Introduction to Texture Analysis Macrotexture, Microtexture and Orientation Mapping
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Introduction to Texture Analysis Macrotexture, Microtexture and Orientation Mapping
Valerie Randle University of Wales, Swansea, UK and Olaf Engler Los Alamos National Laboratory, USA
CRC PR E S S Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data Randle, V. (Valerie) Introduction to texture analysis: macrotexture, microtexture and orientation mapping. p. cm. Includes bibliographical references and index. ISBN 90-5699-224-4 1. Materials—Texture. 2. Materials—Analysis I. Title. II. Engler, Olaf. 620.1′1292
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CONTENTS
PREFACE GLOSSARY PART 1 FUNDAMENTAL ISSUES 1. Introduction 1.1 The classical approach to texture 1.2 The modern approach to texture: Microtexture 1.2.1 Applications of inicrotexture 1.2.2 Electron back-scatter diffraction 1.2.3 Orientation rnicroscopy and orientation mapping 1.3 A guide to the book 2. Descriptors of orientation 2.1 2.2
Introduction Transformation between coordinate systems: The rotation matrix 2.2.1 Coordinate systems 2.2.2 The rotation (orientation) matrix 2.2.3 Crystallographically-related solutions 2.3 The 'ideal orientation' (Miller or Miller-Bravais indices) notation 2.4 The reference sphere, pole figure and inverse pole figure 2.4.1 The pole figure 2.4.2 The inverse pole figure 2.5 The Euler angles and Euler space 2.5.1 The Euler angles 2.5.2 The Euler space 2.6 The anglelaxis of rotation and cylindrical anglelaxis space 2.6.1 Anglelaxis of rotation 2.6.2 The cylindrical anglelaxis space
CONTENTS
2.7
2.8
The Rodrigues vector and Rodrigues space 2.7.1 The fundamental zone 2.7.2 Properties of Rodrigues space Summation
3. Application of diffraction to texture analysis 3.1 3.2 3.3 3.4 3.5 3.6
3.7
Introduction Diffraction of radiation and Bragg's law Structure factor Laue and Debye-Scherrer method Absorption and depth of penetration Characteristics of radiations used for texture analysis 3.6.1 X-rays 3.6.2 Neutrons 3.6.3 Electrons Summation
PART 2 MACROTEXTURE ANALYSIS 4.
Macrotexture measurements Introduction Principle of pole figure measurement X-ray diffraction methods 4.3.1 Generation of X-rays 4.3.2 Pole figure diffractometry in the texture goniometer 4.3.3 Principles of pole figure scanning 4.3.4 X-ray detectors 4.3.5 Correction and normalisation of pole figure data 4.3.6 lnverse pole figures Neutron diffraction methods 4.4.1 Pole figure measurement by neutron diffraction 4.4.2 Time-of-flight (TOF) measurements Texture measurements in low symmetry and multi-phase materials 4.5.1 Peak separation 4.5.2 Multi-phase materials Sample preparation Summation
5. Evaluation and representation of macrotexture data 5.1 5.2
Introduction Pole figure and inverse pole figure 5.2.1 Normalisation and contouring of pole figures 5.2.2 Representation of orientations in the inverse pole figure
CONTENTS
Determination of the orientation distribution function (ODF) from pole figure data 5.3.1 Series expansion method 5.3.2 Direct methods 5.3.3 Comparison of series expansion and direct methods Representation and display of textures in Euler space 5.4.1 Properties of Euler space 5.4.2 Representation and display of orientations 5.4.3 Fibre plots Summation
vii
104 105 109 112 112 113 114 119 121
PART 3 MICROTEXTURE ANALYSIS
125
6. The Kikuchi diffraction pattern
127
6.1 6.2
6.3
6.4 6.5
Introduction The Kikuchi diffraction pattern 6.2.1 Formation of Kikuchi patterns 6.2.2 Comparison between Kikuchi patterns arising from different techniques 6.2.3 Projection of the Kikuchi pattern 6.2.4 Qualitative evaluation of the Kikuchi pattern Quantitative evaluation of the Kikuchi pattern 6.3.1 Principle of orientation determination 6.3.2 Automation of pattern indexing and orientation determination by electron back-scatter diffraction (EBSD) 6.3.3 Automated evaluation of electron back-scatter diffraction (EBSD) patterns 6.3.4 Automated evaluation of transmission electron microscopy (TEM) Kikuchi patterns 6.3.5 Automated evaluation of selected area channeling (SAC) patterns Pattern quality Summation
7. Scanning electron microscopy (SEM)-based techniques 7.1 7.2 7.3 7.4 7.5 7.6
Introduction Micro Kossel technique Electron channeling diffraction and selected area channeling (SAC) Evolution of electron back-scatter diffraction (EBSD) EBSD specimen preparation Experimental considerations for EBSD 7.6.1 Hardware 7.6.2 Experiment design philosophy
127 128 128 132 132 133 134 136 139 142 145 148 148 151 153 153 153 155 157 162 165
16.5 168
CONTENTS
viii
7.7
7.8 7.9
7.6.3 Resolution and operational parameters 7.6.4 Diffraction pattern enhancement Calibration of an EBSD system 7.7.1 Calibration principles 7.7.2 Calibration procedures Operation of an EBSD system and primary data output Summation
8. Transmission electron microscopy (TEM)-based techniques 8.1 Introduction 8.2 High resolution electron microscopy (HREM) 8.3 Selected area electron diffraction (SAD) 8.3.1 Selected area diffraction patterns 8.3.2 Selected area diffraction pole figures 8.4 Microdiffraction and convergent beam electron diffraction (CBED) 8.5 Summation 9. Evaluation and representation of microtexture data Introduction 9.1.1 Statistical distribution of orientation and misorientation data 9.1.2 Orientation and misorientation data related to the microstructure Representation of orientations in a pole figure or inverse pole figure 9.2.1 Individual orientations 9.2.2 Density distributions Representation of orientations in Euler space 9.3.1 Individual orientations 9.3.2 Continuous distributions 9.3.3 Statistical relevance of single grain orientation measurements Representation of orientations in Rodrigues space General representation of misorientation data 9.5.1 Representations based on the anglelaxis descriptor 9.5.2 Intra-grain misorientations Representation of misorientations in three-dimensional spaces 9.6.1 Representation of misorientations in Euler space 9.6.2 Representation of misorientations in the cylindrical anglelaxis space 9.6.3 Representation of misorientations in Rodrigues space Normalisation and evaluation of the misorientation distribution function (MODF) Misorientation distributions between phases
171 176 178 178 180 185 186 189 189 190 191 191 195 198 202
CONTENTS
9.9 Extraction of quantified data 9.10 Summation 10. Orientation microscopy and orientation mapping 10.1 Introduction 10.2 Historical evolution 10.3 Orientation microscopy 10.3.1 Mechanisms to locate sampling points 10.3.2 Specification of grid step size 10.3.3 Data storage, display and retrieval 10.4 Applications of orientation mapping 10.4.1 Spatial distribution of texture components 10.4.2 Misorientations and interfaces 10.4.3 Orientation perturbations within grains 10.4.4 True grain sizelshape distributions 10.4.5 Deformation (i.e. pattern quality) maps 10.5 Summation l l . Crystallographic analysis of interfaces, surfaces and connectivity 11.1 Introduction 11.2 Crystallographic analysis of grain boundaries 11.2.1 The coincidence site lattice 11.2.2 The interface-plane scheme 11.3 Crystallographic analysis of surfaces 11.3.1 Sectioning techniques 11.3.2 Photogrammetric techniques 11.4 Orientation connectivity and spatial distribution 11.5 Summation Synchrotron radiation, non-diffraction techniques and comparisons between methods 12.1 Introduction 12.2 Texture analysis by synchrotron radiation 12.2.1 Individual orientations from Laue patterns 12.2.2 Local textures from Debye-Scherrer patterns in polycrystalline regions 12.3 Texture analysis by non-diffraction techniques 12.3.1 Ultrasonic velocity 12.3.2 Optical methods 12.4 Summation: Comparison and assessment of the experimental methods for texture analysis
X
CONTENTS
PART 4 CASE STUDIES Case study 1: Orientation variants in tungsten wire and superplastically deformed aluminium alloy Case study 2: Crystallographic analysis of grain boundary planes in copper and nickel Case study 3: Use of orientation mapping to investigate grain boundary structure effects on grain growth Case study 4: Crack propagation in a titanium alloy Case study 5: Orientation-related effects in two minerals Case study 6 : Microtexture measurements of multi-phase materials Case study 7: In sitzr investigation of nucleation of recrystallisation in cold rolled boron doped nickel aluminide Case study 8: Deformation and local orientation evolution Case study 9: Formation of the cube recrystallisation texture Case study 10: Texture and oxidation of metals Case study 11: Microtexture determination in iron-silicon by etch pitting Case study 12: Texture and earing in aluminium sheets Appendix 1: Miller and Miller-Bravais indices Appendix 2: Crystallographically-related operations Appendix 3: The 24 crystallographically-related solutions for the four S-texture variants Appendix 4: Spherical projection and the stereographic, equal-area and gnomonic prcjections Appendix 5: X-ray counters and pulse height analysis Appendix 6: Kikuchi maps of fcc, bcc and hexagonal crystal structures References General Bibliography Index
PREFACE
Most solid state materials, including metals, ceramics and minerals: have a polycrystalline structure in that they are composed of a multitude of individual crystallites or 'grains'. This book is concerned with a specific aspect of such materials, the crystallographic orientation of its components or the crystallographic texture, or simply texture, of the polycrystalline compound. The significance of texture lies in the anisotropy of many material properties, that is, the value of this property depends on the crystallographic direction in which it is measured. In most cases grain orientations in polycrystals, whether naturally occurring or technologically fabricated, are not randomly distributed and the preference of certain orientation may indeed affect material properties by as much as 20-50% of the property value. Therefore, the determination and interpretation of texture is of fundamental importance in materials technology. Furthermore, analysis of the texture changes during the therinomechanical treatment of materials yields valuable information about the underlying mechanisms, including deformation, recrystallisation or phase transformations. In geology, texture analysis can provide insight into the geological processes that have led to rock formation millions of years ago. Nowadays there is a selection of techniques available to analyse the texture of materials. The well-established methods of X-ray or neutron diffraction, known as nzacrofextzrre techniques, are now supplemented by methods whereby individual orientations are measured in transmission or sanning electron n~icroscopesand directly related to the microstructure, which has given rise to the term microtextzrre. Microtexture practice has grown principally through the application of electron back scatter diffraction (EBSD) and it is now possible to measure orientations automatically from pre-determined coordinates in the microstructure, which is known generically as orientation mapping. From the full range of texture techniques now available, insights can be gained into materials processing, corrosion, cracking, fatigue, grain boundary properties and other phenomena with a crystallographic component. Over the past 70 years a large number of publications on texture analysis have appeared in the literature. However, there are only a few monographs on the subject area, many of which are highly specialised and with a strong focus on the mathematical aspects of texture. We have written the present book to provide a comprehensive coverage of the range of the concepts, practice and application of the techniques for determination and representation of texture. The mathematics of the subject has been kept to the minimum necessary to understand the scientific principles. For a more complete treatment a comprehensive bibliography directs the reader to more specialised texts. The text is inclined towards microtexture analysis,
xii
PREFACE
reflecting both the growing emphasis on this modern approach to texture analysis and the greater requirement for detailed explanation of the philosophy, practice and analysis associated with microtexture. The book is intended for materials scientists, physicists and geologists - both non-specialists, including students, and those with more experience - who wish to learn about the approaches to orientation measurement and interpretation, or to understand the fundamental principle on which measurements are based to gain a working understanding of the practice and applications of texture. The sequence of the book is as follows. Part 1 on Fundanzentnl Issues addresses the descriptors and terminology associated with orientations and texture and their representation in general. This Part is completed by an introduction to diffraction of radiation, since this phenomenon forms the basis of almost all texture analysis. Macrotexture Analysis, both data acquisition and representation, are covered in Part 2. Part 3 provides experimental details of SEM and TEM based techniques for Microtexture Analysis, followed by a description of how microtexture data are evaluated and represented. The innovative topics of orientation microscopy and orientation mapping are introduced and more advanced issues concerning crystallographic aspects of interfaces and connectivity are treated. The practical application of the methodology described in this book is illustrated by varied Case Studies, which comprise Part 4. We are indebted to a large number of colleagues from whom we have learned, with whom we have discussed and interacted or who have provided thoughtful comments on parts of this book. In particular we would like to acknowledge Gunter Gottstein, Jurgen Hirsch, Martin Holscher, Dorte Juul Jensen, Jerzy Jura, Fred Kocks, Kurt Liicke, Jan Pospiech, Dierk Raabe, Steve Vale, Rudy Wenk and Hasso Weiland. The contribution of OE was made possible by the support of the Center for Materials Science of the Los Alamos National Laboratory with the financial means of the US Department of Energy and by the support of the Alexander von Humboldt Foundation (Bonn, Germany). Olaf Engler Valerie Randle
GLOSSARY
Absolute orientation see crystallographic orientation. Absorption edge Drop in the absorption of X-rays of a given wavelength between certain elements with consecutive atomic numbers, caused by the efficiency of the X-rays in emitting photoelectrons from the electron shells of the sample material. Angle and axis of misorientation (Anglelaxis pair) Transformation between the crystal coordinate systems of two grains via rotation through an angle about a specific axis. Angle and axis of rotation (Anglelaxis pair) Transformation between the crystal coordinate system and the specimen coordinate system via rotation through an angle about a specific axis. Anglelaxis space Three-dimensional space wherein angleiaxis pairs reside (see also cylindrical anglelaxis space). Angular-dispersive diffractometry Conventional technique to separate different reflectors according to their reflection angle for monochromatic radiation. Area detector Two-dimensional position sensitive detector Asymmetric domain of Euler space Portion of Euler space defined such that every interior point in it represents a physically distinct orientation or misorientation. Automated crystal orientation mapping, ACOM see orientation mapping and orientation microscopy. Background Intensities in the diffraction spectra that are caused by incoherent scattering. fluorescence and reflection of X-rays with other wavelengths which do not contribute to the diffracted information. Backscatter Kikuchi diffraction, BKD see EBSD. Bragg angle Specific angle for reinforcement of reflected beams. Camera constant Distance between sample and photographic plate multiplied by electron wavelength in the TEM; measure for the magnification of diffraction patterns. ...
Xlll
xiv
GLOSSARY
Coincidence site lattice, CSL Lattice formed by notional superposition of lattice sites from two crystals and used in the context of texture analysis to categorise grain boundary geometry. Continuous scan Method to scan the pole figure continuously integrating over a range of pole figure angles. Convergent beam electron diffraction, CBED Electron diffraction occurring under irradiation with a convergent rather than parallel electron beam; allows orientation determination with best spatial and angular resolution in the subnanometre range. Crystal coordinate system Coordinate system chosen as the edges of the crystal unit cell. Crystal matrix 3 x 3 matrix to transform the crystal axes of any crystal structure system into an orthonormal frame. Crystal orientation mapping, COM see orientation mapping and orientation microscopy . Crystallographic orientation Orientation of the crystal coordinate system of a volume of crystal with respect to the specimen coordinate system. Crystallographically-related solutions (of an orientation) Equivalent representations of an orientation depending on the crystal system symmetry. Cylindrical anglelaxis space Three-dimensional orientation space wherein anglelaxis pairs reside such that the axis and angle are represented perpendicular and parallel to the long axis of a cylinder respectively. Debye-Scherrer-method Technique for orientation determination in polycrystal volumes which is based on the diffraction of n~onochromaticX-rays. Defocusing error Decrease in recorded intensity in pole figure measurements with increasing tilting angle which is caused by broadening of the reflected peak. Direct methods of ODF calculation Methods to calculate the ODF from pole figure data in real space by an iterative fitting procedure. Disorientation The lowest angle crystallographically-related solution of a misorientation. Electron back-scatter diffraction, EBSD An SEM-based technique for obtaining diffraction information from volumes of crystal typically down to about 200nm diameter. Electron channeling pattern, ECP see selected area channeling, SAC.
GLOSSARY
xv
Energy dispersive diffractometry Technique to separate different reflectors according to the energy (or wavelength) of the reflected radiation for polychromatic radiation. Equal-area projection Projection of a reference sphere onto its equatorial plane such that equal areas on the sphere retain their equality in projection. Etch pits Characteristic polygons that form via etching and whose shape is determined by the crystallographic orientation of the etched grain. Euler angles A description of orientation involving sequential rotations of the crystal coordinate system through three angles with respect to the specimen coordinate system. Euler space Three-dimensional (usually Cartesian) space wherein Euler angles are represented. Fundamental zone of Rodrigues space Volume of Rodrigues space in which the lowest-angle crystallographically-related solution of an orientation (or misorientation) resides. Geiger counter First counter used to detect X-rays in diffractometry, today virtually obsolete. Ghost error Error in ODF calculation by the series expansion method caused by the unavailability of the odd-order C-coefficients. Gnomonic projection Projection of a reference sphere in which the centre of the sphere is the projection point and a plane tangential to the 'north pole' is the projection plane. Goniometer Diffractometer to position the sample and the detector with respect to the incident beam. Harmonic method of ODF calculation see series expansion method. High resolution electron microscopy, HREM Technique for direct imaging of the atoms. Hough transform Transformation of points in an image into sinusoidal curves in an accumulation space. A line in the original image is transformed into a point characterised by the angle cp and the distance p of the line in the original image. Ideal orientation/Miller indices representation Orientation representation having the nearest Miller (or Miller-Bravais) indices of the crystal directions parallel to the specimen Z and X directions respectively.
xvi
GLOSSARY
Interface-plane scheme Methodology for characterising interfaces on the basis of the crystallographic orientation of the planar surface of the interface. Interplanar angle Angle between two crystallographic planes. Interzonal angle Angle between two crystallographic directions Inverse pole figure Angular distribution of a chosen specimen direction with respect to the crystal coordinate system. KP-filter Foil positioned between sample and detector in X-ray diffractometry and made of a material whose absorption edge lies just between the K a and the K$ peak of the X-ray tube. It is used to produce quasi-monochromatic X-radiation. Kikuchi diffraction pattern System of bright and dark lines, the Kikuchi lines, that form by the elastic (Bragg) reflection of diffusely scattered electrons at the atomic planes of a single crystalline volun~ein the TEM. Similar patterns form in the SEM by SAC and EBSD. It is very well suited for orientation determination with the highest angular resolution. Kikuchi line .see Kikuchi diffraction pattern. Laue method Technique for orientation determination in single crystal volumes which is based on the diffraction of polychromatic X-rays. iMackenzie distribution Distribution of disorientation angles and axes, where the latter are located in a stereographic unit triangle, for cubic crystals misoriented at random. Macrotexture An average texture determined from many grains sampled without reference to the location of individual grains within a specimen, or the experimental technique used to obtain this information. Mesotexture The texture between grains, i.e. the texture associated with grain boundaries. Micro Kossel diffraction An SEM-based technique for obtaining crystallographic information from volumes of crystal down to approximately 10 pm. and based on the diffraction of X-ray generated within the sample under investigation. Microdiffraction Generic term for techniques used to generate Kikuchi diffraction patterns in small volumes, with a spatial resolution beyond the limit of SAD, lpm.
-
Microtexture A sample population of orientation measurements which can be linked individually to their location within a specimen, or the experimental technique used to obtain this information.
GLOSSARY
xvii
Misorientation The orientation difference between two individual orientation measurements. usually of the same phase (see a l ~ oorientation relationship). Misorientation distribution function, MODF (MDF) Continuous density distribution of niisorientations. calculated either from discrete data or from a series expansloll method. a/lonochromator Crystal used in X-ray or neutron diffractometry to produce monochromatic radiation of a certain wavelength in accordance with Bragg's law. Normalisation of pole figures Normalisation of the pole figure intensities recorded in numbers of counts to standard units. commonly referred to as 'multiples of a random distribution' independent of the experimental parameters. Orientation see crystallographic orientation. Orientation correlation function, OCF Texture-reduced form of the MODF. derived by division of the MODF by the ODDF. Yields information about the texturereduced spatial correlation of misorientations. Orientation difference distribution function, ODDF Statistical distribution of misorientations in a sample without consideration of the actual location of the orientat i o n ~ .only governed by the texture. It is used to normalise the MODF, but does not contain valuable information on the spatial arrangement of the individual orientations. Orientation distribution function, ODF Continuous density distribution of orientation~.calculated either from discrete data or from a series expansion method. Orientation imaging microscopy, O1PATMsee orientation microscopy. Orientation mapping Depiction of the microstructure in terms of its orientation constituents ( ~ e orientation e microscopy). atrix Cosines of angles between axes of the specimen coordinate system and the crystal coordinate system. arranged to form a 3 x3 matrix. Orientation microscopy Automated measurement and storage of orientations according to a pre-defined pattern of coordinates on the sampling plane of the specimen (see orientation mapping).
idual orientation measureOrientation relationship The difference between two i n d i ~ ments. usually of different phases (see also misorientation). Orientation topography Spatial arrangement of the crystallographic orientations in the microstructure.
xviii
GLOSSARY
Orthonormal coordinate system Right-handed Cartesian coordinate system chosen as three mutually perpendicular unit vectors. Peak separation The means to separate intensities stemming from closely spaced reflectors, either experimentally or numerically. Penetration depth Depth of the specimen from where the diffracted infornlation arises. Pole Normal to a set of crystallographic planes. Pole figure Angular distribution of a chosen crystal direction (usually a set of plane normals) with respect to the specimen coordinate system. Pole figure inversion Calculation of the ODF from the data in (typically several) pole figures. Position sensitive detector, P S D Location-sensitive counter that si~nultaneously records a part of the diffraction spectrum. Proportional counter Counter to detect X-rays that is based on the potential of X-rays to ionise a gas. Pseudo Kossel diffraction An SEM-based technique for obtaining crystallographic information by Kossel diffraction, where the X-radiation is generated in a target layer of an appropriate material deposited on the sample under investigation. Pulse-height analyser Device that distinguishes between electrical pulses of different size and that is used to distinguish between X-rays of different energy, so to reduce background and fluorescent radiation. Radon transform Hough transform under consideration of the grey-values of the individual pixels. Random texture Sample population of orientations having no preferred crystallographic orientation. Reflection geometry Geometrical set-up to study diffraction effects on thick bulk samples where the incident radiation is reflected in the surface layers. Rodrigues space Three-dimensional Cartesian space wherein Rodrigues vectors are represented. Rodrigues vector, R-vector Combination of the tangent of half the angle and the axis of rotation into one mathematical entity.
GLOSSARY
xix
Rodrigues-Frank space, RF space see Rodrigues space. Rotation matrix see orientation matrix. Scintillation counter Counter to detect X-rays that is based on the potential of X-rays to ionise a solid crystal. Selected area channeling, SAC An SEM-based technique for obtaining diffraction information from volumes of crystal down to approximately 10 pm. Selected area diffraction pole figures Technique to derive the orientation distribution of smallest polycrystalline volumes in the TEM. Selected area (electron) diffraction, SAD Characteristic point patterns generated by electron diffraction in thin metallic foils, or the experimental technique to generate those patterns. This was the traditional method for orientation determination in the TEM before the advent of TEMs with microdiffraction facilities. Semiconductor counter Counter to detect X-rays that is based on the possibility of X-rays to ionise a semiconductor crystal. Series expansion method Method to calculate the ODF from pole figure data in Fourier space by fitting pole figures and ODF by a series expansion with spherical harmonic functions. Skeleton line Representation of the relative intensity maxima in the various ODFsections in the Euler space as a function of the corresponding Euler angle. Spallation source Neutron source where the neutrons are produced by spallation, i.e. the interaction of a high energy proton beam with a target material; interesting for time-of-flight (TOF) applications because of the possibility to pulse the neutron beam. Specimen coordinate system Coordinate system chosen as the geometry of the specimen. Spiral scan Method to scan the pole figure on a spiral by a coupled changing of both pole figure angles (outdated). Step scan Method to scan the pole figure in distinct pole figure angles. Stereographic projection Projection of a reference sphere onto its equatorial plane such that equal angles between lines on the surface of the sphere retain their equality in the projection. Structure factor Unit cell equivalent of the atomic scattering amplitude. Summation over the waves scattered by each of the atoms in the unit cell gives the amplitude of the wave being reflected at a given plane.
xx
GLOSSARY
Subvolume of the fundamental zone of Rodrigues space Volume obtained by subdivision of the fundamental zone such that the absolute value of the axis of rotation is given by H > K > L, where HKL are its Miller indices. Synchrotron radiation X-radiation. generated in an electron synchrotron which is characterised by a high intensity and low angular divergence and which can be used for local texture analysis by means of the Laue- and Debye-Scherrer-method. Texture Distribution of the crystallographic orientations of a given sample. Texture fibres Intensity distribution along certain pre-defined paths through the orientation space. Time-of-flight (TOF) measurements Method to separate several reflectors by determining the flight time of the neutrons which depends on their energy or wavelength (see a l ~ oenergy dispersive diffractometry). It requires a pulsed neutron source. Transmission geometry Geometrical set-up to study diffraction effects on thin samples which are penetrated by the incident radiation.
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l. INTRODUCTION
1.1 THE CLASSICAL APPROACH TO TEXTURE This book is concerned with a specific aspect of the structure of natural and technological materials the crystallographic orientation of its component units. The crystallographic orientation, or in this context simply orientation, refers to how the atomic planes in a volume of crystal are positioned relative to a fixed reference. This characteristic applies to all solids whose structure is crystalline, including minerals, ceramics, semiconductors, superconductors and metals. Almost all of these materials are polycrystalline (rather than mono- or hi-crystalline), and their component units are referred to as crystals or 'grains'. Grain orientations in polycrystals, whether naturally occurring or fabricated, are rarely randomly distributed. One of the few examples where grain orientations are randomly distributed is a polycrystalline aggregate which has been made by compression of a powder. However, in most materials there is a pattern in the orientations which are present and a propensity for the occurrence of certain orientations caused firstly during crystallisation from a melt or amorphous solid state and subsequently by further thermomechanical processes. This tendency is known as prejkrred orientation or, more concisely, texture. The importance and significance of texture to materials lies in the fact that many material properties ure texture-specific. Indeed, it has been quoted that the influence of texture on material properties is, in many cases, 20-50% of the property values (Bunge, 1987). Some examples of properties which depend on the average texture of a material are: -
e
Young's Modulus
e
Poisson's ratio
e
strength
e
ductility
e
toughness
e
magnetic permeability
4
INTRODUCTION TO TEXTURE ANALYSIS
e
electrical conductivity
e
thermal expansion (in non-cubic materials)
The effect of texture on properties is exploited in materials technology in order to produce materials with specific characteristics or behaviour. In general, the exact mechanisms by which particular textures evolve is incompletely understood, although the empirical validation is sufficient for many processes to have become established commercial practice. Three examples of texture,/property links follow; the first two examples are well established practice, whereas the third is an example of an application of texture control which is still evolving. Grain oriented silicon iron It is well known that the optimal crystallographic direction of magnetisation in Fe-3%Si steels for power transformers is <001> (Matsuo, 1989). Hence, processing of these steels is directed towards making a product with a high proportion of grains having <001> coinciding with the direction in which the product has been rolled to sheet. Even though technologically this is a vitally important process, the exact mechanisms which control the texture formation are still not completely understood. Recent research has focused on the synergy between precipitates and grain boundary types (Shimizu and Harase, 1989). Strip metal products Very large tonnages of steels and also aluminium alloys for deep drawing applications, such as car bodies or beverage (drinks) cans, are produced each year. For steels, the deep drawing process is greatly facilitated by maximising the amount of (1 11) planes which lie nearly parallel to the sheet surface before the drawing operation (Hatherly and Hutchinson, 1979). For aluminium alloys the required textures are rather more complex, and industrial processes concentrate on efficient use of material by balancing the texture effects so that a homogeneously shaped drawn product results (Hutchinson et al., 1989; Knorr et al., 1994).
The practical application of bulk polycrystalline superconductors such as YBazCu307_sdepends on obtaining a high value of transport critical current density J,. J, has been found to depend crucially on the degree of texturing in the material and, more precisely, on the type of grain boundaries which certain textures produce (Zhu et al., 1991). Manufacture of superconductors is tailored to produce textured materials.
A natural or technological material will often exhibit a con~pletelynew texture after such processes as deformation, recrystallisation and phase transformations. Characterisation of such changes can be used as an experimental probe to investigate the history of the material, especially where the changes involved are well-defined. This widely-used application of texture as a diagnostic tool constitutes the second
INTRODUCTION
5
reason why a knowledge of texture is valuable. Hence, texture research has proceeded on two broad (and related) fronts: Technological control of material processing 0
Further understanding of fundamental materials science and geological sciences
The most established method for measuring textures is by X-ray diffraction using a texture gonioineter (Section 4.3), which gives a measure of the vol-ume fraction of a particular family of planes, e.g. {l 1l ) , {0001), which are orientated for diffraction. Hence the texture that is obtained is an average value for the whole sampled volume, typically con~prisingthousands of grains. The X-ray technique for texture evaluation was first used several decades ago, and much effort has gone during the intervening years into refining it to produce a standard system to provide quantitative texture data in conventionally accepted formats, i.e. polefigures (Sections 2.4. 4.3.2, and 5.2) or orientation distribution fzinctions (ODFs) (Sections 5.3 and 5.4). This Section has established that the importance of texture lies firstly in its direct relationship with material properties and secondly as a method of fingerprinting the history of a material. The dominant approach to texture research and practice prior to the early 1980s was the use of X-rays to probe the average texture of a specimen, and indeed this approach is still valid if knowledge of the average texture is all that is required.
1.2 THE MODERN APPROACH TO TEXTURE: MICROTEXTURE A hallinark of the X-ray approach is that it provides efficiently an overview of the texture. In turn, this texture is compiled from numerous individual grains, each of which possesses a discrete orientation. The X-ray texture tells us what volume fraction of the specimen (obtained from the intensity of diffraction by particular planes) has a particular orientation. However, it does not tell us how these grains are distributed throughout the material. An approach to texture which deals with the orientation statistics of a population of individual grains, and usually encompasses also the spatial location of these grains, i.e. the orientation topogruphj~,has been termed microtexture, that is, the conjoining of microstructure and texture (Randle, 1992). To avoid ambiguity with the traditional view of texture, a texture which reflects an average value obtained from many different grains is often termed 'macrotexture' rather than just 'texture'. In the study of fabricated and natural materials, it is frequently important to have a knowledge of both microstructure and crystallography. The traditional approach has been to make parallel, but separate, investigations: optical microscopy, scanning electron inicroscopy (SEM) (plus chemical analysis) and image analysis to analyse microstructure; X-ray texture determination or diffraction in the transmission electron microscope (TEM) for crystallographic analysis, with some contribution from selected area channeling in the scanning electron microscope (SEM). A typical profile of a material might therefore include the grain size distribution plus the texture as determined by X-rays.
6
INTRODUCTION TO TEXTURE ANALYSIS
The principal disadvantage of the traditional approach has been the lack of a direct connection between the study of microstructure and crystallography. Until a few years ago this meant that although the overall average orientation distribution could be measured. the orientation of individual crystals could only be extracted from this database on a small scale and by tedious means, mostly involving TEM. Hence a wealth of information concerning the spatial component of grain orientations, including the interface regions where grains join, was inaccessible. Furthermore, where phases having identical or similar chemistry, but different crystallography, coexist in the microstructure they could not be distinguished by any simple method. However, with the advent since the early 1980s of innovative techniques in both the SEM and TEM. it is now possible to obtain microtexture measurements in a routine manner. Several recent conferences, or parts of conferences, have been dedicated specifically to microtexture (see General Bibliography).
1.2.1 Applications of microtexture If the spatial location plus the orientation of individual grains in a sample population is known. access to a whole new stratum of knowledge is heralded, since the crystallographic and morphological aspects of structure evolution become fused. The following phenomena, which are all directly linked to microtexture (i.e. individual grain orientation), can be explored: 0
Local property effects
s Interfacial parameters and properties e
Morphological/geometrical grain parameters, i.e. size, shape, location
s Orientation variations within individual grains s Phase relationships
Direct ODF measurement The importance of the way in which grain orientations are distributed can be illustrated by the idealised sketch of a microstructure in Figure 1.l. Here a simplified case is considered where some grains having a strong preferred orie~tationX (with a spread of, say, 1.5") and the remaining grains have a 'random' (i.e. no discern~ble preferred orientation) texture. Two limiting spatial conditions are considered: in Figure l . l a all the X grains are located in a contiguous patch and in Figure l . l b the same number of X grains are spread singly throughout the microstructure. The occurrence of contiguous X grains will affect several aspects of the overall properties, and are illustrated in the remainder of this Section.
Local property effects The effect of property variations with orientation will be magnified where grains occur in clusters rather than singly. For example, slip transmission may be facilitated between grains of favourable orientations, e.g. the contiguous X grains in Figure
INTRODUCTION
7
l . la. Another example is that compressed plates of tantalum take up an 'hourglass' shape due to the occurrence of two different microtextures, which in turn have different deformation characteristics, at the specimen surface and centre (Wright et al., 1994).
