fully computerized system. As the data is collected, the statistical errors decrease, but the time required for the analysis increases. In a fully automated system, this analysis time includes the time needed to select the sample area for analysis, to adjust the contrast and focus, and to correct for background ‘noise’. When using a digitized computer system, the time required to collect the data from an image is just a small proportion of the total time of investigation. Data accumulation is therefore an interactive process, in which the operator cannot rely solely on the computer to make all the decisions. We should avoid statistical bias. Data collection should never be started at a nonrandom point, such as a grain corner. Many microstructural parameters involve a ratio, and it is important to predetermine the denominator and not the numerator. For example, in an areal analysis of volume fraction, the total area sampled should be kept constant from one field of view to another, not the area of the second phase of interest within the section. In the determination of grain or particle size, it is the total length of test grid, not the total number of intersection points, that should be fixed. Statistical errors are most conveniently summarized in terms of the coefficient of variance (CV) defined by: CV ¼

s2 s2 % x20
x2

ð9:13Þ

where s is the standard deviation (the true spread of values in the population being sampled), s is the standard error (the measured spread of values observed in the sample), x0 is the true average value of the property in the population and
x is the measured average value of the property in the sample. It follows that, s and
x are estimates of s and x0.

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Microstructural Characterization of Materials

In terms of Gaussian statistics (the normal or bell-shaped distribution): N 1X x N 1 where N is the number of measurements that have been made, while:


x ¼

s2 ¼

N 1 X ð
x
xÞ2 ðN
1Þ 1

ð9:14Þ

ð9:15Þ

While the normal distribution is a good approximation in many cases, some care is required and standard texts on statistics should be consulted for alternative statistical frequency functions. This is especially important when only a small data set is available. In principle, the normal distribution is limited to large data sets in which the variables may take any rational value (noninteger or integer, positive or negative). For the variables of interest to us, in quantitative microstructural analysis, none may take negative values. In many cases, our data only take integral values. A good example would be the count of a random distribution of surface features on a section, yielding integers above zero. The appropriate statistical function for this case is a Poisson distribution, for which the CV for a sample section is 1/N. If the section is subjected to an areal analysis in order to determine the volume fraction of second phase particles in the bulk, then the CV that is associated with the sectioning of the particles (CVA) must also be included in the errors associated with areal analysis: CVAA ¼

ð1þCVA Þ N

ð9:16Þ

For a random distribution of uniform spheres CVA ¼ 0.2. However, for a random point count of the proportion of points falling within any given area we expect a binomial distribution. If a second phase occupies an areal fraction Aa, then some of these points Pa will fall within the area of the second phase, and the CV for a random point count becomes: CVPR ¼

1 Pa ð1
Aa Þ

ð9:17Þ

This value of CVonly applies to the single feature that has been analysed, and a term needs to be added in order to account for the total number of features in the area sampled. The final result then becomes: CVPR ¼

1 ðP
1Þ þ Pa ð1
Aa Þ P·CVAA

ð9:18Þ

For a random distribution of the sectioned areas of the second phase we can substitute for CVAA and neglect the factor 1/P, leading to an estimated CV for a point count that is given by: CVPR ¼

1 1 ·ð1
Aa Þþ ·ð1þCVA Þ Pa N

ð9:19Þ

where N is now the total number of second-phase features sectioned by the area analysed. Clearly, optimum efficiency requires that Pa %N, that is, for a minimum error attached to a

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485

given ‘effort’ needed to obtain a given number of counts in the data set, the total number of counts collected should be of the order of the total number of visible features in the area that has been sampled. This result is, strictly speaking, limited to a random point count and a random distribution of features. This is therefore a ‘worst case’ scenario. The second-phase features on a twodimensional section can never overlap, and each additional feature has to occupy a smaller available area of the sample. It follows that point counts which are located in an ordered pixel array (so that clusters of points are always excluded), will always give an experimental CV that is smaller than that estimated above. Nevertheless, the primary conclusion remains valid: statistical accuracy is always limited by the number of features sectioned, and there is no advantage in increasing the number of data points counted much beyond this value. It follows that the statistical significance of the volume fraction of a second phase determined from a data set collected by interrogating individual image pixels is optimized when the pixel size is of the order of the resolution, but the pixel separation is of the order of the separation of the second phase particles on the section. The pixel array should cover the maximum area of the sample section chosen, so that as many particles as possible are included in the analysis. Linear analysis of boundary and interface traces can be treated similarly. The grid of lines used to probe a microstructure should be regularly spaced, and the line separation should be of the order of the grain size or the particle separation. Improved accuracy can be obtained by interpolating between the pixels to reduce the uncertainty in the position of a boundary or interface. The statistical accuracy is limited both by the number of interface traces that are sampled, and the number of intercept counts collected. For maximum statistical significance, these two parameters should be of the same order of magnitude. This can be achieved by making the spacing of the test grid approximately equal to the particle spacing or grain size on the section. In the absence of an automated system, counting is by far the easiest visual method of estimating the approximate value of an accessible microstructural parameter: 1. Point features (etch pits, particle density, dislocation intercepts) are best estimated as the number of points per unit area. 2. Linear features (dislocation density, grain size, particle size) are best estimated from the number of intercepts with a superimposed, regular test grid having a line separation similar to the separation of the features in the image. 3. Areal features (volume fraction of precipitates or inclusions) are best estimated using a systematic point count, with the point spacing comparable with the spacing of the features on the section.

9.3.2

Resolution and Detection Errors

While the resolution limit of the microscope constitutes the ultimate limit on the accuracy with which the coordinates of a microstructural feature can be located, it is seldom the controlling factor. In optical microscopy, sample preparation usually determines the detection errors. Chemical etchants develop steps and grooves on the surface of the

486

Microstructural Characterization of Materials

Figure 9.19 Thermal grooving in a polycrystalline ceramic sample typically limits the resolution in an optical micrograph to over 1 mm, but a secondary electron scanning electron microscope micrograph, as shown here, can significantly improve this limit.

polished section that scatter light out of the objective aperture over a region in the image of the order of 1 mm in width. Thermally etched ceramic samples show boundary grooving whose width depends on the heat-treatment conditions (Figure 9.19). Resolution errors are usually associated with the mechanism of contrast. In diffraction contrast images of thin crystalline films taken in the transmission electron microscope, the width of a dislocation image may be of the order of 10 nm, even though the resolution limit for the microscope is better than 0.2 nm. In a diffraction contrast image, the parameter gb plays a dominant role that determines both the width and the apparent position of a dislocation (Section 4.4.5), and it is possible to improve the resolution by using weak-beam, dark-field imaging, in which elastic scattering from the dislocation core region dominates the contrast. In this case the width of the dislocation image is only of the order of 2 nm, while the position of the dislocation image corresponds to the position of the dislocation core. Finally, image recording may also be a limiting factor that degrades the resolution. In digitized images the resolution limit is about 3 pixels, since this is the minimum number needed to record a contrast variation between two microstructural features. In processing digitized image data, it is essential to use sufficient pixels to achieve the required resolution over the full field of view of the image, as well as the required range of contrast. We need to ask: what is the total number of pixels necessary to record all the resolved information in the image? Since there are still severe limitations on the rates of data transmission that broad band and wireless communications systems can support, the answer to this question dominates the electronic transmission of image data over the Internet or by e-mail.

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Compression algorithms are available to optimize the efficiency of image-data transmission and processing, and we will touch on this later (Section 9.4.1). The apparent width of a linear feature in the image of a surface section d can be used to estimate the error in the areal fraction of a microstructural feature that is associated with poor image definition. The areal fraction of the boundary or interface traces on the section will be: AL ¼

p N ·d· 2 L

ð9:20Þ

where N is the number of intercepts and L is the total length of the test line. However, the estimated particle size D determined from a linear analysis is given by: D¼

2La N

ð9:21Þ

where La is the length of the test line lying within the particle sections. Inserting the volume fraction of the particles fa ¼ La/L and assuming the error in fa is given by Dfa ¼ AL, then: Df a pd ¼ fa D

ð9:22Þ

We conclude that the error in determining a volume fraction of a second phase will increase rapidly as the particle size approaches the effective resolution limit. It follows that any attempt to improve the counting statistics for small grains and secondphase particles that are close to the limit of microstructural resolution may be unrewarding, since the accuracy may then be limited by image resolution, rather than by the number of particles that have been sampled or the size of the data set that has been collected. 9.3.3

Sample Thickness Corrections

Errors associated with the sample thickness, either in thin-film electron microscopy or, for a planar section, the sectioning errors in optical microscopy, are a major factor limiting the accuracy of quantitative image analysis. The resolution and detection errors limit the accuracy that we can achieve in determining the x–y coordinates of a feature intersected by the x–y plane of the sample section. Similarly, the sectioning errors limit the accuracy associated with uncertainty in the location of the image plane along the z-axis. Figure 9.20 shows schematically the effect of this section thickness, both for an etched surface viewed in reflection and for a thin film viewed in transmission. There are actually two section thickness corrections required: 1. The increase in the measured area of the section through a second-phase particle that arises from particle projections that lie within the slice, rather than just in the plane of the section. 2. Overlap of particles within the slice that obscures part of the projected area of neighbouring particles. Since the sample is viewed normal to the projected section from one direction only, the contribution from internal surfaces will amount to just half the surface area of the particles

488

Microstructural Characterization of Materials Observed areas True intercepted areas

t

Projected areas

t

True intercepted areas

Figure 9.20 True and observed intercepted areas of a second phase on a sample section are never the same, but are determined by the thickness t of the sample slice from the material that contributes to the projected image. The corrections required are similar for both a reflected image of an etched surface recorded in the optical microscope and in the projected image of a thin film seen in transmission electron microscopy, but significant overlap of microstructural features is only possible in a thin-film, transmission electron microscope image.

that lies within the slice. This correction is, therefore ð1=4ÞS=V·t, where S/V is the surfaceto-volume ratio and t is the slice thickness. Using linear intercept analysis to estimate the surface-to-volume ratio, S/V ¼ 2Na/La, gives a first-order correction for this additional area, so that the corrected volume fraction of the second-phase is now: fa ¼

Aa N a t
 A La 2

ð9:23Þ

This slice thickness correction is particle size dependent, and should be negligible for large particles and small slice thicknesses. As the particle size approaches the slice thickness, particles will either be etched away, as in the case of optical microscope observations made in reflection, or start to overlap in the

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projected image, as in thin-film transmission electron microscopy. This overlap correction will become important for large volume fractions of a second phase. An approximate relation that accounts for both the contribution from surfaces within the slice and the contribution due to particle overlap has been suggested by Hilliard:    Aa s  exp ð9:24Þ 1
f a ¼ 1
A 4Vt This suggested correction is actually an overestimate, since it does not allow for the ‘exclusion’ volume that surrounds each particle, and which significantly reduces the extent of particle overlap estimated on the basis of random positioning.

9.3.4

Observer Bias

It is remarkable the difference that the experience and training of the observer may play in determining the accuracy of even a fully automated and computerized quantitative procedure. While computerized image analysis improves rates of data collection by several orders of magnitude, it has little effect on observer bias, since it is still the observer who selects and prepares the samples and determines the computer settings. It is important to recognize the possible forms that observer bias may take and the following example may help. Let us assume that the grain size of a ceramic is to be determined from a thermally etched sample (Figure 9.19). Observer A is worried that some boundaries may not be clearly visible, so he increases the annealing time to improve the visibility of the boundary grooves. Observer B considers the resolution error, due to the width of the grooves, to be excessive, and so reduces the annealing time accordingly. Observer C wishes to improve the contrast in the image, and to do so coats the thermally etched surface with a thin film of a reflecting metal. We may confidently predict that each of these three observers, using precisely the same microscope and digital recording system, as well as the same data analysis software procedures, will nevertheless arrive at a different average grain size and grain size distribution for the same sample material. Providing sufficient data are collected, these three observers may well be able to prove statistically that the ‘grain size’ they have each determined is, to a high degree of probability, not the same as that measured by their colleagues on precisely the same material! Let us consider one more example. Imagine that the same sample has been supplied to the same three observers, but now the sample has already been prepared. Observer A is careful to keep the magnification of the microscope to a minimum, in order to ensure that a large number of grains are recorded in the field of view to be analysed. Observer B, on the contrary, wishes to ensure that the best possible resolution of the microscope is utilized, and records a series of high-magnification images. Observer C is much more contrastconscious, and decides to use a dark-field objective to reduce background intensity in the image to a minimum and highlight the grain boundary grooves. All of the above choices are completely rational. They reflect legitimate differences in the professional judgement of the individual observers. In practice, it is not easy to identify the precise reasons for the decisions made by an observer. Does the trace of an interface (Figure 9.21) cross and then re-cross a test line, adding two points to the tally of boundary intercepts, or is it just adjacent to the test line (with no increase to the tally of intercept

490

Microstructural Characterization of Materials

Boundary touches test line: No Crossing Points

+

+

+

Boundary crosses test line: Two Crossing Points

Figure 9.21 An interface may be judged to cross a test line by one observer, increasing the tally of boundary intercept points, but not by another who judges the boundary not to be intercepted.

points)? The test line is just one pixel wide, but the boundary position is much less well-defined. There will always be some observer bias, even if it is now located at the observer/computer interface. 9.3.5

Dislocation Density Revisited

We have already noted the complications associated with the estimation of dislocation densities from diffraction-contrast images in TEM (Sections 1.1.3.2 and 9.3.2). In this section we summarize the problems associated with the quantitative determination of dislocation density and the methods which have been used to overcome them. One of the first attempts at quantitative analysis of dislocation substructures was made using X-ray line broadening, well over 50 years ago. The line broadening was associated with both lattice strain (changes in d-spacings) and decreased particle size (sub-grain formation). The difficulty is that an array of dislocations in a slip plane, that is a dislocation pile-up, introduces strain into the lattice, while an array with the same average spacing in a sub-grain boundary results in misorientation across the sub-boundary. It is not straightforward to separate the higher energy dislocation pile-up from the lower energy configuration of the dislocation sub-boundary. X-ray measurements of dislocation density were subsequently followed by a series of etch-pit analyses, pioneered by Gilman working with ionic single crystals. The dislocation arrays were generated by grit particles impacting the surface of the ionic crystal and the subsequently formed etch-pits corresponded to the intersection of the dislocation lines with the sample surface. The dislocation etch-pits were shown to have a distribution that accurately confirmed the predictions of dislocation theory. Although etch-pitting could reveal dislocation substructure in a wide range of semiconductors, ionic materials and metals (including iron), the resolution limit of the method was only a few micrometres. Thin-film TEM led to serious attempts to define dislocation density as a microstructural parameter that could be incorporated into theories of work hardening (the increase in yield strength of a ductile material with increasing plastic deformation). A major prediction from dislocation theory was that the increase in the tensile

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pffiffiffi yield strength should depend on the square-root of the dislocation density, Ds / Gb r, where G is the shear modulus, b is the Burgers vector of the dislocations and r the dislocation density (the dislocation line length per unit volume). The density of dislocations that was revealed by diffraction contrast in a thin film could be estimated, but it proved difficult to account for the relaxation of the dislocations into dense arrays that were poorly resolved by diffraction contrast. It required some 20 years of research effort before researchers finally accepted that the artifacts involved in estimating dislocation density in transmission electron micrographs reflected a basic problem with the general concept of ‘dislocation density’. Today, the importance of determining the morphology of the dislocation substructure and identifying the dominant dislocation interactions in both structural materials and electronic devices is still fully recognized, but the concept of dislocation density is used sparingly. As an alternative to dislocation density, workers in the field of plastic deformation have evoked a complete library of new concepts to describe the complex morphologies that result from dislocation interactions during the plastic flow of ductile polycrystalline materials, for example kink bands, shear bands, cell structure, sub grains, pile-ups and dipoles. For the present, it is quite sufficient that the reader understand why dislocation density is a problematic parameter.

9.4

Automated Image Analysis

Increased computer data storage and handling capacity, combined with reduced prices, has placed automated quantitative image analysis within reach of any research, teaching or industrial laboratory. The first automated systems were constructed in-house over 50 years ago. These were then superseded by commercial systems for image analysis and later, following the digital camera revolution, by computer software packages for image analysis that could be adapted to both optical and electron microscopes using digital charge-coupled device (CCD) or complementary metal oxide semiconductor (CMOS) image data output, or used with scanning electron microscope and optical scanning systems. In principal, there are three options for collecting a digitized data set from a twodimensional projection of a three-dimensional object. These options correspond to the ‘scanning’ of either the source, the object or the image (Figure 9.22): 1. In the scanning electron microscope the electron beam is focused onto the surface of the sample to form a reduced image of the electron source (the probe). This probe is then rastered across the surface of the sample. Since it is a focused image of the source that is being scanned across the surface, this is a source-scanning system and the final, digitized image data set is collected as a function of the electron probe position. The scanning near-field optical microscope also focuses a light beam onto the specimen with an effective probe diameter that is less than the wavelength of the light used. With this technology it is possible to beat the diffraction limit on imaging with visible light and capture scanned image data at resolutions that approach 10 nm. 2. In a few systems, it is the object beneath the probe that is scanned, for example in the (semicontinuous) automated analysis of a powder sample. Some commercial particle analysers

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Specimen

Object Scan

Specimen (c) Image Plane

Image Scan

Specimen

Figure 9.22 Digital image data collection options may be based on scanning of (a) the source, (b) the object or (c) the image.

work on this principle, although they detect the scattering of light from individual particles instead of the data from a focused image. The particles are carried in a stream of a dilute liquid dispersion. They pass through a capillary tube and traverse an aperture that is illuminated by a laser beam. The scattering of the light by each particle is detected and the analysis of particle size is performed in reciprocal space (Section 2.2). Millions of particles

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canbe counted andsized,according tothelightscatteringangles,andthe data presented asa particle size distribution frequency function. Typically the size fractions range from a few tenths of a micrometre up to some tens of micrometres. 3. Finally, focused images, typical of the optical microscope and the transmission electron microscope, can be interrogated in the image plane, by using a suitable digital recording system. A CCD camera is commonly used, although CMOS systems are often found in optical microscopy (Section 3.5). The CCD and CMOS digital image cameras have insufficient response time for dynamic recording systems, and video cameras, which lack the high resolution but have faster recording speeds, are common for timedependent observations. Good video cameras can capture over 106 pixel points at frame speeds suitable for real-time, video recording (16 frames s
1). For still photography, a wide range of both monochrome and colour digital cameras are commercially available. As described in Section 3.5, colour versions are based on a three colour code, usually red, green and blue – the RGB system of primary colours. This can access most regions of the chromaticity triangle (Figure 9.23). The chromaticity triangle is a convenient way of analysing colour. By varying the three intensity levels, any colour within the triangle that is defined by the three primary colours can be simulated, including white, over a wide dynamic range of intensities. In the brief discussion that follows we will assume that scanning is of either the source (as in the scanning electron microscope) or the image (as with a CCD or CMOS camera), and

Figure 9.23 The chromaticity triangle is a convenient representation of colours present in the visible spectrum. A chosen colour within the triangle can be selected by combining different intensities of the three primary colours taken from the corners of the triangle. Colour monitors cover a colour range limited by the response of the available blue, red and green phosphors. Colour printers are limited by the absorption response of the four or more pigments used. (See colour plate section)

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that we are less interested in collecting digital data in reciprocal space (as in a diffraction pattern or particle size analyser). The emphasis in this section is on the quantitative analysis of the digital data (compare Section 3.5). 9.4.1

Digital Image Recording

Several problems have to be solved if digitized data are to be analysed quantitatively and provide unbiased estimates of microstructural parameters. The first condition is that the numerical value assigned to a particular pixel should be linearly related to the physical processes that occur in the section plane of the specimen. In the optical microscope, it is the distribution of the light intensity in the image plane that determines the contrast, while in TEM it is the electron current distribution across the image plane for the thin-film sample that gives the contrast. In scanning electron microscopy (SEM) the signal recorded in the ‘image’ plane, the signal recorded by the collector as the probe beam rasters across the specimen surface, may correspond to secondary electron emission, backscattered electrons or characteristic X-rays. Photographic recording of a light-optical image is a nonlinear process that could only give a linear correspondence between the blackening of the emulsion and the incident intensity over a very limited intensity range (Section 3.2.4.2). High energy irradiation, by either X-rays or electrons, does give a linear photographic response (the number of silver grains in the developed emulsion is then linearly dependent on the incident dose), but only as long as there is no overlap of the silver grains that have been developed. Digitized images, obtained by scanning an archived photographic recording, should be treated with caution, although the linear range of response available in a transparency (a photographic ‘negative’) when viewed in transmission is far better than can be obtained from a printed photograph that has to be viewed in reflection. The resolution of an image-scanning system is usually quoted in dots per inch (dpi) and indicates the maximum density of pixels that the system can record, irrespective of either the density of pixels in the image to be scanned, or the density of pixels needed to retain the best resolution of the microscope. For quantitative image analysis, the integer value assigned to each pixel should be regarded as an estimate of the information present in the pixelated image. The selected pixel density is determined by three factors: 1. The minimum density of pixels that is required to resolve the features of interest. 2. The statistical accuracy desired for the value to be assigned to each pixel, which determines the dynamic range that is selected. 3. The processing speed and storage capacity of the computer and data transmission facilities. Given the low price of storage media and computer RAM, most microscopists tend to use the maximum pixel density that the imaging system can provide. It is important to recognize that the digitized image, in which the numerical intensity value assigned to each pixel is a data point, is only the raw data. Prior to undertaking any quantitative analysis, the raw data may need to be ‘cleaned-up’, and a number of image processing algorithms are available for this. The contrast may be adjusted, background can be subtracted, the data can be ‘smoothed’, edge effects can be enhanced and various mathematical functions are available for more complex manipulation of the digitized raw image data. It is also possible for the operator to ‘manipulate’ the image data by deleting recognized artifacts, such as scratches, or inserting ‘missing’ features, for

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example, unetched grain boundaries. In all such operations there is a significant danger of operator bias, often combined with some unjustified wishful thinking! To summarize: 1. There is no substitute for good sample preparation! 2. The digitized image should be recorded so that the digitized values are linearly related to the physical phenomena that are responsible for image contrast. 3. The digitized image data should be processed to improve the information content before undertaking any quantitative image analysis. 9.4.2

Statistical Significance and Microstructural Relevance

In this chapter we have pointed out repeatedly the difficulties attached to quantifying information contained in a two-dimensional micrograph. The transition that we would like to make from a value judgement, describing a material as ‘fine-grained’, to a quantitative statement, that the material has ‘a grain size of 0.4 mm’, represents additional information. Any attempt to improve the accuracy of microstructural characterization is to be applauded, but the imaging of a specimen is a complex process. The relation between the mechanisms of contrast formation in the image, the bulk microstructural features that are responsible for this contrast, and the engineering properties of the original component are certainly not selfevident. The results of a careful quantitative microstructural analysis may be statistically significant, yet prove to be irrelevant for the engineering problem that is being investigated. Excellent examples of this phenomenon are to be found in the study of mechanical properties that are associated with the presence of ‘defects’ (fracture strength, ductility, notch impact energy and fracture toughness). These properties are all very sensitive to the size of the defects and the angle between the plane of the defects and the axis of an applied tensile load. It is the largest defects that have the greatest effect, and therefore no measurement of either the average size or the size distribution of the defects can possibly be correlated successfully with the mechanical properties. Furthermore, each class of engineering defect, microcracks, soft inclusions, hard inclusions or porosity, affects the properties differently, posing additional problems in establishing the relevance of any quantitative microstructural analysis of the defect content of the material. Despite this, quantifying grain or particle size, measuring the volume fraction of a second phase, and determining the extent of morphological anisotropy are still important objectives of microstructural characterization, certainly in research, but also for material and process development, the improvement of criteria for materials acceptance and for product quality control. Microstructural characterization is an essential part of technological development, and the quantitative characterization of microstructure is a legitimate objective, but is only justified if the results are relevant to the investigation.