Interfacial parameters and properties If individual orientations are known it is possible to calculate the orientation difference or niisorientation between neighbouring grains and hence provide information about the distribution of grain boundary geometry, which is sometimes called mesotexture, i.e. the texture between grains (Chapter 11). This is a very powerful aspect of microtexture since it allows access to grain boundary structure/property links, which is unavailable via a macrotexture approach. In recent years it has been realised that the properties of grain boundaries vary markedly with their structure, and moreover the structure of boundaries can be manipulated to produce boundaries which perform better in service. This subtopic was originally termed grain boundary design (Watanabe, 1984) or, more recently, grain boundar). engineering. With regard to the simple example in Figure 1 . l , the misorientations between grains having orientations X will be low-angle since the orientations are similar, and misorientations between X and other grains and also between non-X grains will be mainly high-angle. Hence the low-angle boundaries, which have different physical properties to those which are high-angle, are spatially connected. This grouping will have a major effect on intergranular transport properties in the material, since some pathways in the grain boundary network will effectively be blocked.
Morphological/geometrical grain parameters The morphological/geometrical characteristics of grains, i.e. grain size, shape and macroscopic location, cannot be accessed by macrotexture measurements. A microtexture approach, however, allows concurrent access to both geometrical and orientational parameters which in turn permits correlations to be made between the two. Referring again to Figure 1.1 the X grains may belong, for example, to a particular size class or be located in a specific way with respect to the specimen geometry, e.g. adjacent to the specimen surface. as shown in Figures l . l c and d respectively. A grain or crystal is formally defined as a unit having a single orientation (but see also below). A direct consequence of this definition is that the grains themselves can be depicted by knowing where in the microstructure orientation changes take place. This leads to 'mapping' of microtexture (Section 1.2.3 and Chapter 10).
Orientation variations within individ~mlgrains
By classical definition a grain has a single, distinct orientation. However in practice the orientation within a grain may vary in a continuous or discontinuous manner, which can be termed orientation perturbation (Randle et al., 1996). In particular there may be perturbations close to interfaces and in deformed structures reflecting the
8
INTRODUCTION TO TEXTURE ANALYSIS
distribution of strain. A microtexture technique is able to characterise these distributions on a local level, which is not a capability of a macrotexture technique.
So far u7ehave considered that the hypothetical grains in Figure 1.1 are all the same phase. In exactly the same manner, multi-phase materials can also be studied to produce concurrently the texture of each phase, the distribution of its orientations and orientation relationships between individual crystals of each phase. For example. coupled growth phenomena in white irons, which contain phases M7C3
Figure 1.1 (caption overleaf)
INTRODUCTION
Figure 1.1 Idealised portrayal of microtext~lreor orientation topography. Grains having orientation X are (a) contiguous in the microstructure: (b) located singly in the microstructure: (c) in a different size class to the rest of the microstructure: (d) located in a special way with respect to the specimen. e.g. near the surface.
and M3C (where M is mostly chromium) have been investigated (Randle and Laird, 1993). Direct ODF measztrenrenz The traditional route for ODF measurement relies on calculation from several pole figures which have been determined by macrotexture methods (Chapter 5). These
10
INTRODUCTION TO TEXTURE ANALYSIS
procedures admit inherent inaccuracies which are obviated when a microtexture technique is used to obtain the orientation distribution. For this case full grain orientations are obtained directly rather than calculated from crystal plane distributions as in the X-ray case. Hence ODFs obtained by microtexture measurements give the true orientation distribution of the sampled grains. Although the schematic illustration in Figure 1.1 might seem rather hypothetical, in practice such microstructures have been observed - e.g. where clusters of similarly oriented grains aggregate into 'supergrains' (Mason and Adams, 1994), a superplastically deformed aluminium alloy (Randle, 1995a), and in various sheet metals and ceramics (Fionova et al., 1993) - and are a focus of research interest. More generally, microtextures may consist of several preferred orientations rather than only one as depicted in Figure 1 . l . However, the preferred orientations tend to be spatially distributed non-randomly which gives rises to anisotropic distributions of grain boundaries and grain properties as described above, and demonstrates the cruciality of obtaining microtexture information in order to address these complexities. In summary, this Section has introduced the concept of microtexture, which is a recent approach to orientation-related analyses. Microtexture focusses texture analysis onto a local basis and allows it to be correlated with the microstructure. This, in turn, allows whole facets of the microstructure to be explored which were inaccessible by macrotexture methods, and has indeed enlarged the view of microstructure to synthesise both the morphological and crystallographic aspects. 1.2.2
Electron back-scatter diffraction
The ability to obtain microstructure-level information implies that the probe size formed by the exploring radiation (i.e. X-rays, neutrons or electrons, Chapter 3: Table 3.1) must be smaller than the size of the microstructural units themselves. Almost always, then, this rules out X-ray diffraction as an experimental tool for microtexture measurement. Electrons are ideal for combined microstructuralj crystallographic studies and indeed until the 1980s TEM was the major technique used for such work (Chapter 8), with some input from selected area channeling in an SEM (Section 7.3). Since that time a more convenient §EM-based technique for microtexture has been developed, known as electron back-scatter difjraction, EBSD (Sections 7.4-7.8). EBSD is now the backbone of most microtexture research. EBSD or, as it is equivalently known, back-scatter Kikuchi diffraction, BKD, is an add-on package to an SEM. The most attractive feature of EBSD is its unique capability to perform concurrently rapid, (usually) automatic diffraction analysis to give crystallographic data and imaging with a spatial resolution of less than 0.5 pm, combined with the regular capabilities of an SEM such as capacity for large specimens, option of chemical analysis, and the ability to image rough surfaces (Venables and Harland, 1973; Dingley, 1984). EBSD is without doubt the most suitable and widely used experimental technique for the determination of microtexture. For this reason, and moreover because EBSD with all its latest ramifications is still a relatively new technique compared to X-ray
INTRODUCTION
11
methods, it is described in detail in Chapter 7 of this book. Briefly, the technique relies on positioning the specimen within the SEM sample chamber such that a small angle, typically 20°, is made between the incident electron beam and the specimen surface. This simple expedient enhances the proportion of backscattered electrons able to undergo diffraction and escape from the specimen surface. The resulting diffraction pattern can be captured and interrogated in real time, and computer algorithms allow the orientation of each diffraction pattern to be obtained and stored, from which raw data a microtexture is constructed serially. At the time of writing, a state-of-the-art EBSD system can position a stationary probe on the specimen, capture a diffraction pattern, index it and store the result at the rate of 5 s-' without any operator intervention. It is not an exaggeration to say that. since its inception, EBSD has revolutionised texture investigations both in research and industry. The technique has passed through various stages of development, and in its most refined form provides the tool for a totally synthesised approach to orientation and microstructure.
1.2.3 Orientation microscopy and orientation mapping There are two modes of operation of an EBSD system: ~ z a ~ u aand l automatic. Manual functioning involves operator interaction with the system during data collection to select locations on the specimen from which orientation measurements are to be made. On the other hand the automated EBSD mode permits sampling locations to be pre-programmed and located via beam or stage control in the microscope. The most sophisticated way in which automated EBSD is exploited is to select a grid of sampling points whose spacing is much finer than the grain size. If a representation of the orientation at each grid point is plotted, a crystallographic map of the microstructure is obtained. This is known variously as crystal orientatiorz ) et al., 1993). mapping (COM) or orientation irmgirzg microscopj. ( 0 1 ~ ~ "(Adams The terms 'automated crystal orientation mapping' (ACOM), 'orientation scanning microscopy' and 'crystal orientation imaging' are other versions which have been adopted. In this book the term orientation microscopy will be used to described the procedures involved in getting the data and or.irntation mapping will be used to describe the output of data (Chapter 10). Orientation microscopy/mapping is a landmark innovation in the field of EBSD and microtexture because it links orientation directly to the microstructure. In other words the crystal orientation map portrays faithfully the orientational components, i.e. those concerned with crystallography, of the microstructure in an analogous manner to the mapping of chemical elements in the microstructure via energy dispersive spectroscopy in an SEM. As the potential and capabilities of orientation mapping are considered in greater depth, it emerges that there are many subtle ramifications to its application, which introduce a degree of quantification into the definition of microstructure which was hitherto not possible. An orientation map can highlight different aspects of the microstructure, depending on the distribution/ step size of the sampling schedule plus the scale and requirements of the inquiry, as specified by the user.
12
INTRODUCTION TO TEXTURE ANALYSIS
1.3 A GUIDE TO THE BOOK This book is intended for the reader who is familiar with crystalline materials and wishes to gain a working understanding of the practice and applications of texture. The following prior knowledge: at an introductory level. is assumed: Basic crystallography (crystal planes and directions) e
Principles of electron microscopy (TEM and SEM)
e
Familiarity with vectors and matrix algebra
Several standard textbooks cover different aspects of this information (see General Bibliography). In the book the mathematics of the subject area has been kept to that required to understand the scientific principles. Many of the examples relate to metallic materials, especially those having a cubic crystal structure. because this group is the largest to be affected by texture-associated phenomena. It is, however, recognised that semiconductors, superconductors and geological materials are growing areas of interest, particularly where microtexture is concerned (e.g. Prior et ul., 1996). The sequence of the book is as follows. We begin in Chapter 2 by addressing the fundamental descriptors and ternlino10,gy associated with orientations and their representation in general. Chapter 3 is an introduction to diffraction of radiation, since this phenomenon forms the basis of almost all texture analysis. This completes Part 1; which is on Fztnduniental Issues. Part 2, i.e. Chapters 4 and 5. describes Macrotextztre Anu1~si.s. both data acquisition (Chapter 4) and data representation (Chapter 5). This Part is shorter than the following Part on Microtexture Anulysis, reflecting both the growing emphasis on the modern microtexture approach to texture analysis and the greater requirement for detailed explanation of the philosophy, practice and analysis associated with microtexture. These are all dealt with in Part 3. evaluation of the basic 'raw data' of microtexture. the Kikuchi diffraction pattern, need to be covered in some detail. and this is accomplished in Chapter 6. Then Chapters 7 and 8 describe experimental details of SEM and TEM based techniques respectively, followed in Chapter 9 by how the statistics of microtexture data are analysed and represented. Orientation microscopy and mapping is an innovative topic, and Chapter 10 is devoted to a treatise on this new subset of microtexture. Chapter 11 is another spin-off of the microtexture approach, and describes more advanced crystallographic issues concerning interfaces and connectivity. The final Chapter in Part 3 (Chapter 12) covers minority techniques for nicrotesture evaluation and comparisons between all the techniques for texture measurement. The practical application of the methodology described in this book is illustrated by varied Case Studies, which comprise Part 4. All the Case Studies have been extracted from items available in the literature, to which the reader is referred for more information.
2. DESCRIPTORS OF ORIENTATION
2.1 INTRODUCTION In this Chapter the main mathematical parameters which are used to describe an orientation are defined and explained. These are: e
Rotation or orientation matrix (Section 2.2)
e
'Ideal orientation', i.e. Miller or Miller-Bravais indices (Section 2.3)
e
Euler angles (Section 2.5)
e
Anglelaxis of rotation (Section 2.6)
e
Rodrigues vector (Section 2.7)
All these descriptors are employed to process and represent different aspects of macrotexture and microtexture measurements. This Chapter will elucidate the theoretical aspects of these parameters and their interrelationships. For graphical representation it is usual to plot orientations, represented as projected poles, Euler angles, anglelaxis or Rodrigues vector, in an appropriate 'space'. These spaces, namely pole figurelinverse pole figure 'projected space', Euler space, cylindrical anglejaxis space and Rodrigues space are explained in Sections 2.4-2.7. These explanations encompass only the basic theoretical aspects - in practice their utilisation is customised according to if they are employed to display and analyse macrotexture or microtexture data. Therefore, these two divisions are discussed separately in Chapters 5 and 9 respectively. The term crystallographic orientation can refer either to a crystal (grain), that is, it is three-dimensional and therefore comprises three independent variables, or to a crystallographic plane which comprises two independent variables. Where the term orierztation is used throughout this book, it refers to the orientation of the full crystal, rather than of a plane, unless it is made clear otherwise.
14
INTRODUCTION TO TEXTURE ANALYSIS
2.2 TRANSFORMATION BETWEEN COORDINATE SYSTEMS: THE ROTATION MATRIX 2.2.1 Coordinate systems In order to specify an orientation, it is necessary to set up terms of reference, each of which is known as a coordinate systen? (Bollmann, 1970; McKie and McKie, 1974; Hansen et al., 1978; Bunge, 1985a). Two are required: one relating to the whole specimen and the other relating to the crystal. Both systems are Cartesian (i.e. having axes at right angles) and preferably right-handed. The axes of the sample or specimen coordir~atesj,stem S = {sis2s3)are chosen according to important surfaces or directions associated with the external form or shape of the specimen. For example, for a fabricated material there are obvious choices defined by the processing geometry. One of the most common of these relates to a rolled product, and hence the directions associated with the external shape are the rolling direction (RD), the through-thickness direction, i.e. the direction normal to the rolling plane (ND) and the transverse direction (TD). These directions are illustrated on Figure 2.1. Other specimens, such as a tensile test piece. a rod or a wire have only uniaxial symmetry and hence it is only necessary to specify one axis in the specimen coordinate system and the other two axes can be chosen arbitrarily. In natural rocks the plane of foliation and a line of lineation within that plane often make a natural choice for specimen axes. Sometimes there are no clearly defined directions associated with the specimen. For these cases the specimen coordinate
Z Normal Direction [OOll
Transverse Direction
lr
X Rolling Direction Figure 2.1 Relationship between the specimen coordinate system XYZ (or RD, TD. N D for a rolled product) and the crjstal coordinate system [100]. [010]; [001] where the (cubic) unit cell of one crystal in the specimen is depicted. The cosines of the angles a,,J,,7,give the first row of the orientation matrix (see text) (Courtesy of K. Dicks).
DESCRIPTORS OF ORIENTATION
Figure 2.2 Orthonormalised crystal coordinate sqsterns for (a) cubic; (b) hexagonal and (c) general (triclinic) symmetries.
system can be chosen arbitrarily and it is colwentional in experimental work for the normal to the 'principal' specimen surface. i.e. that from which diffraction information is collected, to be labelled Z as shown on Figure 2. l . The X and Y axes are both perpendicular to Z and should form a right handed set. The second coordinate system, the crystal coordinate system C = {c1c2c3} is specified by directions in the crystal. The choice of directions is in principle arbitrary, although it is convenient to adapt it to the crystal symmetry. Hence, for example, for orthogonal symmetry (cubic, tetragonal, orthorhombic) the axes [loo], (0101, [001] already form an orthogonal frame i.e. they are mutually perpendicular and are adopted as the crystal coordinate system (Figure 2.2a). For hexagonal and trigonal -
-
16
INTRODUCTION TO TEXTURE ANALYSIS
symmetry an orthogonal frame needs to be associated with the crystal axes. As indicated on Figure 2.2b and A l . l b (Appendix l ) the two obvious choices are Y1 = [T~To], 2 = [0001]
(2. l a )
z= [oooi]
(2. l b )
XI
=
[10iO],
X,
=
[2110], Y,
or =
[oiio],
Subsequently the crystal axes are made orthonormal, i.e. normalised to be all the same length, which is shown on Figures 2.2b and c. This procedure is carried out, in the general case, by premultiplying a zone axis, referenced to the crystal coordinate system, by a matrix L having the following elements (Young and Lytton, 1972): Ill = a 112= b COS y = ccos ,8 l,l = 0 l,, = b sin y = COS a =0 =0
-
cos P cosy)/ sin -y
/33= c ( 1 + 2 c o s a c o s ~ c o -s ~c o ~ ~ a - c o ~ ~ ~ - c o ~ ~ ~ ) ~ ' ~ / s i n y are the interzonal angles. In this way all where a, h, c are lattice parameters and a , crystal systems have the same form of crystal coordinate system, i.e. orthonormal. The general matrix given in equation (2.2) can be simplified in all but the triclinic case. For example, for hexagonal crystals the transformation matrix L becomes
Conversely, to transform from orthonormal coordinates back to the crystal reference system, it is necessary to premultiply by the inverse crystal matrix L-l. Directions and planes in hexagonal or trigonal crystals can be described using either Miller or Miller-Bravais indices (Appendix 1).
2.2.2 The rotation (orientation) matrix Having specified the specimen and crystal coordinate systems, an orientation is then defined as 'the position of tlze crystal coor.dinate system with respect to the specimen coordinate system', i.e.
DESCRIPTORS OF ORIENTATION
17
where Cc and CS are the crystal and specimen coordinate systems respectively and g is the orientation. g can be expressed in several different ways, and these are listed in Section 2.1. The fundamental means for expressing g is the rotation or orientation matrix, which embodies the rotation of the specimen coordinates onto the crystal coordinates. The orientation matrix is a square matrix of nine numbers and is obtained as follows. The first row of the matrix is given by the cosines of the angles between the first crystal axis, [loo], and each of the three specimen axes, X, Y, Z, in turn. These three angles, a1, PI, 71, are labelled on Figure 2.1. The second row of the orientation matrix is given by the cosines of the angles C Yp2, ~ , 79 between [OlO] and X, Y, Z in turn. Similarly, the third row of the matrix comprises the cosines of the angles cu3, pj, 7 3 between [OOl] and X, Y, Z. Hence the complete matrix is cosal cos pi cos% c o s a cosp: cos-2) cosa3 c0sp3 cos 7 3
=
g11 g12 g13 (g21 g22 g13) g31 g32 g33
=
(
0.768 0.384 -0.512
-0.582 0.753 -0.308
0.267 0.535 0.802 (2.5)
The orientation matrix allows a crystal direction to be expressed in terms of the specimen direction to which it is parallel, and vice versa. Equation (2.5) shows the orientation matrix in terms of the direction cosines, elements and a numerical example. Both the rows and columns of the matrix are unit vectors, that is, the matrix is orthonormal and the inverse of the matrix is equal to its transpose. Since a crystal orientation needs only three independent variables to specify it (Section 2.1), it is clear that the matrix, having nine numbers, contains non-independent elements. In fact the cross product of any two rows (or columns) gives the third and for any row or column the sum of the squares of the three elements is equal to unity. These properties can be checked using the example in equation (2.5). The pertinence of the orientation matrix is that it is the mathematical tool for calculation of all the other descriptors of orientation, as shown on Figure 2.3.
2.2.3 Crystallographically-related solutions We have stated that the crystal coordinate system and specimen coordinate system are related by the orientation (rotation) matrix. However, specification of both these coordinate systems is not usually unique, and a number of solutions can exist depending on the symmetry of both the crystal and the specimen. To consider the crystal symmetry, there are 24 different ways in which a crystal with cubic symmetry can be arranged. Consequently, there are 24 solutions for an orientation matrix of a material having cubic symmetry. The 24 crystallographically-related solutions for the example matrix given in equation (2.5) are listed in Table 2.1, and it can be seen that the rows of the matrix, and the signs of the elements, interchange. The full set of solutions are obtained by premultiplying the orientation matrix by each of 24 'symmetry matrices', which are listed in Appendix 2, in turn. These matrices describe the symmetry operations - 2 rotations of 120" about each of the four < I l l > , 3
INTRODUCTION TO TEXTURE ANALYSIS
18
Figure 2.3 Relationship between the orientation matrix and the most commonly used orientation descriptors. Table 2.1 24 crystallographically-related solutions for (123)[634] Ideal orientation
Matrix
(123)[63T]
0.768 -0.582 0.384 0 753 -0.512 -0.308
0 267 0.535 0.802
-0.512 0.354 -0.765
-0 308 0.753
120.90
0.582
0.802 0.535 -0.267
-0.768 0.384 0.512
0.582 0.753 0.308
-0.267 0.535 -0.802
155.3: 0 271
0.933
0.237
0.512 0.308 -0.802 0.384 0.753 0.535 0.768 -0.582 0.267
74.6' 0.579
0.814
-0 039
wip361
(i23)[634]
(121)[436]
(13?)[6a]
(171)[634]
(132)[643]
(213)[3a]
Angle, axis pair 48 6' 0.563
-0.028
Rodrigues vector
-0.520
-0 644
-0.915
-0.403
0.768 -0.552 -0 512 -0.308 -0.384 -0.753
0.267 0.802 -0.535
122 5" 0.922
0.768 -0.582 -0.384 -0.753 0.512 0.308
0.267 -0.535 -0.802
153.3: -0.937
0.272
-0.220
0.768 -0.582 0.512 0.308 0.384 0.753
0.267 -0.802 0.535
72.2" -0.816
0.061
-0.574
-0 567
0.824
0.384 -0.768 -0.512
0 753 0.535 0.582 -0.267 -0.308 0.802
67.1" 0.022
Euler angles
0.254
-0.235
-0 291
301.0
36.7
26.6
-0.048
-1.613
-0.711
232 9
105.5
56 3
1.239
4 263
1.081
121 0
143.3 333.4
0 441
0.620
-0 030
52.9
74.5 303.7
-0.075
333.0
122.3
-3.944
1.146 -0.925
121.0
143.3 183.4
-0 595
0.045 -0.419
153.0
57 7
161.6
301 0
36.7
116.6
1.679 -0.704
18.4
-0 386 -0.041
0.015 -0.378
0.550
DESCRIPTORS OF ORIENTATION
19
Table 2.1 (continued) (i53)[m]-0 768
(213)[363]
(231)[346]
(312)[463]
- -
-
(231)[346]
(n2)[463]
-0384 -0.512
0.552 -0.753 -0.305
-0.267 -0.535 0.802
149.3' -0.222
-0.384 0.768 -0.512
-0.753 -0 582 -0.308
-0.535 0.267 0.802
I25 6" 0.354
0.014
0 384 0.753 -0.512 -0 308 0.768 -0.582
0.535 0.802 0 267
109 2" 0732
0 124
-0.512 -0.30X 0.768 -0.582 0 384 0.753
0.802 0.267 0.535
141.2' -0.388
-0.334
-0.859
-0 753 -0.535 -0.308 0 802 0.582 -0.267
168.4" 0.545
-0.582
-0.600
71.7-0.537
0.625
0.567
0.512 0.308 -0 802 0.768 -0.582 0.267 -0.384 -0753 -0.535
143.3' 0.854
0.350
-0.385
0.384 0.512 -0.768
106.7" -0 722 -0 6x0
-0.384 -0 512 -0 768 0.512 -0.768 0.384
0.308 0.582 0.757
-0802 -0.267 0.535
-0 807 -0.240
-0.872
3.440
301 0
36 7 206.4
301.0
36 7 296.6
0945
0.688
0 027
1.030
0.174
-1.820
-0.935 0.942
52.9
74.5
33.7
57.7
71.6
0.470 -0.948
-2442
153.0
5.413 -5.748
-5.920
232.9
105.5 326.3
153.0
57.7 251.6
-1.102
-0.388
0.451
0.409
2.577
I055
-1.163
333.0
122.3 288.4
-0.914
0.169
232.9
105.5 146.3
52.9
74 5 213.7
333 0
122.3 108.4
-~
(312)[463]
(23)[34$]
(?31)[346]
(31?)[4631
(213)[364]
0.753 0.535 0.308 -0.802 0 582 -0.267 -0.753 0.308 -0.582
-0.535 -0.802 0.267
113 9" -0.120
-0.512 -0 308 -0.768 0.582 -0 384 -0.753
0.802
-0 267 -0.535
137.10 0.357
-0.871
178.6" -0.832
-0.457
-0.384 0.512 0 768
0.354 0.753 0.768 -0.582 0.512 0.308
0 535 0.267
0.582 -0.308 0.753
-0.267 0.802 0 535
140.4' 0 038
-0.512 -0.384 0.768
-0 308 -0.753 -0.582
0.802 -0.535 0.267
177.3' 0.493
-0.384 -0.768 0.512
-0 753 -0.535 0.582 -0.267 0.308 -0.802
(m)[<43]-0 768
0.582 0.512 0 308 -0.384 -0.753
-0.267 -0.802 -0.535
175.9' -0.339
0.512 0.308 -0.384 -0.753 -0 768 0.582
-0.802 -0.535 -0.267
138.9' -0.880
(321)[436]
(?13)[%4]
(m)[436]
-0.185
1.094 -1.062
0 7 1 2 -0.692 0.907
-2.216
0.860
0.338 -67 953
-37 363 -25.710
121 0 143.3
63.4
-0.315
-0.802
-0 768 -0.512 0.384
(T32)[G3]
-0.971 0.126
143.3' -0.482
0.107 0.511
20.621 -0.351
0.876
1.421
2.387
153 0
33.260
52.9
57.7 341.6
0.859 -14.661
74.5
123.7
0 796
-1 453
2 644
0.039
121.0 143.3 243.4
-9.362
-22.343
13.305
333.0
122.3 198.4
-2269
0068
1.407
2329
105.5 236.3
0 013
-0 808 0.481
0.026 0 527
INTRODUCTION TO TEXTURE ANALYSIS
rotations of 90" about each of the three < 100> and one rotation of 180" about each of the six < l 10>, plus the identity matrix. Crystal systems other than cubic have fewer crystallographically-related solutions (Schmidt and Olesen, 1989); for example the twelve and four solutions for hexagonal and orthorhombic crystals are also listed in Appendix 2. It is important to stress that each of the matrices in Table 2.1 describes the same orientation and so in general any one can be chosen to represent the orientation. However, depending on what method is chosen to represent an orientation (i.e. Miller indices, anglelaxis pair, Rodrigues vector, Euler angles or misorientation, as described in the remainder of this Chapter) the selection of a particular crystallographically-related solution may both facilitate representation and allow insight into the physical meaning of the data. These points are expanded in Sections 2.6, 9.6.1 and 11.2.
2.3 THE 'IDEAL ORIENTATION' (MILLER O R MILLER-BRAVAIS INDICES) NOTATION A practical way to denote an orientation is via the Miller indices. Formally, the direction cosines in both the first column (the specimen X direction expressed in crystal coordinates) and the last column (the specimen Z direction expressed in crystal coordinates) of the orientation matrix are multiplied by a suitable factor to bring them to whole numbers, then divided by their lowest common denominator and conventionally written as (lzkl)[uvw], or {hkl)
if non-specific indices are quoted, for example (l 12)[11T] and { l 12)<11 l > . These are the Miller indices. Hence for the example in equation (2.5) the ideal orientation is (123)[634]. It should be remembered that it is generally only for cubic materials that the indices for a plane and a direction are identical and hence it is less straightforward for non-cubic symmetries to extract the ideal orientation from the rotation matrix (Partridge, 1969). In practice, the direction cosines from the orientation matrix are 'idealised' to the nearest low-index Miller indices. For example the following orientation is expressed both in direction cosines and the nearest family of Miller indices and is 2" and 6" away from (hkl) and [uv~']respectively:
This shorthand method of representation is very popular in metallurgy. Most usually single digit Miller indices are used to formulate the ideal orientation, although the components of the direction cosines could also simply be multiplied by one hundred or one thousand. An ideal orientation is also commonly formulated for hexagonal metals, for example (1~10)[10T0]. The crystallographically-related solutions for an orientation feed through to the ideal orientation notation, as shown in the 24 equivalent solutions for (123)[634] listed in Table 2.1. However, there are also orientations which, when denoted in the ideal orientation notation, can appear confusingly similar. For example the
DESCRIPTORS OF ORIENTATION
21
orientation (123)[634] is not a crystallographically-related solution of (123)[634] and it does not therefore appear in the list in Table 2.1. Another way to express this is that the matrix associated with (123)[634] is
which cannot be transformed into the matrices in Table 2.1 by column swapping, i.e. multiplying by the matrices in Appendix 2. In the cubic system there are, for the most general case, four different orientations which have the same Miller indices families, (hkl). These are (Izkl)[zw@] (hkl) [mw] (khl) [vu@] (khl)[FEW] Appendix 3 lists all 24 crystallographically-related solutions for each of these, using (123)[634] (which is sometimes called the S-orientation) as an example, illustrating that there is no duplication among the 24 X 4 solutions (Davies and Randle, 1996). These four different sets of solutions are related through symmetries of the specimen coordinate system (Sections 2.5.2, 5.4.2 and 9.4). Case Study No. 1 demonstrates how these variants can be recognised in practice using microtexture techniques. The main advantage of the ideal orientation notation is that it highlights important planes and directions which are parallel to principal directions in the specimen. These are often of great practical importance in texture analysis (Hatherly and Hutchinson, 1979). Furthermore, for cubic materials it is usually possible to extract the ideal orientation 'by eye' from the orientation matrix and it is easy to visualise the particular orientation. For example, Figure 2.4 is a schematic diagram of the (011)<100> orientation which is called the Goss orientation and is very important for power transformer technology (see Section 1.1). Here (01 I) lies in the plane of the sheet. i.e. the direction normal to this plane is parallel to ND, and
Figure 2.4 Schematic illustration of the relationship between the crystal and specimen axes for the j110) <001> orientation, i.e. the normal to j110) is parallel to the specimen ND, or 2, axis and <001> is parallel to the specimen RD, or X, axis.
22
INTRODUCTION TO TEXTURE ANALYSIS
<100> lies parallel to RD. For hexagonal materials the relationship between the ideal orientation expressed as Miller-Bravais indices and the orientation matrix is more complex (Section 2.2.1). An example is
2.4 THE REFERENCE SPHERE, POLE FIGURE AND INVERSE POLE FIGURE The direction of any three-dimensional vector in a crystal, i.e. a crystallographic direction or the normal to a crystal plane, can be described as a point on the unit reference sphere, that is a sphere with radius 1 notionally residing around the crystal (Appendix 4). As an example, Figure 2.5 shows the (0001) plane in a hexagonal crystal. The point of intersection of the normal to this plane, i.e. its pole, (0001), with the reference sphere is a measure for the arrangement of this plane in the crystal. Provided that the reference unit sphere is attached to an external frame, the position of the pole on the sphere also provides information on the crystallographic orientation of the crystal with respect to this frame, although the crystal has still one degree of freedom by rotating around this particular axis. Representation of the three-dimensional orientation information on the unit sphere in a two-dimensional plane requires a projection of the sphere onto that plane. In crystallography and metallurgy, most commonly the stereographic projectiorz is used,
north pole
reference direction
Figure 2.5 Orientation of the basal plane (0001) in a hexagonal crystal. The position of the (0001) pole o n the unit sphere with regard to an external reference frame is described by the two angles cr and 3. However, since the crystal can still rotate about the (0001) pole. for an unequivocal definition of the orientation of the crystal more information, here the position of the (1070) pole; is required.
DESCRIPTORS OF ORIENTATION
Figure 2.6 Presentation of the 1100) poles of a cubic crystal in the stereographic projection. (a) crystal in the unit sphere; (b) prosection of the j100) poles onto the equatorial plane; (c) j100) pole figure and definition of the pole figure angles a and 9 for the (100) pole.
the principle of which is shown in Figure 2.6 and explained in more detail in Appendix 4. In geology the eqtlul-areaprojectiot?,which is also explained in Appendix 4, tends to be more popular. Derivatives of the stereographic or equal area projection (pole figures and inverse pole figures, see below) are discussed further with respect to macrotexture and microtexture in Sections 5.2 and 9.2 respectively. 2.4.1 The pole figure Poles are projected from the reference sphere onto a pole Jgzrre as follows. The position of a given pole on the sphere is commonly characterised in terms of two angles (e.g. Hansen et al., 1978). The angle ol describes the azimuth of the pole, where
24
INTRODUCTION TO TEXTURE ANALYSIS
a = 0" is the north pole of the unit sphere. The angle P characterises the rotation of the pole around the polar axis, starting from a specified reference direction (Figure 2.5). To characterise the crystallographic orientation of the crystal under investigation, the spatial arrangement of the corresponding poles in terms of the angles a and P has to be determined with respect to an external reference frame, i.e. the sample or specimen coordinate system S (Section 2.2.1). For instance, for rolling symmetry the sheet normal direction N D is typically chosen to be in the north pole of the sphere, so that a = 0" for ND, and the rotation angle P is 0" for the rolling direction RD or, less frequently, the transverse direction TD. For other deformation modes or specimen geometries an appropriate three-dimensional, preferably right-handed, coordinate system has to be established (Section 2.2.1). Thus, the reference system of the pole figure is the specimen coordinate system S given by the specimen axes {s1s2s3},and the crystal orientation, given by the axes of ), into this frame. If R is a the crystal coordinate system C = { c ~ c ~ cis~ projected vector parallel to the pole of interest (XYZ), then it can be expressed in the two frames S and C according to:
and
(XYZ) are the coordinates of the pole in the crystal frame, e.g. (1 1 l), and N is a constant with N = J(x2 Y2 Z 2 ) to normalise R to unity. Scalar multiplication of equations 2.10a and 2. lob in succession by the three vectors S, under consideration of the definition of an orientation (equation 2.4) yields:
+ +
[
sin a cos p
1
=
[
g11 g21 g31 g12 g22 g32 g13 g23 g33
1( '
(2.11)
(note that gT = g-l appears here). Equation 2.1 1 yields 9 equivalent expressions to derive the pole figure angles a and for a given pole (XYZ) from the orientation matrix g (Hansen et al., 1978). So far, only individual poles have been considered. However, as already mentioned above, one pole does not yield the entire orientation information, since the crystal can still rotate about this particular pole. For this reason, the distribution of the c-axes in a hexagonal material, i.e. its (0001) pole figure, does not provide an unambiguous representation of the texture of the sample, but other poles have to be considered to represent unambiguously an orientation. In the above example the additional information about the position of say the (1070) pole would determine the orientation in an unequivocal manner (Figure 2.5). Whereas in this example the orientation is fully characterised by two poles, in general - depending on the symmetry of the crystal and/or the poles - three poles are necessary to derive
DESCRIPTORS OF ORIENTATION
25
completely the orientation matrix g in equation 2.1 1. Usually, the additional information is provided by other poles of the same family of planes (Section 2.2.1) i.e. different poles (hkl) of the family {hkl). For instance, the hexagonal crystal has three equivalent (1 120) planes, (1120), (1210) and (21 10), so that each orientation in the corresponding { l 120) pole figure is exactly characterised by the corresponding three poles. Accordingly a given orientation in pole figures, like { l 121) and { l 101), is defined by six equivalent poles each, which describes unambiguously this orientation. In cubic crystals, a given orientation is described by (not counting the same poles with negative signs) three {loo}, four { l 1l ) , six (1 101, twelve (0123, {l 121, { l 131, etc., and - in the most general case twenty-four {hkl) poles, which means that all pole figures of cubic crystals comprise enough poles to describe unambiguously an orientation. -
2.4.2 The inverse pole figure Rather than representing the orientation of the crystal coordinate system in the specimen coordinate system, i.e. in a pole figure, vice versa the orientation of the specimen coordinate system can be projected into the crystal coordinate system. This representation is called the inversepole$gure. Thus, the reference system of the inverse pole figure is the crystal coordinate system C, and the 'orientation' is defined by the axes of the specimen coordinate system S, e.g. RD, TD and ND. In analogy to equation 2.10a, where the pole figure angles a and P of a unit vector parallel to the crystallographic axis (XYZ) have been considered in the frame S, now the angle y and S of a vector parallel to a specimen axis S, in the coordinate system Cmust be introduced: S, = cl
sin yi cos Si
+ c2 sin T~sin Si+ c3 cos hi
(2.12)
Scalar multiplication now leads to (see equation 2.11):
These expressions describe the elements of the orientation matrix g in terms of the position of the specimen axes. Inverse pole figures are often used for axial symmetric specimens, where only one of the axes is prescribed. For instance, for compression or tensile specimens the orientation changes of the compression or tensile axis are plotted in the crystal coordinate system. According to the crystal symmetry it is not necessary to show the entire pole figure, but one unit triangle (Appendix 4) suffices. In the case of cubic crystal symmetry, the well known unit triangle , <110> and < 111> is used.