9.5

Tomography and Three-Dimensional Reconstruction

We have noted that there is a wide range of three-dimensional microstructural data that cannot be interpreted quantitatively from a two-dimensional section, unless we make very

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serious simplifying assumptions about the microstructural geometry of the bulk material. A good example would be the interpenetrating two-phase structures that are often typical of binary eutectics and some spinodal systems. Interconnecting pore structures that are characteristic of the earliest stages of sintering have similar topological properties, as do many materials used as particulate filters, catalyst substrates and porous anodes for solidstate capacitors. For all such cases, it would be highly desirable to be able to view the microstructure in three dimensions, rather than to rely on cross-sections that have been selected for twodimensional microstructural characterization. Biologists have been leaders in developing sectioning and data-processing techniques that are designed to provide insight into the structure of three-dimensional objects, both for the study of anatomy on the macroscale, and for histology (the microstructure of soft tissues and cells). A term commonly used for these methods of three-dimensional microstructural investigation is tomography, from the Greek tomo meaning a section, and graphos meaning a picture. In the present scientific usage of the term, tomography is the development of a three-dimensional image data file that can be interrogated by a computer software program, in order to generate three-dimensional images and image sections in any selected projection or orientation and from any selected slice taken from the three-dimensional data set. 9.5.1

Presentation of Tomographic Data

Tomographic data can be collected from a very wide range of imaging technologies, but in practice the data that constitute the raw material for tomographic analysis are always related to a sequence of two-dimensional projections and constitute a series of consecutive, twodimensional data sets. A simple case would be the optical imaging of a rough surface in reflection from a through-focus series. The image data are analysed using an algorithm that filters the spatial frequencies in each image, and then combines the images. Those regions that are out of focus in any one image will appear blurred and can only contribute to the low spatial frequencies of that image. These low frequencies can be removed from a Fourier transform of the pixelated intensity data for each image. The remaining high-frequency, infocus portions of each image can be combined, colour coding or contour mapping each data set to provide a striking, high-contrast image in which the z-resolution perpendicular to the x–y plane is determined by the size of the defocus steps, rather than by the numerical aperture of the optical microscope objective lens (Figure 9.24). Confocal microscopy, in which a focused and scanned light probe generates a scattered signal from features in a slice within an otherwise transparent sample, can be used to collect a series of data sets, so that each set constitutes an ‘optical slice’ taken from different depths in the specimen. Soft tissues that have been suitably labelled by a fluorescing molecule can be imaged using the fluorescent excitation. Resolutions well below the exciting wavelength have been achieved. Again, the data sets from the series of confocal optical slices can be combined into a single three-dimensional data set and processed to provide quantitative information on the bulk microstructure. Using a second, picosecond laser the fluorescent emission can be partially quenched, improving the resolution for the excited regions to of the order of 30–40 nm, well below the diffraction limit and dramatically improving the image resolution available for a histological examination.

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Figure 9.24 Example of a reconstructed optical microscope image from a defocus series of a gold pad on a circuit board processed using Fourier filtering. (Courtesy of Syncroscopy).

The availability of powerful X-ray synchrotron sources has been used to collect radiographic projection data of samples. The data have been translated and rotated to generate two-dimensional data sets for suitably oriented projections which can be combined into a three-dimensional image, using software identical to that employed for computer-aided tomographic scans (CAT scans) in medical imaging. Many years ago a micro-focus X-ray source was used to probe the bulk structure of an interpenetrating eutectic, Al–Sn bearing alloy by micro-radiography, but at that time the computer and data processing facilities were not available to extract quantitative three-dimensional information from the micro-radiograph. Figure 9.25 shows a more recent example of a dendritic structure taken using a synchrotron source. Tomographic data can be presented in several ways. Using computer graphics the threedimensional data set can be rotated, either continuously or in steps, so that a complete projection sequence is visible to the observer on a monitor screen. Alternatively, slices of selected thickness, taken from the data in a chosen projection, can be viewed, and the slice displaced in steps, either in the x–y plane or in the z direction, perpendicular to the viewing screen, in order to explore the full range of the data set. It is also possible to ‘zoom’ into the data set, in order to enlarge a specific volume element that may be of particular interest. It is not always the intensities of the individual pixels in the three-dimensional data set that provides the best view of the data, and in some cases the gradient of intensity is highlighted, by differentiating the intensity over a set of neighbouring pixels in order to derive a smoothed intensity gradient. This may be particularly useful when it is the phase boundaries in the microstructure, rather than the second-phase particles that are of interest. This may also be the case if we wish to ‘see through’ the second-phase particles, and in many cases it may be useful to make the particles semi-transparent, so that we can clearly

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Figure 9.25 Synchrotron micro-radiograph of a Sn–Bi alloy, recorded in-situ during solidification, with a resolution of a few micrometres. Reprinted from B. Li, H.D. Brody, and A. Kazimirov, Synchrotron Microradiography of Temperature Gradient Zone Melting in Directional Solidification, Metallurgical and Materials Transactions A, 37, 1039–1044, 2006, with permission from The Minerals, Metals, & Materials Society.

distinguish the particle volume, while simultaneously placing the particle boundary in high contrast. This means, combining the readout of the different intensity values for the pixels that fall within the particles, together with a high contrast for the pixels at the phase boundaries, identified by the intensity differences between the two phases. 9.5.2

Methods of Serial Sectioning

Automated serial sectioning has now been applied to just about every possible method of microstructural characterization that has been employed in materials science. In the present section we will consider these methods for serial sectioning and, especially, the minimum spacing of the slices in each case. We will also discuss the origin of artifacts and the attainable three-dimensional resolution. Mechanical cutting and grinding have always been the primary tools for sectioning engineering components, while mechanical polishing and chemical etching have been the preferred methods of surface preparation for optical reflection microscopy. Early attempts to prepare parallel serial sections by mechanical grinding were handicapped by lack of control of the thickness removed at each stage. Automated, computer-controlled polishing systems have resulted in major improvements, and the control of thickness removal to of the order of a few micrometres has been claimed for serial sections taken 10–100 mm apart. Diamond microtomes (‘ultra-microtomes’) have been used for many years to prepare serial sections of embedded, histological samples with thicknesses down to about 10 nm, and recent reports have used the same method to prepare serial sections for atomic force microscopy (albeit, only at low resolutions and for materials of low elastic modulus). Focused ion beam (FIB) milling (Section 5.4.4) is proving to be a remarkably flexible technique, not just for sample preparation, but also for serial sectioning, with fine-tuning of

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the slice thickness to of the order of 10 nm. The FIB-milled sections can be viewed in situ using secondary electron scanning images and two-dimensional digital data sets can be recorded from successive layers. In dual-beam FIB instruments, continuous scanning electron microscopy has been used to monitor the progress of the milling process and, if required, successive slices can be removed from orthogonal planes in the sample. Such FIB-generated three-dimensional data sets are therefore obtained from separate volume elements that have been sectioned along different coordinate axes, effectively eliminating any bias associated with interpolation of two-dimensional data collected perpendicular to a single axis aligned along the milling direction. The major disadvantage of ion beam milling is associated with the side effects of radiation damage. These are primarily due to the injection of point defects that condense into clusters and dislocation loops, and can result in diffusion-induced composition and phase changes. The damage also includes displacement of individual atoms by focusedcollision sequences over distances of the order of 10 nm. Alloying elements are dispersed, both by enhanced diffusion and by such knock-on collisions. Thin layers that have been deposited on a substrate may experience considerable blurring of the original, as-deposited concentration gradients. Depth profiling in X-ray spectroscopy and Auger electron spectroscopy also relies on sputtering under ion bombardment to remove successive atomic layers (Sections 8.1 and 8.2), but only Auger spectroscopy has the spatial resolution needed to collect twodimensional image data from the individual layers. In principle, adequate calibration of the depth removed at each stage would make it possible to record and process a threedimensional digital-data image file, but this has not yet been accomplished. Surface roughness due to the sputtering process and radiation damage induced blurring of the concentration gradients destroy the planarity of the surface and limit the spatial resolution in the third dimension. Similar problems have prevented secondary-ion mass spectrometry from achieving good three-dimensional resolution. Atom probe tomography (Section 7.3.2), is the only serial sectioning method of threedimensional, nanostructural characterization that accurately identifies atomic species and variations in concentration at the resolutions needed to study boundary segregation and the earliest stages of second-phase nucleation. Unfortunately, the engineering materials that can be studied by atomic probe tomography have to possess some electrical conductivity and must be able to withstand the mechanical stresses imposed by the very high electric field strengths at the specimen tip. It should also be noted that, since the ion collection efficiency is usually of the order of 50–60 %, approximately half of the atoms in the volume sampled by the data set are missing. 9.5.3

Three-Dimensional Reconstruction

Collecting the digitized, two-dimensional image data from a set of serial sections is the first stage in preparing a three-dimensional data set for subsequent viewing and analysis. The second stage is to bin the data in such a way that the three-dimensional pixels each represent elements of known volume in the material being studied. Since the spacing of the serial sections is often significantly larger than the lateral resolution, it may be necessary to interpolate ‘virtual’ data points by averaging intensity or counts from sets of pixels in neighbouring sections. For some techniques, fiducial markers are used in order to ensure

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that successive x–y sections are accurately positioned perpendicular to the z-axis. Hardness or micro-hardness indentations are a convenient form of fiducial marker for slice thicknesses down to 1 mm or even less since they are readily located. The depth and width of the impression allows slice thicknesses to be monitored, with thicknesses ranging up to as large as 0.1 mm. Additional indentations can be made as the sectioning proceeds. Most manipulation of the three-dimensional data sets involves rotation, translation and ‘zoom’ functions (magnification), and these functions can of course be combined. An important case is that of the atom probe, and the radius of the sample tip increases as atoms are field-evaporated from the surface, reducing the average magnification. Adjusting the initial magnification scale of the two-dimensional data sets to the uniform magnification of the three-dimensional grid only takes care of part of the problem, since the magnification of a projected field-evaporated ion image also reflects variations in radius of curvature over the sample tip that are due to local crystallographic features. The densely packed, low index planes at the surface generally result in a larger local radius, that is, lower curvature, than the less densely packed regions. Published tomographic images usually, but not always, incorporate these corrections for local variations in magnification. The restructured, three-dimensional data set taken from a sequence of two-dimensional data sets that has been derived from serial sections can be viewed in several ways. Two classes of manipulation are basic to viewing comfort on a computer screen. The first concerns the processing of the data set itself. The removal of background, smoothing the data by averaging the contents of the three-dimensional pixels, or selecting intensity gradients, in order to define the phase or grain boundaries are all basic operations needed when interpreting the three-dimensional image. The second form of data manipulation concerns the viewing of the data on a monitor. Rotation, translation and zoom are basic operations, but for very large data sets it may also be useful to add perspective to the image, for example by reducing the magnification and intensity for data points that are ‘distant’ from the observer in order to widen the field of view for the more ‘distant’ features, while at the same time reducing their contrast. In this final section on three-dimensional reconstruction we have not mentioned quantitative analysis, mostly because very little has been published on the subject. One good reason for this is statistical, since even the millions of ions collected in the atom probe do not usually result in accurate statistics for particle size or phase volume fraction. As dualbeam FIB technology becomes more widely available, we are likely to see increasing use made of this technique for serial sectioning of integrated electro-optical and electronic devices, and for the microstructural analysis of polyphase materials, especially composites and cast or welded structures. The ‘inaccessible’ microstructural parameters that were modelled approximately from the ambiguity of a single two-dimensional section will soon be visible in three-dimensional images generated by reconstruction.

Summary The quantitative analysis of image data requires an understanding of the stereological relations between the two-dimensional data, recorded from a projection image and the three-dimensional microstructure of the bulk sample before sectioning. To this knowledge of stereology must be added an appreciation of the statistical errors that are

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associated with sample selection, sectioning of the sample and digital-image, data collection. The objective is always to optimize the statistical significance of image data with respect to both the effort involved in collecting the data and the statistical accuracy needed for the analysis. The isotropy and homogeneity of the material are important microstructural features. The properties of most engineering materials are, to some extent, usually both anisotropic and inhomogeneous, and this is reflected in their microstructure. Crystallographic anisotropy is associated with the preferred alignment of specific crystallographic planes and directions with respect to the coordinates of the component, while morphological anisotropy refers to the spatial alignment of morphological features (grains or particles, or ordered arrays of particles and inclusions). Inhomogeneity may be chemical, associated with local variations in composition, or morphological, associated with local variations in grain or particle size. A single, planar section is often insufficient to characterize a material microstructure. The position and the orientation of any chosen sample section must be known, and should be selected with respect to the principal axes of the component. The microstructure may also vary across the selected section (inhomogeneity), as well as with the angle that this section makes with the principal axes of the component (anisotropy). The results of a statistical analysis of any microstructural parameter may reflect macroscopic variability in the bulk of the component, mesoscopic variability over any given cross-section, or microscopic variability within the observed field of view visible at a given magnification in the microscope. Statistical variations in an analysis arise from several sources. That of primary interest is generally the inherent variability of the parameter being measured: the spread of grain size or the non uniform distribution of a second phase. Sampling errors, over which some control is possible, are associated with the number of features sampled, the number of recorded measurements, the location of the samples selected and the plane chosen for the section. There will also be experimental errors involving the quality of the specimen preparation (polishing and etching procedures), the resolution of the microscope, and the recording of the image (the total number of pixels in the image and the dynamic range of their measured intensity values). It is important to distinguish between accessible and inaccessible microstructural parameters. Accessible parameters can be determined unambiguously from a twodimensional section, without making any stereological assumptions about the microstructural morphology. Inaccessible parameters require some geometrical model in order to interpret the bulk structure (for example, by assuming spherical particles). The volume fraction of a second phase is a straightforward example of an accessible parameter that can be determined from a surface section to any required degree of accuracy, limited only by the resolution of the imaging system. Providing grain size and particle size are defined solely in terms of surface-to-volume ratio, these parameters are also readily accessible. The surface-to-volume ratio is an accessible parameter, but it is independent of particle shape. Needles, platelets and dendritic shapes are not distinguished, and measures of particle shape, derived from two-dimensional image data, require a stereological model of the bulk geometry for their interpretation in three dimensions. The same applies to the interpretation of particle or grain size distributions, as well as to dislocation arrays. In the case of second-phase particles, it is sometimes useful to assume convexity, that is, all regions have positive curvature, since no convex particle can then intercept a planar section more

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than once. Composite materials, especially those that include highly textured, woven reinforcement, can present a serious descriptive problem. Errors of quantitative analysis that are associated with specimen preparation and the recording of image data can be minimized by careful experimental work that is based on an awareness of the origin of the possible artifacts. Counting errors are usually reduced by ensuring that the sample selected is large enough to ensure that any bulk variability in the microstructure is readily detectable above the level of the random, statistical background. Since both the microscopic variability, within a given area, and the mesoscopic variability, from one area to another, contribute to the statistical errors, it is important to ensure that several areas of the specimen are sampled. For example, the average number of particles or grains that are observed in a specific area should be comparable with the number of sample areas selected, so as to ensure the minimum statistical variance for any given counting effort. The resolution errors of the method of microstructural investigation will limit the accuracy if the size of the features of interest approaches the resolution limit of the microscope. Specimen preparation always introduces a thickness or sectioning error that is associated with the depth of the surface layer over which the sample contributes contrast to features in the projected image. In optical microscopy this slab thickness is of the order of the resolution, since the depth of field of the objective lens is also of the order of the resolution. In both scanning and transmission electron microscopy the depth of field is orders of magnitude greater than the resolution limit of the microscope, and thickness corrections for quantitative analysis can be very important. Observer bias is also a major source of variability in quantitative analysis, and may reflect more than a varying level of experimental competence. Two observers can arrive at significantly different quantitative estimates of the same microstructural parameter, as a result of entirely justifiable differences in professional judgement. Digital data collection and automated image analysis have not eliminated observer bias. The settings for any computer program are always based on the professional judgement of the operator. Fully computerized systems have increased the rate of data collection by orders of magnitude, making quantitative stereological analysis of microstructural morphology readily available, reducing the effort required to collect large data sets and achieve statistical significance. However, the wide choice of computer software programs can lead to some confusion, since some programs fail to explain clearly the stereological assumptions on which they are based. The ease of data processing may also obscure the steps involved in treating digital data. This may degrade the statistical significance during digital data processing and the presentation of the final results of analysis. Major errors may be due to careless specimen selection and preparation, and no amount of subsequent automated image analysis can ‘correct’ for such experimental sloppiness. Three-dimensional imaging is now a reality for a range of magnifications and resolutions that include simple serial-sectioning, by automated grinding and polishing, ion beam milling, with controlled thickness removal to an accuracy of 10 nm, and culminates in the three-dimensional reconstruction of sample chemistry on the nanoscale by atom probe tomography. The assembly of two-dimensional digital data sets from a series of sections of known spacing into a three-dimensional matrix of volume element pixels requires manipulation of the raw, two-dimensional data to ensure that the three-dimensional matrix of image data is free of both background noise and spatial distortion. The reconstructed

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three-dimensional image must be visualized on a computer screen by employing software tools that are able to rotate, translate and magnify the image in order to allow the observer to select specific data subsets and to introduce perspective. In threedimensional imaging it is proving possible to study many features that are interconnected in the bulk microstructure. When sectioned by a planar sample surface, such features become inaccessible and, until recently, have been beyond the reach of quantitative microstructural analysis.

Bibliography 1. J. C. Russ, The Image Processing Handbook, 2nd Edition, CRC Press, London, 1995. 2. J. C. Russ and R. T. Dehoff, Practical Stereology, Kluwer Academic/Plenum, New York, 2000. 3. J. E. Hilliard and L. R. Lawson, Stereology and Stochastic Geometry (Computational Imaging and Vision), Kluwer Academic, New York, 2001. 4. A. Baddeley and E. B.Vedel Jensen, Stereology for Statisticians, Chapman & Hall/CRC, Boca Raton, FL, 2005.

Worked Examples Three examples of size measurements for microstructural features on different length scales taken from different microstructures, will demonstrate the principles of quantitative measurement at the meso, micro and nano morphological levels. The first example is the size of the alumina grains in a sintered body, expected to be in the micrometre range. For these measurements we employ SEM. The second example is provided by the much smaller size of aluminium grains developed by nucleation and growth during chemical vapour deposition. The final example explores the limit of detection of TEM in the measurement of ordered domains in a disordered matrix of Pb (Mg1/3,Nb2/3)O3. Figure 9.26 shows two micrographs of sintered alumina; the first after sintering at 1400  C for 2 h, and the second after sintering at 1600  C for 10 h. Sintering of ceramics is always a compromise: on the one hand, the density of the sintered product should be maximized, which generally requires a high sintering temperature and long sintering times. On the other hand, grain growth should be minimized, usually by limiting both the sintering temperature and the sintering time. Optimizing the sintering process involves measuring the residual porosity and the grain size as a function of sintering time and temperature. As far as the grain morphology is concerned, two basic questions need to be answered: how does the average grain size depend on the sintering parameters, and what is the shape (aspect ratio) of the alumina grains? To quantify the grain size we apply the linear intercept method (discussed in Section 9.2.1.2). Figure 9.27 shows the processed images of Figure 9.26, and the average values of the grain intercept. As expected, a large increase in grain size has occurred after sintering at 1600  C for 10 h, as compared with sintering at 1400  C for 2 h. By quantifying the grain size as a function of the process parameters, we can characterize the sintering mechanisms empirically, and so optimize the sintering process.

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Figure 9.26 SEM micrographs of thermally etched alumina, after sintering for (a) 2 h at 1400  C and (b) 10 h at 1600  C.

We now turn to our chemical vapour deposition aluminium samples. We need to determine the size of the aluminium grains as a function of the deposition time, in order to assess the influence of some of the process parameters. SEM micrographs of the aluminium grains formed on two different TiN substrates, as a function of the deposition time, are shown in Figure 9.28. Figure 9.29 shows an example of a processed image, in which the aluminium grains are distinguished from the TiN background by their contrast. The nominal aluminium grain size, measured as the projected area of each grain, is a function of the deposition time for both TiN substrates and is given in Figure 9.30. As in the case of the previous, alumina sample, employing a computer program can dramatically increase the sample size, and hence improve the statistical significance of the results. A final example, from a completely different material system, demonstrates the combination of quantitative microstructural analysis combined with elemental detection limits. The material is a Pb(Mg1/3, Nb2/3)O3 (PMN) sample that has been doped with

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Figure 9.27 Micrographs from Figure 9.26 after image processing for analysis. The results of the linear intercept method for grain size measurements are included. Note that the two grainsize distributions are in no way self-similar with respect to either the grain size or the grain shape.

varying amounts of lanthanum. PMN has the cubic perovskite structure shown in Figure 9.31. In the ideal structure, cations of type A occupy the corner sites of the unit cell with coordinates 0,0,0, while cations of type B occupy the body-centred position 1/2,1/2,1/2, and oxygen anions are located in the face-centred sites of the cubic unit cell with coordinates of type 1/2,0,0. In disordered PMN there are two types of cations located at the type A sites (Mg and Nb), while Pb occupies the type B sites. However, under certain conditions chemical ordering can take place to form a superlattice, in which distinctive {111} planes, containing either Mg or Nb cations, form a new face-centred cubic unit cell

506 Microstructural Characterization of Materials

Figure 9.28

SEM micrographs of aluminium grains deposited on two different TiN substrates, as a function of the deposition time.

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Figure 9.29 (a) Original micrograph and (b) the resultant processed image. After processing, the aluminium grains are clearly distinguished from the TiN background by their contrast.

with a larger lattice parameter of 2a0, where a0 is the lattice parameter of the original, disordered PMN lattice, shown in Figure 9.32. The difference between the disordered and ordered crystal structures is easily detected in the transmission electron microscope by selected area diffraction after orienting the

508

Microstructural Characterization of Materials 3.5x105

Integrated

3.0x105

Air exposed Area (nm2)

2.5x105 2.0x105 1.5x105 1.0x105 5.0x104 1 0

10

20

30

40

50

60

70

Deposition time (s)

Figure 9.30 Nominal aluminium grain size, measured as the projected area of each aluminium grain, as a function of the deposition time for the two TiN substrates that were used.

specimen into a [110] zone axis, as shown in Figure 9.33. If the ordered regions are sufficiently large, then dark-field diffraction contrast images can be recorded using the {111} ordered reflections. Figure 9.34 demonstrates such a dark-field image in which the ordered regions, separated by anti-phase boundaries, are clearly visible. PMN crystals may also form small ordered regions that have length scales of the order of nanometres. Increased doping levels of lanthanum in these crystals seem to increase the size of the nano-ordered regions. To confirm this hypothesis, we can use TEM and image processing in order to analyse the effect quantitatively. Our first step is to select a method to observe the nano-ordered regions. The minimum size of these regions is just a few nanometres, so any good transmission electron microscope should have sufficient resolution. However, we have to detect and distinguish the ordered PMN regions from the disordered background. We can compare two TEM methods

Figure 9.31 The disordered PMN unit cell has the cubic perovskite structure.

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Figure 9.32 Cation positions in the face-centred cubic unit cell of fully ordered PMN. (The oxygen ions have been omitted for clarity).

quantitatively, namely dark-field, diffraction contrast imaging and high-resolution transmission electron microscopy using phase contrast to obtain lattice images. We have already noted that large, ordered regions can be detected in a dark-field image taken with the ordered {111} reflections, and nano-ordered regions can be detected in a similar way. Micrographs showing the influence of varying lanthanum concentrations on the scale of the ordered regions are shown in Figure 9.35, in which the bright regions

Figure 9.33 Selected area diffraction patterns, taken from a [110] zone axis, of the (a) ordered and (b) disordered PMN crystal structures.

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Microstructural Characterization of Materials

Figure 9.34 Dark-field TEM micrograph recorded using the {111} ordered PMN lattice reflection. Large fully ordered regions are separated by clearly visible anti-phase boundaries.

are the ordered domains. To quantify the size of these ordered regions, we must first process the image to remove the background due to the disordered matrix. This gives us the image shown in Figure 9.36. We then determine the projected area of each ordered region. The data can now be summarized as the average projected area of an ordered region and the corresponding standard deviation, plotted as a function of lanthanum concentration (Figure 9.37). From Figure 9.37 we immediately note a problem with the data. The average domain area does show a tendency to increase with increasing lanthanum content, but the standard deviation of the data is too large for the effect to be judged statistically significant. A major reason for the large standard deviation is that the ordered regions are smaller than the thickness of the TEM specimen, with frequent overlap of the ordered regions, invalidating any quantitative conclusions. We could collect data only from the thinnest regions of the specimen, providing we could determine the thickness accurately, but instead we prefer to use high resolution TEM, and then optimize the microscope parameters to select the ordered regions preferentially. What do we mean by preferential selection of the ordered regions? The periodicity and contrast in high resolution TEM lattice images depends not only on the crystal structure, but also on the microscope contrast transfer function (CTF). In order to detect and differentiate the ordered regions, the microscope operating conditions should maximize the intensities of the superlattice reflections. Figure 9.38 shows the CTF for the microscope at the Scherzer defocus, as well as the relative values of the structure factor for the various crystallographic planes in the PMN crystal superlattice. Most of the diffracted beams which contribute to the [110] lattice image are from the {202}, {222}, and {004} matrix lattice, while the relative structure factors of the ordered, superlattice, reflections {111} and {113} are of much lower intensity than those of the matrix reflections. Thus, although the point resolution is optimized by setting the objective lens current to the Scherzer defocus value, the detection

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Figure 9.35 Series of dark-field TEM micrographs of PMN showing the influence of lanthanum concentration on the size of the nano-ordered regions: (a) 5% La; (b) 3% La; (c) 1% La; (d) 1%La þ22.5 PT. The ordered regions appear in bright contrast.

of the ordered superlattice planes will be extremely difficult if we use this setting. However, we can modify the CTF by changing the object lens defocus and increase the contribution of one or more of the superlattice reflections to the image, at the same time suppressing the dominant, disordered matrix reflections. This is illustrated in Figure 9.39, in which the CTF has been optimized for the {111} ordered reflections. This optimized defocus has been used to record the lattice image in Figure 9.40(a). Although Figure 9.40(a) shows ordered PMN domains under optimized conditions, it is still difficult to differentiate the ordered from the disordered regions, but this may be improved by Fourier filtering of the image data. Figure 9.40(b) is a Fourier transform (FT) of the image in Figure 9.40(a). By superimposing a mask to all frequencies in the FT other than those corresponding to the periodic reflections from PMN (both ordered and disordered crystal regions), we can remove background noise in the image [Figure 9.40(c)]. Now the ordered regions are easily visible. If we further mask all frequencies in the FTother than the {111} ordered reflections, the contrast from both the background noise and the disordered

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Microstructural Characterization of Materials

Figure 9.36 A dark-field TEM image of PMN (a) processed in order to remove the background due to the disordered matrix (b) and obtain an image suitable for quantitative analysis.

matrix is suppressed [Figure 9.40(d)]. From images such as [Figure 9.40(d)]. we can measure the nano-ordered, projected area and compare the results with the nano-ordered regions seen in a dark-field diffraction contrast image (Figure 9.37). The final results are shown in Figure 9.41. These once more give the average ordered area and standard deviation as a function of lanthanum content, but by optimizing high resolution TEM lattice imaging, we have significantly reduced the standard error and there is now no doubt that the lanthanum doping has increased the size of the ordered regions.