2.5 THE EULER ANGLES AND EULER SPACE Both the orientation matrix and the ideal orientation notation overdetermine the orientation, since only three variables are needed to specify an orientation. The most
INTRODUCTION TO TEXTURE ANALYSIS
26
well-established method of expressing these three numbers is as the three Eziler angles, which reside in a coordinate system known as E d e r space.
2.5.1 The Euler angles The Euler angles refer to three rotations which, when performed in the correct sequence, transform the specimen coordinate system onto the crystal coordinate system - in other words specify the orientation. There are several different conventions for expressing the Euler angles. The most commonly used are those formulated by Bunge, as shown on Figure 2.7 (Bunge, 1965, 1985a). The rotations are:
1. p1 about the normal direction ND, transforming the transverse direction T D into TD' and the rolling direction R D into RD'; 2. cD about the axis RD' (in its new orientation); 3.
92
about ND" (in its new orientation)
where 31, Q, 4 2 are the Euler angles (Bunge notation). The effect of the operation sequence of these three rotations can be followed on Figure 2.7.
Figure 2.7 Diagram showing how rotation through the Euler angles. in order 1. 2, 3 as shown, describes the rotation between the specimen and crystal axes (Courtesy of K. Dicks).
DESCRlPTORS OF ORIENTATION
Analytically, the three rotations are expressed as
1 0 0 cos Q 0 -sin@
sin@ cos@
By multiplication of these three matrices in order. an expression is obtained which links the rotation matrix to the Euler angles:
The elements of the matrix in terms of the Euler angles are therefore given by
g31 = sin 91 sin @ g32 g33
=
-
COS
p1 sin (D
= cos (D
The ideal orientation (123)[634. which is used as an example throughout this Chapter, can equivalently be expressed as Euler angles p1 = 301.0". Q = 36.7". p7 = 26.63.Table 2.1 shows the effect of the crystallographically-related solutions on these Euler angles. From the definition of the Euler angles it follows that they are periodic with period 2n. Thus, it holds: g(pl
+ 227, D + 2
~9 2,
+ 227) = g(pl.(D. 13:)
(2.17 )
Moreover. there is the identity
representing a reflection in the plane Q = ;r with a simultaneous displacement through n in both p1 and 4 2 . This means that there is a glide plane in the Euler angle space. It seems that in the most general case the Euler angles are defined in the range
28
INTRODUCTION TO TEXTURE ANALYSIS
0" 5 p1, p2 < 360' and 0" 5 Q, 5 180". The sine and cosine functions in equation 2.16 are only defined in the range -90" < cpl, Q,, p2 < 9O0, however. Therefore, determination of the Euler angles from the rotation matrix (equation 2.16) is not straightforward, since the range of the angles has to be considered too. As an Q ) be example, let Q = 35" and g31 = 0.439. In that case, p1 = a r c ~ i n ( g ~ ~ / s i ncan both 50" and 130". The correct value has to be decided by taking the information in other matrix elements into account, e.g. g32. Besides the convention of Euler angles according to Bunge two other definitions are in use. Independently of Bunge, Roe (1965) proposed a set of Euler angles Q, 0, Q, which differs from Bunge's definition by the second rotation being performed about the transverse direction rather than the rolling direction. More recently, Kocks (1988) defined a set of so-called symmetric Euler angles Q, 0, 4. Whereas the rotations of Q and 0 are identical to those introduced by Roe, 4 was defined as to be 7r - Q,Roe. In this set, the orientation matrix g becomes its transpose upon an interchange of Q and 4, which gives a symmetry between the 'reference' and the 'rotated' frame, which is not present in any of the other conventions. The equivalence between these two sets and the Bunge angles is: Bunge-Roe: p1 = Q? 7r/2 Q=0 q 2 = Q - 7i/2 (2.19a)
+
Bunge-Kocks: p1 =
Q +7r/2
Q=0
p2 = 7r/2
-
0
(2.19b)
The Euler angles formed the backbone of texture research and practice for many years and have been particularly associated with X-ray analysis of textures. Details of their use in practice are given in Sections 5.4. 9.3, 9.6.2 and Case Study No. 9. Although the Euler angles are less intuitive than the ideal orientation nomenclature, they do present a fully quantitative description of the texture which is suitable for graphical representation.
2.5.2 The Euler space Any orientation expressed in terms of its Euler angles can be represented unequivocally as a point in a three-dimensional coordinate system whose axes are given by the three Euler angles (Figure 2.8). The resulting space is referred to as Euler space (which is one example of an orientation space or g-space). Although the Euler space suffers from several disadvantages, it is by far the most common orientation space in which to represent macrotexture data and hence we will encounter it in several Sections throughout this book. In particular. the quantitative analysis of X-ray macrotextures is mainly based on the representation of textures in Euler space, and this issue will be discussed in detail in Section 5.4. Accordingly, Euler space is also used to represent individual orientations and orientation distributions which are determined by techniques for single grain orientation measurements (Section 9.3). For the representation of misorientations other spaces are more advantageous, but nonetheless, misorientation distributions can be represented in Euler space, which warrants a short discussion (Section 9.6.1).
DESCRIPTORS OF ORIENTATION
Figure 2.8 Representation of orientations in a three-dimensional orientation space defined by the Euler angles. Each orientation g corresponds to a point in the Euler space whose coordinates are given by the three Euler angles p,,Q, 9 2 describing the orientation (Bunge's convention).
As shown in Section 2.5.1 in the most general case of triclinic crystal symmetry and no sample symmetry the Euler angles are defined in the range O0 5 pl, p 2 5 360" and 0" 5 5 180°, which in turn defines the maximum size of the Euler angle space, the so-called asymmetric unit. However, as already discussed earlier (Sections 2.2 and 2.3), symmetries of the crystal and/or the specimen frame result in a number of different, but equivalent, descriptions of a given orientation. Such symmetries lead to a reduction in the size of the Euler space which is required to represent unequivocally all possible crystallographic orientations and, consequently, leads also to a number of symmetrically equivalent subspaces (see Table 2.2). The necessary transformation laws for the various symmetry operations can be found in Hansen et al., 1978; Bunge, 1982; Matthies et al., 1987 and Kocks. 1998.
Table 2.2 S ~ z eof the Euler space necessary to represent unequivocally orientations for different sample and crystal symmetries Crystal structure
Cubic Tetragonal Orthorhombic Hexagonal Trigonal Monoclinic Triclinic
Crystal symmetry
90" :L 90" 90" 90" 90" 90" 180"
Sample synlmetry Orthotropic
Monoclinic
None (triclinic)
90"
180"
360"
90" *
90" 180" 60" 120" 360" 360"
p p
" 120" symmetry 1s not included
30
INTRODUCTION TO TEXTURE ANALYSIS
Different sunzple s~mrizetriesaffect the range of the angle PI. Whereas in the most general case, that is no sample syinmetry (sometimes called triclinic saniplr symmetry), the full range 0" < p1 < 360" is required to represent all possible orientations, any symmetry in the sample coordinate system reduces the size of the Euler space. In samples deformed by rolling or more generally under a plane strain deformation state - it is usually assumed that there is a two-fold symmetry axis (diad), or the normal of a mirror plane, parallel to each of the three sample axes, the rolling direction, the transverse direction and the sheet normal direction. In this cases of ortlzotropic or ortlzorhonzhic sanzple symmetry the range of the Euler angle p1 is reduced to one quarter, i.e. to 0" < pi < 90". When only one symmetry element is present. the Euler space is halved to 0" < p1 < 180". This so-called monoclinic sample sjwwnetrj. is given e.g. for samples deformed by shear. For instance, a specimen deformed in torsion contains a two-fold axis in the radial direction of the torsion cylinder. Monoclinic sample syinmetry also applies for the surface layers of a rolled sheet. In contrast to the plane strain state at the sheet centre, which has orthotropic sample symmetry, the surface layers only have one symmetry element, a mirror plane perpendicular to the transverse direction (Section 5.4.2). For that reason, when a pole figure obtained from the surface of a rolled sheet is projected from the transverse direction, it resembles those of torsion experiments (Kocks, 1998). Crystal symmetry further reduces the size of the Euler space by affecting the range of the angles @ and 92 In general, an rz-fold symmetry axis reduces the Euler space B, p2 i 36O0/n) by a factor of n. This means that the Euler angles (p1.Q, p2)and (p1, are equivalent. Thus, the 3-fold axis in trigonal crystal syinmetry reduces the range by a factor of 3, which means that a range of 0" < p1 < 120" is sufficient to represent all possible orientations. Accordingly, for tetragonal (4-fold axis) and hexagonal crystals (6-fold axis), 9 2 can be reduced to 90' and 60°, respectively. An additional 2-fold symmetry or mirror planes affects Q by reducing the range from 180" to 90°, which is given in all but triclinic crystal syinmetry (Table 2.2). For cubic crystal symmetry it has already been discussed in Section 2.2.3 that there are 24 possibilities to describe one and the same orientation of a cube in any reference frame, so that the Euler space can be subdivided into 24 equivalent subspaces. In accordance with the above considerations, the three 4-fold axes (i.e. 90°/<100> rotations) and the six 2-fold axes (i.e. 180°/<100> rotations), reduce the necessary Euler angle space to the range 0" < p1 5 360" and 0" < @, p2 < 90°, which means that there are only eight equivalent subspaces. However, this reduction does not consider the 3-fold axes, i.e. the 120°/<111> rotations. Since a further reduction under consideration of the 3-fold symmetry would lead to a complex shaped subspace, this operation is not usually carried out and, therefore, for cubic crystals each orientation appears three times in the reduced Euler angle space. Figure 2.9 shows the symmetry elements in the Euler space for cubic crystal symmetry and orthotropic sample symmetry (Hansen et al., 1978). If we now consider r?zisorieiztations between two crystals, as opposed to the orientation of a crystal with respect to an external reference frame, the Euler space is further reduced by applying the crystal symmetry of the second crystal rather than sample symmetry. Furthermore, the Euler space is reduced by an additional factor of 2 due to the equivalence of the misorientation and the inverse misorientation, -
DESCRIPTORS OF ORIENTATION
Figure 2.9 Symmetry elements in the Euler space for cubic crystal symmetrq and orthotropic sample symmetry.
because with respect to the misorientation the two crystallites are indistinguishable. Thus, for cubic crystal symmetry one has 24 X 24 X 2 equally valid descriptions of a given misorientation, which contain 24 generally different rotation angles 8.One of the resulting 1152 equivalent subspaces of the Euler space is shown in Section 9.6.1. 2.6 THE ANGLEIAXIS O F ROTATION AND CYLINDRICAL ANGLEIAXIS SPACE In Section 2.2 we showed how an orientation can be described by the concept of three rotations which transforms the coordinate system of the crystal onto the reference, i.e. specimen, coordinate system. The same final transformation can be achieved if the crystal coordinate system is rotated through a single angle, provided that the rotation is performed about a specijic axis. This angle and axis are known as the angle of rotation and axis of rotation, or more briefly the anglelaxis pair. Figure 2.10a illustrates the anglelaxis of rotation between two bodies, in this case cubic coordinate systems. The left-hand cube (axes X1YIZ1) is chosen as the fixed, reference position and the right-hand cube (axes X2 Y2Z2)would need to be rotated through 50" about an axis which is vertical in the diagram and parallel to the common [OOl] in order to achieve the reference orientation. It is important to notice that the rotation axis [001] is the same direction in both cubes. Similarly, Figure 2.10b shows anglelaxis of rotation between two hexagons. If we now consider the orientation of crystal lattices, X I Y I Z l is replaced by the specimen coordinate system as a reference, e.g. RD, TD, ND, and X2Y2Z2 is replaced by the crystal axes [loo], [010], [OOl]. The three independent variables which embody an orientation comprise one variable for the rotation angle B and two
32
INTRODUCTION TO TEXTURE ANALYSIS
i
Common Axis
K 001 l
Angle SO0
Figure 2.10 Diagram showing the angleiaxis of rotation between (a) two cubes (Courtesy of K. Dicks) and (b) two hexagons (Adapted from Vicens et al.. 1994).
variables for the rotation axis s, since s is normalised such that rl + r i + r i = 1 and so the third index is redundant. The anglelaxis of rotation is extracted from the orientation matrix g as follows (Santoro and Mighell, 1973):
DESCRIPTORS OF ORIENTATION
If Q = 180°, then r is given by:
For example, the orientation matrix given in equation (2.5) yields an anglelaxis pair of 48.6"/(0.562 -0.520 -0.644). The reference orientation, chosen to coincide with the specimen axes, is described by the identity matrix. It is not usually necessary to ascribe a handedness to the rotation, since conventionally the rotation is described as a right-handed screw operation and hence Q is always a positive angle (Mykura, 1980). A negative angle is equivalent to changing the sign of r. An example of where it is necessary to know the sense of a rotation is when measuring the orientation of adjacent subgrains (Sections 9.5.2, 10.4.3 and Case Study No. 8). If the rotations are all in the same sense, then the total rotation is cumulative; otherwise adjacent rotations of the same magnitude cancel each other out (Hansen and Juul Jensen, 1994; Randle et al., 1996). The transformation from 0/r back to the orientation matrix is given by: g11 = r:(l - cos 0) + cosQ l cos Q) - r3 sine g12 = r ~ r z ( g13 = rlr3(1 - cos Q) r2 sin Q g21 = ~ r 1 (-1COS Q) r3 sin Q
+ + COS 0) + COS e
g 2= ~ r;(i g23 = r2r3(1 - COS Q)- 1.1 sin 6' g31 = q r l ( 1 - COS Q)- r, sin Q g32 = r3r2(1- COS Q)+ sin 0 g33
= ri(1
-
cos 0) + cos 0
The crystallographically-related solutions of a single orientation were discussed in Section 2.2. It follows then that these solutions will give rise to the same number of axis/angle pairs, each of which is calculated from the crystallographically-related solution of the matrix (Appendix 2). Table 2.1 includes the 24 angle/axis pairs which accompany the ideal orientation (123)[634]. Inspection of Table 2.1 shows that, even though each of these anglelaxis pairs describes exactly the same orientation, this could not have been known from the format of the anglelaxis pairs themselves numerically they appear to be quite unrelated. The relationship only becomes clear from concurrent inspection of the matrix or ideal orientation.
-
2.6.1 Anglelaxis of rotation There is a second choice for designation of a reference orientation. Instead of choosing the specimen axes for the reference orientation, it could be the axes, i.e.
34
INTRODUCTION TO TEXTURE ANALYSIS
orientation, of another grain in the specimen, usually the neighbouring grain. The orientation of grain 2, i.e. a grain having crystal axes parallel to X2 Y2 Z2, is then expressed relative to the orientation of grain I, which is similarly a grain having crystal axes parallel to X I Y 1Z1.For the examples in Figures 2.10a and 2.10b, the anglelaxis of misorientation is 50°/<001> and 9O0/<1T00> respectively, where the latter axis is given in Miller-Bravais indices. The purpose of such a descriptor of orientation is to characterise the orientation relationship (between different phases) or the misorientation (between two grains of the same phase) (Lange, 1967; Santoro and Mighell, 1973). For the latter case specification of the misorientation gives access to the grain boundary crystallography, which is a very important subset of microtexture (Randle, 1993). Extraction and analysis of grain boundary parameters from the misorientation are described in Chapter 11. A misorientation is calculated from the orientations of grain 1 and grain 2 by:
where M12is the matrix which embodies the misorientation between g2 and g,, where g1 is arbitrarily chosen to be the reference orientation. The anglelaxis of rotation (usually called the anglelaxis oj'misorientation to distinguish it as a grain or phase boundary parameter) is calculated by using equations (2.20) to (2.22) and substituting the elements of the misorientation matrix M12 for those of the orientation matrix g. In summary the essential difference between an anglelaxis pair description of the orientation of a single grain and an anglelaxis pair description of the misorientation between the crystal lattices of two grains is simply the choice of reference axes. This can be illustrated graphically on the stereogram, as shown in Figure 2.11. (The stereographic projection is explained in Appendix 4). In Figure 2.1 1a the location of r, common to both the reference orientation and the orientation of grain 2, is plotted on the stereogram along with the angle H through which [100],, [010],, [OOl], must be rotated in order to make them parallel with the reference axes. The reference axes are hence the specimen axes, whereas in Figure 2.1 1b the reference axes are the crystal axes of grain 1. Thus, Figure 2.11a depicts the orientation, sometimes called the absolute ovientution, of grain 2 and Figure 2.1 1b depicts the misorientation between grain l and 2, both expressed as an anglelaxis pair. Figure 2.3 includes the most common parameters which are derived from the misorientation matrix. As it stands, the angle/axis pair is not a very useful representation of an orientation since it does not convey easily understood physical information about the relationship between the orientation and important axes in the specimen. The value of the anglelaxis description of oricntation is as a starting-point for the Rodrigues vector (R-vector) formulation, which leads to a superior way of representing orientations and misorientations (Section 2.7). On the other hand. the anglelaxis pair is an extremely effective means for representation of a misorientation, since the description actually relates physically to the grain or phase boundary misorientation angle and axis, which in turn is used for categorisation and subsequent analysis (Section 11.2). Frequently, the angle and axis of misorientation are represented separately, which is discussed in Section 9.5.1. The Rodrigues vector
DESCRIPTORS OF ORIENTATION
Figure 2.11 Stereographic representation of (a) an orientation and (b) a misorientation. In (a) the < 100 >-crystal axes of two grains 1 and 2 are represented relatibe to the specimen axes XYZ. The rnisorientation axis is given by the intersection of great circles which bisect pairs of the same crystal axes in both grains. The angle between the poles of two of these great circles gives the misorientation angle (Append~x4). In (b) the misorientation axis (cross) and angle (arcs of small circles) are expressed relative to the crystal axes of one grain. This particular example is a twin (Adapted from Randle and Ralph, 1988).
36
INTRODUCTION TO TEXTURE ANALYSIS
formulation is also used with great effect to convey and represent misorientations. In summary, the formulation of the angle/axis of misorientation is an end in itself and further used to calculate R-vectors, whereas the anglelaxis of rotation for orientations is not a particularly effective parameter except to illustrate the principles of the R-vector, as shown in Section 2.7.
2.6.2 The cylindrical anglelaxis space Orientations and inisorientations can also be represented in a three-dimensional space formed by the parameters of the angle/axis pair description. As the anglejaxis representation is easy to visualise and convenient for analysis and interpretation of orientation correlations, the anglelaxis space will mostly be used for the representation of misorientations, but the basic principles discussed in the following also hold for the presentation of orientations. As described in Section 2.6.1, both orientations and misorientations can be expressed in terms of rotational parameters, i.e. the rotation axis r and the rotation angle 8. To represent the orientations or misorientations in a suitable angle/axis space, the rotation axis r = (I.,, r;, r,) is expressed in terms of two angles 8 and g: I., = cos
4sin 8
I;. = sin)I sin t9
rz = cos t9
(2.25)
Thus. the anglelaxis rotational parameters are now given by the three angles 8, 3 and g. The necessary analytical descriptions connected with this way of representing orientations and misorientations are given by Pospiech (1972) and Pospiech et al. (1986). In calculations, the relations between Euler angles (gl,Q, p2) and the rotational parameters (0,8, v) might be useful:
All possible orientations can be presented in the interval (0" < 8 5 360°, 0" < 19 < 90°, 0" < d 5 360") into a space with the simple shape of a cylinder, i.e. cylindrical arzgle/axis space (Figure 2.12). When the cylinder-axis is chosen parallel to the 0-axis, the base plane of the cylinder correspoilds to a stereographic projection of the rotation axis r. Thus for each misorientation angle 8 the corresponding rotation axis r can readily be derived from its position in the stereographic triangle (Appendix 4) which is a very familiar way to evaluate misorientation data (Section 9.5.1). Examples of misorientation distribution functions (MODFs) in the cylindrical angle/ axis space will be shown in Sections 9.6.2 and 9.8. 2.7 THE RODRIGUES VECTOR AND RODRIGUES SPACE An orientation can be expressed as a three-dimensional vector, since such a vector requires three variables to specify its position in space and similarly an orientation requires three variables for its specification. Such an 'orientation vector' is most easily described and visualised from the anglelaxis descriptor: the rotation axis gives
DESCRIPTORS OF ORIENTATION
Figure 2.12 Representation of an orientation g given by the anglelaxis description parameters i. and 8 in a cylindrical orientation space defined by the rotational parameters 19,d and Q.
the direction of the vector and the rotation angle its magnitude. The Rodrigues vector R combines the angle and axis of rotation into one mathematical entity thus:
For example, the first angle/axis pair entry in Table 2.1 gives a Rodrigues vector with elements R1R2R3given by
The attraction of this parameterisation is that the three components of the Rvector define a vector which lies in a Cartesian coordinate system whose axes can be chosen to correspond to either the specimen or the crystal axes. Hence, a population of R-vectors, each of which represents an orientation or misorientation, can be represented in a three-dimensional space known as Rodrigues-Frank (RF) space or Rodrigues space. Since the R-vector is derived directly from the anglelaxis pair, it can be described by a number of different crystallographically-related solutions, with the precise number of solutions depending on the crystal system and symmetry (Section 2.2.3). The R-vector is a function of Q, and so it follows that the smallest value of the vector is calculated from the smallest 0. Such R-vectors will lie closest to the origin of
38
INTRODUCTION TO TEXTURE ANALYSIS
R F space and therefore the crystallographically-related solution of an orientation or misorientation having the smallest angle (e.g. the first row in Table 2.1) is the most convenient one to select for representation in Rodrigues space. Use of Rodrigues space is a relatively new field compared with the long-established Euler space approach for representation of rotations (Frank, 1988). The explanation for the late introduction of Rodrigues space is that its application is intimately connected with microtexture, i.e. discrete orientation measurements, whereas representation of texture in Euler space relates to the representation of continuous orientation distributions as calculated from pole figures (Sections 2.5 and 5.2). Microtexture data can be. and frequently are, represented in either Euler space or Rodrigues space (Chapter 9) whereas macrotexture data tend to be represented only in Euler space because commercial systems are configured for this approach (Chapter 5). The geometrical features of Rodrigues space offer considerable benefits for microtexture analysis. Here we will describe the essential features of Rodrigues space and the methodology of its use in Sections 9.4 and 9.6.3. Mathematical formalisms are kept to a minimum since these are not essential to the understanding and use of Rodrigues space, and moreover details can be found elsewhere (Becker and Panchanadeeswaran, 1989; Neumann, 1991a,b: Randle and Day, 1993; Field, 1995).
2.7.1 The fundamental zone The entirety of Rodrigues space encompasses each one of the crystallographicallyrelated solutions of a rotation (Section 2.2.3). Since it is simpler for each orientation from a sample population to be represented only once, it is convenient to formulate the R-vector from only one crystallographically-related solution. That solution with the lowest rotation angle is chosen because it feeds through to give the smallest Rvector. By use of the lowest angle description all orientations, represented by the endpoints of their R-vectors, can be made to reside in a polyhedron known as the firrzdumentul o n e qf'Rodrigues space. which contains the origin of the space. Hence any orientations which lie outside the fundamental zone can be re-expressed as equivalent points lying within it by re-choosing the crystallographically-related solution to be the lowest angle one. Exceptionally. an orientation may lie on the surface of the fundamental zone and will be matched by an equivalent point on the opposite face. The shape of the fundamental zone is governed by the crystal symmetry of the material and the geometry of these polygons has been derived for each crystal system (Morawiec and Field, 1996: Kumar and Dawson, 1998) and for combinations of lattices (Heinz and Neumann. 1991). For cubic symmetry, the shape of the zone is a cube with truncated corners such that the bounding surfaces are six octahedra and eight equilateral triangles as shown in Figure 2.13. For the representation of orientations, the axes of the zone are aligned with the specimen axes. Alternatively the axes of the zone can be aligned with the crystal axes, which is convenient for the representation of misorientations. The distance of the point where the reference axes intercept the zone surface is represented, in terms of crystal coordinates, by the anglelaxis pair 45"/<100> which gives R = (0.4142,0,0). The R-vectors of the centres of the equilateral triangles which
DESCRIPTORS OF ORIENTATION
Figure 2.13 The fundamental zone of Rodriguer space for cubic sqmrnetry, showing also the decomposition of the space into 48 subvolumes (Day, 1994).
Table 2.3 Parameters of Rodrigues space according to crystal system (from Morawiec and Field, 1996) Crystal system
Distance of the main face from the origin
Description
Sub\ olume fraction
Cubic
tan
1/48
Hexagonal
tan ~ ' 4
6 octagonal faces normal to the 4-fold axes 8 triangular faces normal to the 3-fold axes 2 dodecagonal norinal to the 6-fold axis 12 square faces normal to the diad axes 2 hexagonal faces normal to the 3-fold axis 6 square faces normal to the diad axes 8 triangular faces perpendicular to the 3-fold axes (i.e. a regular octahedron) 6 faces normal to the diads (i.e. a square) 12 pentagonal fixes normal to the 5-fold symmetry axes (i.e. a regular dodecahedron).
tan
7i
7i
8
4
Tetragonal
tan 7i16
Orthorhoinbic Icosahedral
tan tan
ri 7i
4 l0
1/12 l16
1,24 1,s 1:120
form the truncated corners of the zone, are given by the ang1e;axis pair 60"/< 111> and a corresponding R-vector (0.333. 0.333. 0.333). Another point which defines the shape of the fundamental zone for cubic crystals is an apex of the equilateral triangle, given by 62.S0/<1, 1, - l)>, which represents the greatest possible misorientation between two cubes, and yields an R-vector of (0.4124, 0.4124. 0.1716). The shapes of fundamental zones for all crystal systems are described in Table 2.3. It can be seen that the shape of the zone is equivalent to the crystal symmetry. For nionoclinic and triclinic symmetries the fundamental zone corresponds to the entire Rodrigues space, and for this case the space has no boundaries, which is a disadvantage to the representation of orientations having this symmetry. However, most materials of interest from a texture point of view fall into other symmetry categories and can be represented readily in Rodrigues space. e.g. trigonal A1203 (Morawiec and Field, 1996).
(a
2.7.2 Properties of Rodrigues space The most powerful aspect of Rodrigues parameterisation for the representation of orientations and misorientations is that only rectilinear, i.e. straight-line geometry, is
40
INTRODUCTION TO TEXTURE ANALYSIS
involved which renders the space easy to handle and visualise. The rectilinear geometry of Rodrigues space is manifested by the following: e
The axis of rotation gives the direction of the R-vector. Hence rotations about the same axis of rotation lie on a straight line which passes through the origin
e
The angle of rotation gives the length of the R-vector. Hence small angle boundaries cluster close to the origin. These first two points are especially relevant to the analysis of misorientations (Section 9.6.3)
e
Orientations which include a common direction, e.g. a fibre texture, lie on a straight line which in general does riot pass through the origin
e
The edges of zones in Rodrigues space are straight lines, and the faces of zones are planar
In summary, every straight line in Rodrigues space represents rotations about a certain, fixed axis. These are called geoclesic lines and remain straight even if the origin of the space is shifted (Frank, 1988; Neumann, 1991a,b). Another advantageous aspect of Rodrigues space is that it is nearly lioniochoric, i.e. a distribution of untextured orientations will be distributed almost uniformly throughout the space. The invariant orientation volume is actually within a factor of 2 (Neumann, 1991a,b; Kumar and Dawson, 1998) which is generally considered low enough for orientation distributions in the space to be evaluated in a straightforward manner. The fundamental zone can be further divided into identical subvolumes, as shown in Figure 2.13 (Randle, 1990). There are 48 of these for the cubic system, and the subvolume fraction for other crystal systems is shown in Table 2.3. Subvolumes are used for the display of misorientation data.
2.8
SUMMATION
This Chapter has established the terminology associated with analysis, representation and display of orientation and misorientation data. All the descriptors of orientation and their associated spaces which are commonly encountered in the field of texture - the orientation matrix, 'ideal orientation'lMi1ler indices, Euler angles, anglelaxis of rotation and Rodrigues vector plus the misorientation counterparts, where appropriate - have been explained. There is some variation in the popularity and patterns of usage for each descriptor. Briefly, the orientation matrix is the mathematical calculation instrument, the ideal orientation notation and Euler angles have been traditional descriptors of texture and associated with X-ray diffraction, whereas the angle/axis of rotation and Rodrigues vector have come to the fore much more recently and are used more for misorientations. However, a modern approach to texture can draw upon any of the texture descriptors, and so an introductory knowledge of each is desirable in order to choose the most convenient descriptor for a particular task.
3. APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS
3.1 INTRODUCTION The vast majority of techniques for texture analysis is founded upon the diffraction of radiation by a crystal lattice, and so it is vital to understand this phenomenon in order to appreciate the principles upon which the various techniques for experimental texture measurement are based. Radiation which is diffracted by crystallographic lattice planes is able to provide information on their arrangement and, consequently, on the orientation of the sampled volume of material with respect to some fixed reference axes (Section 2.2). To instigate diffraction of radiation at lattice planes, the wavelength of the incident radiation must be smaller than the lattice spacing, which for materials of interest is typically tenths of a nanometre. Table 3.1 shows the wavelengths, in addition to other characteristics, of various radiations which are commonly used for texture measurements. Data for light is included only for comparison, since X-rays, neutrons and electrons are diffracted by lattice planes whereas light is not. The various kinds of radiation interact with matter in different ways. This manifests itself by substantial differences in the absorption of the radiation by matter and, with regard to texture analysis, by the depth of penetration of radiation in the sample material. To illustrate this, the scale of penetration depths for the various radiation is also given in Table 3.1. Thus, the particular characteristics of the Table 3.1 Average diffraction properties of radiation used for texture measurement by diffraction, with light also included for comparison
Wavelength [nm] Energy [eV] Charge [C] Rest mass [g] Penetration depth. absorption length [mm]
Light
Neutrons
X-rays
Electrons
400-700 1 0 0
0.05-0.3 1o - ~ 0 1.67 X l 10-100
0.05-0.3 1o4 0 -0 ~ ~ 0.01-0.1
0.001-0.01 10j -1.602 X 1 0 - l ~ 9.11 X 10-" 10-?
~
42
INTRODUCTION TO TEXTURE ANALYSIS
radiation have a strong impact on the application. This Chapter addresses the fundamental aspects of diffraction, the characteristics of various radiations and how these relate to the experimental set-ups for texture measurement. It is pertinent to have some knowledge of subjects such as structure factor (Section 3.3) and characteristics of radiation (Section 3.6) in order to understand subsequently the principles of texture determination. Further details on these and related topics can be found elsewhere (e.g. Bacon, 1975; Cullity, 1978; Thomas and Goringe, 1979; Barrett and Massalski, 1980; Goodhew and Humphreys, 1988).
3.2 DIFFRACTION O F RADIATION AND BRAGG'S LAW Electromagnetic radiation like light, X- or y-rays is diffracted by elastic scattering of the incident waves at the atoms of the sample material. Particle beams. like electrons and neutrons, can also be considered as waves of radiation, with their wavelength given by the de Broglie relation:
where X is wavelength, lz is Planck's constant and m, v, E k i n are mass, velocity and kinetic energy of the particles respectively. When a plane wave of radiation hits an atom, this acts as a source of spherical waves of the same wavelength (Figure 3.1). The efficiency of an atom in scattering radiation is usually described in terms of the atomic scattering amplitude (or scattering length) f . Since the intensity of a wave is the square of its amplitude, f 2 is a measure of the intensity of the scattered wave in dependence on the sort of the atoms, the scattering angle .3, the wavelength X of the incident beam and on the kind of radiation (Doyle and Turner, 1968; Cullity, 1978; International Tables for Crystallography, 1983).
planar wave front
Ih
Figure 3.1 Scattering of a planar wavefront at a point source giving rise to the formation of a Huygens spherical wave.
APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS
Figure 3.2 Angular variation of the atomic scattering amplitude f of copper for X-rays, electrons and neutrons (note different scales for electrons). The data for X-rays and electrons were taken from Doyle and Turner (1968) and those for neutrons from Bacon (1975). The reflection angles of the upper axis were calculated with X = 0.07107 nm for X-rays and neutrons and with X = 0.00251 nm for electrons (seeTable 3.1).
X-raj.s are scattered by the shell electrons of the atoms through an interaction between the charged electrons and the electromagnetic field of the X-rays. Since the size of atoms is of the same order as typical X-ray wavelengths, different waves that are scattered at the various electrons of the atom interfere, which gives rise to a strong dependency of the scattering amplitude fx on the atomic radius R and the scattering angle 19. Whereas near the incident beam, i.e. for small angles 6, the individual scattered waves will all nearly be in phase and reinforce each other, for large angles 8 they are out of phase and, hence, reinforce each other much less. Accordingly, jx decreases rapidly with increasing scattering angle 6 or, more precisely, with an increasing value of sin.3/X, as shown in Figure 3.2. Electrons interact with both the shell electrons and the nuclei of the scattering atoms (Section 3.6.3). Their scattering amplitude fE decreases with sin.LY/X similarly as f x does, though more rapidly, but the value of fE is by about 4 orders of magnitude greater than fx (Figure 3.2). Thus, electrons scatter much more intensely than X-rays, and this is the reason why electrons can provide such high resolution in electron microscopy. Neutrons, in contrast, mainly interact with the nucleus of the atom (except for magnetic materials where magnetic scattering by the shell electrons takes place, see Section 3.6.2). Since in neutron diffraction experiments the size of the atomic
INTRODUCTION TO TEXTURE ANALYSIS
planar wave front
1
Figure 3.3 Reinforcement of scattered waves producing diffracted beams of different orders (Barrett and Massalski, 1980).
nucleus is negligible compared to the typical wavelengths, the scattering amplitude for neutrons Lv is virtually independent of the reflection angle 29 (Figure 3.2). When radiation interacts with matter, rather than with individual atoms, the individually scattered waves interfere to form a secondary wave (Huygens-Fresnel's principle), as shown schematically in Figure 3.3. In most cases, the different waves will be out of phase, which leads to an annihilation of the reflected intensity to zero. Only at specific angles will the wave fronts be in phase, which means that at those angles diffraction of the incoming radiation can be observed. For this to occur, three conditions must be fulfilled: The atomic arrangement must be ordered, i.e. crystalline e
The radiation must be monochromatic, i.e. consist only of one wavelength A
e
This wavelength must be of the same order of magnitude (or smaller) than the diffracting features
Under these circumstances, independent of the kind of radiation, specific diffraction maxima are observed. From Figure 3.3 it appears that the angles 0 at which diffraction occurs depend on both the wavelength X and the spacing of the scattering atoms d. To derive this dependency, diffraction of radiation at the individual atoms in a crystal can be considered to be reflection of radiation at a set of semi-transparent 'mirrors', separated by a distance d. Bragg (1913) showed that these mirrors are formed by atomic planes, i.e. the lattice planes { h k l ) which are considered to be geometrically smooth. Figure 3.4 shows a section through a lattice with three atom layers A, B and C with rays incident upon these planes in the direction LM at the angle Q. A small portion of the incident radiation will be reflected at an angle 20 at plane A, whereas the rest
APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS
Figure 3.4 Diffraction from lattice planes, indicating the geometry which leads to the derivation of Bragg's law.
continues travelling into the lattice until it will be reflected at layers further below. The line L-L2 is drawn perpendicular to the incident beam to indicate one of the crests of the approaching in-phase waves. In order to get a reinforced, reflected beam in the direction MN, the waves must again be in-phase along the line N2.4-N2C.I n order to achieve this, the path lengths for beams reflected at different. successive layers in the crystal must differ by an integral number of wavelengths. In the example shown in Figure 3.4 for instance this means that the path difference between the two reflected beams L-M-N and L1-MI-NIB,i.e. the distance PMIQ, is either one wavelength X or a multiple of it, i.e. nX. It can be seen that:
It is clear from Figure 3.4 that the other reflected beams bear the same geometrical relationships and that the condition for reinforcement of all reflected beams can thus be written as:
where n is the order of reflection and d is the interplanar spacing. This is Bragg's law and is usually written as:
since often only first order diffraction (n = 1) is considered. Accordingly, the specific angles at which diffraction is observed are termed Bragg angle5 OB. (A similar relation was derived by Laue from consideration of scattering at individual atoms.) Bragg's law is fundamental to texture work since, for radiation of known wavelength, lattice planes can be identified from a measurement of the Bragg angles through which the waves are diffracted. Although the phenomenon which occurs is diffraction and not reflection, diffracted beams are frequently referred to as
46
INTRODUCTION TO TEXTURE ANALYSIS
beam incident
Figure 3.5 Extinction of the 100 reflection by anti-phase reflection at the intennediate 1200)-plane in a body-centred crystal.
'reflected beams' and the lattice planes as 'reflecting planes' or 'reflectors'. Note that it is convenient to distinguish between the reflecting planes, written with parentheses {hkl), and the corresponding reflected beam hkl written without parentheses. For most crystal structures, reflection through the respective Bragg angle is not observed for all possible sets of lattice planes. To illustrate this, we consider the reflection by the (100)-planes in a body-centred crystal, as shown schematically in Figure 3.5. For 100-reflection to occur, the Bragg angle must be set so that the (100)-planes are in reflection position, i.e. for lattice spacing d l o o However, the = dloo/2 are situated exactly half way between (200)-planes with spacing the (100)-planes (in Figure 3.4 this would mean we consider reflection at the planes A and C with the plane B inbetween). For these intermediate planes the path difference P-M1-Q in Figure 3.4 is exactly X12 (equations 3.3, 3.4), rather than an integral multiple of X as required to reinforce reflection. Hence, reflection by the (100)-planes is rendered completely extinct by the anti-phase reflection at the (200)planes inbetween. This cancellation will not occur if the crystal is irradiated at the Bragg angle for second-order reflection (i.e. n = 2), as then the scattering from both the (100)- and (200)-planes would be in phase. In general, extinction occurs when there is an equivalent plane halfway between the planes that are in the Bragg position for reflection. For body-centred crystal structures this is always given when the sum of the Miller indices h k + l is odd, or, conversely, reflection is obtained from the planes (1 101, (200), (21 l), (3 10) etc. For face-centred structures reflection is observed when the individual Miller indices h, k and I of the reflecting planes are either all odd or all even, i.e. (1 1l ) , (2001, (220). (31 1) etc. Note that if the scattering power of the intermediate layer is not equal to that of the plane under consideration - because of a different number or kind of atoms - there will be only a weakening but no complete extinction of the reflection. Accordingly, in pure nickel the 100-reflection is absent, but it appears, though weak, in the intermetallic compound NilAl with a very similar crystal structure. In general, the relative intensity of a given reflector and, consequently, the rules of extinction, can be deduced from calculation of the structure factor, which is shown in the next Section.
+
APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS
47
3.3 STRUCTURE FACTOR Diffraction of radiation at the various atoms in a crystalline structure is governed by the structure factor F, which is the unit cell equivalent of the atomic scattering amplitudef. Each atom j within a given unit cell scatters radiation with an amplitude proportional to j; for that atom and with a phase difference with regard to the incident beam which is determined by the position (X,,y,, z,)of that atom in the unit cell. Summation of the waves scattered by each of the atoms in the unit cell gives the amplitude of the wave being reflected by the plane ( I ~ k l )which , is called structure factor F:
(i is the imaginary constant, i.e. i = G ) Since . the intensity of a wave is proportional to the square of its amplitude, the intensity of the reflected radiation is proportional to and, hence, reflection only occurs if the reflected waves are all in phase, i.e. if F f 0. The differences between the various types of radiation used for texture analysis, i.e. X-rays, neutrons and electrons, are represented by the atomic scattering amplitude ]'(Section 3.2), but the geometrical considerations associated with the structure factor hold for all kinds of radiation. In the following, we calculate the structure factors for several simple crystal structures and thus derive the rules of extinction introduced in the previous section. The results are summarised in Table 3.2. If there is only one atom at the position (X. y , z) = (0, 0, 0) in the unit cell, the structure factor F is independent of { h k l ) since for all values of h, k and l, F = f. Thus, in a crystal with primitive structure all lattice planes would reflect radiation. For body-centred crystals, we have two atoms per unit cell, one at (xl, j l , z,) = (0,0,0) and the other at (X?,y2, z2) = (112, 112, 112). Substituting these values for (X, y. z) in equation 3.5, the structure factor F becomes:
To derive the rules of extinction, the following relations with complex numbers are helpful:
+ +
It is seen that if (11 k l) is odd, the exponential term in equation 3.6 becomes - 1, so that F = 0. However, if (h k I) is even, then F = 2f, which means that, in accordance with the qualitative considerations in Section 3.2, reflection in bodycentred crystals only occurs at planes for which (l1 + k + l ) is even. For face-centred crystals, four atoms have to be considered: (0,0, O), (112, 112, O), (1/2, 0, 1/2), (0, 112, 112). In this case, the structure factor F is:
+ +
F = f .[l + exp(xi ( h + k ) ) + exp(ni ( h + l ) ) + exp(7ii ( k + l))]
(3.7)
INTRODUCTION TO TEXTURE ANALYSIS Table 3.2 Geometrical rules for the structure factor F Crystal structure
Observed reflection
Structure factor F
Primitive Body centred Face centred Hexagonal closed packed
all h, k, I (h k + I) even h, k, l all odd or even h + 2k = 3rz, I even h+2k=3nil,lodd h + 2k = 3n 3 1, Ieven
f (l atom per unit cell)
+
2f (2 atoms per unit cell) 4f (4 atoms per unit cell) 2f (e.g. {0002)) &f (e.g. {oiii}) f (e.g. {olio})
Considering the possible combinations of h, k and I, it turns out that if all three are either odd or even, then all of the exponential terms are exp(2nni) = 1. In those cases, the structure factor F equals 4f and reflection occurs. In the other possible cases, however, two of the three phase factors will be odd multiples of T , giving two terms of - l in equation 3.7 and, therewith, F = 0 (Table 3.2). Note that the above considerations are valid regardless of what crystal structure the body- or face-centred crystal belongs. The hexagonal close packed (hcp) unit cell contains two atoms with coordinates (0, 0, 0) and (113, 213, 112). Inserting these values in equation 3.5 yields
The results for F for the possible combinations of 11, k , I are listed in Table 3.2. It turns out that the rules mainly depend on whether or not lz + 2k is a multiple of 3. For instance, the (1120) and (1122) reflections will be strong, whereas (1121) is absent. Likewise, (3030) is strong but (1010) and (1011) are weak. So far, only reflection of radiation in pure metals has been considered, which means that the unit cell only contains atoms of one sort. For unit cells containing different sorts of atoms, e.g. intermetallic phases, ordered structures, etc.. the scattering factors f, of the different atoms vary, so that the different phase factors do not necessarily cancel out. This leads to the observation of superlattice peaks in the case of ordered phases. As an example, in the ordered intermetallic phase Ni3A1mentioned above, the Al-atom can be considered to be on the (0,0,O) site and the 3 Ni-atoms on the centres of the faces (L12-structure). Hence, the structure factor is given by:
+ k)) + exp(.sri( h + I ) ) + e x p ( ~(ik + l ) ) ] (3.9) In analogy to the face-centred lattice, reflection occurs with F = (fAl + 3fxl) if h, k, F =f
+
.
~ l f h ~ ~[exp(ni( h
1 are all either odd or even. However, for mixed h, k , l, the structure factor is not zero but F = (,f~l- fN,), SO that the missing peaks occur as well, although with weaker intensity.
3.4 LAUE AND DEBYE-SCHERRER METHOD In principle, crystallographic orientations can be studied by means of two different approaches, which have been developed respectively by Laue et al. (1912/13) and
APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS
49
Figure 3.6 Laue diagram of an Al-l.g%Cu-single crystal (reflection technique)
by Debye and Scherrer (1916, 1917) shortly after Rontgen (1895) discovered X-radiation. Accordingly, the approaches are strongly linked to this kind of radiation, though their principles hold for other radiation as well. In Section 8.3.2 we introduce an application of the Debye-Scherrer method in electron diffraction, namely, the determination of pole figures from small volumes in the TEM. Until 1912, both the periodic lattice structure of crystals and the electromagnetic character of X-radiation were only hypotheses, before Laue et al. (1912113) were able to prove both presuppositions by one single experiment. They irradiated crystals of zinc blende and copper sulphate with a finely collimated, polychromatic (i.e. 'white') X-ray beam and recorded the transmitted intensities on a film. Rather than finding homogeneously scattered intensities, they observed a large number of spots that were symmetrically arranged in a characteristic pattern (e.g. Figure 3.6), which looked as if the crystal contained an arrangement of 'mirrors' at different angles. A few months later Bragg (1913) showed that these mirrors are formed by the various lattice planes in the crystal, which led to the formulation of Bragg's law as described in Section 3.2. To utilise the Laue method for orientation determination, a finely collimated X-ray beam is focused on the single crystal or the grain of interest in a polycrystalline sample. As in this method white, i.e. polychromatic, X-radiation is used, for each set of lattice planes ( h k l ) X-rays will exist which fulfil Bragg's condition for diffraction (Figure 3.7). The diffracted beams result in a point for each set of lattice planes in the Laue diagram, e.g. Figure 3.6. Provided the crystal structure is known, the spatial arrangement of the corresponding planes and, thus, the crystallographic orientation can be evaluated from the position of the diffraction spots.
INTRODUCTION TO TEXTURE ANALYSIS
P sample
(a) transmission geometry
(b) reflection geometry
Figure 3.7 Measurement of individual orientations according to the Laue technique; (a) transmission technique: (b) reflection technique.
Probably the discovery which has made the most momentous contribution to modern texture analysis was made by Debye and Scherrer (1917), who observed interference of X-rays in fine grained polycrystalline powders which were irradiated with monochromatic X-radiation. In a powder sample typically all possible orientations are present. so that always some crystals will have lattice planes that are oriented at the corresponding Bragg angle with respect to the incident beam (whereas in the Laue method the lattice planes 'select' the proper wavelength out of the white X-radiation). The diffracted beams lie on a cone with half apex angle 28 about the direction of the incident beam. To produce a Debye-Scherrer pattern, the sample is placed in the centre of a cylindrical camera and the diffraction cones are recorded on an X-ray sensitive film situated at the inner wall of the camera (Figure 3.8),
incident
Figure 3.8 (a) Debye-Scherrer technique to determine the crystal structure and texture of a polycrystalline sample. Diffraction cones from three lattice planes are shown intersecting the film. In (b), the whole diffraction pattern is shown on an unrolled fiim.
APPLICATION O F DIFFRACTION TO TEXTURE ANALYSIS
51
where the cones form segments of circles of radius 28 which through Bragg's law is related to the lattice spacing d. Powder samples having a random orientation distribution produce patterns with homogeneous Debye-Scherrer rings, as the number of reflecting planes is equal in all spatial directions. By contrast, in samples with a preferred orientation distribution the number of reflecting planes varies in different directions. As a result, the intensity distribution in the Debye-Scherrer rings becomes inhomogeneous, and in heavily textured samples some parts of the rings may even be completely missing (examples of such Debye-Scherrer diffraction patterns are shown in Figure 12.2 and, for electron diffraction, in Figure 8 . 4 ~ )Wever . (1924) utilised this fact to generate the first pole figures of heavily cold rolled aluminium, before pole figure goniometers and Geiger-counters were introduced in the 1940s to enable quantitative texture analysis to be performed. This is discussed in Chapter 4.
3.5 ABSORPTION AND DEPTH O F PENETRATION In Section 3.2 we saw that the various kinds of radiation - X-rays, neutrons and electrons interact with atoms in different ways. With regard to texture analysis of large crystalline arrangements, these interactions manifest themselves as substantial differences in the absorption of radiation by matter (Table 3.1). Upon transmission of radiation through a sample with thickness t the incident intensity I,,is attenuated according to: -
where p is the linear absorption coefficient (commonly given in cmp1) which depends on the type of radiation, the investigated material and the wavelength of the radiation. Often, the absorption coefficient is related to the specific weight p to yield the mass absorption coefficient pip (cm2gp1).Table 3.3 gives data of p/p for X-rays with different wavelengths as well as for neutrons in various pure metals. For alloys, (pip) can be calculated by a summation of the absorption coefficients (pip), of the individual components i with weight factors W , :
The mass absorption coefficients for X-rays are typically of the order of 100 cm2gp1 (Table 3.3), so that it readily follows from equation 3.10 that X-rays are heavily absorbed by matter. As an example, the intensity of X-radiation with wavelength 0.17902 nm (CoKa) that passes through a 10 pm thick zinc foil (p = 7.13 g/cm3) is reduced to about 50% of its initial intensity, and a 100 pm sheet would almost completely absorb the X-rays (I/Zo< 1%). Thus, the depth of penetration of X-ray in matter is limited to layers of 10-100 pm, which has a strong impact on the design of techniques for texture determination by means of X-rays (Chapter 4). Neutrons are absorbed much less by matter, and with few exceptions the mass absorption coefficients are of the order of 0.01 cm2gp1 (Table 3.3). Thus, in order to attenuate
INTRODUCTION TO TEXTURE ANALYSIS Table 3.3 Mass absorption coefficients ( p / p ) for X-rays and neutrons for different wavelengths (data for X-rays were taken from Barrett and Massalski (1980) and for neutrons from Bacon (1975)). Examples of absorption edges are underlined (Section 3.6.1).
Material (atom~c number, Z )
X-ray
X = O.07lO7nm (MoKa)
X = 0.15418nn1 (CuKa)
Neutrons
X =0.17902nm (CoKa)
X
= O.lO8nm
neutron radiation to 50% of its initial intensity as in the above example, a zinc sample with a thickness of 17cm is necessary, which means that the penetration depth of neutrons is larger by about 4 orders of magnitude than that of X-rays. Electrons, on the other hand, are much more strongly absorbed than X-rays. Here, mass absorption coefficients of are common, so that the penetration of electrons is limited to depths as low as 1 pm (Table 3.1).
3.6
CHARACTERISTICS O F RADIATIONS USED FOR TEXTURE ANALYSIS
3.6.1 X-rays
X-ray diffraction is the most established technique for texture measurement, and reveals the integral macrotexture of a volume of material from a flat specimen by measuring the intensities of diffraction maxima. As will be discussed in detail in Section 4.3, application of X-ray diffraction to measure macrotextures in a material of known lattice parameter involves the use of a monochromatic beam of radiation of a given wavelength with the Bragg angle fixed for a chosen set of lattice planes { h k l ) . If a polycrystal of this material is rotated in space, when grains in the sampled volume become oriented such that their { h k l ) planes coincide with the Bragg condition for their interplanar spacing, a diffracted intensity is measured. The direct output from an X-ray texture goniometer is a chart showing the diffracted intensity with respect
APPLICATION O F DIFFRACTION TO TEXTURE ANALYSIS
MOK a
53
Cu K a
Figure 3.9 Variation of the mass absorption coefficient p / p with wavelength X in platinum, showing the K- and L-absorption edges (Barrett and Massalski, 1980).
to the rotation angles of the specimen, which can be represented in a pole figure or used for calculation of the three-dimensional ODF (Section 5.3). In Section 3.5 we noted that the high absorption of X-rays by matter is a very important aspect of texture analysis using X-ray radiation, as it limits the penetration depth to about 100pm and, as discussed in Chapter 4, it also influences the choice of an appropriate X-ray tube. The mass absorption coefficients of most metals are in the range l 0 to 350 c m 2 g ' (Table 3.3). Beryllium has a notably lower absorption of about l cm2g-' and is therefore used as window in X-ray detectors. It is seen in Table 3.3 that the absorption generally increases with rising atomic number Z of the analysed sample material and with the wavelength X of the radiation used. For a given wavelength, however, there are steep drops in absorption by up to one order of magnitude between certain elements; some examples are underlined in Table 3.3. These absorption edges are caused by the efficiency of X-rays in causing photoelectrons to be emitted from the electron shells of the sample material. For a given material, if the wavelength of the radiation is too long (i.e. the energy is too low) to eject photoelectrons, the radiation passes through the material with quite low absorption. Waves with shorter wavelengths may have sufficient energy to eject an electron from one of the shells of the absorbing atom, however. The absorbed energy which is then released by the recombination of the electron hole is emitted asJuorescence, which leads to a much higher absorption of such radiation.
54
INTRODUCTION TO TEXTURE ANALYSIS
Accordingly, ejection of an electron from the K-shell results in K-excitation, and the discontinuity in absorption is called the K-absorption edge. As an example, the Kand L-absorption edges in platinum are shown in Figure 3.9. This effect is utilised to produce (quasi) monochromatic radiation by eliminating unwanted radiation from the X-ray spectrum by appropriate filters (Section 4.3.1). For texture analysis high absorption is undesirable, however. and hence the corresponding combinations of X-radiation and sample material should be avoided.
3.6.2 Neutrons Neutron diffraction is used to obtain an average macrotexture in a very similar manner to that for X-rays (Section 4.4). However, the low availability of the specialised facility required (mostly a beam line on a nuclear reactor) means that neutrons are only used for texture work where they can offer advantages over X-rays. Such situations include: e
Where the greater depth of penetration is an advantage, e.g. irregularly-shaped or large grained specimens, small volume fractions of second phases or porous materials
e
Where the speed of measurement related to the greater volume is an advantage, e.g. for dynamic studies of texture development For samples with low crystal symmetry or multi-phase systems
Figure 3.10 Variation of the neutron scattering amplitude with atomic number Z due to the superposition of resonance scattering and potential scattering. The regular increase for X-rays is shown for comparison (Bacon, 1975).
APPLICATION O F DIFFRACTION TO TEXTURE ANALYSIS a
55
Where the sample must not be prepared for X-ray texture analysis, i.e. nondestructive texture analysis
The elastic scattering of thermal neutrons by crystalline matter consists of two contributions, nuclear and magnetic scattering. Nz~clear.scattering is caused by the interaction between the neutrons and the atomic nuclei. These interactions lead to diffraction effects akin to the diffraction of X-rays and are utilised for analysis of the crystallographic texture. However, there are some characteristic differences between X-ray and neutron diffraction which warrant mentioning at this stage. Since neutrons are mainly scattered by the atomic nucleus, interaction is much weaker than the interaction of X-rays with the shell electrons. As we have seen in Section 3.5, this leads to a much lower absorption and much higher penetration depth. Furthermore, for X-rays the scattering amplitude fx increases in proportion to the atomic number Z (Figure 3.10), whereas for neutrons no such simple dependency exists. Here, the nuclear scattering consists of the potential scattering (scattering of a and the resonance planar wave at a rigid sphere), which increases with scattering, which yields an irregular dependence on Z (Figure 3.10). Because of resonance scattering different isotopes of the same elements may have different scattering amplitudes and can consequently be separated with neutron diffraction. On the other hand, signals from light elements are of similar magnitude to those from heavy ones, so that they can also be detected by neutron diffraction. Whereas the scattering amplitude of X-rays depends strongly on the diffraction angle 0, that of neutrons is virtually independent of G (Figure 3.2), though the overall intensity still decreases somewhat with 0 because of thermal vibration and other effects. Therefore, peaks at higher diffraction angles can be evaluated better by neutron diffraction. Furthermore, most neutron diffraction instruments provide better spectral resolution than X-rays which is advantageous for deconvoluting complex diffraction patterns with overlapping peaks such as in multi-phase systems or low-symmetry materials (Section 4.5). Magnetic scattering results from the classical dipole-dipole interaction of the magnetic momentum of the neutrons with the electrons of the atomic shell when these have a resulting magnetic momentum (e.g. Bacon, 1975; Szpunar, 1984; Bunge, 1989a). In general, the intensity of the magnetic scattering is much weaker than that of the nuclear scattering and can be neglected for texture analysis. However, in some magnetic materials, e.g. Fe and Mn, magnetic scattering can give rise to peaks in the diffraction pattern, which can be exploited to derive magnetic pole figures, i.e. the distribution of the magnetic dipoles in the sample. However, the magnetic contribution to diffracted neutrons is always superin~posedon the contribution from the nuclear scattering, i.e. the crystallographic texture, which means that the magnetic pole figures have to be separated from the crystallographic pole figures (Miicklich et al., 1984; Zink et al., 1994; Birsan et al.: 1996). 2 ' 1 3 ,
3.6.3 Electrons Electron diffraction is employed in a fundamentally different way from X-ray or neutron diffraction for studying texture. This is a consequence of firstly the fact that
56
INTRODUCTION TO TEXTURE ANALYSIS
electrons are the only radiation listed in Table 3.1 to carry a charge and secondly their penetration depth and interaction volume is small enough to allow diffraction from individual grains rather than volumes containing many grains as for X-ray/ neutron diffraction. The charge on electrons allows them to be deflected by magnetic lenses and scanned in a raster, and so to be used to produce highly magnified images in the transmission electron microscope (TEM) and scanning electron microscope (SEM). This means that diffraction information from discrete sample volumes which are sub-micron in size can be obtained concurrently with images of the same region. Hence, electron diffraction is used to obtain the orientations of individual grains, i.e. microtexture, rather than the macrotextures obtained by X-raylneutron diffraction experiments. The wavelength X of the electron radiation is controlled by the accelerating voltage U of the electron microscope; it can be computed from the de Broglie relation (equation 3.1) under consideration of the relativistic mass increase of the electrons: X=-
k JSU=
h J2m0 eU(1
+ eU/(?rno c'))
(3.12)
where h is Planck's constant, e is the charge on an electron, m is the electron mass, nzo is the electron rest mass and c is the velocity of light. For typical accelerating voltage values of U = 20 kV in an SEM and U = 200 kV in a TEM the wavelengths are X = 0.00859nm and X = 0.00251 nm respectively (Table 3.1). Because of these small wavelengths - and the strong dependence of the diffracted intensity on the scattering angle (Section 3.2, Figure 3.2) - the Bragg angles BB for reflection of electrons (equation 3.4) are typically very small. Even in the case of high-index planes the angles reach only approximately two degrees. This means that the reflecting lattice planes are always situated approximately parallel to the primary electron beam. When the electron beam in an electron microscope is directed onto a metallic sample, different scattering events and interactions with the sample material give rise to a variety of different electron and X-ray signals (Figure 3.1 1). These signals are used to produce the images in electron microscopes, and they also provide information on the chemical composition and, with regard to the subject of this book, on the crystallography and crystallographic orientation of the sampled volume. Part of the primary electron signal is scattered elastically with no energy loss. Elastically backscattered electrons are used for imaging purposes, as their intensity yields information on the atomic number Z of the sample material (Zcontrast) and thus on the chemical composition of the sampled region, e.g. of particles, etc. In transmission, coherent elastic scattering of electrons at crystalline objects gives rise to selected area diffraction (SAD) patterns which can be used for orientation determination (Section 8.3). Electrons which are inelastically scattered at the atomic shell are subject to energy loss and absorption or they induce emission of secondary electrons, Auger electrons or X-rays. Secondary electrons with an energy less than 50 eV come from a depth of only a few nanometres and are consequently very well suited for imaging the sample topography in an SEM with high spatial resolution. Auger electrons with energies
APPLICATION OF DIFFRACTION TO TEXTURE ANALYSIS incident electron beam
Back reflection (SEW
backscattered electrons (EBSD-, SAC-pattern,
secondary electrons (imaging) Auger electrons (surface analysis)
Sample
Transmission (TEW
(Kossel-technique, chemical analysis)
absorbed electrons
J
elastically scattered electrons (SAD-point diagrams, dark field imaging)
\
1
I
X-rays (Kossel-technique, chemical analysis)
inelastically scattered electrons (Kikuchi-diagrams)
transmitted electrons (bright field imaging) Figure 3.11 Summary of the various signals obtained by interaction of electrons with matter in an electron microscope.
up to 2 keV are formed in the furthest surface layers (-1 nm). As their energy depends on the emitting element they can be used for surface analysis. X-rays are typically used for chemical analysis, but they also yield information on the crystallographic orientation since they can be diffracted at the crystal lattice to give Kossel diffraction (Section 7.2). Multiple scattering of electrons in somewhat thicker sample regions gives rise to Kikuchi patterns, which are very well suited for orientation determination in the TEM, as discussed in Section 8.4. In the SEM, similar diagrams can be produced by selected area channelling (SAC, Section 7.3) and by electron back-scatter diffraction (EBSD, Sections 7.4-7.8).
3.7 SUMMATION Most techniques for texture analysis rely on diffraction and Bragg's law. Since the angle between the incident and diffracted beam cannot exceed 90°, it essentially follows from Bragg's law that the wavelength X of the incident radiation must be of the same or smaller order as the lattice spacing d or, more precisely, X 2d (equation
<
-
INTRODUCTION TO TEXTURE ANALYSE
58
Neutrons
X-rays (synchrotron)
I
Electrons
,
I
I diffractorneter
r
diffractometer
I DebyeScherrer
:
Macrotexture
Laue
A
SAC
EBSD
7-7
SAD
Kikuchi
Microtexture
Figure 3.12 Categorisation of the mainstream techniques for texture determination according to the radiation used as a probe. Macrotexture and microtexture methods are on the left- and right-hand parts of the diagram respectively.