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30

Area (nm2)

25

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0 0

1

2

3

4

5

6

7

8

(atom %) La Figure 9.37 Average projected area of the ordered regions in PMN and their standard deviation, determined by quantitative image analysis, as a function of the lanthanum dopant concentration.

The calculations required to determine the optimum defocus, the Fourier filtering and the image processing, may appear complex, but many computer programs now exist for such calculations, both commercial and ‘free-ware’. Use of these computer programs is not especially difficult, and can even be a lot of fun. However, it must be remembered that computerized image processing still requires the microscopist’s understanding of the mechanism of image formation. 1.0 ∆ f=–50 nm 0.5

(nm–1 )

0.0

-0.5

008

10 426

9 444 515

8 206 335

7 404 315

6 333

5

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4

313

3

004

2

202

1

111 002

0

113

-1.0

Figure 9.38 The CTF for the high resolution transmission electron microscope used to study the nano-ordered regions in PMN at the Scherzer defocus, as well as the relative values of the structure factor for the various crystallographic planes in the PMN superlattice.

514

Microstructural Characterization of Materials 1.0 ∆ f=–65 nm 0.5

(nm–1)

0.0

-0.5

008

10 426

9 444 515

8 206 335

7 404 315

6 333

5

224

4

313

3

004

2

202

1

111 002

0

113

-1.0

Figure 9.39 The CTF for the same microscope, adjusted to study the nano-ordered regions in PMN at an optimum defocus for the detection of these regions, as well as the relative values of the structure factor for the various crystallographic planes in the PMN superlattice.

Problems 9.1. A pixel in a digitized image is assigned specific x,y coordinates, while the resolution of a printer is commonly given in dots per inch (dpi). What dpi would you choose for your printer to ensure that the intensity assigned to an image pixel is accurate to within 10 % when the separation of the pixel points in the digitized image is 0.2 mm? 9.2. Given that the resolution of the human eye is about 0.2 mm and that images are most comfortably viewed from a distance of about 200 mm, estimate the number of pixels needed to ensure that a digitized image does not appear ‘discontinuous’ to the eye. 9.3. Distinguish between morphological and crystallographic anisotropy, and describe some experimental tests that can be used to identify both forms of anisotropy. 9.4. Define the term sampling error. The number of microstructural features in a given field of view and the number of fields of view selected for quantitative microscopy both contribute to the statistical errors of measurement. Explain how you would plan a quantitative analysis of the microstructure in order to ensure maximum statistical accuracy for minimum effort. 9.5. The number of particles of a second phase per unit volume of the sample cannot be estimated from a planar section without making some assumption about the particle shape. What is the assumption usually made and why is it necessary? 9.6. The volume fraction of a second phase and the surface-to-volume ratio are both described as accessible microstructural parameters. What is meant by this term?

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Figure 9.40 (a) Lattice image of the ordered PMN domains under optimized high resolution TEM imaging conditions. (b) A FT of the lattice image shown in (a). (c) By applying a mask to all frequencies in the FT, other than the periodic reflections from the PMN lattice (both ordered and disordered), we can remove all background noise in the image. (d) If we further mask all frequencies in the FT other than the {111} ordered reflections, the contrast from both the background noise and the disordered matrix is suppressed.

9.7. How is the surface-to-volume ratio related to grain size, and what factors may contribute to the ambiguity of the concept of ‘grain size’? 9.8. The aspect ratio of a microstructural feature (particle, grain, inclusion, cluster, etc.) on a surface section bears no obvious or simple relationship to the shape of the feature in three dimensions. Give three examples of microstructural features that illustrate this unfortunate fact. 9.9. Sectioning errors occur in the quantitative analysis of both polished sections and thin films. If an etched boundary appears twice as wide as the depth of etch, can you model the sectioning error as a function of grain size by assuming a boundary curvature equal to the grain size? (There is no one ‘correct’ answer to this question, and various assumptions about what constitutes an intersection of the boundary with a test line are possible.)

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Microstructural Characterization of Materials 30

Area (nm2)

25

20

15

10

5

0

0

1

2

3

4

5

6

7

8

(atom%) La Figure 9.41 The average ordered area and standard deviation as a function of lanthanum content, measured from optimized high resolution TEM lattice images of PMN.

9.10. List the errors associated with estimating the density of dislocations observed by diffraction contrast in a thin-film TEM sample. Compare the errors associated with a count of the intersections of the dislocation images with a test grid, as opposed to a count of the number of dislocation intersections with the top and bottom surfaces of the thin film.

Appendices Appendix 1: Useful Equations We present here some sets of equations relating to crystallography that should be useful to the reader. Interplanar Spacings The following relationships give the interplanar spacing d (sometimes referred to as the d-spacing) of the (hkl) planes in the crystal lattices of different crystal structures: Cubic

1 h2 þ k2 þ l2 ¼ a2 d2

Tetragonal

1 h2 þ k 2 l2 ¼ þ a2 c2 d2

Orthorhombic

Hexagonal

Rhombohedral

1 h2 k2 l2 ¼ 2þ 2þ 2 2 a c b d


 1 4 h2 þ hk þ k2 l2 þ ¼ a2 c2 d2 3

1 ðh2 þ k2 þ l2 Þsin2 a þ 2ðhk þ kl þ hlÞðcos2 a
cosaÞ ¼ 2 a2 ð1
3 cos2 a þ 2 cos3 aÞ d

Monoclinic


2  1 1 h k2 sin2 b l2 2hl cos b ¼ þ þ
c2 ac d 2 sin2 b a2 b2

Microstructural Characterization of Materials - 2nd Edition
2008 John Wiley & Sons, Ltd.

David Brandon and Wayne D. Kaplan

518

Microstructural Characterization of Materials

1 1 ¼ ðS11 h2 þ S22 k2 þ S33 l2 þ 2S12 hk þ 2S23 kl þ S13 hlÞ d2 V 2

Triclinic

In the equation for triclinic crystals, V is the cell volume, and: S11 ¼ b2 c2 sin2 a S22 ¼ a2 c2 sin2 b S33 ¼ a2 b2 sin2 g S12 ¼ abc2 ðcos a cos b
cos gÞ S23 ¼ a2 bcðcos b cos g
cos aÞ S13 ¼ ab2 cðcos g cos a
cos bÞ Unit Cell Volumes The unit cell volume V for the various unit cells of the different crystal structures is given by the following relationships: Cubic

V ¼ a3 V ¼ a2 c

Tetragonal Orthorhombic

V ¼ abc

pffiffiffi 2 3a c ¼ 0:866a2 c Hexagonal V ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rhombohedral V ¼ a3 1
3 cos2 a þ 2 cos2 a Monoclinic Triclinic

V ¼ abc sin b

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ abc 1
cos2 a
cos2 b
cos2 g þ 2 cos a cos b cos g

Interplanar Angles The following equations give the angle f between the pole of the crystal plane (h1k1l1) with a spacing d1 and the crystal plane (h2k2l2) with a spacing d2, where V is the unit cell volume:

Cubic

h1 h2 þ k 1 k 2 þ l 1 l 2 cos
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh21 þ k21 þ l21 Þðh22 þ k22 þ l22 Þ

Appendices 519

h

 þ lc1 l22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tetragonal cos
¼ r  2 2  h21 þ k21 l21 h2 þ k2 l22 þ þ 2 2 2 2 a c a c 1 h2

þ k 1 k2 a2



Orthorhombic

 þ kb1 k2 2 þ lc1 l22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos
¼ r  2 ffi h21 k21 l21 h2 k22 l22 þ þ þ þ a2 c2 a2 c2 b2 b2 h1 h2 a2

h1 h2 þ k1 k2 þ 12 ðh1 k2 þ h2 k1 Þ þ 3a 4c2 l1 l2 cos
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2    2 2 2 2 3a2 2 h1 þ k21 þ h1 k1 þ 3a 4c2 l1 h2 þ k2 þ h2 k 2 þ 4c2 l2 2

Hexagonal Rhombohedral

" # 2 a4 d1 d2 sin aðh1 h2 þ k1 k2 þ l1 l2 Þ þ cos
¼ V2 ðcos2 a
cos aÞðk1 l2 þ k2 l1 þ l1 h2 þ l2 h1 þ h1 k2 þ h2 k1 Þ

 d1 d2 h1 h2 k1 k2 sin2 b l1 l2 ðl1 h2 þ l2 h1 Þ cos b þ þ 2
Monoclinic cos
¼ 2 c b2 ac sin " b a2 # d1 d2 S11 h1 h2 þ S22 k1 k2 þ S33 l1 l2 cos
¼ 2 Triclinic V þ S23 ðk1 l2 þ k2 l1 Þ þ S13 ðl1 h2 þ l2 h1 Þ þ S12 ðh1 k2 þ h2 k1 Þ Direction Perpendicular to a Crystal Plane The formulae in Table A1 define the conditions that need to be met in order for a crystal direction [uvw] to be perpendicular to a crystal plane (hkl). Table A1 Directions perpendicular to a crystal plane for the different crystal systems. Crystal system Cubic Tetragonal Orthorhombic Hexagonal Rhombohedral

[uvw], given (hkl ) u v w ¼ ¼ h k l u v w c2 ¼ ¼ h k l a u 2 v 2 w 2 a ¼ b ¼ c h k l u v 2wc2 ¼ ¼ 3la2 2k þ h h þ 2k u ¼ h sin2 a þ ðk þ lÞðcos2 a
cos aÞ v ¼ k sin2 a þ ðl þ hÞðcos2 a
cos aÞ w l sin2 a þ ðh þ kÞðcos2 a
cos aÞ

(hkl ), given [uvw] h k l ¼ ¼ u v w  h k l a 2 ¼ ¼ u v w c h k l ¼ ¼ ua2 vb2 wc2 h k l ¼ ¼ 2u
v 2v
u 2wðc=aÞ2 h ¼ u þ ðv þ wÞcos a k ¼ v þ ðw þ uÞcos a l w þ ðu þ vÞcos a (Continued)

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Microstructural Characterization of Materials

Table A1 (Continued) Crystal system

[uvw], given (hkl ) u

Monoclinic

hb2 c2
lab2 c cos b v ¼ 2 2 ka c
sin2 b w

¼

(hkl ), given [uvw] h ¼ ua2 þ wca cos b k vb

2

¼

l uca cos b þ wc2

la2 b2
hab2 c cos b u ¼ hS11 þ kS12 þ lS13 v ¼ hS12 þ kS22 þ lS23 w hS13 þ kS23 þ lS33

Triclinic

h ¼ ua2 þ vab cos g þ wca cos b k uab cos g þ vb2 þ wbc cos a

¼

l uca cos b þ vbc cos a þ wc2

Hexagonal Unit Cells While we prefer to work with four indices when describing directions and planes for hexagonal structures, in order to reflect the hexagonal symmetry, there are many cases in the literature that use only three indices. Table A2 summarizes the relationships needed to convert between three and four indices.