3.4). Comparing the wavelengths of different types of radiation in Table 3.1, it is clear that X-rays, neutrons and electrons are diffracted by lattice planes whereas llght is not. Besides a categorisation of the techniques for texture determination with respect to the type of radiation used as a probe X-rays, neutrons or electrons - the techniques can also be classified on the basis of whether it is a macrotexture. i.e. averaged orientation data from many grains (Section 1.1), or a microtexture, i.e. single orientations (Section 1.2), method. Figure 3.12 shows all the mainstream techniques for texture measurement by diffraction. There exists also a few peripheral techniques such as acoustic, ultrasonic, magnetic or optical methods, some of which are described in Chapter 12. With regard to the usage of the techniques featured in Figure 3.12, X-ray goniometry and EBSD are by far the most popular choices for macrotexture and microtexture analysis respectively. All the techniques in Figure 3.12 are discussed in this book, with the extent of the coverage depending on the relevance to modern texture analysis. The choice of method(s) used to obtain orientation information will depend ultimately on factors such as the scale and resolution of the inquiry, the nature of the information sought, and the equipment available. -
PART 2 MACROTEXTURE ANALYSIS
4. MACROTEXTURE MEASUREMENTS
4.1 INTRODUCTION In this Chapter, we introduce the common techniques to derive the distribution of crystallographic orientations present in a crystalline sample, i.e. its macrotexture, by means of X-ray and neutron diffraction. As mentioned in Section 3.4, X-ray diffraction was first employed in 1924 by Wever to investigate preferred orientations in metals by evaluating the inhomogeneous intensity distribution along the DebyeScherrer rings. With the introduction of the texture goniometer and the use of Geiger counters by Decker et al. (1948) and Norton (1948) pole figures could directly be recorded, and in particular the fundamental work by Schulz (1949a,b) initiated modern quantitative X-ray texture analysis. The principles of pole figure analysis by diffraction methods and the techniques and procedures to obtain quantitative texture data by means of X-ray diffraction are described in Sections 4.2 and 4.3. Since the 1960s neutron diffraction has also been employed regularly to determine crystallographic textures, and the corresponding methods are discussed in Section 4.4. The principles of texture analysis by X-ray or neutron diffraction apply for all materials, independent of their crystal structure. However, when investigating materials with low crystal symmetry and/or multi-phase systems, some specific additional problems arise which are addressed in Section 4.5. 4.2 PRINCIPLE O F POLE FIGURE MEASUREMENT The principle of pole figure measurement by means of diffraction techniques is based on Bragg's law for reflection of radiation, X-rays or neutrons, at the crystal lattice planes (equation 3.4). As each set of lattice planes has a different lattice spacing d, reflections from various sets of lattice planes can be distinguished by setting the detector to the corresponding angle 28 with respect to the incident radiation. To illustrate this. Figure 4.1 shows the diffraction spectrum of a titanium sample with almost random texture. Such spectra. that are well-known from powder diffractometry, can be obtained by recording the reflected intensities as a function of the diffraction angle for gradually increasing angles 0 and 20, Provided the crystal
INTRODUCTION TO TEXTURE ANALYSIS
loo
0
reflection angle 8 Figure 4.1 8/28 diffraction spectrum of a titanium powder sample (CuKa tube; BG: background, see Section 4.3.6).
structure of the material is known. the individual peaks can readily be indexed with the help of Bragg's law plus consideration of the rules of extinction (Section 3.3). To derive the crystallographic orientation of a given crystallite, the arrangement. with respect to an external reference frame, of a set of lattice planes (hkl), has to be determined. First we consider reflection at a single crystallite as schematically shown in Figure 4.2. The crystal is irradiated with monochromatic radiation at the proper Bragg angle for reflection at the lattice planes ( M ) ,and the detector is set at the angle 20 with respect to the primary beam. Of course, a reflected intensity is only measured if the corresponding lattice planes are arranged such that they lie parallel to the sample surface, i.e. their normal is the bisector of the angle between incident and reflected beam, as shown in Figure 4.2b, which means that in most cases no reflection is obtained (Figure 4.2a). In order to ensure reflection from other lattice planes, the sample has to be rotated and/or tilted until the lattice planes are in reflection condition, i.e. parallel to the sample surface. The necessary rotation and tilt angles are a measure for the arrangement of the lattice planes within the crystal, which means they are characteristic of the orientation of the crystal with respect to
MACROTEXTURE MEASUREMENTS source
counter
&X
Iathce planes (hkl)
source
counter
Figure 4.2 Sketch to illustrate the effect of sample rotation of the arrangement of the lattice planes. (a) untilted position (a = R = OC);(b) sample tilted such that the lattice planes are in Bragg condition (a > 0'. B > 0'): (c) reflection peaks of a Goss-{llO) < 001 > oriented crystal in the stereographic projection at (i) a = 35", B = 903, (ii) a = 35", ,B= 270", (iii) a = 90'. 8 = 0".
the external sample frame. As an example, when we irradiate a Goss - i.e. oriented crystal at the Bragg angle for reflection from the (200){110)<001> planes, reflected intensities will be obtained for a sample tilt of 35" plus an additional rotation by either 90" or 270" (a third peak would to be expected at a sample tilt of 9OC,but in that case the incident beam is parallel to the sample surface so that no reflection can occur; Section 4.3.2). In order to determine an unknown crystal orientation in practical applications, the sample is systen~aticallyrotated in a texture goniometer about well-defined angles in such a way that all possible lattice planes are successively brought into the reflection condition and the reflected intensities are recorded as a function of these rotation angles. As we will see later, the rotation angles are directly related to the pole figure angles o (radial) and P (azimuthal), so that the reflected intensities can readily be represented in a pole figure (Section 2.4.1). For polycrystalline samples, which are composed of an array of single crystallites, the intensity recorded at a certain sample orientation is directly proportional to the number of crystallites -more precisely, to the volume fraction of crystallites - which currently is in reflection condition (we need to qualify this statement later in this section). Thus, the pole figure of a polycrystalline sample reflects the distribution of orientations in that sample, i.e., its crystallographic texture. As an example, Figure 4.3 shows the (111) pole figure of aluminium, after rolling to 97% thickness reduction, plotted in the stereographic projection together with the definition of the pole figure angles cu and P (Section 2.4.1). Two different geometrical set-ups of texture goniometers are conceivable: transmission and reflection geometry. Determination of texture can be performed on thin samples which are penetrated by the neutrons or X-rays; this tvansmissiorz -
INTRODUCTION TO TEXTURE ANALYSIS
Figure 4.3 (111) pole figure of 97% cold rolled aluminium with the definition of the pole figure angles a and 3. The orientation densities are given by iso-intensity lines in multiples of a random orientation distribution.
8
=
nter
(a) transmission geometry
(b) reflection geometry
Figure 4.4 Diffraction in a four-circle texture goniometer with definition of the instrument angles
geometry is shown in Figure 4.4a (Decker et al., 1948; Schulz. 1949b). Note that in Figure 4.4 we use a different nomenclature for the rotation axes than in Figures 4.2 and 4.3. The nomenclature with the axes Q, 4, X and W is standard in single crystal diffractometry and marked on most diffractometers. We will see in Section 4.3.2 how these angles are related to the pole figure angles a and 9. As discussed in Section 3.5,
MACROTEXTURE MEASUREMENTS
65
neutrons are readily able to transmit samples of several centimetres thickness and, therefore, texture analysis by neutron diffraction is generally performed in transmission geometry. X-rays, in contrast, are strongly absorbed by matter, and so the transmission method is generally restricted to materials with low absorption, e.g. magnesium or aluminium, and to extremely thin samples with thickness below approximately 100 Fm. However. with sufficiently strong X-ray tubes sheets with thickness up to several millimetres can transmit X-rays, which has been utilised for on-line texture control of steel sheets in industrial applications (Kopinek, 1994). Besides such exceptions, X-ray texture analysis is mostly performed on thick bulk samples with a plane surface in reflection geometry, as shown in Figures 4.2 and 4.4b (Schulz, 1949a). In the above discussion of the principle of pole figure measurements we have stated that the intensity recorded at a certain sample position is proportional to the volume fraction of crystallites in reflection condition. This statement is oversimplified, however, and needs three supplements: 0
The pole figure data need several corrections which will be described in Section 4.3.5
o
As we have seen in Section 2.4.1, to characterise unambiguously a given orientation in a pole figure, the positions of at least two poles are required. In turn this means that a pole figure does not provide the complete orientation distribution but only the distribution of the poles
e
The reflections hkl and hkl from the opposites sides of a given lattice plane have equal intensities. This is caused by the fact that the diffraction averages over the volume and it is irrelevant whether the beam impinges a lattice plane from the front or the back side, which is known as Friedel's law (Friedel, 1913). Therefore, pole figures measured with diffraction techniques are always centro-symmetric, i.e. the symmetry contains an inversion centre, even if the crystals are noncentric. As we will see later, this also causes the 'ghost error' during computation of the ODFs from pole figure data (Section 5.3.1)
4.3 X-RAY DIFFRACTION METHODS A complete X-ray texture goniometer system is composed of the following main units: 0
X-ray generator and X-ray tube
o
goniometer with a sample stage
0
detection system
Electronics and computer control to control the rotations and read the recorded intensities are also required. Figure 4.5 shows a photograph of the set-up which has been developed at the Institut fiir Metallkunde und Metallphysik, RWTH Aachen (Hirsch et al., 1986). Nowadays, fully functional texture systems are commercially
66
INTRODUCTION TO TEXTURE ANALYSIS
Figure 4.5 Texture goniometer at the Institut fur Metallkunde und Metallphysik, RWTH Aachen
available from several vendors. In the following parts of this Section, the various units will successively be addressed. As already mentioned earlier, X-ray pole figure measurements are mostly performed in reflection geometry today. Accordingly, we will mainly focus on reflection geometry, and transmission geometry is only occasionally addressed for completeness.
4.3.1 Generation of X-rays In most applications, conventional X-ray tubes with a particular metal as anode material are used as an X-ray source. Rotating anode generators produce 5 to 10 times higher intensities than ordinary vacuum tubes but are much more expensive, and require considerable maintenance. Hence, they are not often used for diffraction experiments. X-ray tubes produce a continuous wavelength spectrum (Bsemsstsahlung) which - at sufficiently high accelerating voltages of several tens of Kilovolts - is superimposed by several sharp intensity maxima. As an example, Figure 4.6 shows the spectrum of a MO-tube. Since these peaks are extremely narrow and since their
MACROTEXTURE MEASUREMENTS
wavelength h [A] Figure 4.6 Emission spectrum of a molybdenum X-ray tube and absorption spectrum of a zirconium filter (Barrett and Massalski. 1980).
wavelengths are characteristic of the particular anode material. but do not depend on the accelerating voltage, they are called characteristic lines. Table 4.1 lists the wavelengths of the characteristic KO and K/3 radiation of some typical anode materials. It is seen in Figure 4.6 that the Ka peak has an intensity which is about l00 times higher than the wavelengths of the adjacent continuous spectrum, and Table 4.1 Characteristics of various X-ray tubes and appropriate filters (data from Int. Tables for Xray Crystallography. 1985) Wavelength [nm]
Kd-filter
Anode material
Ka
K3
Material
Edge wavelength [nml
Thickness for K3IKa = 1 : 500 [pm]
Loss in Km [%l
Cr Fe CO Cu Mo Ag
0.22909 0.19373 0.17902 0.15418 0.07107 0.05609
0.20848 0.17565 0.16208 0.13922 0.06323 0.04970
V Mn Fe Ni Zr Rh Pd
0.22690 0.18964 0.17433 0.14880 0.06888 0.05338 0.05092
17 18 19 23 120 90 92
51 53 54 60 71 73 74
68
INTRODUCTION TO TEXTURE ANALYSIS
about 5-10 times higher than the KP peak. This clearly suggests the use of the characteristic K a radiation for X-ray diffraction experiments. For most techniques of texture analysis monochromatic radiation is required, but, besides the strong Ka peak, the X-ray spectrum also contains the KO peak as well as the continuous spectrum. The K,3 peak can substantially be reduced by applying a filter in front of the detector, which is made of a material whose K-absorption edge (Section 3.6.1, Figure 3.9) lies just between the K a and the K3 peak of the tube material. For a molybdenum tube, zirconium represents the right filter material, and the absorption edge of zirconium as well as the resulting filtered X-ray spectrum are included in Figure 4.6. Suitable filter materials for other tubes are given in Table 4.1. As an example, following the discussion in Section 3.5, for the commonly used copper tube the intensity of the K,!?radiation is reduced to less than 1% if radiation passes through a 23 pm nickel foil (pip = 275cm2g-l), whereas K a is only attenuated to 40%. Considering further that the intensity of K,!? has only 10-20% of the intensity of the K a peak in the first place. this means that an appropriate K,3 filter reduces the ratio of K@ to Ka to about 1:500, which gives sufficiently 'monochromatic' radiation for most texture applications. Thicker filters would further improve the ratio of Kp to Kol, but of course they would also attenuate the intensity of the desired K a radiation. Continuous contributions to the X-ray spectrum remain at wavelengths larger than Kn and at very short wavelengths (Figure 4.6). Although these intensities are relatively small compared to the Ka intensity, they may become significant in the case of very sharp textures, e.g. for diffraction from single crystals or from thin films that are superposed on those from a substrate crystal. This is illustrated for a 400 nm thin film of the high temperature superconductor yttrium-barium-cuprate (YBaCuO) which had been deposited on a lanthanum-aluminate (LaA103) substrate crystal (Wenk, 1992). The pole figure measured with truly monochromatic CuKnradiation (Figure 4.7b) contains much less peaks than the pole figure of the same
Figure 4.7 (102) pole figure of a 400nm YBaCuO film on a LaAIOi substrate measured with (a) K3filtered CuKa-radiation; (b) truly monochromatic CuKcu-radiation. Note additional peaks in (a) which are due to diffraction with different wavelengths on the substrate (Courtesy of H.R. Wenk).
MACROTEXTURE MEASUREMENTS
sample measured with nickel-filtered CuKa-radiation (Figure 4.7a). The additional poles arise from diffraction by the substrate with some wavelengths of the continuous spectrum, and thus they are artefacts. Truly monochromatic radiation can be obtained by using single crystal monochromators, which yield radiation of a certain wavelength in accordance with Bragg's law (equation 3.4). With a mosaic graphite rnonochromator, intensities of the effective beam are only slightly lower than for &%filtered X-rays. As a monochromator also reduces the background intensity, the signal-to-background ratio is even better than for filtered radiation. The monochromator is best inserted in the primary beam between the X-ray tube and the collimator, since such a geometry causes no problems with defocusing during pole figure measurements. If the monochromator is inserted in the reflected beam, i.e. between sample and detector, the X-ray path must be controlled by Soller slits which reduce the intensity. As already addressed in Section 3.5, X-ray texture analysis is strongly influenced by the high absorption of X-rays in matter, as this accounts for the penetration depth of X-rays in the sample and it also influences the choice of an appropriate anode material of the X-ray tube. In general, absorption increases with rising atomic number Z of the analysed sample material (Table 3.3). Furthermore, it increases with the wavelength of the X-ray tube used, which means that X-rays with short wavelengths (silver, molybdenum or copper anode) are typically less absorbed than those with large wavelengths (cobalt, iron or chromium anode). However, there are some combinations of sample and target materials which yield anomalously high absorption, as in these instances the X-ray photons have a sufficiently high energy to yield fluorescence (Section 3.6.1). Since high absorption is generally undesirable for texture analysis, such unfavourable combinations of X-ray tubes and sample material should be avoided. For example, steels or materials containing cobalt and manganese should not be analysed with CuKa-radiation (Table 3.3). Another important aspect of absorption is the depth of X-ray penetration. The mass absorption coefficients for X-rays are typically of the order of 100cm2g-' (Table 3.3), and for transmission geometry we have already discussed in Section 3.5 how the intensity of X-rays passing through a material with a given absorption coefficient can be derived (equation 3.10). In reflection geometry the situation is more complicated. Here the question is: from what depth of the specimen does the diffracted information come? For the untilted sample position, the fraction G, of the total diffracted intensity which is contributed by the surface layer of depth x can be estimated from (Cullity, 1978): G,
=
1 - exp(-2pxl sin 8)
(4.1)
(The factor 2 considers that the radiation has to pass through the layer twice, namely, before and after the reflection.) If we now arbitrarily define that a contribution from the surface layer of 95% (or 99%) of the total diffracted intensity is sufficient, then x is the depth of penetration. It is seen that the penetration depth increases with Bragg angle 0, and it would also decrease with increasing sample tilt a. Figure 4.8 shows the evolution of G, with depth X in aluminium for three different X-ray tubes i.e. three different absorption coefficients (Table 3.3) for a Bragg -
-
INTRODUCTION TO TEXTURE ANALYSIS
Figure 4.8 Evolution of G, with depth x in aluminium for three different X-ray tubes (reflection angle B = 20°, normal incidence).
Table 4.2 Depth x (in pm) which leads to G,-values of 95% for different X-ray tubes and materials analysed in reflection geometry at a Bragg angle B = 20"
angle Q = 20" and normal incidence. i.e. for a non-tilted sample (a = 0"). If we consider CoKa- or CuKa-radiation. it turns out that 95% of the reflected intensity comes from a layer with thickness of 25 pm or 39 pm respectively, which typically in recrystallised samples would mean that only one or two grain layers are recorded. Furthermore, this information is again heavily weighted; when 95% of the recorded information stems from a depth of 39 pm, then 50% of that information originates in the first 8.4 pm. Some further examples of sample thicknesses calculated for G-values of 95% are listed in Table 4.2. Note that fluorescence leads to very low G-values of only 2 pm if iron is analysed with CuKa-radiation.
4.3.2 Pole figure diffractometry in the texture goniometer
A pole figure goniometer essentially consists of a four-axis single crystal diffractometer. The goniometer positions the detector with respect to the incident X-ray beam at the proper Bragg angle 28. In the widely used Eulerian cradle (Figures 4.4, 4.5) the sample is positioned relative to the X-ray beam by rotations about the
MACROTEXTURE MEASUREMENTS
71
three perpendicular axes Q,X and w, the W-axiscoincides with Q (Figure 4.4). Stepper motors, controlled by a personal computer, allow positioning of the gonioineter to any arbitrary angular position of the four axes 28, w, X and (within a certain range to avoid mechanical collisions). In reflection geometry, a sample with a flat surface is mounted on the sample holder with its normal direction parallel to the axis of the $-rotation (Figure 4.4b). Then, it is rotated in its plane about the &axis, so that the angle q5 corresponds to the azimuth p of a pole in the pole figure. After one full rotation, the sample is tilted about the x-axis, but as X and the pole figure radial angle a are defined in the opposite direction, the relation is a = 90" - X. For most applications w is kept constant at 0°, so that the X-circle is symmetrical between incoming and diffracted beams, i.e. it is positioned at the Bragg angle Q. This arrangement is referred to as the Bragg-Brentano focusing condition for reflection geometry. w-scans (rocking curves) can be used to assess the quality of crystals and to derive exact peak shapes (e.g. Heidelbach et al., 1992; Sous et al., 1997). For special applications, e.g. for texture measurements in thin films, it may be advantageous to use a low incident beam angle to increase the path length of the Xrays in the specimen (grazing beam technique, e.g. Bowen and Wormington, 1993; Van Acker et al., 1994). In that case, the X-circle is no longer syminetrical and w is the deviation from the bisecting position. Although the limiting value for a in the Bragg-Brentano focusing condition is obtained when both the incident and reflected beams are parallel to the sample surface, so that a,,, < 90°, there are several aspects that usually limit a,,, to values of 60-85". Even when we analyse a sainple with a perfectly uniform 'random' texture, the intensity of the reflected peak strongly decreases with a . With increasing tilting angle a, the intersection of the incident X-ray beam with the sainple - i.e. the projection of the beam onto the sample surface becomes increasingly elongated; furthermore, different incident angles Q add an additional distortion of the projected beam (Figure 4.9a). Figure 4.9b shows the area irradiated by an originally squareshaped X-ray beam on the sample surface for different angles a and d. Due to the penetration depth of the X-rays into the sample the true geometry of the reflecting volume is even more complicated. The projected area on the sample surface. and therewith the region from where X-rays are reflected, increases with increasing sample tilt a . It also increases with an increasing deviation of the Bragg angle Q from the Q = 90" position and, therefore, pole figures should not be measured at 0 < 10". To achieve optimum angular resolution, diffractometers are constructed such that reflection of the - originally divergent incident X-ray beam at the plane surface of an untilted sample ( a = 0") brings the diffracted beam back into a focal point that coincides with the receiving slits of the detector. Tilting of the sample to angles a > 0" has the effect that half of the sample is behind and half of it in front of its original plane. For those sample regions, however, the focusing condition is no longer exactly fulfilled; which leads to a decrease in intensity simultaneously with a broadening of the reflected peaks. Accordingly, this effect is called defocusing error. Figure 4.10a shows the peak broadening for two diffraction peaks (1 11 and 200) of a randomly oriented copper powder sample for three different tilting angles a. Although the integral intensity of the peak remains constant the detector, which is -
-
INTRODUCTION TO TEXTURE ANALYSIS
sample
Figure 4.9 (a) Sketch to visualise the distortion of the irradiated spot on the sample surface; (b) Shape of the irradiated spot in dependence on tilting angle a and Bragg angle 8 for a square-shaped incident beam.
equipped with a system of receiving slits, records only the intensities diffracted from a small, constant area of the sample surface, so that the recorded intensity decreases with increasing sample tilt (Figure 4. lob). Measures to correct for the intensity drop due to defocusing will be given in Section 4.3.5. Besides the defocusing error, absorption of the X-rays within the sample has to be considered; with increasing sample tilting the path length of the X-rays in the sample increases, which means that the reflected intensity decreases due to absorption. However, for thick samples this increase in absorption is exactly balanced by the increase in irradiated volume that contributes to the diffracted intensity (Schulz, 1949a). Note that 'thick' samples means that the sample is thicker than the X-ray penetration depth which, as we have seen above, is usually only of the order of 100 pm. However, this does not hold for
MACROTEXTURE MEASUREMENTS
incident
sample
detector
Figure 4.10 (a) Peak broadening with increasing tilting angle a (copper powder sample, CuKa radiation); (b) Sketch to visualise the defocusing error.
very thin samples, e.g. for thin films, and in such cases absorption has to be corrected for (Section 4.3.5). In transmission geometry, the sample holder is turned 90" such that the sample, e.g. a thin sheet, is mounted normal to the x-axis (Figure 4.4a). Here, the angle $ is kept constant at 0" and the sample is tilted about the W-axis, which corresponds to the complement of the pole distance (90" - a ) in the pole figure. The azimuthal angle P in the pole figure is obtained from a rotation of the sample about the x-axis. In a 90' are transmission geometry, the outer parts of the pole figure with a,,,
< <
74
INTRODUCTION TO TEXTURE ANALYSIS
covered. As described for reflection geometry, the recorded intensity strongly decreases with sample tilt LJ, so that the practical minimum of a is reached before the theoretical limit, which is given when the incident or reflected beam is parallel to the sample surface, so that W < 90" - Q or a.,,i, = B. In transmission geometry. the increase in absorption with sample tilt is not compensated for by the increase in the reflecting volume (Decker et ul., 1948; Schulz, 1949b), so that an absorption correction is always necessary (Section 4.3.5). We have seen that for pole figures measured in reflection geometry the errors strongly increase with a . The inner circles of a pole figure (e.g. a < 45") can practically be obtained without corrections whereas the outer parts need strong corrections ( a > 45") or are even not accessible at all ( a E 90"). In transmission geometry vice versa the outer parts of a pole figure can readily be determined and the inner circles are missing. Accordingly, conzplete pole ,figures can be obtained by combination of two pole figures, one of which is measured in reflection and one in transmission geometry. Subsequently, the pole figure data are overlapped, which often requires some intensity adaptations to bring the two measurements to the same intensity scale. However, because of the laborious preparation of samples for transmission texture measurements, nowadays in most cases only incomplete pole figures are measured in reflection with tilting angles up to typically 60"-85". If necessary, complete pole figures are derived by re-calculation from the threedimensional orientation distribution functions (Section 5.3). Besides the rotations to scan the pole figure described so far, the sample can be subjected to an additional translation in its plane to increase the sampling area (Bunge and Puch, 1984; Section 4.6).
Older texture goniolneters were driven by only one motor, and the pole figures were scanned by changing both angles a and P simultaneously on a spiral (Figure 4.1 la).
Figure 4.11 (a) Spiral and (b) circular, equal angle scanning grid for pole figure measurements (note that for clarity both scanning schemes are shown in steps of 20").
MACROTEXTURE MEASUREMENTS
75
In modern stepper motor driven goniorneters, the pole figures are mostly scanned on equidistant concentric circles of constant a in steps of typically 5" both in a and ,3 (Figure 4.1 1b). For a complete pole figure this results in ( ax,3) = 19 X 72 = 1368 points. At counting times of typically one second per point this yields a total measuring time of less than 30min per pole figure. For very sharp textures, the standard 5" X 5" grid may be too coarse and finer grids, e.g. 2.5" X 2.5", are required. Also, for calculation of ODFs according to the series expansion method (Section 5.3) with high series expansion coefficients I, finer grids may be necessary to yield a sufficient number of data points. For each point ( a ,,3) the reflected intensity is measured by an electrical counter (Section 4.3.4) which 'sees' a certain angular range of the pole figure. This polefigure window is defined by the aperture systems of X-ray tube and counter, and especially by the receiving slits of the latter, but it also depends on the geometry of the set-up, i.e. reflection or transmission geometry, and on the settings of Bragg angle Q and sample tilt a. Usually, the pole figure window has an elongated shape, as schematically shown in Figure 4.12. With regard to the pole figure scanning, two different modes are possible - step scanning and continuous scanning (Bunge and Puch, 1984). In the step mode the sample is rotated by a certain angle Ap, typically 5", either on a spiral or o n concentric circles, and then at this position the reflected intensity is determined by counting for a given time, e.g. one second. In the continuous mode the intensities are recorded continuously, thus integrating over an angular interval A,6; and subsequently the measured intensity is attributed to the centre point P of that interval. In modern texture goniometers the rotations are performed by stepper motors, which can be moved in small steps of say 0.01" so that with a suitable computer control both scanning modes can readily be accon~plished. In should be noted that in the step mode intensities are missed and sharp peaks may even entirely be skipped, as the step width is typically larger than the pole figure window A,6 (Figure 4.12). Continuous scanning, on the other hand, records all intensities in a certain range of AD. As the length Acr of the pole figure window
Figure 4.12 Shape of the pole figure window for reflection and transmission geometry
INTRODUCTION TO TEXTURE ANALYSIS
180" Figure 4.13 Equal area scanning grid for pole figure measurements (Adapted from Brokmeier. 1989).
concurrently integrates over a range A a , continuous scanning ensures a rather comprehensive coverage of the entire pole figure. However, continuous scans are restricted to homogeneous scanning grids like spirals or concentric circles, whereas more complex scanning grids. e.g. equal area scans (see below), require step scanning. In the standard equal angle ( a X P) 5" X 5" grid the density of measuring points per area is very inhomogeneous, as the pole figure centre is covered with a much higher density in 2 than the outer circles (see Figure 4.1 lb). To cover the pole figure more uniformly, an equal area scan can be applied, and various schemes to achieve an equal area coverage have been proposed in the literature (e.g. Welch, 1986; Will et al., 1989; Morris and Hook, 1992; Matthies and Wenk, 1992). All these schemes have in common that they are close to a true equal area coverage (Appendix 4) and reduce the number of measurement points by about one half (e.g. Figure 4.13). Thus, for a given angular resolution, the recording time can be halved - which is particularly interesting if measuring time is of concern (e.g. in neutron diffraction, Section 4.4) or, allowing the same total measuring time, the angular resolution can be doubled. If necessary. the data for the conventional 5" X 5" grid, or for any other grid, are subsequently obtained by interpolation. 4.3.4 X-ray detectors In modern computer-controlled texture goniometers, the reflected X-ray intensities are recorded by means of an appropriate electrical counter. There are four different types of counters suitable for X-ray diffraction experiments: e
Geiger counters
e
Scintillation counters
MACROTEXTURE MEASUREMENTS
e Proportional counters e
Semiconductor counters
All these counters depend on the power of the X-rays to ionise atoms, either of a gas (Geiger and proportional counters) or of a solid (scintillation and semiconductor counters). The working principles and characteristics of these four types of X-ray counters are described in Appendix 5. Geiger counters were the first electrical counter used to detect radiation, but are now virtually obsolete in diffractometry. In all counters that are commonly used today - scintillation, proportional and semiconductor the output voltage is proportional to the energy of the triggering X-ray quanta. Therefore, electrical units that distinguish between pulses of different size, so-called pulse-height analysers, can distinguish between X-rays of different energy. As discussed in more detail in Appendix 5, with the use of pulse-height analysers background and fluorescent radiation with different wavelengths can efficiently be reduced, so that the peak-tobackground ratio can markedly be improved. When we analyse the diffraction spectra of materials with high crystal symmetry, cubic or hexagonal, the diffraction lines are usually widely spaced (e.g. Figure 4.1). Thus: even considering some peak broadening at large tilting angles (defocusing error, Sections 4.3.2 and 4.3.5), the reflected peaks can generally easily be separated. In some cases, finer receiving slits may be necessary to improve angular resolution (e.g. to separate the 0002 and 1011 reflections in Figure 4.1). However, this is not necessarily true in samples with line-rich diffraction patterns, e.g. materials with low crystal symmetry or multi-phase alloys, where peaks may partially. or even completely, overlap (Section 4.5.1). In such cases, the peaks can no longer be separated, and peak profile analysis becomes necessary to attribute unambiguously the recorded intensities to the individual hltl-reflections. To register the peak profiles with conventional counters, the 0120-spectrum must successively be scanned with a small receiving slit for each pole figure point ( a , 3) to achieve sufficiently high angular resolution. This is an extremely time-consuming procedure, which can be accomplished much more easily with the help of a location sensitive counter that records the entire peak profile simultaneously (e.g. Heizmann and Laruelle, 1986; Wcislak et al., 1993). A typical one-dimensional position semitive detector; PSD, suitable for texture analysis consists of a location sensitive proportional counter with a resistance wire anode (Figure 4.14a). An incoming X-ray quantum generates an electrical charge Q at the point x of the anode. This electrical charge Q has the possibility to flow off in both directions to the left and the right preamplifier. However, due to the resistivity of the wire the partial charges Q[ and Q, flowing to either side generate different voltage pulses U, and U, with the ratio 0;./U, = x / L - X, where L is the active length of the resistance wire. Thus, the location x of the incoming X-ray quantum can be detected electronically and the electrical pulse is accumulated in the corresponding channel of a multi-channel analyser (MCA). So, eventually, the contents of the various channels of the MCA reflect the intensity distribution of the diffraction peak at different locations of the PSD. In most applications, short linear detectors are used which are mounted tangential to the 20-goniometer circle (a set-up where the PSD -
INTRODUCTION TO TEXTURE ANALYSIS X-rays Be-window
//
cathode
+ right
left + signal
11
signal anode
gas inlet
sample rotation +axis
incident
sample tilt x-axis
Figure 4.14 (a)Principle of a one-dimensional location sensitive proportional overflow, overpressure gas ionisation counter (Adapted from Wcislak et al.. 1993); (b) Pole figure measurements with a position sensitive detector (PSD).
records intensities along the a-circle is described in Section 4.4.1). Such PSDs typically provide a viewing angle of about 10" with an angular resolution of -0.01" which is sufficient to record individual peaks, inclusive of peak broadening at high tilting angles, and this also permits separation of overlapping peaks. Long, curved detectors with viewing angles of up to 120" are also available (Wolfel. 1983; Bessiires et al., 1991). However, these are mainly used to record diffraction spectra for phase
MACROTEXTURE MEASUREMENTS
79
analysis rather than for texture measurements; since with increasing deviation from the Bragg-Brentano diffraction condition two problems arise, as described below. In the Bragg-Brentano diffraction condition the Eulerian cradle is set in such a way that the X-circle coincides with the bisector of the angle between incident and diffracted beam, and any direction on the pole figure can be brought parallel to this direction by two rotations 90" - X and p, corresponding to the pole figure coordinates a and 9 (see above). With a PSD this is true only for one position which is usually the middle of the detector, whereas for all other positions the X-circle is inclined by W to the bisector (Figure 4.14b) (in the PSD geometry U is equal to AB, the deviation of the X-circle from the bisecting position). This means that the focusing condition is violated, which leads to some peak broadening, and this effect increases with increasing distance from the exact Bragg-Brentano reflection condition (Heizmann and Laruelle, 1986; Wcislak and Bunge, 1991). Furthermore, when several pole figures are measured simultaneously, a given setting (LL),X, 4)of the texture goniometer corresponds to different pole figure angles ( a ,0). Thus the (U, X, $1 coordinates need to be transformed into the pole figure angles (a, $1, and the corresponding transformation laws have been derived for transmission geometry (Bunge et al., 1982; Heizmann and Laruelle, 1986) and for reflection geometry (Wcislak and Bunge, 1991). However, poles with a < cannot be brought to coincidence with the bisector, which results in a non-measured 'blind' area in the centre of the pole figure. With increasing deviation from the exact Bragg-Brentano condition, i.e. with increasing w,the size of this blind area increases, which means that the pole figures become less meaningful. Thus, for these two reasons the violation of the focusing condition and the blind area - too large deviations from the exact Bragg-Brentano condition should be avoided. Whereas a one-dimensional PSD records the intensities linearly, mostly along a circle in the equator plane, a two-dimensional (area) detector would simultaneously register a certain part of the sphere. Accordingly, a conventional single detector can be considered as a zero-dimensional point counter. As area detectors yield a high angular resolution in two dimensions, they enable one to measure sharp textures with high accuracy. If materials with peak-rich diffraction spectra are investigated, peak profile analysis can be performed and, under certain circumstances, the high angular resolution even allows determination of the orientations of individual crystals in a polycrystalline aggregate (Bunge and Klein, 1996). In principle, a pliotog~aphicjlnzpositioned so as to record part of the diffraction sphere is a two-dimensional area detector (Wassermann and Grewen, 1962). Recently, image plates have been used for texture analysis (e.g. Broklneier and Ermrich, 1994) which are far superior to films in terms of dynamic range, sensibility and efficiency. However, they also require separate registration and readout periods of several minutes which prevents an interactive, on-line analysis of the results. To overcome this shortcoming, charged-coz~pleddevice (CCD) based detector systems have been developed for recording diffraction patterns generated by synchrotron radiation (Bourgeois et al., 1994; Garbe et al., 1997; Section 12.2). For real-time texture analysis in conventional X-ray texture goniometers, Siemens developed a two-dimensional nzulti~Yreproportional counter filled with a high-pressure xenon-gas mixture (Smith and Ortega, 1993; Klein et al., 1997). The detector is equipped with a concave beryllium-window with a diameter of 11.5 cm. The reflected intensity data -
INTRODUCTION TO TEXTURE ANALYSIS
Figure 4.15 Representation of Bragg's law for reflection at the various lattice planes (lzkl) in copper. The vertical line A represents the energy-dispersive and the horizontal line B the angular-dispersive diffraction experiments.
are received as a 1024 X 1024 pixel frame which can be transferred into a computer for storage and evaluation. Similarly as discussed for the one-dimensional PSD. the results have to be transformed into the standard pole figure grid. In all applications mentioned so far, the counters were used in an angular dispersive way, i.e. the diffracted intensities for monochromatic radiation were recorded in dependence on different angular settings of the sample, and energy discrimination is only used to reduce background and fluorescence by setting a narrow window of the pulse height analyser (Appendix 5). Figure 4.15 is a representation of Bragg's law Q =f(X) for reflection at the various lattice planes (Izkl) in copper. To determine the reflection spectrum in conventional angular-dispersive experiments with monochromatic radiation, 0/20-scans are performed with X-rays of a given wavelength, e.g. X = 0.15 nm (line B in Figure 4.15), and diffraction peaks are detected at each intersection of line B with the Bragg curves for the lattice planes (Izkl). Alternatively, the energy or wavelength of the reflected radiation can be exploited as well (energy dispersive diffi.actonzetry, e.g. Giessen and Gordon. 1968; Buras et al., 1975). If we introduce the energy of the radiation into equation 3.4, we can write Bragg's law in a different way:
MACROTEXTURE MEASUREMENTS
81
It is seen that at a constant setting with Bragg angle B, the energy of the diffracted radiation is a measure for the lattice spacing dhkiof the reflecting crystal planes (hkl). Thus, for polychromatic radiation covering a whole range of wavelengths, a detector at a fixed scattering angle, e.g. B = 45" (line A in Figure 4.19, would again record the various Izkl-diffractions which now were distinguished by different wavelengths. To perform energy dispersive diffractometry, the diffracted intensities of polychromatic radiation are recorded by an energy dispersive detector and then analysed by a multi-channel analyser (MCA) that transforms the pulse-height spectrum into a wavelength (or energy) spectrum. To achieve maximum energy resolution, use of a semiconductor counter is recommended, but nonetheless, the resolution of neighbouring diffraction lines is inferior to that of conventional angular dispersive techniques. The diffraction spectrum is obtained by plotting the contents of the various channels of the MCA and, as an example, Figure 4.16 shows a diffraction spectrum obtained from a polycrystalline steel sheet measured with a chromium tube at a constant Bragg angle B = 15". The fluorescence lines at the low-energy side of the spectrum are characteristic of the sample material, whereas the diffraction lines disclose its crystallographic structure. To utilise the results obtained by energy dispersive diffractometry for deriving the texture of the sampled volume, the wavelength spectrum must be convoluted into the contributions from the different reflector planes. By rotating the sample in a goniometer as in conventional angular dispersive applications, a whole range of lattice planes can be registered simultaneously (e.g. Szpunar and Gerward, 1980). This means that the measuring time is virtually independent of the number of pole figures recorded, which makes this method particularly attractive for low-symmetry
0
1024
2048
3072
4096 MCA channel
Figure 4.16 Diffraction spectrum of a steel sheet with random texture, obtained by energy dispersive diffractometry. During the measurement the sample was rotated about its normal direction; the time necessary to acquire the diffraction spectrum was 200 s (28 = 30°, CrKa-radiation, liquid nitrogen-cooled Ge-semiconductor counter; adapted from Shimizu et al., 1986).
82
INTRODUCTION TO TEXTURE ANALYSIS
materials with peak-rich diffraction spectra. Furthermore. as mentioned for the PSDs, overlapping peaks can be separated by peak profile analysis. Alternatively, the diffraction spectra can be exploited for a given, fixed sample position, which means that the sample does not have to be moved during texture analysis. The resulting texture information corresponds to an inverse pole figure of the sample surface (Szpunar et al., 1974; Shimizu et al., 1986; Section 4.3.6). Texture data from four inverse pole figures have been used to calculate a three-dimensional O D F with a reasonable accuracy (Szpunar, 1990). Texture analysis at a fixed sample position is also of great interest for in situ experiments (Gerward et al., 1976) as well as for on-line quality control in production plants. In the arrangement described by Kopinek (1994), a system for on-line texture analysis with energy dispersive detectors has been installed at the exit of a continuous annealing line of steel sheets. Four detectors, each recording the reflected intensities of five lattice planes, provide enough information to compute the low-order C-coefficients which can be used to calculate average physical properties of the final sheet product, including r-value, Young's modulus and magnetic properties (Bunge et al.. 1989).
4.3.5 Correction and normalisation of pole figure data Before the pole figures can be evaluated, or used for subsequent ODF calculation, several corrections have to be applied. In this Section. we discuss measures to correct for background and for errors introduced by defocusing and absorption, and we address also the influence of counting statistics. Finally, the texture data which are given as counts per pole figure point need to be normalised to standard units - e.g. multiples of a random orientation distribution that do not depend on the experimental parameters. The first error which has to be corrected for is the backggl.ound error (Figures 4.1, 4.17). Background intensities are caused by incoherent scattering and fluorescence in the sample. Minor contributions may also result from interaction of the X-ray beam with any material in the path of the X-rays e.g. collimator, beam stop and air as well as from electronic noise. Especially when the experiments are performed with 'monochromatic' radiation produced by a K3-absorption filter, reflection of X-rays with other wavelengths of the continuous spectrum strongly contributes to the overall background intensity. Several measures to improve the peak-to-background ratio have already been addressed in earlier sections: To prevent fluorescent radiation an appropriate X-ray tube with wavelength larger than the K-absorption edge of the sample material should be used (Section 4.3.1). but the short-wave components of the continuous spectrum may still excite fluorescence. A monochromator (Section 4.3.1) or an energy dispersive detector (Section 4.3.4) both substantially reduce background intensities with different wavelengths. With conventional counters, the background can be reduced by choosing a small window of the pulse-height analyser (Appendix 5). Background is also diminished by using smaller receiving slits, but this would enhance the 'defocusing error', i.e. the drop in peak intensity with increasing tilting angle a . Although background intensities can - and of course should be reduced by such measures, they cannot completely be eliminated in practice and, therefore, the -
-
-
-
MACROTEXTURE MEASUREMENTS
- BG,
Figure 4.17 Determination of the background intensity BG in a Bl20-spectrum. Case A illustrates a strong dependence of the reflected intensity on 8.whereas in case B the background is ala~ostindependent of B.
diffraction data have to be corrected by subtracting the background from the measured intensity (Figures 4.1, 4.17). For correcting pole figure data, the background intensity BC is determined and subtracted from the measured pole figure intensity I,ll,,,:
Within a given pole figure, the background changes with increasing tilting angle a, but for sufficiently large samples does not depend on the sample rotation ,8. Thus, a background correction curve BG(a) can be obtained by measuring a complete pole figure away from the diffraction peak B and integrating over the pole circles, i.e. over 3. At small angles 8 the background typically strongly decreases with B and, hence, it is necessary to measure the background at either side of the Bragg peak and subsequently interpolate the results (case A in Figure 4.17). Measuring at one side may be sufficient if the background intensities remain more or less constant with B (case B in Figure 4.17). Of course, in both cases it must be ensured that the distance from the Bragg peak is large enough to allow for peak broadening at large sample tilts a but that it does not interfere with neighbouring peaks. The decrease of the background intensity with a is fairly independent of B. Therefore, it is sufficient to measure the background intensity only at one angle a: typically a = OC,and then derive the background values for other a from a pre-determined correction curve. Not only the background intensity but also the reflected peak intensity strongly decreases with a. As described in Section 4.3.2, increasing sample tilt leads to a broadening of the reflected peaks (Figure 4.10a). The detector, however. is equipped with a system of receiving slits and hence 'sees' only a small, constant area of the sample surface, so that peak broadening results in a marked decrease in recorded -
-
INTRODUCTION TO TEXTURE ANALYSIS
Figure 4.18 (a) Intensity drop with increasing sample tilt a , normalised to 100% for a = On. Data were obtained experimentally by recording the { l 11) (0 = 21.7") and (31 1) (B = 45.1") pole figures of a copper powder sample with CuKa-radiation; (b) Evolution of I with a for three different sample thicknesses, computed according to equation 4.6 and again normalised to 100% for a = W.Whereas for the 2min thick sample the increase in absorption is exactly balanced by the increase in diffracting volume, for the thinner samples an absorption correction is necessary.
intensity (Figure 4. lob). A typical intensity drop obtained for copper with increasing a, normalised to 100% for a = 0°, is shown in Figure 4.18a for two different Bragg angles 0, i.e. two different pole figures. To correct for this defocusing error, a correction function U ( a ) must be applied which for any value of a normalises the intensity of a random sample to the intensity at a = 0":
As described in Section 4.3.2. the defocusing error is mainly determined by the sample position in terms of the two angles a and 0 and, therefore, it appears reasonable to obtain the correction curve from geometrical considerations (e.g. Gale and Griffiths, 1960; Tenckhoff, 1970). However, the intensity drop also depends on the alignment of sample and goniometer as well as the size of the collimator and the receiving slit. The smaller the collimator size and the larger the receiving slit, the smaller is the defocusing error, though large receiving slits decrease the angular resolution. Thus, empirical correction curves U(a), as derived from measurements of a random sample with the particular experimental set-up, usually give better results. For that purpose, pole figures of a sample with a random texture are measured at the Bragg peak, integrated over the rotation angle P, and normalised to unity, i.e. U(OO)= 1 (Figure 4.18a). It should be mentioned here that the preparation of standard samples with a completely random texture is not trivial. In most cases, powder compacts are used which may be slightly compacted or embedded e.g. in
85
MACROTEXTURE MEASUREMENTS
epoxy. Isostatically pressed powders can reveal too pronounced textures when the stress was not purely hydrostatic. On the other hand, in too loosely bonded powder compacts the reflected intensity may deviate from the curves obtained for 100°h dense bulk samples. However, it is not necessary to have powder samples for all materials measured. As the defocusing correction U(a) only slightly depends on the Bragg angle Q, correction curves of other materials with similar Bragg angles can be used if no powder sample is available. Thus, textures obtained in different steels or aluminium alloys can generally be corrected with the correction curves derived from a powder sample from pure iron and aluminium, respectively. Another error which has to be considered is caused by absorption. When a sample analysed in transmission geometry is tilted, the path length of the X-rays within the sample increases much more than the increase in the diffracting volume, resulting in a strong decrease in diffracted intensity. According to Schulz (1949b) and Cullity (1978), the intensity of the reflected X-ray, in dependence on the Bragg angle Q and the tilting angle W ( a = 90" - W),can be derived from: Id/Id&=
+
cos Q[exp{-pt/ cos(Q- W ) ) - exp{-pt/ cos(Q W))] ,ut exp{-pt/ cos Q)[cos(Q- W ) / cos(0 W) - l ]
+
(4.5)
where p is the linear absorption coefficient, Section 3.5. In the case of reflection of X-rays at thick samples, the increase in absorption is exactly balanced by the increase in diffracting volume, such that the reflected integrated intensity remains constant and a special correction is not necessary. In the case of very thin samples, e.g. thin films, however, the volume increase is dominating and an absorption correction becomes necessary. The ratio between the reflected intensities from a sample of thickness t and an infinite thick sample is given by (Schulz, 1949a):
In Figure 4.18b the evolution of I = I(t)/I, with a is shown for three different sample thicknesses, again normalised to 100% for a = 0" (see Figure 4.18a). For the 'infinitely' thick sample (or a material with a very high absorption coefficient) the correction factor stays constant at 1. With decreasing sample thickness, and especially for the 0.2 pm thin film, the intensity increases greatly with sample tilt however, and a correction of the absorption/volume effect (equation 4.6), combined either with an empirical or a theoretical defocusing correction, is necessary (e.g. Ghateigner et al., 1994). Finally, it should be mentioned that any pole figure measurement is influenced by counting statistics which imposes an uncertainty on each measurement with a standard error of the net intensity (Cullity, 1978: Humbert, 1986). The number of counts N derived per time unit at a given sample position is subject to statistical scatter which, for sufficiently large N , follows a Gauss law with the average value No and a standard deviation 0 = In the presence of a background the situation is slightly more complicated (Figure 4.17). If N p and NBG are the number of counts
m.
86
INTRODUCTION TO TEXTURE ANALYSIS
obtained in the peak and the background, respectively, then the relative standard deviation a is
For instance, for N p = 1000 and NBG= 100, equation 4.7 yields a relative standard deviation of 3.7%. Note that the error only depends on the number.ofpulses counted, but not on the mte. Thus, the accuracy of the measurements can be improved by prolonging the counting times. In the above example, doubling the measuring time would yield ATp = 2000 and NBc = 200. and this would reduce the relative error to 2.6%. After pole figure measurement and subsequent correction of the data with respect to background intensity, defocusing error and, if necessary. absorption, the pole figure data are available as number of counts, or counts per second. per pole figure point (a:9).For representation of the pole figures and for a subsequent evaluation, however, it is necessary to normalise the intensities to standard units that are not dependent on the experimental parameters. The commonly used convention is to express the data in multiples of a random distri1,~ltion. For that purpose. a normalisation factor N is derived by integrating over the measured and corrected intensities I,,,,(a, p) over the full pole figure and weighting the intensities with regard to their area1 contribution in the pole figure: 1 I ,,,,,, (0. 3) = - . L,,,(a. 3): N = N
S
I,,,,(a, 6)sin a ,
(4.8) sin a,
Therewith, the integral over the full pole figure range becomes l , which means that the pole figure of a randomly oriented standard sample would be 1 at all points. Pole figure regions with densities larger than 1 indicate that more lattice planes are aligned in those directions than in a random sample, and vice versa. For incomplete pole figures a true normalisation is not possible. but a pseudonormalisation factor N' can be derived by summation over the measured pole figure range:
A more accurate normalisation factor can be derived during a subsequent ODF analysis. 4.3.6 Inverse pole figures In most cases, inverse pole figures - the presentation of the sample axes with respect to the crystal axes (Section 2.4.2), not to be confused with 'pole figure inversion'
MACROTEXTURE MEASUREMENTS
87
(Section 5.3) - are recalculated from the three-dimensional O D F that has been derived from conventional pole figure data of the corresponding sample (Section 5.3). However, inverse pole figures can also directly be measured by means of diffraction methods (e.g. Harris, 1952; Bunge and Roberts. 1969). The sample is prepared perpendicular to the direction of interest usually RD, TD or ND - and mounted in reflection geometry. Inverse pole figures can be derived from the 20diffraction spectrum with fixed sample geometry, i.e. no rotations in X or d~ are necessary, so that a standard powder diffractometer without Eulerian cradle can be used. Thus, with a PSD or an energy dispersive detector the necessary data can readily be acquired (Section 4.3.4). The resulting peak intensities are normalised to the intensities of a standard sample with random texture and then the inverse pole figure can be constructed. However, this technique only yields the intensities of reflections lzkl that are in the accessible Q-range, i.e. Q is smaller than 90°, and for which no extinction rules apply. Furthermore, most experimental points lie on symmetry lines so that this method is much inferior to the conventional pole figures with a continuous coverage of the pole sphere. As an example, Figure 4.19 shows the inverse pole figure of a rolled steel sheet which consists of only 8 reflections. The results are superposed on the inverse pole figure that has been recalculated from the O D F of the same sample, indicating fairly good agreement between both methods. It is seen that whereas for high symmetry materials only a couple of diffraction peaks can be considered, the coverage is better for materials of low crystal symmetry with line-rich diffraction -
Figure 4.19 Inverse pole figure of the sheet normal direction ND of a commercially produced stabilised steel sheet in the temper rolled condition. Measured points: inverse pole figure measured as described in the text: intensity lines: inverse pole figure recalculated from the three-dimensional ODF (Adapted from Bunge and Roberts, 1969).
88
INTRODUCTION TO TEXTURE ANALYSIS
patterns. Application of an energy dispersive detector is advantageous, as this allows one to consider high-index planes which correspond to the short wavelength components of the X-ray spectrum (Szpunar et al., 1974; Okamoto et al., 1985). However, the decreasing intensity of high-index reflections and problems with peak separation still limit the application of this method for texture analysis. Finally, it should be mentioned that from several inverse pole figures, e.g. obtained for different sample directions, the ODFs can be calculated in a similar way as that used for pole figures (Section 5.3).
4.4 NEUTRON DIFFRACTION METHODS The first texture measurements by means of neutron diffraction were performed in 1953 by Brockhouse at Chalk River National Laboratory. In the 1960s, this technique was further developed by Tobisch and coworkers (e.g. Kleinstiick and Tobisch, 1968; Tobisch et al., 1969; Kleinstiick et al., 1976) and the advantages of neutron diffraction for texture analysis have been demonstrated (Section 3.6.2). Nowadays, the application of neutron diffraction for texture analysis is well established, see e.g. the reviews by Szpunar (1984), Welch (1986), Bunge (1989b), Wenk (1991), and Brokmeier (1997). In principle, pole figure analysis by neutrons is equivalent to X-ray pole figure measurements (Section 4.4.1). However, more sophisticated methods like time-of-flight (TOF) measurements (Section 4.4.2) offer additional advantages over X-ray diffraction techniques.
4.4.1 Pole figure measurement by neutron diffraction Texture investigations by means of neutron diffraction are mostly performed in research reactors where nuclear fission provides a supply of fast neutron. The neutrons are slowed down by interaction with appropriate moderators, until they eventually approach an approximate Maxwell velocity distribution with a maximum wavelength in the desired range from 0.05 to 0.3 nm which is suitable for diffraction experiments (Figure 4.20). However, no pronounced maxima analogous to the characteristic X-ray peaks form, so that application of a monochromator crystal is required to receive monochromatic radiation. Often copper (1 11) or graphite (0002) monochromator crystals are used. The advantages of time-of-flight (TOF) measurements (Section 4.4.2) have encouraged interest in pulsed neutron sources, and this has drawn attention to using linear accelerators for generating neutron beams. When pulses of electrons accelerated to energies of 10OMeV fall on a target of a heavy element, pulses of neutrons are generated through the intermediary of y-rays, which can be used for diffraction experiments (Bacon, 1975). A more efficient way is to use high energy proton beams which produce neutrons by a process called spallation (e.g. Bauer, 1989; Lisowski et al., 1990). When high energy protons with energies of several 100 MeV hit the atomic nuclei of a given target material, neutrons and a variety of other spallation products will be produced. The spallation products may react with other target nuclei, producing a cascade of particles with about 10 to 50 times more neutrons emitted than
MACROTEXTURE MEASUREMENTS
Kp K,
Figure 4.20 Wavelength spectrum of a thermal neutron source. compared with the spectrum of a copper X-ray tube with characteristic lines.
incident protons. To utilise these processes for generating a neutron beam, a high energy (up to l GeV) proton beam is generated in a particle accelerator and focused on a heavy metal target, e.g. lead or tungsten. The energy of the resulting neutrons is too high (i.e. the wavelength is too small) for diffraction experiments, however, which means that moderation to thermal Maxwell distribution is necessary. The principles of pole figure analysis by neutron diffraction are equivalent to X-ray pole figure measurements. The neutron detector is set to the corresponding Bragg angle 28 for the selected lattice planes ( h k l ) . The pole densities of that lattice plane in different sample directions are scanned with a goniometer by rotating the sample around two axes, q5 and X , corresponding to the pole figure angles 9 and a, to cover the entire orientation range (see Section 4.3.2). The main difference between neutron and X-ray diffraction is the much lower, almost negligible, absorption of neutrons by matter (Table 3.3). Therefore, large samples, usually several centimetres in size, can be analysed in transmission geometry. As shown by Tobisch and Bunge (1972), roughly spherical samples are completely sufficient for neutron experiments. In contrast to X-ray experiments, where the X-ray beam must not leave the sample surface, for neutron diffraction experiments the sample must not leave the neutron beam. Then. no intensity corrections are necessary and complete pole figures, i.e. up to a = 90°, can be obtained in one scan. The number of grains encountered is much larger, usually by 4-5 orders of magnitude, which means that grain statistics are much better. Because of the low absorption, environmental stages can be used for in .ritu investigations during heating, cooling, straining, etc. of the samples (e.g. Hansen et al., 1981; Elf et al., 1990). One drawback of neutron diffraction is that even in high flux reactors the intensity is much lower compared to X-ray experiments, so that counting statistics (equation
90
INTRODUCTION TO TEXTURE ANALYSIS
4.7) become an important consideration. A conventional pole figure scan takes between 6 and 48 hours, compared to about 30 minutes in X-ray diffraction (Section 4.3.3). Since furthern~oreaccess to neutron beam time is generally limited, techniques to accelerate the pole figure measurement are of great interest. To speed up neutron diffraction experiments, frequently the faster equal area scanning grids (Section 4.3.3) are used. Measuring time can be cut down very efficiently by using position sensitive detectors which record the diffracted intensities of several pole figures simultaneously. In most applications, a curved onedimensional position sensitive detector is mounted to record a certain range of the 28 spectrum (Bunge et al., 1982; Will et al., 1989), similarly as discussed in Section 4.3.4 for X-ray texture analysis (Figure 4.14). Unlike in X-ray diffractometry, the samples studied by neutrons are typically small compared to the goniometer dimensions and, therefore, the focusing condition is still fulfilled for larger deviations from the exact Bragg-Brentano condition. This allows determination of continuous diffraction profiles from which integrated intensities can be extracted and used for pole figure determination, peak separation, etc. An alternative arrangement with a vertical detector fixed at the position 28 = 90" is installed at the neutron source at Risa National Laboratory, Denmark (Juul Jensen and Kjerns, 1983). This unique arrangement simultaneously records the diffracted intensities corresponding to a range of positions within a given pole figure, rather than one point of several pole figures, so that fast texture analysis is possible. With recording times of about 15 min per pole figure this system can be used for in situ investigations, e.g. of recrystallisation kinetics or phase transformations (e.g. Juul Jensen and Leffers, 1989). To measure different pole figures, the wavelength of the incident neutrons has to be varied, which is achieved by a special set-up that allows continuous rotation of the entire texture unit - the monochromator, the beam path between rnonochromator and sample and the Eulerian cradle about the centre of the monochromator. -
In analogy to the energy dispersive X-ray diffractometry described in Section 4.3.4, an alternative method to measure large numbers of pole figures simultaneously utilises the polychronzatic nature of the neutron source. However, as there are no energy sensitive detectors for neutrons available. another way had to be found. The method is to determine the flight time of the neutrons, which in a given experimental set-up is a measure for the energy or wavelength of the neutrons, giving rise to the term time-qf-flight (TOF) (e.g. Szpunar, 1984; Feldmann, 1989; Wenk, 1994). It follows from the de Broglie relation (equation 3.1) that energy and wavelength of the neutrons are related to their velocity through:
(mAy,v v are mass and velocity of the neutrons respectively). Thus, for neutrons with an energy spectrum between 10p3 and lOeV (corresponding to wavelengths
MACROTEXTURE MEASUREMENTS
pulsed
electronic
-3 / detect:
Figure 4.21 Schematic arrangement to perfo~mTOF-measurements The total flight time T of the neutrons is determined by t h e ~ r~ e l o c i t )and the length L , between source and sample and L2betneen sample and detector
between X = 0.5 nm and 0.01 nn1) the velocities would range from vy = 10' mssl to 4 X 104ms-l. In a given experimental arrangement such velocities can be determined in terms of the flight times of the neutrons. Figure 4.21 shows a schematic arrangement to perform TOF-measurements. The total flight time T of the neutrons is given by their velocity v\. and the distance they travel, which is the length L1 between source and sample plus the length L2 between sample and detector. Thus, the relation between time-of-flight T and the energy E - or wavelength X - is given by:
This means that in a typical set-up wlth flight lengths L1 + L2 of approximately 10-50 m time differences of the order of milliseconds have to be resolved. To apply TOF in practice for determination of diffraction spectra, a pulsed neutron source is required, i.e. TOF-experiments have to be conducted either at a pulsed reactor or at a spallation source. The detector signal is stored in a multichannel time analyser which is started synchronously with each neutron pulse. The recorded spectrum is successively built up in dependence on the neutron flight time. which means that the first signals that correspond to the fastest (i.e. high energy/low wavelength) neutrons are stored in the first channels and so on. Considering the possibilities of both position sensitive detectors and TOF it appears that a combination of both would substantially speed up pole figure
92
INTRODUCTION TO TEXTURE ANALYSIS
measurements, and this has indeed been accomplished at the Intense Pulsed Neutron Source (IPNS) at Argonne National Laboratory and at the Los Alamos Neutron Science Center (LANSCE) (Wenk et al., 1991). With these set-ups, coverage of the entire hemisphere of reciprocal space -- i.e. complete pole figures for a number of different reflections hkl plus background data requires about 27 different sample settings as opposed to the 1368 points for one pole figure measured in conventional angular dispersive diffractometry (Section 4.3.3). A more advanced user facility with an increased number of detectors (10-20 times more than at present) is currently under development at LANSCE. -
4.5 TEXTURE MEASUREMENTS I N LOW SYMMETRY AND MULTI-PHASE MATERIALS Of course, the principles of texture analysis by X-ray or neutron diffraction described so far apply for all materials, independent of their crystal structure. However, in the case of multi-phase materials and/or structures with low crystal symmetry there are some specific additional problems which have to be considered (e.g. Bunge, 1985b; Brokmeier, 1989): e
Problems with peak separation caused by peak-rich diffraction spectra in both multi-phase and low symmetry materials (Section 4.5.1)
e
Volume fraction of the various phases in multi-phase materials (Section 4.5.2)
e
Anisotropic absorption of the reflected X-rays in multi-phase materials (Section 4.5.2)
In general, neutron diffraction offers some advantages over X-rays in analysing both low symmetry and multi-phase materials.
4.5.1 Peak separation Both multi-phase materials and low symmetry materials are characterised by linerich diffraction spectra; Figure 4.22 shows examples for two-phase alp brass with 2 0 ~ 0 1 %8-phase (Engler and Juul Jensen, 1994) and for calcite with trigonal crystal structure. In such line-rich diffraction spectra the various diffraction peaks may be situated very close together which can cause the problem of a proper separation of the contributions of the corresponding reflections/phases. In the above examples, the ( l l 1)-peak of the a-brass is situated very close (0.5") to the (01 l)-3-peak (Figure 4.22a). In particular with peak broadening at high sample tilting during pole figure analysis this can lead to a substantial overlapping of such closely spaced diffraction peaks, which makes texture analysis in complex crystal structures, e.g. high temperature superconductors, a very challenging task (e.g. Wenk et al., 1996). In contrast to the purtiul coirzcidence of reflection peaks in the above examples, diffraction peaks can also completely overlap. For instance, in cubic metals the two sets of lattice planes (330) and (411) have the same lattice spacing d, namely
MACROTEXTURE MEASUREMENTS
1.0
2.0
2.5
30
3.5
1
lattice spacing d [ A ]
reflection angle 0 (a) cdo brass
1.5
93
(b) Calcite
Figure 4.22 Examples of line-rich diffractioll spectra: (a) two-phase brass (Ct@lO%Zn, X-ray 8/28 spectrum; Engler aud Juul Jensen, 1994); (b) calcite (neutron TOF spectrum; Lutterotti et al., 1997).
~i~~~ = d411= a/&8 (a is lattice constant). Consequently, the Bragg angles are identical too, which means that these two diffraction peaks completely coincide (Figure 4.16). Whereas in the cubic structures this is not really a problem the example was chosen only as a simple illustration of the effect - there are more severe cases where all diffraction peaks of a given phase are systematically overlapped by another phase which is usually related by a phase transformation. Examples include ferrite and martensite in dual phase steels (Schwarzer and Weiland, 1988) and the primary a-phase and the martensitic 0"-phase in two-phase titanium alloys (Dunst and Mecking, 1996). Enantiomorphic materials, i.e. materials consisting of lefthanded and right-handed crystals, can be considered as a special case of two-phase materials with complete coincidence of all peaks so that the textures of the two phases cannot be determined from macrotexture pole figures (Bunge, 1985b). In such cases of complete peak coincidence texture analysis has to be performed by electron diffraction techniques which determines the textures of the various phases independently. In the case of a partial coincidence, peak separation can slightly be improved by using a smaller receiving slit of the detector (Section 4.3.4). Use of an X-ray tube producing X-rays of a longer wavelength (Table 4.1) would shift the reflections towards larger Bragg angles 0 and, in particular, would increase the distance A0 between neighbouring peaks. However, this can eventually result in the loss of diffraction peaks at too high reflection angles (0 > 90") and, furthermore, the problem of fluorescence must be considered (Section 4.3.1). Likewise, second-order peaks at larger 0 are more widely spaced, but they suffer from poor signal-to-noise ratio (Figure 4.22a). Neutron diffraction is superior to X-ray diffraction as neutron diffractometers in general offer a higher spectral resolution, especially at large Bragg angles (Section 3.6.2). However, in many cases only a few peaks are affected, e.g. in a/?-duplex steels only the {l 11)-peak of the face-centred cubic (fcc) 7-austenite and the (01 1)-peak of the body-centred cubic (bcc) a-ferrite overlap (similar to the -
94
INTRODUCTION TO TEXTURE ANALYSIS
a,'8-brass in Figure 4.22a). In such cases usually enough non-overlapping pole figures are available to compute the O D F so that the overlapping pole figures can eventually be recalculated (Section 5.3), if necessary. A very efficient way to separate partially overlapping peaks is to record the entire diffraction profile of the overlapping peaks by means of a PSD or energy dispersive detector (Section 4.3.4). The peaks are fitted with appropriate functions which allows derivation of the integral intensities of the corresponding pole figures (Jansen et al., 1988; Wenk and Pannetier. 1990; Antoniadis et al., 1990). Figure 4.23 shows an example of separation of the (020) and (001) peaks in feldspar. Dahms and Bunge (1989) developed an analytical deconvolution scheme which achieves peak separation by an iterative procedure of ODF-calculation and recalculation of the corresponding pole figures until a proper peak separation is obtained. This method even allows determination of the textures of two phases in cases where some peaks conzpletely overlap, where all other methods described so far would necessarily fail. Examples of successful peak separation have been given for low symmetry materials (natural quartz (Dahms, 1992)), for multi-phase systems (slytitanium aluminides (Dahms et al., 1994) and two-phase titanium alloys (Dunst and Mecking, 1996)), as well as for a combination of both low symmetry and multi-phase materials, namely paragneiss, a natural mineral consisting of quartz, feldspar and mica (Heinicke et al., 1991). A similar scheme to separate overlapping peaks based on the direct WIMV-method (Section 5.3.2) has been developed by Kallend et al. (1991b). Rather than extracting information from single diffraction peaks, it would be more efficient to use the whole diffraction spectrum and extract texture information
I
10"
I
I
11"
12"
Bragg angle 0 Figure 4.23 Separation of two partially overlapping feldspar diffraction peaks recorded by a linear PSD (adapted from Brokmeier, 1989).
MACROTEXTURE MEASUREMENTS
95
in a similar way to that by which crystallographers extract structural information from a powder pattern with the Rietveld technique (Rietveld, 1969). A method recently proposed by Wenk et al. (1994) relies on an iterative combination of crystallographic Rietveld profile analysis and quantitative ODF-calculation (RITA, Rietveld texture analysis). At the time of writing, this method has been applied to several synthetic test cases (Matthies et al., 1997; Wang et al., 1997) as well as to a calcite sample (Lutterotti et al., 1997; Von Dreele, 1997) and in all cases resolution was very good, even for the highly overlapped regions of the diffraction spectrum. Thus, the RITA-method can be expected to improve quantitative texture analysis of low symmetry compounds and multi-phase materials with overlapping peaks, to enable efficient data collection by requiring only a small part of the pole figure and to make maximal use of data obtained by means of energy dispersive methods that record complete spectra with many diffraction peaks, especially at neutron T O F facilities (Section 4.4.2).
4.5.2 Multi-phase materials Besides the problem of overlapping peaks treated in the preceding section, in the case of multi-phase materials two further problems have to be considered: (i) The reflected intensity is proportional to the volume fraction of the considered phase and hence it becomes difficult to determine properly pole figures of phases with small volume fractions (Bunge, 1985b). In Figure 4.22a it is obvious that in the case of 20% P-brass the diffraction peaks of the minor component are much smaller than those of the a-phase - resulting in a rather poor signal-to background ratio for the former - though in this case the textures of both phases could still properly be determined (Engler and Juul Jensen, 1994). In general, 5 ~ 0 1 % can be estimated to be the lower limit for texture analysis by X-ray diffraction techniques. Because of the much higher penetration depth of neutrons a much larger volume contributes to the reflected intensities in neutron diffraction, yielding sufficient counting statistics also for phases with small volume fractions. Brokmeier and Bunge (1988) obtained reliable texture data in aluminium with only 1% copper and even the diffraction peaks of as little as 0.05% copper in aluminium could be recorded (Brokmeier, 1989). (ii) A given phase may be strongly elongated. In particular in deformed materials the various phases are typically highly aligned along the delormation direction (e.g. Figure 4.24a). Differences in the absorption of X-rays in the different phases can cause errors in X-ray texture measurements (Bunge, 1985b). Depending on the sample setting - i.e. Bragg angle Q and pole figure angles a and P the intensity reflected in a given phase has to pass through different lengths of the other phases and, hence, is differently attenuated by absorption (Figure 4.24b). In the case of large differences in absorption coefficients in the different phases this can cause large errors in the intensity distribution in the pole figures obtained. For neutron diffraction, absorption is much lower (Section 3.6.2) and - independent of the sample setting - the neutrons are scattered from the total sampled volume so that neutron absorption is virtually not affected by a possible alignment of the various phases (Bunge, 1985b; Brokmeier, 1989; Engler and Juul Jensen, 1994). -
96
INTRODUCTION TO TEXTURE ANALYSIS phase II
\
reflected
Figure 4.24 (a) Microstructure of cold rolled two-phase Cu4O%Zn illustrating the alignment of the phases with deformation; (b) schematic representation of the reflection of X-rays during pole figure measurements illustrating the occurrence of anisotropic absorption in layered two-phase structures.
In conclusion, for texture measurements in multi-phase materials neutron diffraction is advantageous compared to X-ray diffraction because of the much higher penetration depth in matter of the former. Smallest volume fractions can be determined with sufficient counting statistics and the anisotropic absorption as described for X-rays can be neglected in the case of neutron diffraction.