Table A2 Transforming between three and four crystal indices for hexagonal crystal systems. Three indices

Four indices

[UVW] (HKL)

[uvtw] (hkil) Planes

HKL

U ¼ 2u þ v V ¼ 2v þ u W¼w

Directions

h¼H k¼K l¼L i ¼
(H þ K) u ¼ 1/3(2U
V) v ¼ 1/3(2V
U) t ¼
1/3(U þ V) w¼W

Appendices 521

The Zone Axis of Two Planes in the Hexagonal System The zone axis of two planes in the hexagonal system is given by: u ¼ l2 ð2k1 þ h1 Þ
l1 ð2k2 þ h2 Þ v ¼ l1 ð2h2 þ k2 Þ
l2 ð2h1 þ k1 Þ t ¼
ðu þ vÞ w ¼ 3ðh1 k2
h2 k1 Þ

Appendix 2: Wavelengths Relativistic Electron Wavelengths For an electron of energy E (kV) and a wavelength l (nm): 

12 eE 0:03877 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ h 2me eE 1 þ 2 2me C Eð1 þ 0:9788 · 10
3 EÞ and thus: E (kV)

l (nm)

100 200 300 400 1000

0.0037 0.002507 0.001968 0.001643 0.0008715

X-Ray Wavelengths for Typical X-Ray Sources Table A3 provides a list of wavelengths from sources typically used in X-ray powder diffractometers. Table A3 Target element and X-ray wavelengths for Ka1 and Ka2 (in nm). Target element Fe Co Ni Cu Mo

Ka1

Ka2

0.1936042 0.1788965 0.1657910 0.1540562 0.0709300

0.1939980 0.1792850 0.1661747 0.1544390 0.0713590

Index Page numbers in italics indicate figures. Page numbers in bold indicate tables. 3D imaging see tomography Abbe equation 130–131, 130, 132 aberrations electron lenses chromatic 265 spherical 188–189 correction 189, 190, 212 function 210 optical lenses 127–128 absorption coefficient 65, 350 edges 70, 350 effects on contrast 228 of visible light 149 of X-rays 70 cubic lattices 65 spectral correction 347–349 accessible parameters 467–468 achromat 128, 137 acicular, definition 19 AFM see atomic force microscopy alloys heat treating 7 sample preparation 147–148. see also steel alumina 106 electron energy loss spectroscopy (EELS) 376– 381 electron microscopy 238–241 energy-dispersive spectrometry (EDS) 375–376, 377 grain size determination 503 optical microscopy 175–176 powder diffraction 105 scanning electron microscope (SEM) 318–321 transmission electron microscope (TEM) micrograph 240 aluminium as electron diffraction calibrant 109, 110 electron diffraction pattern 113 grain size determination 504 Microstructural Characterization of Materials - 2nd Edition Ó 2008 John Wiley & Sons, Ltd.

reflectivity 149 American Society for Testing Materials (ASTM) 21 amorphous phases 30 amplitude-phase diagram 82, 213 electron diffraction 213–215 analyte see sample anion 26 anisotropy crystallographic 75, 89, 459–461, 479 morphological 459–461 optical, and polarized light 157–163 annular dark-field detector 235 APB (anti-phase boundary) 217–218 aperture electron microscope, and diffraction 209 function 210 optical microscope 132–133, 137 and image brightness 137 in interference microscopy 154 and polarized light 159 apochromatic lens 128, 137 applications atomic force microscopy (AFM) 407–408 scanning probe microscopy (SPM) 391 X-ray photoelectron spectroscopy (XPS) 431 APT (atom probe tomography) 413–414, 416, 418 areal analysis 469, 471 and volume fraction 484 aspect ratio of grains and particles 477–481 of scanning electron microscope (SEM) images 166 astigmatism, electron lens 191 ASTM grain size 21 atom probe 401 atom probe tomography (APT) 413–414, 416, 418 atomic absorption spectrometry (AAS) 423 atomic force microscope 18, 403 tapping mode 408 atomic force microscopy (AFM) 409 as scanning tunnelling microscope (STM) 409 David Brandon and Wayne D. Kaplan

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Index

atomic force microscopy (AFM) (Continued) compared with scanning tunnelling microscopy (STM) 407 development 403 instrument 403–405 atomic number and Auger excitation 439 and inner shell electron excitation energy 69 and secondary ion yield in secondary ion mass spectrometry (SIMS) 443 imaging and contrast 277 atomic number correction, X-ray quantitation 345–347 atomic scattering factor 80, 205 Auger electrons 281, 425 effect of atomic number on excitation 439 emission 431–432 Auger electron spectroscopy (AES) 431–440 overview 431–433 worked example 448–452 Auger excitation, potential image resolution 433 Auger imaging 438 automated image analysis 491 background correction in electron energy loss spectroscopy 363 electron microscope 15 reduction, image analysis 511 subtraction, wavelength-dispersive spectrometry (WDS) 380 X-ray analysis 335, 353–354 thin films 355 X-ray photoelectron spectroscopy (XPS) 426 removal 425. see also noise backscattered electrons 277–280, 278. see also Rutherford backscattering diffraction 278, 289 patterns 290 bend contours in transmission microscopy 215 binocular viewing 139 biological samples, in transmission electron microscope (TEM) 237 bit depth (of digital image) 168 Bitmap (BMP) digital images 168 Bloch waves 226–227 and stacking faults 227–230 body-centred cubic (BCC) lattice 59, 64 bonding conduction 28 covalent 27–28 ionic 26–27, 26 secondary 29–30 two-body model 392–393 boron 26

boundary grain 241, 378 special 293 particle 175 phase 232–233, 233 Bragg equations 58–59, 94, 204 and diffraction grating 131 vector representation 61 Bragg-Brentano diffractometer 72, 74, 86 counting time 105–106 Bravais lattices 32 scattering 57–59 Brermmstrahlung 67 Brewster angle 149 bright-field electron detector 235 bright-field optical image 150 bright-field transmission image 204 brightness, and intensity in digital images 166–167 Burgers vector 21 of a dislocation 21 calibration atom probe tomography (APT) 417 electron diffraction 109–111 secondary ion mass spectrometry (SIMS) 444–445 caliper diameter 19 camera length 96 electron microscope 96 cantilever probes 405 in scanning probe microscopy 405 carbon 381 carbon coating 201 Castaing, Raymond 199 cathodoluminescence 288 cation 26 CCD see charge-coupled device cementite 46–47 ceramics 125 sample preparation 148, 149 characteristic (spectral peak) intensity 344 characteristic X-ray excitation 67, 269 characteristic X-ray images 271 charge compensation, at low beam voltages 287 charge-coupled device (CCD) data storage 167 in field-ion microscopy 414 optical microscopy 139 chemical analysis Auger electron spectrometry 431–440 overview 424–425 secondary ion mass spectrometry 440–446 X-ray spectrometry 343–357, 424–431 spectral corrections 343–353

Index 525 chemical binding sites, and Auger electron spectroscopy (AES) 435 chemical concentration map 273–275, 276 chromatic aberration electron lens 189–191 optical lens 127–128 chromaticity triangle 493 CMOS (complementary metal oxide semiconductor) 139 and data storage 167 coefficient of variance 483 coherence envelopes, electron microscope 210, 211 coherency strains 219–221 colour in chemical concentration maps 273–275 in digital images 168, 493 filtering 165 in scanning electron microscope (SEM) images 261 colour centres 28 colour coding, preferred orientation 293 complex plane 83 composites morphological anisotropy 459–460 sample preparation 148 electron microscopy 198 comparative performance: transmission and scanning microscopy 92 compression, digital images 168–169 computer aided tomography 497 computer simulation, lattice images 231–232 condenser aperture 135 in optical microscope 135 condenser lens, optical microscope 135 conductive coatings 295 for scanning electron microscope samples 295 confocal microscopy 152 and tomography 496 contact region (of two surfaces) 394, 395 contamination (of sample) and carbon spectral peaks in elemental analysis 381 electron microscope 193, 195, 275–276, 317 X-ray photoelectron spectroscopy (XPS) 429 contrast electron microscopy backscatter 279–280, 291 diffraction 204, 205–207, 213–215, 221–227 from dislocations 218–219 from lattice defects 215 point defects 219–221 from stacking faults and anti-phase boundaries 216–218 enhancement, in scanning electron microscopy 295

from dislocations 477 mass-thickness 205 overview 203–205 phase 207–213 optical 148–150, 170 bright-field 150–151, 150 interference microscopy 152–157 polarized light 159 contrast transfer function (CTF) 209–211, 211, 212, 510, 513 and lattice imaging 230–231 convergent beam electron diffraction 96 coordination number (of ion) 26 copper crystals 33, 33–34, 35 unit cell 83 covalent bond in materials 27 critical illumination in optical microscopy 150 cross-section 197 preparation for transmission electron microscopy 197 cryomicroscopy 138 crystal directions 36 crystallite see grain crystallography, equations 517–521 crystal orientation distribution function 89 crystal systems 32 crystals scattering 56–59 and contrast 204 structure bonding 25–30 determination 31–38 grain see grain lattice 31–42 defects 205–207 and electron beam phase 215–216 contrast 216–221 definition 31 hexagonal 41–42 preferred direction 36–38, 479–481. see also anisotropy space groups and symmetry 33–36 spacings 75 unit cells 47–49 cubic 38 face-centred 35–36, 59–60 curvature, definition 473 dark-field images electron microscope 215–216 optical microscope 150–151 data collection, atomic force microscopy (AFM) 406 presentation, tomography 497–498

526

Index

data (Continued) processing, scanning electron microscope (SEM) 264 recording, optical microscopy 139 storage 166–167 databases, powder diffraction spectra 103 de Broglie relationship 14, 185, 90–91 Debye interaction 395 defocus error 87 deformation texture 461 depth of field, optical microscope 133–134 depth of focus in optical microscopy 133 depth profiling, Auger electron spectroscopy (AES) 437 detection limit 133, 353 in microanalysis 353 in optical imaging 133 detector atomic force microscopy (AFM) 405 energy-dispersive spectrometry (EDS) 342–343 scanning electron microscope (SEM) 263 and secondary electron emission 284–285, 286 cathodoluminescence 288 induced current 289 X-ray 271 scanning transmission electron microscopy (STEM) 235 secondary ion mass spectrometry (SIMS) 442–443 transmission electron microscope (TEM), X-ray analysis 355 X-ray 71, 338–339 development (of photographic film) 141–142, 142 diamond 83 and electron energy loss spectroscopy (EELS) calibration 362 differentiated signal spectrum 434 diffracted intensity 84 diffraction contrast 203 column approximation 206 from crystal lattice defects 205 electron 90–98, 102 patterns backscatter 289–291 definition 100 Kikuchi 96–98, 99, 100, 101 ring 94–96, 95 worked example 108–114 grating (optical) 131 spectra 55, 56 spectrum 55 standard 55 X-ray 31, 102 methods 63–67

patterns 76–79 spectra 79–90 diffractometer electron microscope as 207–213, 208 X-ray 67–73, 74, 86, 101–102 diffusion depth for high-energy electrons 267, 277 digital data processing 165 digital data recording 142, 494 digital data storage 166 digital images data collection 492 dynamic range 167–170 optical 165–170 dimpled feature 8, 9 dimpling of transmission electron microscope specimens 196 direction (of crystal lattice) 36–38. see also preferred orientation directions perpendicular to crystal planes 519 disc of least confusion 127 electron lens 188 dislocation contrast in transmission electron microscopy 218 dislocation density determination 21, 477, 490 dislocations 21–24, 22, 23 density 476–477 estimation errors 490–491 DLVO theory of surface forces 396 double diffraction 94 ductile failure 8 dynamic electron diffraction 221 dynamic range digital images 184 eye 167 EBSD. see electron backscatter diffraction edge dislocations 218 EDS see energy-dispersive spectrometry EELS see electron energy loss microscopy elastic scattering 3, 4, 4, 5 X-rays 70 electrical resistivity 7 electrochemical thinning of electron microscope samples 198 electromagnetic focusing: thick lens effects 187 electromagnetic lens astigmatism 191 electromagnetic lenses 185 electron see scanning electron microscopy; transmission electron microscopy electron back scatter diffraction (EBSD) 264, 277–278 diffraction patterns 289–291 image contrast 279–280, 280