4.6 SAMPLE PREPARATION One of the advantages of neutron diffraction experiments is the flexibility in the sample geometry suitable for neutron texture analysis, so that in most cases only a minimum of sample preparation is required. Because of the large penetration depth of neutrons. bulk samples of centimetre size can readily be investigated in transmission geometry (Section 4.4.1). Neutron texture analysis was greatly advanced by the spherical sample method since in this geometry no absorption correction is necessary, yielding highest accuracy of the texture results (Tobisch and Bunge, 1971). The deviations from the exact sphere may be quite large, however, so that cylindrical or even cubic samples, maybe with rounded edges. are usually sufficient (Welch, 1986). Thus, roughly spherical bulk samples can generally be analysed without special sample preparation, which makes neutron diffraction attractive for geological studies. For analysis of too thin specimens, e.g. sheet material, a cylindrical composite sample may be produced by stacking several discs cut from the sheet, tied together e.g. with screws or a wire (preferably of a material with different diffraction peaks). Gluing is less suitable since the hydrogen in the glue produces a significant amount of neutron scattering. One may wish to first electropolish or etch the sheets so as to reduce the influence of a possible surface texture (Juul Jensen and Kjems, 1983). In contrast, the low penetration depth of X-rays in most materials entails a certain amount of sample preparation for X-ray texture analysis. Texture measurements in transmission geometry require very thin parallel samples with thickness typically below 100 pm, so that this method is restricted to foils or thin wires which can be
MACROTEXTURE MEASUREMENTS
97
penetrated by the X-rays. Thicker samples may be thinned by grinding and etching. Today, the vast majority of texture measurements is performed in reflection geometry, however, and in the following we will address some ways for an appropriate preparation of the samples for X-ray texture analysis in reflection geometry. To obtain the textures of wires, a bundle of parallel wires is mounted on a flat holder (e.g. Wenk, 1998). In most cases, production of the wires results in very heterogeneous textures with the mantle of the wire being different from the core. Provided the penetration depth of the X-rays used is not too large, the surface texture can be measured from the wires as they are, i.e. without any preparation. Subsequently the surface can be removed by chemical etching and the texture of the core determined (e.g. Montesin and Heizmann, 1992). When only the average texture is of interest, texture measurements can be conducted on the cross-section of a bundle of wires. However, for analysis of axial-symmetric textures in general an arrangement with the symmetry axis being parallel to the sample surface, rather than perpendicular to it, is advantageous with a view to separation of crystallographic texture and artefacts cause by geometrical corrections (Wenk and Phillips, 1992). Preparation of bulk samples, including sheet materials, for X-ray pole figure measurements in reflection geometry is usually easy and can be performed with standard metallographic techniques. Samples can be round or rectangular shaped, with their size typically ranging from l0mm to 30 mm and a thickness of 0.5 mm up to several mm. The sample surface must be flat, and the samples should carefully be ground down to 1000 grit. Finer grinding or additional polishing is generally not necessary. However, the sample surface must not show any deformation as this would alter the texture of the affected layers and, therewith, the results obtained. Therefore, chemical etching or, if appropriate, short electropolishing of the surface is essential. For most metals standard etching reagents which homogeneously attack the surface yield satisfactory results; some examples are: For aluminium and many alumiizium-alloj~sbest results are obtained by etching for 10-20 min in about 50% NaOH. Heating of the solution to -- 60 "C ensures a more homogeneous etching, but should be avoided for deformed samples to prevent recovery. After the etch, many aluminium alloys (in particular coppercontaining ones) have a grey or black layer which can readily be removed in -10% HNO3. Copper can be etched in diluted HNO3. For copper-alloys too high concentrations (e.g. exceeding 50%) should be avoided as they might selectively etch different alloying elements. Fevitic steels, including low-carbon steels and iron-silicon transformer steels, can be etched for l-2min in a solution of l00ml H 2 0 2 and 5-15 m1 HF. In alloys with high chromium contents l0ml H N 0 3 and 15 m1 HCl can be added. Etchants suitable for some other bcc metals are l00 m1 HC1, 90 m1 H F , l00 m1 H N 0 3 , 30 m1 H202 (tungsten); 100 m1 H2SO4, 40 m1 H N 0 3 , 40 m1 H F (tantalum); 100ml H N 0 3 , 35 m1 HF, 1-5 m1 H2SO4(rziobium) and 100ml H N 0 3 , 8 m1 H F or 100 m1 HCl, 30 m1 H 2 0 2(molybdenum). 20-30 minutes etching time removes 5 pm (Raabe, 1992).
98 e
INTRODUCTION TO TEXTURE ANALYSIS
Kelly et al. (1996) developed an etching technique for tantalum and tantalun7ulloys, which can be used for preparation of samples for X-ray analysis. It turned out that their technique is suitable for a surprising variety of materials, including other bcc metals and alloys as well as titarzizm and a variety of titanium-alloys.
For ceramics and geological materials sawing and mechanical grinding are generally adequate. The bulk samples are usually cut with a diamond saw and then further ground with silicon carbide, alumina or diamond abrasive powders down to a grit size of 10 hm.
Figure 4.25 Measures to increase the sampling area during X-ray texture analysis: (a) meander scan: The sample is translated in x and 4 direction, and at the end of each x-stroke of 42mm a lateral movement of 4 m m is carried out. After 8 of these lateral shifts the direct~onis reversed (Hirsch et d.. 1986); (b) sample with slightly slanted surface averaging over several through-thickness layers (not drawn to scale); (c) 'sandwich' sample stacked from several rotated pieces.
MACROTEXTURE MEASUREMENTS
99
Owing to the small penetration depth of X-rays, grain statistics often is a problem for X-ray texture analysis. If we assume a penetration depth of 0.1 mm and an area of 10 X 10mm2 being irradiated by the X-ray beam, the recorded volume is of the order of 10mm3. For a material with a grain size of 25pm more than lo6 grains contribute to the reflected intensity which definitely yields good statistical evaluation. For a grain size of 250 pm, in contrast, only slightly more than 1000 grains are recorded, which is anticipated to be the lower limit for sufficient grain statistics (Section 9.3.3). The number of grains that contribute to the reflected intensity can be enlarged by a translation movement of e.g. l0mm in the plane of measurement (Bunge and Puch, 1984), provided the sample is sufficiently large. For samples with elongated grains, e.g. deformed materials. it is advantageous to perform this translation perpendicular to the direction of the grain elongation (i.e. parallel T D for rolled sheets) as this enlarges the number of encountered bands. To enlarge further the sampling area it has been proposed to perform a meandershaped scan of the sample surface (Hirsch et al., 1986). By this, a sampling area of as much as 50 X 40mm2 can be completely scanned (Figure 4.25a), so that textures of coarse grained materials could be determined. but this method requires a rather complicated control of the goniometer head with the possibility for an xly-sample positioning. Rather, it is much easier to measure pole figures of several standard sized samples and then average the resulting pole figure data (e.g. Engler et al., 1996a). This procedure is easy to perform and the sampling size is at least theoretically unlimited. It must be emphasised that these methods to enlarge the sampling area require a homogeneous texture, i.e. no pronounced texture gradients should exist throughout the positions from which the different scans are taken. For rolled sheets with strongly elongated but very thin deformed grains the measuring statistics can also markedly be improved by preparing the sample surface with a slightly slanted surface (Figure 4.25b). If we allow for a maximum angle of l", which is tolerable in most cases, at a sample of length 25mm a depth profile of 450 p would be sampled, which for most materials is significantly more than the average penetration depth of X-rays (Table 4.2). The 'extreme case' of a sample with a slanted surface would be a sample that is prepared perpendicular to the reference direction, i.e. rotated by 90" about one of the reference directions. For example, in the case of hot rolled bands preparation of a sandwich sample from 5 pieces stacked together gives very satisfactory results (Figure 4.25~).Note that in these cases the resulting textures integrate over the sample thickness, i.e. they reveal an average over the different layers of the sheet. In any case, some preliminary knowledge on the microstructure and, if possible, on the texture of the material investigated is valuable to select the best way of sample preparation and, therewith, to achieve optimum experimental results. -
-
4.7
SUMMATION
Measurements of pole figures by means of X-ray diffraction started in the 1940s by introducing texture goniometers to scan the pole figures and electrical counters to record the reflected intensities. Since the 1960s pole figures are also regularly
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INTRODUCTION TO TEXTURE ANALYSIS
determined by neutron diffraction. Today, automated computer-controlled systems are available and the corresponding measuring techniques and the necessary correction methods which have been introduced in this Chapter are very well established, so that diffraction experiments can routinely be performed to yield reliable, quantitative results for macrotexture. Because of the much easier access to X-ray goniometers than to neutron sources, the vast majority of pole figure measurements is performed by means of X-ray diffraction. Neutron diffraction offers some advantages, however, which are mainly caused by the much lower absorption of neutrons by matter. The much greater penetration depth of neutrons in the sample material allows investigation of spherical samples of centimetre size in transmission geometry, which yields the opportunity to perform non-destructive texture measurements or in situ experiments. Complete pole figures can be obtained without intensity corrections. The grain statistics are better, which is of particular interest in coarse grained samples or for detection of low volume fractions of a second-phase. Texture analysis can be greatly advanced by the use of position sensitive detectors or energy dispersive detectors (in neutron diffraction: time-of-flight, TOF, experiments). In contrast to the conventional point detectors, these detectors record a part of the diffraction spectrum, so that the intensities of several pole figures, or of parts of one pole figure, can be measured simultaneously. This can be used for fast texture analysis, e.g. in in situ experiments or in on-line applications. Furthermore, position sensitive and energy dispersive detectors greatly facilitate texture analysis in low symmetry and multi-phase materials with line-rich diffraction patterns, where overlapping peaks can be separated by peak profile analysis.
5. EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
5.1 INTRODUCTION In Chapter 4 the common techniques to determine macrotextures experimentally by means of X-ray and neutron diffraction were introduced. The direct output of such texture measurements is the reflected intensity in the form of the two pole figure angles a and P, that is, the distribution of the poles in the traditional pole figure (Case Study No. 10). Hence the pole figure is the obvious choice for a critical assessment of experimental texture results, and this method will be discussed in some detail in Section 5.2. The projection of the three-dimensional orientation distribution onto a twodimensional pole figure causes a loss in information. so that in turn it is not possible to determine the orientation density of crystallites in a polycrystalline sample, i.e. its texture, from the experimental pole figures without a certain ambiguity. Thus, for quantitative analysis of macrotexture a three-dimensional description - e.g. in terms of the three-dimensional orientation distribution function (ODF) - is required (Case Study No. 9). However, unlike pole figures, ODFs cannot directly be measured by means of diffraction techniques, but need to be calculated from the pole figure data, typically from a set of several pole figures obtained from a given sample. In Section 5.3 we will introduce the most common approaches to accomplish this pole Jigure inversion and address their main advantages and disadvantages. In order to display the three-dimensional ODF, a representation in an appropriate three-dimensional frame is required. In Chapter 2 several suitable orientation spaces have been introduced, namely Euler space (Section 2.5), cylindrical anglelaxis space (Section 2.6) and Rodrigues space (Section 2.7). Among them, the Euler space is most often used to represent macrotexture data, and in Section 5.4 we will address the properties of Euler space as well as discuss suitable techniques for displaying textures in Euler space. A 'statistical' nzisorientation distribution function, MODF, can be derived from the ODF and displayed in Euler space. However, since the topic of misorientation distributions is more relevant to microtexture, the statistical MODF is discussed in Section 9.7 rather than in this Chapter. The cylindrical
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anglelaxis space and Rodrigues space are mostly used to represent microtexture data (Chapter 9), but have not found wide applications in conventional macrotexture analysis so far.
5.2 POLE FIGURE AND INVERSE POLE FIGURE In this Section the principles of representing textures in pole figures (Section 2.4.1) and inverse pole figures (Section 2.4.2), which form the backbone of macrotexture analysis, will be introduced. We will address means of normalising and contouring the orientation densities in a pole figure, and illustrate use of the inverse pole figure, whlch finds applications in the case of axial symmetric textures (e.g. in tensile tests).
5.2.1 Normalisation and contouring of pole figures To compare pole figures of different samples it is necessary to normalise the orientation densities to standard units that are not dependent on the experimental set-up. The commonly used convention is to express the data in multiples of a random or uniform distribution of orientations, which means that the pole figure of a random standard sample would be ' X l ' at all points (Section 4.3.5). Pole figure regions with intensities higher than this indicate that more lattice planes are aligned in those directions than in a sample with random texture, and vice versa. The most comn~onlyused way of contouring the orientation density in a pole figure is to draw lines of equal intensity, iso-density lines, as known from geographical maps. The values of the various contour lines do not need to be linearly spaced. Often, it is more instructive to use logarithmic progression to reveal texture details (Figure 5.la). For an easy visual comparison of different pole figures it is of
Figure 5.1 (a) {l 1l} pole figure plotted with contour lines (logarithmic progression: regions with intensities lower than 1 are dotted): (b) same pole figure as in (a) plotted in greq shades (commercial purity aluminium AA1145, rolled to a strain of 3, recrystallisation annealed for 1000s at 350°C).
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considerable help to use standard distances, preferably starting at the same minimum level. In other words, levels in a given percentage of the maximum intensity, say 10%, 30%. 50% and 75% of maximum intensity, should be avoided. In some cases, in particular for weak textures, contour lines may be ambiguous, which means that intensity variations may not unmistakably be resolved. In such cases the individual contour lines need to be labelled with their values (e.g. Figure 5.2) or clearly marked by different (mininmm 3) line styles or colours. In many cases those ambiguities can be avolded by simply marking pole figure regions below the minimum level, or above the maximum one, by shadings or by dots (e.g. Figures 5 la, 5.2). Alternatives to the contour lines are different grey-shadings or colours in accordance with the intensity of the corresponding pole figure regions (e.g. Figure 5.lb). Such pole figures are more complicated to reproduce, however.
5.2.2 Representation of orientations in the inverse pole figure Figure 5.2 shows the inverse pole figure of the extrusion axis of extruded titanium, which was plotted with iso-intensity lines in the hexagonal standard triangle formed by the axes [OOOl], [l 1201 and [IoTo]. The texture displays a concentration of the extrusion axis in the vicinity of [10T0] which means that the basal plane normals are perpendicular to the extrusion axis, which is the common texture of hexagonal materials deformed by extrusion (Rollett and Wright, 1998). However, although it is generally anticipated that in uniaxially deformed samples a fibre texture is present, this information cannot be derived from Figure 5.2 since the inverse pole figure only shows the orientation of one reference axis - here the extrusion axis - but rotations about this axis are not considered. Thus, for a complete representation of the threedimensional orientation distribution the orientations of tn.0 reference axes would be required. Although for analysis of the macrotexture of fibre textures this may not be necessary, for the evaluation of misorientations the full orientation information must be considered. The implications of this will be addressed in more detail in Section 9.6. In conclusion, for the representation of macrotexture data inverse pole figures are usually inferior to the pole figures.
Figure 5.2 Inverse pole fieure of titanium deformed by extrusion to a von Mises equivalent strain of 1.75 (inverse pole figure of the extrusion axis). (Adapted from Rollett and Wright, 1998.)
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INTRODUCTION TO TEXTURE ANALYSIS
5.3 DETERMINATION O F THE ORIENTATION DISTRIBUTION FUNCTION. (ODF) FROM POLE FIGURE DATA The projection of the three-dimensional orientation distribution onto a twodimensional projection plane results in a loss in information, so that in turn the three-dimensional orientation distribution, i.e. the texture, cannot be derived from a given pole figure without some uncertainty. Only in the case of very simple textures consisting only of a small number of orientations - e.g. textures of single crystals or the cube texture in recrystallised aluminium or copper - may the pole figures be sufficient to yield unan~biguouslythe entire texture information in a quantitative manner. In general, however, the intensities of the individual poles which form a given orientation cannot clearly be assigned to that orientation, in particular when the poles of different orientations overlap, which renders a clear separation of the volumes associated with the individual orientations complicated or even impossible. Furthermore, most experimentally obtained pole figures are incomplete pole jjgures, which means that not the entire pole figure range has been recorded (Section 4.3.2). This further reduces the texture information available in such pole figures. To overcome these ambiguities and thus to permit a quantitative evaluation of the textures, it is necessary to describe the orientation density of grains in a polycrystal in an appropriate three-dimensional representation, i.e. in terms of its orientation distribution function (ODF). As shown schematically in Figure 5.3, a pole which is defined by the direction y , e.g. the two angles a and R, in a given two-dimensional pole figure P/,(y) corresponds to a region in the three-dimensional O D F f (g) which contains all possible rotations with angle 7 about this direction J in the pole figure. Written in a mathematical manner this means:
Figure 5.3 Representation of the fundamental equation of pole figure inversion in a (100) pole figure.
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105
Thus, macrotexture ODFs cannot be measured directly, but they need to be calculated from pole figure data. In general, the ODF is not completely determined by one pole figure, and the missing information must be provided by additional pole figures. The integral equation 5.1 represents the fz~ndarnerztalequation of ODF conzputation, which has to be solved to calculate the ODF. This is called pole figure inversion. An analytical solution of the problem of pole figure inversion is not possible. however. and in the literature several mathematical approaches to solving equation 5.1 have been proposed. In the following sections, we will briefly introduce some of the most common methods for pole figure inversion. These sections are by no means intended to provide a rigorous mathematical description of the various approaches, but try to outline their underlying ideas and to compare their main advantages and disadvantages. 5.3.1 Series expansion method The basic premise of the series expansiorz method is the assumption that both the measured pole figures and the ensuing ODF can be fitted by a series expansion with suitable mathematical functions. Appropriate functions to use in a spherical coordinate system are the 'spherical harmonic functions' and therefore this method has also been called the harrnorzic method. A complete scheme for texture analysis by series expansion has been developed independently by Bunge (1965) and Roe (1965), and in the following, the method as proposed by Bunge will briefly be outlined (for more details, see Jura and Pospiech, 1978; Bunge, 1982; Bunge and Esling, 1985). Tn the series expansion method, the ODF is expanded in a series of (symmetrised) generalised spherical harmonic functions T ( g ) :
The spherical harmonic functions TYn(g)can be calculated for all orientations g and are usually stored in Libraries (e.g. Pospiech and Jura, 1975;Wagner et al., 1982). Then, the ODF is completely described by the series expansion coefficients C;"". This is the general principle of a series expansion, which is well-known in the form of Fourier analysis, where the special functions are sine or cosine functions. The harmonic functions T(g) are more complex three-dimensional functions, but the principle is the same. The pole figures can also be expanded in a series expansion. As the pole figure is characterised by the two angles a and P. an expansion in (symmetrised) spherical harmonic functions K;'(y) is sufficient:
The two-dimensional series expansion coefficients F;' are related to the C';"coefficients through: l 47r F;' ( h ) = ----- . C yK:"' (l?) 21+ 1 m=-1
C
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INTRODUCTION TO TEXTURE ANALYSIS
(The asterix in equation 5.4 indicates the complex-conjugate quantity.) Equation 5.4 forms a system of linear equations which can be solved to derive the C;""coefficients. Usually, a larger number of pole figures is determined and used to calculate an ODF than is mathenlatically necessary. Whereas for cubic and hexagonal materials respectively 3-4 and 5-6 pole figures suffice. in the case of less symmetric crystal structures more pole figures are required. The experimental data are then averaged by a least-square scheme which reduces the impact of measuring errors and, in particular, allows use of incomplete pole figures for the ODF-calculation. Similarly, inverse pole figures can be used to compute an ODF. Vice versa, from the ODFs both pole figures and inverse pole figures can readily be recalculated. An ODF computed as described above contains several errors, even under the assumption of ideal pole figure data. One error is caused by the necessity to truncate the series expansion of pole figure and ODF expansion. This tru~mztion error. leads to broadening of the texture peaks as well as to minor artefact intensities in the vicinity of strong texture components (e.g. Liicke et al., 1981). All summations in equations 5.2 and 5.3 are controlled by the index 1 which means that the truncation error is merely determined by the maximum value of In general, = 22 or, for for cubic materials the order of the series expansion is limited to I, sharper texture, to I, = 34, which yields sufficiently accurate results with an acceptable computation effort. Another, more severe, problem of ODF calculation is the so-called ghost error. In the 1970s, Lucke and coworkers (summarised in Lucke et al., 1981) observed that intensities of the O D F may be missing (negative ghosts), or wrong intensities may appear (positive ghosts). Matthies (1979) proved that these ghosts are caused by the lack of the odd-order series expansion coefficients C?. Because of the centrosymmetry of the pole figures derived by diffraction experiments (Section 4.2), they contain less information than necessary to reproduce unambiguously an ODF. Written in a formal manner, the complete ODF f(g) is composed of two parts (Matthies, 1979):
The first part, J(g), which is defined by the even-order C-coefficients, can be derived from diffraction pole figures, whereas the second part. z ( g ) given by the oddorder C-coefficients, cannot. Because of the missing odd partf(g) it may seem that as much as one half of the information is missing. However, since the odd C-coefficients are zero for low l and quite small for large l, the impact of the odd C-coefficients is limited in practice and the errors are typically about 20%. To visualise the impact of the ghost error on the ODFs, Figure 5.4a shows the rolling texture of an AlFeSi alloy as obtained from the experimental pole figures, i.e. the even part f (g), plotted in sections through the three-dimensional Euler space (Section 5.4). Then, the odd C-coefficients have been obtained according to the method by Dahms and Bunge (see below) and the complete - i.e. ghost-corrected O D F f ( g ) is shown in Figure 5.4b. Although the general texture features are similar, it is obvious that the overall sharpness of the ghost-free O D F is higher by about 20% and that -
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA P<
V7=c0n~f
FUNE ODF
P
V~=CO~S~
107
compl ODF
Figure 5.4 Rolling texture of an AI-Fe-Si alloy in the as cold rolled state (a) reduced ODF as obtained from the experimental pole figures; (b) complete O D F f ( g ) . i.e. ODF from (a) plus subsequent ghost correction according to the method by Dahms and Bunge (1988, 1989).
some small intensities of the experimental ODF (Figure 5.4a) are missing, which means that these are ghosts. Thus, to derive complete, ghost-corrected textures by means of the series expansion method, the part&) given by the odd-order C-coefficients must be determined under certain assumptions. Most procedures to derive the odd C-coefficients make use of the condition that the O D F must be non-negative for all orientations. i.e.f(g) > 0. In the zero-range method proposed by Bunge and Esling (1979) and Esling et al. (1981), ranges in the pole figures with zero intensity yield an approximation to the indeterminable part of the ODF. From the pole figure zero-ranges the corresponding zero-ranges in the orientation space can be derived by geometrical considerations. For these ranges it readily follows:
Thus, in the zero-ranges of the orientation space the function?(g) is known and can be expressed in terms of the series expansion. The corresponding C-coefficients are now considered as an approximate solution o ~ J ( in ~ )the entire orientation space. It is obvious that this method can only yield satisfactory results for pole figures with well defined zero-ranges, i.e. in the case of sharp textures. In the quadratic method (Van Houtte, 1983) the O D F f (g) is expressed in a quadratic form h2(g) or alternatively in an exponential form (Van Houtte, 1991) - which -
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INTRODUCTION TO TEXTURE ANALYSIS
cannot be negative. The function h(g) which produces a satisfactory description of f(g) is found in an iterative procedure. This method disadvantageously alters the (known) even C-coefficients which introduces some residual errors. As in the case of the zero-range method, the quadratic method is only applicable if well defined zeroranges exist. The positivity method by Dahms and Bunge (1988. 1989) is a generalisation of the zero-range method. At the n-th step of this iterative procedure the O D F f (g) is given by:
where the (complete) correction function the (n - l)-th function fn-l ( g ) (Figure 5.5):
approximates the negative parts of
The initial solution (rz = 0) is the experimental ODF, i.e. the O D F directly calculated froin the pole figure data. Note that only the odd of the correction function is added to the previous solution in equation 5.7, so that the (known) even part of the O D F J(g) is never modified. This procedure is repeated until the entire O D F is positive or until the remaining negative regions are sufficiently small. This method was improved by introducing a small positive offset r in equation 5.8 (Dahms and Bunge, 1989, Wagner et al., 1990). By choosing different values of r the width of the range of possible solutions can be estimated, and the
Figure 5.5 Ghost correction according to the method by Dahms and Bunge (1988.1989) at the nth iteration step in a one-dimensional representation: &,(g): ODF at the prececing iteration step, (g),?(g) odd part of the f;:(g): (complete) correction function to approximate the negative ranges of correction function f;:(g).
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109
solution with maximum r corresponds to the solution with maximum isotropic background, or 'phon' (Matthies, 1984). This generalised positivity method offers a fast, efficient way to obtain the complete ODF and thus to correct for the ghost error, which was used e.g. for the example shown in Figure 5.4. In a similar context the concept of maximum entropy has been developed (Wang et al., 1988, 1992; Schaeben, 1988, 1991). Out of the possible range of complete ODFs, the solution with maximum 'entropy' has the smoothest distribution with minimum additional information to avoid artefacts, e.g. ghosts, for which there is no evidence in the pole figures (the term 'entropy' is not meant in a thermodynamic sense here). This method has the advantage of providing ghost correction also in cases where no negative ODF-ranges occur. Liicke et al. (1981, 1986) proposed to deconvolute the experimental ODF into a set of several, typically 5-10, individual orientations. From this set of orientations a model ODF is calculated by associating each orientation a Gauss-type scatter in Euler space. This method, which will be discussed in more detail in Section 9.3.2, yields a complete ODF, i.e. both odd- and even-order C-coefficients are obtained. By subtracting the even-order part of the model O D F from the experimental ODF, a difference ODF is obtained which can then be used to improve the fit between the model ODF and the experimental O D F by shifting the position of the components or by adding new ones. The complete O D F is finally derived by combining the even Ccoefficients of the experimental and the odd C-coefficients of the model ODF. By this method the entire experimental information is retained but still a substantial reduction of the ghost error is achieved, even if the model fit is poor. As a rule-ofthumb, a fit of the difference O D F to 20% (which in most cases is easy to achieve) or, better, to 10% would respectively reduce the ghost error to 4% or even as little as 2%. It should be emphasised that although the assumption of Gauss-type orientat i o n ~e.g. , in case of weak textures or pronounced fibre textures, is not necessarily justified, this method yields very satisfactory results with regard to a reduction of the ghost error. This method has successfully been applied to a huge amount of experimental data by the Aachen texture group. Furthermore, the information of the volume fractions and scatter widths of the main Gauss components can be used for a very condensed, quantitative description of the ODFs (e.g. Engler et al., 1994a). The major disadvantage of this method is that the iterative derivation of the model ODF has not been automated so far, so that the method is rather time-consuming to apply.
5.3.2 Direct methods Besides calculating the ODF by means of the harmonic method in a Fourier-space, other algorithms have been developed to derive the O D F values f(g) directly from the pole figure data, and therefore those methods are summarised under the term direct methods. Such methods consider the pole figures P h b ) at a finite number of individual points g, and. accordingly, the ODF f (g) at a finite number of individual orientations gj. Typically, both the pole figures and the orientation space are subdivided into a regular grid with a spacing of 2.5" or 5". The relation between the individual pole figure points and the corresponding cells of the orientation space are
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INTRODUCTION TO TEXTURE ANALYSIS
established under consideration of the crystal geometry. Then the integral equation 5.1 can be replaced by a finite summation for each pole figure cell i:
Equation 5.9 defines a set of linear equations which under appropriate conditions can be solved to yield the ODF f(g). After an initial estimate of the ODF, the fit between the ODF-values and the value of the associated pole figure points is improved by means of an iterative procedure. In all direct methods the positivity condition is directly taken into account. In the following, the principles of the so-called WIMV-method will be outlined, which is implemented in the Los Alamos texture software package popLA (Kallend et al., 1991a). This method goes back to Williams (1968) and Imhof (1977) and has since been improved by Matthies and Vine1 (1982), which led to the acronym WIMV. For a more comprehensive review the reader is referred to Matthies and Wenk (1985) and Kallend (1998). The initial f(g) values of each ODF-cell (e.g. in a 5" X 5" X 5" grid) are estimated by the geometric mean of the values in the associated cells in the experimental pole figures. Thus, as required, zero-values in the pole figure automatically lead to zeros in the orientation space. The O D F is then refined by a series of so-called inner iterations (Figure 5.6). If P" denotes the recalculated pole figure after the 12-thiteration step, for each cell of the ODF a correction factor is derived as the ratio of the geometric mean of the corresponding cells in the experimental pole figure to the one in the recalculated pole figure P". The next estimate is then derived by multiplying the value of each O D F cell with this correction factor: -
(Iis the number of measured pole figures, Ml is the multiplicity of pole i, and N is the normalisation.) If, for example, the ODF-value in a given cell is too large, then the corresponding recalculated pole figure values would also be too large. Consequently, the correction factor would be less than 1 so that the corresponding ODF-value will be reduced in the next step. Additionally, an outer iteration can be applied to limit the minimum value of the ODF to f(g) > f(g),n,, which means that the phon is raised. By this any potential negative ghost will be filled which in turn reduces the corresponding positive ghosts as well. In practice, the algorithm converges rapidly, and typically a satisfactory solution is obtained after about 10 iterations. Besides the WIMV-method, there are several further direct methods for O D F computation which are listed here for completeness: e
Based on the WIMV-method, Pawlik (Pawlik, 1986; Pawlik et al., 1991) developed the ADC (arbitrary defined cells) method. Whereas most direct methods consider projection lilzes in relating the pole figures to the ODF cells. ADC uses projection
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
a Experimental data
First estimate of ODF
Estimated ODF
I Calculate correction factors
Recalculate pole figures
/\ unsatisfactory
recalculated pole figures with experimental data
<
ODF output
Figure 5.6
Flow chart of ODF-calculation according to the WIMV algorithm (Kallend, 1998)
tubes. The cross section of these tubes is determined by the shape of the corresponding pole figure cells. Computation of the intersections of the tubes with the O D F cells allows to derive the volume fraction of an O D F cell that contributes to the pole figure and leads to a better smoothing in the final result. e
In the vector method both the pole figures and the orientation space are subdivided into small subregions, which are considered as components of vectors of high dimensions (Ruer and Baro, 1977; Vadon and Heizmann, 1991). The two vectors are coupled by a huge, though sparse, matrix which contains the geometrical relations between the pole figures and the orientation space. The texture vector representing the final O D F is derived from the pole figure vectors through an iterative fitting procedure. The method is straightforward and easy to apply, but handling of the large matrices causes problems with computation efficiency which reduces the resolution of this method.
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INTRODUCTION TO TEXTURE ANALYSIS
In the component method by Helming and Eschner (1990) the O D F is described by a set of a finite number (mostly less than 30) of generalised Gausscomponents. The orientations and some parameters to describe these generalised Gauss-components are directly derived from the pole figures by an iterative nonlinear optimisation procedure. Though this method is rather tedious to apply, it provides a very condensed description of textures with physically meaningful quantities. Furthermore, this method is well suited to derive ODFs from incomplete pole figures with uncommon non-measured areas, as e.g. SAD-pole figures (Helming and Schwarzer, 1994; Section 8.3.2).
5.3.3 Comparison of series expansion and direct methods Comparing the series expansion method with its different approaches to derive the odd part of the ODF with the various direct methods reveals some characteristic advantages of each, which are summarised in the following: The series expansion method is a fast, reliable method which is rather insusceptible to experimental scatter in the pole figures. Its main output the C-coefficients represent a condensed manner with which to characterise the texture which is superior to storing the entire ODF-data. The C-coefficients also disclose valuable additional information on texture-related properties, e.g. elastic modulus. elastic and plastic anisotropy, etc. Furthermore, changes in the reference coordinate system can be accommodated by manipulation of the C-coefficients, so that texture changes during phase transformations (Kallend et al., 1976; Park and Bunge, 1994) or recrystallization (Pospiech and Liicke, 1979: Liicke and Engler, 1990) can readily be analysed. Finally, normalisation of the experimental pole figures (Section 4.3.5) is easier. In the direct methods the condition of non-negativity and the handling of the ghost problem are usually implicitly fulfilled, and no truncation error arises. Analysis of incomplete pole figures as well as different sample and/or crystal symmetries is easier to accomplish than in the harmonic method. In general, direct methods require fewer pole figures to obtain satisfactory solutions, which is of particular interest when low-symmetry materials are analysed. All current approaches to compute ODFs from pole figure data have now been developed to a state that, provided good experimental data are available, they are able to yield routinely reliable and reproducible quantitative texture results (e.g. Wenk et al., 1988; Matthies et al., 1988). Nonetheless, because of the systematic differences between different methods it is strongly recommended not to change the method within a series of experiments, and in particular not to switch between series expansion and direct methods. -
5.4 REPRESENTATION AND DISPLAY OF TEXTURES I N EULER SPACE The quantitative description of a given texture in terms of its ODF, i.e. the orientation density in terms of the three Euler angles, necessitates an appropriate three-dimensional orientation space for its representation. The first orientation space was introduced by Perlwitz et al. (1969) by defining three angles p,, p2 and p3 to
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113
represent the orientation distributions of rolled copper and brass which were determined by electron diffraction in a TEM (Pitsch et al., 1964). This space was chosen so that the most important rolling texture components all came to lie in a two-dimensional section with p3 = 0" through the three-dimensional space, which strongly facilitated the visualisation and evaluation of textures in this space. In the present Section we will focus on the orientation space which is most frequently used today, that is the Euler angle space. This orientation space is formed by three Euler angles such that an orientation given by a triple of Euler angles corresponds to a distinct point in Euler space whose coordinates are defined by these three angles. Because of the necessity to represent the orientation information in a threedimensional orientation space, texture analysis is often considered a complicated problem which requires a large ability to visualise things. This task can greatly be facilitated by suitable ways to display and, if possible, to condense the texture data. We will address the common methods used to represent textures in the Euler space as well as appropriate means to condense textures and thus to facilitate texture analysis.