Index 527 electron beam amplitude, and phase 213–215 diffraction see diffraction, electron energy and secondary electron emission 283 and X-ray emission 335–336 focussing 185–186 penetration depth 267 scanning electron microscope (SEM) 265–266 scattering 56 electron beam-induced current 288 electron diffraction, resolution limit 188 electron energy loss spectroscopy (EELS) detection limit 359 overview 357–359 schematic arrangement 359 schematic diagram 359 summary 374 worked example 376–381 electron energy transitions in energy loss spectroscopy 360 electron microscope 4–5 contrast 203–218 as diffractometer 207–213, 208 transmission, schematic diagram 180 see also scanning electron microscope; transmission electron microscope electron range in solids 267 electron source Auger electron spectroscopy (AES) 433 beam coherency, and contrast 231 scanning electron microscope (SEM) 265–266 transmission electron microscope 180–182, 181 electron spectroscopy for chemical analysis (ESCA) 424 electron spin resonance 424 electrons Auger 281, 425, 431–432 backscattered 277–280, 278, 286 diffraction see diffraction emitted in scanning electron microscope (SEM) use 281 energy levels in scanning electron microscope (SEM) 281–283, 281 in scanning tunnelling microscopy (STM) 411–412 free 28 secondary 5, 184, 426 type 1 319–320 in X-ray photoelectron spectroscopy (XPS) 424–425 wave properties 91–94, 185–187 electron transitions, radiation 68 electron wavelength 91, 185, 521

electrostatic force 395 electrostatic lenses 185 elemental analysis see atomic number; chemical analysis elemental maps 273–275, 276 emission probability, and Auger electron spectroscopy (AES) quantitation 436 energy-dispersive spectrometry (EDS) 271–277, 341–343 sample contamination 275 worked example 375–376 energy-filtered transmission electron microscopy (EF-TEM) 367–369, 368, 369, 371 energy loss near-edge structure 360 envelope function 210 equiaxed polycrystal microstructure 19 errors and resolution 485–487 electron diffraction 95 fractography 297 lens aberrations see aberrations optical microscopy 133, 144 dislocation density 477, 490–491 grain counting 483–485 observer bias 489–490 statistical 466–467 X-ray analysis 353 diffraction measurements 85–90 ESCA (electron spectroscopy for chemical analysis) see X-ray photoelectron spectroscopy etching 146–147, 175 focused ion beam (FIB) microscopy 307, 309 and surface roughness 398–399 excitation energy for characteristic X-rays 69, 335 extended energy loss fine structure 366–367 extinction thickness 94 and electron diffraction 225 eye 11–13, 129 dynamic range 167 image formation 129 optical sensitivity 140 eyepiece, optical microscope 127 fabrication, focused ion beam (FIB) microscopy 308–310 face-centred cubic (FCC) lattices 35–36, 59–60, 64 failure analysis 8, 295–298, 296, 297, 299 Faraday cup, in determining scanning electron microscope (SEM) beam diameter 266 far-edge fine structure in energy loss spectroscopy 366 ferrous alloy see steel field-emission gun for electron microscopy 181, 265 field-emission microscope 401

528

Index

field evaporation 414–416 field-ion atom probe 401 field-ion microscope 401, 413–414 film, photographic 141–142 fine structure in Auger electron spectra 432 in extended energy loss spectra 366–367 fluorescence spectroscopy 424 fluorescence (X-ray spectral) correction 349–352 fluorescent screen 182 focal length, optical microscopy 127 focused ion beam chemical vapor deposition 307 for cross-sections 307 for serial sectioning 499 for thin film specimen preparation 310 machining 304, 306 radiation damage 304 focused ion beam (FIB) microscopy 301 dual-beam 306 overview 301–302 principle of operation 302–304 focussed ion beam (FIB) milling 498–499 forbidden reflections in diffraction 59 Fourier transform 511–512 Fowler-Nordheim equation 401, 411 fractal analysis 466 fraction analysis 24, 25, 469–472, 470 fracture analysis (fractography) 8, 295–298, 320– 321, 321 fracture toughness 7 and fracture strength 10 free electrons 28 frequency function 466 full width at half-maximum (FWHM) beam diameter 266 criterion 266, 337 gamma, recording response curve 142 geometrical optics 125 GIF image files 168 goniometer 71, 73 grain aspect ratios 477 boundaries 241, 378 special 293 shape 477–478 size 19–21, 20, 21 definition 473 determination 472–476 worked example 503–513 great circle 40 grey levels in digital recording 168 grinding 144–145 grit size for grinding media 145

Guinier-Preston zone 219 Harris method 89 H-bar method (sample preparation) 310–311, 311, 312 hexagonal system, zone axes 521 hexagonal unit cells, alternative indices 520 hexapole aberration correction in transmission microscopy 190, 212 high-angle annular dark-field (HAADF) detector 235 homogeneity 461–463 human eye, structure and performance 129 image contrast, back-scattered electrons 279 image contrast in optical microscopy 148 image contrast see contrast imaging 457 Auger electron spectroscopy (AES) 438–440 image acquisition 491–494 image recording 494–495 secondary ion mass spectrometry (SIMS) 445–446 inaccessible microstructural parameters 467, 476 inaccessible parameters 476–481 inelastic energy losses 266 inelastic scattering 3–4, 4, 5, 268 and Bloch waves 226 and diffraction analysis 91–92, 92 scanning electron microscope (SEM) 266–268 information limit 212 infrared spectrometry 423 inhomogeneity 461–463 integration time, eye 11 intensity (of signal) and brightness 166–167 and data storage 167 interatomic potentials 393 interfacial dislocation 221 interference electron microscope 207, 207–213 optical 153–155 and sample surface topology 156–157 intergranular failure 9 intermediate lens 140 optical microscope 127, 140 intermetallics, sample preparation 148 International Centre for Diffraction Data 103 International Tables for Crystallography 34–35, 47–48 interplanar spacings in crystals 517 inverse pole figure in preferred orientation 89 ion channelling 305 ionic bonding 26–27

Index 529 ion milling 199–201, 200, 440–441 ions coordination number 26 ion source, focused ion beam (FIB) microscopy 301, 302 ionization and elemental analysis 346 as a result of electron excitation 269 cross-section 346 energy 269 isotropy 459–461, see also anisotropy Israelachvilli, Jacob 396 Joint Committee of Powder Diffraction Standards (JCPDS) 103 JPEG (Joint Photographic Experts Group) digital images 168 Kikuchi patterns 96–98, 99, 100, 101 electron diffraction 223, 224 kinematic theory of diffraction contrast in transmission microscopy 213 K-line spectra 67–69 lanthanum hexaboride electron sources 181 lattice defects diffraction contrast in transmission microscopy 215 lattice image alumina 242 aluminium 246 computer simulation 231 in high resolution transmission electron microscopy 230, 232 lattice vacancy 220 lattice vectors in crystals 36 Laue diffraction 76–78, 78, 95 Laue equations 56–57 Laue zones 95 lens aberration see aberration electron 185–191, 210 probe 183, 262–263, 265 eye 128 long working distance in optical microscopy 138 optical 126–128, 127 objective 138 lift-out method (TEM sample preparation) 311–314, 313, 314 limiting sphere 60–61, 62 electron beam 92–94 limit of detection electron energy loss spectroscopy (EELS) 360 eye 12 in energy loss spectroscopy 361

optical microscope 133 X-ray microanalysis 338 X-ray wave-dispersive analysis 354. see also resolution linear analysis 470–472 statistical analysis 485 linear intercept analysis 21, 470 line-scan 272, 274 Linnik interferometer 154–155, 155 lithium-drifted silicon X-ray detectors 339 long-range surface forces 394 long-wavelength radiation detection 343 Lorentz polarization factor 84 macrostructure definition 11 sample variability 463–464 stress 76 variations 482 magnetic domain structures 409 magnification level optical microscope 14. see also resolution Markov process 466 mass-absorption coefficient 70 mass spectrometry secondary ion (SIMS) 440–446 time-of-flight 414–416 mass-thickness contrast 203 mass-to-charge ratio, in secondary ion mass spectrometry (SIMS) 443–444 materials, examples 6 matrix dissolution 469 mechanical thinning for thin film preparation 195 mesostructure definition 11 homogeneity 464 sampling 465 metallic bond 28 metals 28–29 homogeneity 461 optical microscopy 123 reflectivity 149 sample preparation electron microscopy 198–199 optical microscopy 147–148 Metals Handbook 175 mica, as scanning probe microscopy (SPM) substrate 397–398 microanalysis 3 of thin films 354 micromilling, focused ion beam (FIB) microscopy 306–307 microscope column focused ion beam (FIB) microscopy 303 scanning electron microscope (SEM) 263

530

Index

microscopy atomic force 18, 18 electron see electron microscope field-ion 16 ion beam 301–306 optical 13–14 components 135 confocal 152 eyepiece 139–140 image contrast, overview 148–150 interference multi-beam 155–156, 156 two-beam 152–155 light source 134–136 overview 123–125 phase contrast 163–165 reflection 123 sample preparation 143–148 specimen stage 136 transmission 123 types 2 UV and X-ray 14 microstress 76 microstructure definition 10, 11 scale 10–19 Miller indices 36, 37–38, 58 minerals, sample preparation 149 Moire´ fringes 221, 222, 243 Monte Carlo simulations, scanning electron microscope (SEM) 266–268, 267 morphological anisotropy 459 morphology, surface see topology Mu¨ller, Erwin 16 multi-beam interference 154 multiplicity (of crystal planes) 84 n-type semiconductors 28 NA see numerical aperture nanostructure, definition 11 National Institute of Standards and Technology (NIST) 392 near-edge fine structure in electron energy loss 365 near-field microscopy 132 needle-like particles 478 neutrons, diffraction 56 Newton’s colours 159–161, 161 nickel, powder diffraction spectrum 104 noise and digital data storage 167 and digital images, compression artefacts 169 and optical image detection 142 and quantitation 336–337. see also background in image data 458

non-contact region 394, 395 Normarski contrast 164–165 nuclear magnetic resonance 424 numerical aperture (NA) 132–133, 133 and image brightness 137 interference microscopy 154 Nyquist criterion 167 objective aperture 136 objective lens dark-field 138 in optical microscopy 136 optical microscope 127 oblique illumination in optical microscopy 151 observer bias in quantitative analysis 489 occupancy (of crystals) 33, 36 optical anisotropy in crystals 157 optical emission spectrometry 423 optical light sources for microscopy 134 optical microscopy image formation 125–130 objective lens 136–138 worked examples 173–175 optical wedge in optical microscopy 159 optics 125–130 order of reflection 58 orientation distribution 481 orientation imaging microscopy (OIM) 289 and preferred orientation 292–294 applications 292 resolution 291 orthochromatic emulsion 140 p-type semiconductors 28 panchromatic film 141 parallax measurements, scanning electron microscope (SEM) 298–301, 300 particle (microscopic) aspect ratio 459, 477–478, 478–480 boundary 175 size and yield strength 8. see also grain determination 472–476, 503–513 in powders 73 peak broadening 86–87 peaks, spectral diffraction patterns 100 intensities, measurement errors 87–88 Pearson symbol 33 Pearson’s Handbook of Crystallographic Data for Intermetallic phase boundary contrast in transmission electron microscopy 216 phase plate in optical microscopy 164

Index 531 Phases 33, 46–47 photodiode detectors 405 photoelectron attenuation length, energy dependence 427 photoelectron emission, mechanism 426 photoelectron spectroscopy applications 431 chemical binding states 428 combined with other techniques 430 contamination effects 429 penetration depth, electron beam 267 percolation 476 Petch equation 8 phase boundaries, lattice images 232–233, 233 phase volume 24, 25, 469–472, 470 and particle counting 479–480 and sample thickness 488 phonon 360 photographic emulsion 139, 140–141 optical sensitivity 140 piezoelectric position control 404 pixels 166 plasmon 361 plate-like particles 478 PMN 504–508, 509 point count 24 point analysis 470, 470–472, 471 point dilatations 219–221 point lattice 31 point resolution 211 polarity, liquids 397 polarization forces 29–30, 395–396, 398 polarized light 158–159 analysis 159, 160, 161–162 Polaroid camera 141 polarized light, reflection 162 polishing 145–146 computer-aided 498–499 and surface roughness 398–399 polishing, chemical, for sample preparation 145 polishing, composite materials 148 polishing, electrolytic, for sample preparation 145 polishing, mechanical, for sample preparation 145 polishing, semiconductors and ceramics 148 polishing, soft metals and alloys 147 polycrystalline materials 19 polyethylene 124 polymers, sample preparation 149 porcelain glazes 29 porous materials 24 position control system, atomic force microscopy (AFM) 404–405

powders analysis 491–492 diffraction of X-rays 73 electron diffraction 73–76 patterns 96, 98 worked examples 103–114 sample preparation, electron microscope 194 preferred orientation (of crystal lattice) 75, 89, 292–294, 459 pressure, operating see vacuum requirements printers 169–170 probability 466–467 of electron emission 436 probe atomic force microscopy (AFM) 403–404 cantilever 404–405, 405, 406 tip 405 chemical analysis 425 size, transmission electron microscope (TEM) and scanning electron microscope (SEM) 356 types 2 probe lens, scanning electron microscope (SEM) 183, 262–263, 265 process defects 323 projection (of crystal structure onto 2 dimensions) 38–42, 48–51 proportional counters for X-rays 70 quadrupole mass analyser 442 qualitative analysis 2, 334–335 Auger electron spectroscopy (AES) 436 electron energy loss spectroscopy (EELS) 361, 364–365 of contrast 230 interference microscopy 157 lattice images 233 secondary ion mass spectrometry (SIMS) 445 tomography 500 X-ray spectrometry 343–357 quantitative analysis, statistical significance in 495 quantitative microscopy, accuracy and sources of error 481, 482, 485 quantitative phase analysis 89 radiation damage channeling 305 focused ion beam (FIB) microscopy 304–306, 305 in crystals 22 Raleigh 12, 131–132 Raman spectrometry 424 rastering 272 ray diagram 129

532

Index

reciprocal lattice unit cell 63 reciprocal space 60–63 reciprocity failure 141 reflecting sphere 65, 66 electron diffraction 93, 94 reflections from cubic lattices 59–60, 60, 64 of light 149 of polarized light 162 refractive index in optically transparent materials 126 refraction 125–126, 126 replica specimens 202–203 residual stress 75, 294 resistance, electrical 7 resolution and detection errors 485–487 Auger electron spectroscopy (AES) 432, 433 definition 12 electron energy loss spectroscopy (EELS) 358, 361–364 electron microscope 14–15, 187 limitations 188 scanning electron microscope (SEM) 317, 338 imaging 266 elemental map 277 secondary electron emission 287 eye 12 image-scanning systems 494 optical microscope 13, 130–133, 170 orientation imaging microscopy (OIM) 291 Raleigh 13 scanning probe microscope 399–400 scanning tunnelling microscope 397–400 secondary ion mass spectrometry (SIMS) 442–443 imaging 445–446 spectral, wavelength-dispersive spectrometry (WDS) 337, 343 transmission electron microscope (TEM) 179 X-ray photoelectron spectroscope (XPS) 427 ring patterns, electron diffraction 94–96, 95 rotating crystal X-ray diffraction 79 Ruska, Ernst 16 Rutherford backscattering 424 salt (common) 83–84 sample damage Auger electron spectroscopy (AES) sputtering 437 focused ion beam (FIB) microscopy 304 density, and secondary electron emission 284

electron microscopy 194–203 contamination 193 orientation 89 oxidation, and Auger electron spectra 449 preparation and image analysis 482 electron microscopy metals 199–201 scanning electron microscope (SEM) 318–319 cleaning 323 thin films 321 sputter coating 201 transmission electron microscope (TEM), using focused ion beam 310–315 failure analysis 297 focused ion beam (FIB) microscopy 304–306 milling, ion beam 306–310 optical microscopy 143–145, 172 polishing 145–148, 175, 176 alumina 176 replicas 202–203 scanning electron microscopy (SEM) 294 sectioning see sectioning transmission electron microscope (TEM) 194–203, 238–240, 239 metallic films 243 region selection 463–466 scanning electron microscope (SEM) 192–193 failure analysis 297–298 size and counting time 483–485 transmission electron microscope (TEM) 192 X-ray diffraction 72 stage see specimen stage thickness and diffracted electron beam intensity 225 and transmission loss 88 counting error correction 487–489, 488 electron energy loss spectroscopy (EELS) 364 transmission electron microscope (TEM) 237 and defocus effects 247 X-ray analysis 355 sampling stereology 463–466 sampling errors 485 sampling procedures 143, 463 for coatings 464 sapphire 47–48 scale levels, microstructural 12 scanning electron microscope (SEM) 6, 192–193 backscattering 277–280 cathodoluminescence 288 column 383

Index 533 compared with transmission electron microscope (TEM) 192–194 components 262–264 electron beam 264–268 image analysis 491 orientation imaging 289 overview 183–184, 261–264 sample preparation 294–298 schematic diagram 183 secondary electron imaging 280–288 X-ray spectra 269–277 scanning image 184 scanning probe microscopy (SPM) overview 391, 400–403 probe tip radius 399 probe types 402 scanning system, scanning electron microscope (SEM) 266 scanning transmission electron detectors 356 scanning transmission electron microscopy (STEM) 234–235, 234 and scanning electron microscope (SEM) sample preparation 311 scanning tunnelling microscopy (STM) 410–416 atomic force microscopy 403–409 overview 410–411 summary 420 scattering by atoms 80–81 equations, beam angles 57–59 from unit cell 81–82 inelastic 3–4, 4, 5 and diffraction analysis 91–92, 92 scanning electron microscope (SEM) 264–265 Scherzer focus 211–212, 231 Schottky emission 182 secondary electrons 5, 184, 426 collection efficiency 285 detector Everhart-Thornley 286 in-lens 285 emission 280–285, 330 and image contrast 286–288 scanning electron microscope (SEM) 319–320 escape distance 282 in focused ion beam (FIB) microscopy 304 in X-ray photoelectron spectroscopy (XPS) 424–425 type 1 319–320 secondary emission coefficient, incident energy dependence 283 secondary-ion mass spectrometry calibration 444 imaging 445

mass sensitivity 441, 442 mass spectra 444 resolution 444 secondary ion mass spectrometry (SIMS) 440–446 ‘shadowing’ 441 secondary phases, detection 241 sectioning electrochemical 198–199 focused ion beam (FIB) microscopy 307–308, 316 mechanical 195–198, 196, 197 optical microscopy 143–144 particles 20–21, 475, 475 scanning electron microscope (SEM) 314–315 serial 195 3D scanning electron microscope (SEM) imaging 264 and tomography 498–499 using focused ion beam (FIB) microscopy 325–326, 327 sections, limitations 467–468 selected area electron diffraction 108, 109, 507–508 SEM see scanning electron microscopy semi-contact region 394–395 semiconductors 28–29 optical microscopy 125 sample preparation 148 electron microscopy 198 sensitive tint plate 161 sensitivity see limit of detection; resolution serial sectioning 195, 264 and tomography 498–499 shadowing (of detectors) 383, 441 Shannon’s sampling theory 167 shear forces, in extruded metals 461–462, 461 short-range surface forces 395 signal strength backscattered electrons 278 X-rays in scanning electron microscope (SEM) 271 signal-to-noise ratio 336 signal-to-background see background significance (statistical) and microstructural homogeneity 464 in quantitation 495 of X-ray signal 276 single crystal diffraction 76–79 size distributions 479 slack, definition 136 small circle on a stereogram 40 software 3D imaging 499–500 error correction and quantitation 344–345 image analysis 491–494 quantitation 481–483 tomography 497–498, 513

534

Index

solids, interatomic bonding 25–30 solid-state X-ray detectors 271, 355 Soller slits 72 solubility limits determining 380–381 WDS 378 space groups 32–36 spatial coherence envelope, electron microscope 210, 211 spatial frequency, of information in transmission 210 special boundaries 293 specimen see sample specimen stage optical microscope 136 transmission electron microscope (TEM) 309 transmission electron microscopy 182 X-ray photoelectron spectroscopy (XPS) 430 specimen mounting, optical microscope 144 specimen preparation, transmission electron microscopy 194 spectra absorption 269–270 Auger electron spectroscopy (AES) 434–435 capture, electron energy loss spectroscopy (EELS) 358 differentiated 434 diffraction 55, 56 X-ray 67–73, 79–90 electron energy loss spectroscopy (EELS) 360–361, 360 fine structure 365–367 emission, Auger 434–435 mass spectral 414–416 spectroscopy, X-ray 269–277 spherical aberration 128 electron lens 188–189 corrector 189, 190, 212 spherical triangle 41 spherulites, in polymers 124 sputtering and optical image contrast 149 in Auger electron spectroscopy (AES) 437, 449–451 in secondary ion mass spectrometry (SIMS) 440–441 ion milling 199–201 secondary ion mass spectrometry (SIMS) 442 sputter coating 201, 295 sputtering yield in secondary-ion mass spectrometry 441 stability, voltage and current in electron microscopy 193 stacking fault 216–218, 229

and dynamic diffraction 227–230 steel energy-dispersive spectrometry (EDS) 381–385 optical image 124 optical microscopy 173–175, 174 sample preparation 147 STEM see scanning transmission electron microscope stereographic projection (of crystal structure) 38–42, 39, 44, 48–51 noncubic crystals 112 stereology definition 458 process overview 458–459 stereoscopic imaging 10 optical 140 scanning electron microscope (SEM) 261, 298, 298–300, 317. see also tomography steric hindrance 396, 398 stereogram, for crystallographic analysis 39 stochastic properties 19 stopping power, high energy electrons in solids 346 stress 76 and orientation imaging microscopy (OIM) 292–294, 294 effect on crystal lattice 75–76 structure factor 83–84 structure-property relationships 7 sub-grain boundary 22 surface force measurements 396 surface forces 392–396 as function of interatomic separation 393 effect of a double layer 397 surface morphology, restructuring 411 surface morphology see topology surface probe microscopy (SPM) 18–19 surface to volume ratio, particles 472 symmetry (of crystals) 33–36, see also anisotropy tapping mode, scanning probe microscopy (SPM) 419 television camera 142–143 television raster 142 TEM see transmission electron microscope temperature and etching rate 175 and optical microscopy 138 effect on diffraction peak intensity 88 temporal coherence envelope, electron microscope 210, 211 thermal conductivity 7 thermionic electron gun 181 thermionic emission, of electrons 181 thickness extinction fringes, in transmission electron microscopy 216 thin films 105–107

Index 535 grain size 504 sample preparation 198–199 scanning electron microscope (SEM) 321–326 X-ray analysis 354–356 thin lens approximation, in geometrical optics 129 three-dimensional imaging see tomography three-dimensional reconstruction, of microstructural morphology 495, 499 TIFF (Tagged Image File Format) 168 time-of-flight mass spectrometer, in secondary ion mass spectrometry (SIMS) 443 tint plate 161–162 titanium 49–51, 49, 50 tomography 495–500 and serial sections 499–500 data presentation 497–498 topology topographic contrast 151, 287 and image contrast 287–288 and optical interference fringes 156–157 and phase contrast microscopy 163–164 and scanning probe microscopy 397–400 and secondary electron emission 284–285, 284 and specimen preparation in scanning electron microscope (SEM) 294–298 measuring with atomic force microscopy (AFM) 407–408 scanning tunnelling microscopy (STM) 411–412 transgranular failure 9 transmission electron microscope (TEM) biological samples 237 compared with scanning electron microscope (SEM) 192–194 components 180 contrast see contrast history 179 lens aberrations 187–191 overview 179–185 principle of operation 185–187 sample preparation 238–240, 239 transmission imaging and transmission diffraction comparison 208 transmission intensity, X-rays 70 two-beam diffraction, electron beam 221–225 two-beam interference, in optical microscopy 152 underfocus, electron lens 210 unit cell hexagonal 44 in reciprocal lattice 63 of crystal lattice 31 scattering 81–82 volume, and scattering 57–59

unit cell (of crystal), copper 35–36 unit cell volume of crystals 518 uranium X-ray spectra 69 vacuum requirements 193 Auger electron spectroscopy (AES) 433 scanning electron microscope (SEM) 262 and Auger electrons 281 X-ray photoelectron spectroscopy (XPS) 429 van der Merwe dislocations 221 van der Waals forces 29, 395–396 and steric hindrance 398 visual spectrum 11–14 voltage stability, electron microscope 193–194 volume fraction 24, 25, 469–472, 479–480, 484 wavelength dispersive spectrometry (WDS) 263–264, 340–341 ‘shadowing’ 383 wavelengths for common X-ray sources 521 wavelengths of relativistic electrons 521 wave-particle duality 90–91, 185–187 wedge optical 159–161 samples 196 white radiation 67, 68 Widdmanstatten structure 124 wire-bonding 323–326, 326, 327 work function (of surface), and secondary electron emission 283, 425 working distance in scanning electron microscope 26 optical microscopy 138 Wulff net 41, 43 Wyckoff generating sites 33 X-ray diffractometer 67 X-ray goniometer 71 X-ray line-scan 272 X-ray mapping 273 resolution 277 X-ray microanalysis 334 detection limit 338 resolution 338 X-ray monochromator 71 X-ray photoelectron spectroscopy (XPS) 424–431 chemical binding states 428 instrumental requirements 429–430

536

Index

X-rays absorption and fluorescence 351 adsorption edges 270 counting times 276 diffraction spectra 79–90 excitation 334–338 in scanning electron microscope (SEM) 263–264, 269–277 long wavelength 343

spectrum 272 wavelength 67, 70–71 yield strength 7 ZAF equation 352–353 zero curvature interface 473 zero loss peak 360 zone axis, electron diffraction 111

Figure 5.23

Figure 1.20

Figure 5.24

Figure 5.25

Figure 6.27

Figure 7.18

Figure 7.22

Figure 9.23