5.4.1 Properties of Euler space The Euler space is a three-dimensional orientation space whose axes are formed by the three Euler angles. In most cases the three Euler angles are chosen to form an orthogonal coordinate system resulting in a Cartesian Euler angle space (Morris and Heckler, 1968). With regard to the axes of this space, in Section 2.5.1 three different sets of the Euler angles were introduced, namely the angles pl, Q, p2 (Bunge), 9,0, (Roe) or Q, 0,4 (Kocks). Although all these conventions likewise define an orientation space, in the reminder of this Section we will confine the discussion to Bunge's angles g,,Q , cpz, as these are by far most commonly used in the literature. The principal considerations hold for all conventions, however. As already pointed out in Section 2.5.2, an orientation given by a triple of Euler angles corresponds to a distinct point in Euler-space whose coordinates are unequivocally given by these three angles (Figure 2.8). We have also seen how sample and crystal symmetry influence the size of the Euler angle necessary to represent all possible orientations in an unequivocal manner (see Table 2.2), and it turns out that the widely used method of representing textures in a cubic Euler space with 0" < (p,, Q, p2) < 90' inevitably assumes orthotropic sample symmetry. It should be emphasised that even in cases where orthotropic sample symmetry applies in principle - e.g. in samples deformed in plane strain - this reduction can lead to loss in information and even can cause errors. It is clear that asymmetries in the textures which point to deviations from the idealised plane strain state cannot be resolved in such reduced ODFs (Sections 5.4.2, 9.6). More severely, in all cases where the initial texture of a sample did not depict orthotropic symmetry prior to deformation, reduction of the Euler angles under the assumption of orthotropic sample symmetry is misleading, although orthotropic sample symmetry does apply. For instance, single crystals in most cases have no orthotropic symmetry (with the exception of crystals with highly symmetrical orientations like cube {001)<100> and Goss
INTRODUCTION TO TEXTURE ANALYSIS
{011)<100>) so that important texture features like direction and amount of rotations cannot properly be resolved. In accordance with the definition of the Euler angles (Section 2.5.1), for Q, = 0" (and @ = 180") the two remaining angles p, and 9 2 describe rotations about the same axis, namely, the normal direction, which gives rise to strong distortions in the Euler angle space. Figure 5.7a shows the position of the three equivalent variants of the orientation (123)[634] in the reduced Euler angle space. For each central orientation (marked by a dot) an isotropic scatter of 10" is marked. It is obvious that the resulting spheres in the Euler angle space are far from being isotropic, but rather show strong distortions which become most pronounced for small angles Q. An orientation with @ = 0" even deteriorates to a line with the condition p1 + p? = constant. These distortions become most evident by representing the texture in the (unusual) @-sections: an example will be shown in Section 9.6.1. With regard to the cube recrystallisation texture, the distortions of Euler space result in the fact that RD-rotations of the cube orientation {001}<100> can be found in the 9 2 = 0" section by following the angle Q at p1 = O0,whereas TD-rotations are visible in the same section along at p1 = 90". In Section 5.4.2 we will see that these distortions can be reduced by representing the textures in polar rather than Cartesian coordinates.
-
5.4.2 Representation and display of orientations
A representation of an O D F in a three-dimensional space as in Figure 5.721 is not generally suitable for publication on the printed page. Therefore, it is common practice to represent ODFs in form of sections through the orientation space. As already discussed for the pole figures, the intensity distribution in the individual sections can be displayed by different colours, different grey values or, most commonly, by isointensity lines (Figure 5.7b). Usually, equal distance sections along one of the angles in 5"-steps (sometimes 10"-steps) are used. In principle, sections of all three angles can be produced and evaluated alike, though surveying the literature shows that ODFs of fcc and hexagonal materials are usually presented in p?-sections. whereas the textures of bcc materials are traditionally shown in pl-sections, though p?-sections offer some advantages (see below). @sections are hardly ever used, in contrast. As already mentioned in Section 5.4.1. the distortions in the Cartesian Euler space at small angles Q can greatly be reduced by representing texture data in polar rather than the usual Cartesian coordinates (Williams, 1968: Hansen et al., 1978; Wenk and Kocks, 1987). For that purpose, the angles p1 and are plotted in polar coordinates just as in a pole figure and p2 is represented as the axis of a cylinder perpendicular to the polar plot, as shown on Figure 5.8a. Analogous to the q2-sections through the Cartesian Euler space (Figure 5.7b), a set of sections through the resulting cylindrical space at constant values of 9 2 is adequate to represent texture in a two-dimensional image (Figure 5.9). Following the definition of the Euler angles (Section 2.5), for fixed values of p1 and Q, the angle p2 defines a rotation about a distinct [001]-axis. Thus, the projection of the O D F in the plane with 9 2 = 0" corresponds to a (100) pole figure (Figures 5.8a, 5.9). 92-sections have been labelled cqlstul orientation distribution (COD) to indicate that in this representation the crystal
-
-
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
1 15
Figure 5.7 Spatial arrangement of the three components of the orientation (123)[63a] with isotropic scatter of 10" in the reduced Euler angle space showing strong distortions at small angles Q. (b) Same texture as in (a) plotted as iso-intensity lines in p?-sections ( - 1=~5')~ through the Euler space.
orientations are expressed in a sample coordinate system (Wenk and Kocks, 1987). Alternatively, one may wish to represent the angles Q and in polar coordinates with y1 as cylinder axis (Figure 5.8b). As for fixed angles p 2 and Q, cpl indicates a rotation about the sample normal direction ND: a projection of the ODF in the
l 16
INTRODUCTION TO TEXTURE ANALYSIS
Figure 5.8 Representation of the orientation space in a cylindrical Euler space. (a) Cylinder axis parallel the projection of texture on the base (pz = 0") corresponds to a pole figure (crystal orientation to v,-?; distribution, COD): (b) Cylinder axis parallel to pi; the projection of texture on the base (pi = 0") corresponds to an inverse pole figure (sample orientation distribution, SOD).
plane with 91 = 0" corresponds to an inverse pole figure of the ND. For that reason, yl-sections may be referred to as sample orientation distribution (SOD) (Wenk and Kocks, 1987). The texture presentation in a cylindrical space in sections (p: = constant is particularly suitable for demonstration of the influence of sample symmetry on the necessary range of orientation space. As outlined in Section 2.5.2, sample symmetry affects the range of 91.Thus. for the three cases orthotropic, monoclinic and triclinic sample symmetry respectively a quarter circle (i.e. (pinax = 90°), a semicircle ( c p y = 180') and a full circle ((p? = 360") is necessary to present the texture. As an example, Figure 5.9a and b show the texture of a hot band of the aluminium alloy AA6016 at the band centre and at the surface, respectively. In both cases, the textures are represented in the cylindrical Euler space in 9:-sections (i.e. CODS).At the centre layers of the sheet, the assumption of orthotropic sample symmetry is very well fulfilled, which results in symmetric semicircles representing the different pz-sections as well as in a symmetric projection, i.e. pole figure (Figure 5.9a). Accordingly, a quarter circle would suffice to display all texture details. At the band surface, in contrast, the different semicircles evidently are not symmetric (Figure 5.9b), representing the deviation of the strain state from orthotropic sample symmetry. It is seen in Figure 5.9 that the representation of ODFs in polar coordinates still reveals some problems at small angles, i.e. at the centre of the circular sections, which may particularly affect representation of the cube-orientation. These distortions can further be refined by introducing oblique 0-sections (with a = (p1 9412; Bunge, 1988; Matthies et al., 1990), which will not be discussed here any further. The main disadvantage of the representation of textures in the cylindrical Euler space is that determination of the orientations is more con~plicatedthan in an orthogonal space.
+
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
117
a projection
ND
Figure 5.9 Texture of hot rolled aluminium AA6016 represented in the cylindrical Euler space (Qsections, COD). (a) Rolling texture at the centre layer of the sheet depicting orthotropic sample symmetry. The sections p,-2= 0", 45" and 90" as well as the (projected) pole figure are symmetrical with respect to TD. Also, note symmetries between sections p2 = 15' and 75" and p2 = 30' and 60"; (b) Shear texture at the sheet surface depicting monoclinic sample symmetry. The various semicircles show no symmetry with respect to TD, representing the deviation of the strain state from orthotropic sample symmetry.
INTRODUCTION TO TEXTURE ANALYSIS
l18
In many studies, a series of comparable textures is analysed. e.g. rolling textures at different levels of deformation or recrystallisation textures obtained after annealing at different temperatures. To compare the usually similar - textures in such cases it may be useful to confine the representation to the sections of the ODF which display the characteristic texture changes, which typically affect only one or two sections rather than the entire Euler angle space. (Yet another way, the representation of intensity distributions along certain paths through the Euler space, will be addressed in Section 5.4.3). For instance, textures of bcc materials can very compactly be represented by plotting the section with 9 2 = 45" that contains two fibres which are characteristic of many rolling and recrystallisation textures of bcc materials, that are the a-fibre comprising the orientations (hkl) which have a common <011> direction parallel to the R D and the 7-fibre having orientations ( l l l) with a common { l 11) normal in N D (Hutchinson, 1984; Raabe and Liicke, 1994). Despite 1-sections the fact that bcc rolling and recrystallisation textures are usually shown in 9 (Figure 5. lOa), it is seen that the most important texture features can be found in the section with 9 2 = 45' (Figure 5. lob, c). Many interesting features of recrystallisation textures in fcc materials are visible in the p2 = 0" section (e.g. Engler et al., 1995a). -
rvwwK (P2
(P, =const.
8
8
compl. ODF
Q
Q
(b) cp2=4tj0-section
1
@
f;7
@m@
Levels: i 2 3 4 5
r
(a) v,-sections Figure 5.10 Texture of cold rolled Fe-C steel with 0.450ioC in the Euler space (75% rolling reduction). (a) Conventional representation of the ODF in p,-sections: (b) $2 = 45" section displaying the intensity distribution of the a-fibre and ^.-fibre orientations; (c) schematic representation of the most important orientations in bcc materials in the yz = 45' section.
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
5.4.3
119
Fibre plots
Instead of displaying the entire ODFs, textures can be represented in a very condensed way by plotting the orientation intensity along certain characteristic paths or distinct crystallographic fibres through the orientation space versus an angle which defines the position along this path or fibre (Bunge and Tobisch, 1968; Liicke et al., 1985). As already mentioned in Section 5.4.2, the rolling and recrystallisation textures of many bcc materials are characterised by two crystallographic fibres, the a-fibre with <01 l>//RD and the y-fibre with { l 1l)IjND; these and other, usually less important, fibres of bcc materials can be found in Table 5.1. Therefore, the characteristic texture changes during rolling and recrystallisation can readily be visualised by plotting the orientation density along these fibres. Figure 5.11 shows an example of the evolution of the a- and y-fibres during cold rolling of a ferritic low carbon steel starting from the hot band up to sheets with 90% thickness reduction. Provided that the texture information not covered by these fibres is of minor importance. such diagrams evidently represent a very convenient way of displaying characteristic textural features compared to O D F plots. In the rolling textures of fcc samples the orientations also accumulate in the vicinity of several fibres, which are summarised in Table 5.1. The most important ones are the a- and the P-fibre: at low deformation degrees the a-fibre with (01 l}// R D is observed, which runs from the Goss-orientation {011)<100> to the Borientation {011)<211>. At higher deformation degrees the rolling textures are mainly characterised by the B-fibre, which runs through the Euler space from the C-orientation {112)<111> through the S-orientation {123)<634> to the Borientation {011)<21 l > where it meets the a-fibre (Hirsch and Liicke, 1988). Thus, specific differences in the rolling textures of different fcc materials can conveniently be displayed by plotting the intensity distribution along these fibres. Note that the path of the 9-fibre is not fixed in the Euler space, but it connects the relative intensity maxima in the corresponding ODF-sections along the fibre, giving rise to the alternative term skeleton line. Therefore, besides analysing the intensity distribution Table 5.1 Characteristic fibres in bcc and fcc metals and alloys
Material
Fibre 0;
bcc
7
5 3
fcc
0; I
Fibre axis
Euler angles*
: 'RD < 111>;:ND /'RD ,';ND < 110>,"TD
0".O c , 45"-0". 90",45"
*8
60". 54.7". 45'-90". 54.7". 45" O0, O0, 0"-0". 45", 0" O", 45". 0'-90". 45". 0' 90", O", 45"-90". 90". 45" On, 35". 45"-9OC, 54.7', 45" 0". 4 j 0 , 0"-9O0, 45". 0" 60". 54.7". 45"-90", 54.7", 45" 90"; P, 45'-90". 90°, 45" 90". 35", 45"-35". 45". 90'
* Typical values without symmetry considerations. ** Defined by the rnaximunl intensity rather than lographic position.
by exact crystal-
120
INTRODUCTION TO TEXTURE ANALYSIS
(a) a-fibre //RD
(b) y-fibre {l 11)//ND
Figure 5.11 Development of the rolling texture fibres of a low-carbon steel with increasing rolling reduction (Holscher et al.. 1991).
along the @-fibre,subtle details in the position of the P-fibre in the Euler space can be evaluated, which may e.g. yield valuable information with regard to underlying deformation mechanisms (Hirsch and Lucke, 1988; Engler et al., 1989). In many fcc metals and alloys there is a characteristic dependence of the rolling texture on the material, especially on its stacking fault energy ( S F E ) . As an example, Figure 5.12a shows the ,!?-fibresof rolled pure copper and copper with 5% and 37% zinc after 95% rolling reduction. With increasing zinc-content, i.e. with decreasing §FE, the P-fibres show a systematic decrease of the intensities close to the C- and Sorientations to the advantage of the B-orientation, which reflects the characteristic rolling texture transition from the copper- or pure-metal type to the brass or alloy type (Wassermann and Grewen, 1962; Hirsch and Lucke, 1988). Rather than changing the SFE, alloy additions may precipitate in the form of second-phase particles, and different precipitation states affect the deformation behaviour by interaction of the slip dislocations with the particles in dependence on particle size and volume. Figure 5.12b shows the rolling textures obtained in an aluminiumcopper alloy after different pre-ageing treatments, where it turned out that the resulting different precipitation states significantly influenced the intensity distribution along the 8-fibre (Engler et al., 1989; Lucke and Engler, 1990). Other
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
121
Figure 5.12 (a) P-fibres of pure copper and two different copper-zinc alloys, showing the influence of stacking fault energy on the rolling textures (95% rolling reduction; Hirsch and Liicke, 1988); (b) 9-fibres of AI-1.8%Cu cold rolled after different ageing treatments, showing the influence of precipitates on rolling textures (95% rolling reduction; Engler et al., 1989).
parameters like initial texture (Mao e f ul., 1988), initial grain size (Engler, 1995) and deformation temperature (Engler et al., 1993a; Maurice and Driver, 1997) have also been shown to affect the intensity distribution along the various fibres, though generally less dramatically than alloy composition.
5.5 SUMMATION In this Chapter the most common methods to represent and evaluate macrotexture data have been introduced and their main advantages and disadvantages have been addressed. The most well known way of representing macrotextures is to project the crystal orientations into the sample coordinate system in a pole figure. Since the density distribution in a pole figure is the direct output from most techniques to determine textures experimentally, pole figures are well suited to a critical assessment of the experimental data. Inverse pole jigures, the projection of the sample coordinate system into the crystal coordinate system, are used frequently to represent the orientation distribution of the main sample axis in the case of uniaxial sample symmetry. Whereas for analysis of fibre textures, representation of one reference direction may suffice, for the evaluation of misorientations the full orientation
122
INTRODUCTION TO TEXTURE ANALYSIS
information must be considered, however, which renders inverse pole figures inferior to the pole figures for the representation of texture data. The projection of the three-dimensional orientation onto a two-dimensional pole figure results in a loss in information, so that for a quantitative evaluation of macrotextures a three-dimensional representation, i.e. the orientation distribution function (ODF), is required. However, ODFs cannot be measured directly in X-ray or neutron diffraction experiments but have to be calculated from several pole figures. known as pole figure inversion. An analytical solution of this problem of pole figure inversion is unfortunately not possible, and a variety of numerical approaches to derive the ODF has been developed, which can basically be subdivided into two groups. The harmonic method is based on a series expansion of ODF and pole figures. This method inherently suffers from the so-called ghost error, which is caused by the impossibility to derive the odd-order C-coefficients from diffraction experiments. This means that the odd-order C-coefficients must be determined based on additional assumptions. e.g. the condition that the O D F must be nonnegative for all orientations. In combination with an appropriate method to correct for the ghost error, the series expansion method is a fast, reliable method which is rather insusceptible to experimental scatter in the pole figures. The resultmg C-coefficients represent a condensed manner to characterise the texture and, furthermore, they disclose information on texture-related properties. 0
In direct methods the ODF is computed in an iterative procedure directly in the orientation space, i.e. not in a Fourier space as in the harmonic method. The condition of non-negativity and the handling of the ghost problem are usually implicitly fulfilled and no truncation error arises. Analysis of incomplete pole figures as well as different sample andior crystal symmetries is straightforward to accomplish.
A comparison of the various methods to compute ODFs from pole figure data proves that nowadays all methods are able to yield reliable and reproducible quantitative results, but because of the systematic differences between methods it is suggested to stay with a given method within a series of experiments. To represent the orientation density in dependence on the three Euler angles, i.e. the ODF, an appropriate three-dimensional orientation space is necessary. Traditionally, textures are represented in sections through an orthogonal orientation space whose axes are formed by the three Euler angles, that is the Euler space. However, this Cartesian Euler space depicts strong distortions at small angles Q, which renders a visualisation of the orientations and textures complicated. These distortions can be reduced by representing the textures in polar rather than orthogonal coordinates similar as in pole figures which results in a cylindrically shaped Euler space. Texture analysis can greatly be facilitated by suitable ways to condense the data. In many cases texture representation can be confined to characteristic regions of the Euler space which contain the texture information of interest rather than showing the entire ODF. For instance. if material processing affects only a few sections of -
-
EVALUATION AND REPRESENTATION OF MACROTEXTURE DATA
123
the Euler space, the representation of these sections may be sufficient to visualise important texture changes. Many textures can be represented in a very condensed way by plotting the orientation intensity along certain characteristic paths or distinct crystallographic fibres through the orientation space versus an angle defining the position along this path or fibre.
MICROTEXTURE ANALYSIS
6. THE KIKUCHI DIFFRACTION PATTER
6.1 INTRODUCTION In the previous Chapters we have seen that there is a basic divide between macrotexture techniques (based on X-ray or neutron diffraction, Chapters 4 and 5 ) and microtexture techniques (based usually on electron diffraction, Chapters 7-1 l), which relates to the scale of the inquiry and thence affects the nature of the primary output data. For macrotexture techniques the primary output is a profile of diffracted intensities which is characteristic of a large contiguous sample volume, whereas for microtexture i.e. individual grain techniques the primary output is a diffraction pattern from each sampled volume. Such a pattern embodies the complete crystallographic information inclusive of the orientation of the respective sampled volume, which usually is an individual crystallite wherein the orientation can be taken to be uniform. For the vast majority of microtexture work in both the TEM and §EM this type of pattern is a Kikuchi diffraction pattern. Other techniques occasionally used are selected area diffraction, which still has applications in TEM (Section 8.3) and the micro-Kossel technique in the SEM (Section 7.2). In this Chapter we will address first the formation of Kikuchi patterns (Section 6.2). Since the principles of pattern formation, and the crystallographic information embodied in them, is equivalent for TEM or §EM generation, they can all be evaluated according to the same principles. The interpretation of a Kikuchi pattern, i.e. the determination of the crystallographic orientation of the sampled volunle, is performed in two consecutive steps: -
-
e
Firstly, the pattern is indexed by identifying the crystallographic indices of the poles and bandsilines in the pattern
e
Secondly, the relative position of the poles or bands/lines with respect to an external reference frame is determined
Although interpretation of Kikuchi patterns is theoretically straightforward, consistent indexing and subsequent orientation calculation is a cumbersome, timeconsuming procedure and care is needed to avoid errors. Appropriate computer codes strongly facilitate the procedure, and the present Chapter introduces various
128
INTRODUCTION TO TEXTURE ANALYSIS
levels of dedicated computer support from assistance with matrix handling during manual, off-line microtexture analysis up to fully automated systems for pattern indexing and orientation determination in on-line applications (Section 6.3). Another parameter which can be derived from the Kikuchi pattern is a measure of its 'quality' or diffuseness, which is significant because it can provide information on plastic deformation in the lattice (Section 6.4). Although this is not strictly texture analysis, it is included here because the pattern quality is obtained concurrently with the orientation, and can augment interpretation of the data (Section 10.4.5).
6.2 THE KIKUCHI DIFFRACTION PATTERN A Kikuchi diffinction pattern arises by application of the following electron diffraction techniques (e.g. Dingley, 1981; Gottstein and Engler, 1993): e
Selected area channelling (SAC) in the SEM (Section 7.3)
e
Electron back-scatter diffraction (EBSD) in the SEM (Section 7.4)
a
Microdiffraction or convergent beam electron diffraction (CBED) in the TEM (Section 8.4)
Nowadays it is common for Kikuchi patterns to be analysed automatically, especially in the SEM, often using commercially available software packages. Whereas this is a tremendous benefit since data can be acquired very rapidly by an operator who needs to have only a rudimentary knowledge of crystallography, such a 'black box' approach can be detrimental if it should be necessary to check data or to understand the principles of the calibration routine. For these reasons a description of the formation, meaning and interpretation of the geometry of Kikuchi patterns, which relates directly to the crystal orientation, is included here. 6.2.1 Formation of Kikuchi patterns In order to understand the principles of orientation determination, the formation of Kikuchi lines can be explained in terms of a simplified model which considers only their geometrical aspects (Kikuchi, 1928). Although Kikuchi diffraction occurs both in the TEM and SEM, the geometry is easier to illustrate for the TEM case and so this will be considered first. When an electron beam enters a crystalline solid, it is diffusely scattered in all directions. This means that there must always be some electrons arriving at the Bragg angle BB at every set of lattice planes, and these electrons can then undergo elastic scattering to give a strong, reinforced beam. Figure 6. l a illustrates the situation for just one set of lattice planes. Since diffraction of the electrons through the Bragg angle is occurring in all directions, the locus of the diffracted radiation is the surface of a cone (Kossel cone) which extends about the normal of the reflecting atomic planes with half apex angle 90" - &. The source of electron scattering can be considered to be between lattice planes, as shown on Figure 6.la, and hence two cones of radiation result from each family of planes viewed simplistically as one from 'either side' of the source.
THE KIKUCHI DIFFRACTION PATTERN
(a)
129
incident cones
diffuse scatterin
(c)
incident electron beam
excess- defect-
4
Kikuchi-line
Figure 6.1 Kikuchi lines in transmission geometry in a thin TEM specimen. (a) Origin of Kikuchi lines by inelastic scatter of the electrons. giving Bragg diffraction at source S on lattice planes (lzkl);(b) Kikuchi pattern from nickel (CBED, accelerating voltage 120kV); (c) formation of excess and defect Kikuchi lines.
From substitution of typical values for electron wavelength (equation 3.12) and lattice interplanar spacing into Bragg's law (equation 3.4), the Bragg angle BB is found to be about 0.5". Consequently the apex angle of a diffraction cone is close to
INTRODUCTION TO TEXTURE ANALYSIS
130
180°, i.e. the cones are almost flat. If some sort of recording medium - a phosphor screen interfaced to a camera or a piece of cut film - is positioned so as to intercept the diffraction cones, a pair of parallel conic sections results, which are so nearly straight that they generally appear to be parallel lines. These are Kikuchi lines, and it can be seen that their spacing is an angular distance of 2QB which in turn is proportional to the interplanar spacing. Thus, the whole Kikuchi pattern consists of pairs of parallel lines where each pair - or 'band' has a distinct width and corresponds to a distinct crystallographic plane. The intersection of bands corresponds to a zone axis (pole), and major zone axes are recognised by intersection of several bands. The Kikuchi pattern therefore essentially embodies all the angular relationships in a crystal - both the interzonal and interplanar angles - and hence implicitly contains the crystal symmetry. Figure 6.lb is a TEM Kikuchi pattern from nickel. The orientation of the pattern and hence of the volume from which it has arisen is evaluated by 'indexing', i.e. identifying the poles and bands in the pattern, and calculating the relationship between these and some chosen reference axes. This process will be described in detail in Section 6.3. It is seen from Figure 6.lb that the two lines forming a given Kikuchi band in general have different intensities, and typically the line that is closer to the primary beam is darker (the 'defect' line) than the background, whereas the other one is brighter (the 'excess' line). The formation of dark and bright Kikuchi lines is illustrated in Figure 6 . 1 ~In . general, the angle 8,between the primary beam and the diffracted beam that gives rise to the excess line is smaller than the angle dd between the primary beam and the beam diffracted towards the defect line. As discussed in Section 3.2, the scattering amplitude of electrons strongly decreases with increasing scattering angle 8.Thus, in the direction of the incident electron beam, where Bragg's law is satisfied, low-intensity defect lines will occur since there the electrons are scattered away and hence do not contribute to the transmitted intensity. Lines of higher intensity, i.e. the excess lines, are obtained in the direction into which the electron bean1 is diffracted since there the electrons add to the background intensity. In Figure 6.1 the diffracted electron beams are shown passing through the specimen, which is the case for TEM. For EBSD in the SEM, diffraction occurs from the interaction of primary 'backscattered' electrons with lattice planes close to the specimen surface (Sections 7.4-7.8). Tilting the specimen by angles of typically 60-70" allows more electrons to be diffracted and to escape towards the detector or recording medium. Figure 6.2a shows the modified diagram for formation of one pair of Kikuchi lines for the tilted specimen EBSD geometry, rather than the TEMtransmission geometry (Figure 6. la). Kikuchi patterns can also be produced in the SEM by changing the direction of the incident beam, e.g. by rocking the electron beam at a given sample site, which is selected area channelling (SAC) (Section 7.3). Although the exact formation mechanisms of the resulting bands are slightly different (pseudo-Kikuchi bands), both their geometry and their crystallographic information are equivalent to the Kikuchi patterns and, consequently, they can be evaluated according to the same principles (Section 6.3). -
THE KIKUCHI DIFFRACTION PATTERN
(4 incident electron beam
phosphor screen
Figure 6.2 EBSD Kikuchi patterns. (a) Origin of Kikuchi lines from the EBSD (i.e. tilted specimen) perspective; (b) EBSD pattern from nickel (accelerating voltage 20 kV).
132
INTRODUCTION TO TEXTURE ANALYSTS
6.2.2 Comparison between Kikuchi patterns arising from different techniques
Figure 6.2b is an EBSD Kikuchi pattern from nickel. If this is compared with the TEM Kikuchi pattern from the same material (Figure 6.lb). there are two differences: e
The capture angle is about five tlmes greater for the EBSD pattern than the TEM pattern This is a consequence of the experimental set-up m both cases and faellltates patteln indexing and identification of the symmetry elements for EBSD compared to TEM
e
The Kikuchi lines are sharper in the TEM case than in that of EBSD, such that defect and excess lines can be distinguished in the former. This is a consequence of the different electron transfer functions in the two cases and means that the greater precision in measurements from the diffraction pattern is obtained in TEM.
Figure 6.3 shows a comparison between Kikuchi patterns from silicon obtained by SAC and EBSD respectively (Wilkinson, 1996). Only a small portion of each pattern around the <103> zone axis is shown to illustrate that more fine detail is visible in the SAC than in the EBSD pattern. Although this detail is redundant as far as orientation measurement is concerned, it is an advantage for other applications such as strain and stress measurements (Dingley, 1981; Troost rt d., 1993; Joy, 1994; Wilkinson, 1996).
6.2.3 Projection of the Kikuchi pattern The Kikuchi diffraction pattern is essentially a projection onto a flat surface, i.e. a film or screen. of the angular relationships in the crystal. The projection is visualised with the aid of a reference sphere, as shown in Figure 6.4. Appendix 4 comprises a description of the use of projection techniques in texture analysis. We imagine that the sampled volun~eof crystal resides at the centre of a sphere with radius ON. where ON is the distance between the specimen and the projection plane which is positioned so as to be the tangent plane at the 'north pole' N. N is referred to as the d$fj.uction pattern centre. Diffracted rays from the specimen intersect the sphere and from there project out to the recording medium. Since the projection source point is at the centre of the sphere, the position of a point P on the projection plane will be given by ON(tan r),where T is the angular displacement of P from the pattern centre, as shown on Figure 6.4. For large values of r, i.e. > 60", the projection takes on a 'stretched' appearance because the tangent function increases rapidly as it approaches 90". This effect is not apparent in TEM or SAC patterns where the angular range of the patterns is small, but it is encountered in EBSD patterns. The type of projection illustrated in Figure 6.4 is called a gnon~onicprojection (Phillips, 1971). Appendix 6 shows standard Kikuchi-maps constructed from the most prominent reflectors for fcc, bcc and hexagonal crystal structures. Such maps are very helpful during manual indexing or to check automatic indexing of Kikuchi patterns (Section 6.3).
THE KIKUCHI DIFFRACTION PATTERN
133
Figure 6.3 Comparisons between Kikuchi patterns from silicon obtained by different techniques in the SEM (same zone axis and same capture angle). There is more fine detail in (a) the SAC pattern than in (b) sthe EBSD pattern (Courtesy of A. Wilkinson).
Figure 6.4 Illustration of a Kikuchi pattern as a gnomonic projection. showing the reference sphere (radius r ) and a projected pole P. The projection point is 0 and the origin of the projection (pattern centre) is N.
6.2.4 Qualitative evaluation of the Kikuchi pattern A Kikuchi diffraction pattern contains a wealth of information, as indicated in the previous Sections. Whereas the pattern can be quantitatively interpreted to obtain an exact orientation (Section 6.3), other information. which may be valuable to the investigation, is available from rapid qualitative, i.e. visual, evaluation of the pattern. This information includes:
134
INTRODUCTION TO TEXTURE ANALYSIS
Lattice strain can be identified by diffuseness in the diffraction pattern. wh~chis a consequence of lattice plane bending. In general, the pattern blurredness is used as a guide to strain in the lattice, e.g. to pick out recrystallisation nuclei from a coldworked matrix or to formulate one type of orientation map (Section 10.4.5).There has been some success in evaluating quantitatively the amount of lattice strain from pattern diffuseness (Wilkinson, 1996; Troost et al.. 1993). Methods to describe the pattern quality in a more quantitative way will be described in Section 6.4. Graitz/phase boundaries can be identified from the change in the real-time diffraction pattern which accompanies a traverse of the sampling probe over the specimen surface. With some experience, tasks such as distinguishing low-angle boundaries from high-angle types, checking the integrity of single crystals or estimating the grain size distribution, can be carried out.
Identification of certain orientations, e.g. the cube orientation (001)<100>. can be rapidly accomplished by visual appraisal, facilitated by overlaying a computer simulation of the target diffraction pattern for the exact orientation on top of the real-time patterns.
6.3 QUANTITATIVE EVALUATION O F THE KIKUCHI PATTERN As described in Section 6.2.1, the Kikuchi lines are linked directly to the arrangement of the reflecting lattice planes in the crystal and hence Kikuchi patterns are perfectly suited for determination of the crystallographic orientation. Interpretation of the Kikuchi patterns, which gives the determination of the crystallographic orientation of the sampled volume. is performed in two consecutive steps. First. the pattern has to be indexed, which means that the crystallographic indices of the Kikuchi bands and poles (more precisely. of the corresponding lattice planes) have to be determined. After the indexing, in the second step the relative positions of the bands or poles with respect to an external reference frame, i.e. the crystallographic orientation of the sampled volume with respect to the specimen axes (Section 2.2.2): are derived. Figure 6.5 illustrates this procedure for the TEM case. The basic principles of orientation calculation are outlined in the Section 6.3.1 for the case of manual evaluation of TEM Kikuchi patterns. A more rigorous approach to derive the orientation matrix will be introduced in Section 6.3.2 together with the nlethods for automatic indexing of EBSD patterns. With the advent of EBSD for large scale microtexture analysis the demand for support by dedicated computer codes increased, which eventually led to the development of fully automated, on-line systems for pattern indexing and orientation determination (also Figure 7.6). Nowadays, such systems are comn~erciallyavailable from several vendors. In order of dependence on the degree of automation - and the availability of the increasingly sophisticated software - the operation modes for evaluating Kikuchi patterns are: o
Manzral, non-automated. Traditionally, Kikuchi patterns were recorded on photographic films and then evaluated 'off-line' after developing the films. In most
THE KIKUCHI DIFFRACTION PATTERN
incident electron beam
sample
I
1
7'7 RD
Figure 6.5 Diagrams illustrating the evaluation of a TEM Kikuchi pattern. (a) Correlation between the reference frame of the specimen and a Kikuchi pattern; (b) indexed Kikuchi pattern, drawn to scale. from a-hich the crystallographic orientation is derived (see text for details).
INTRODUCTION TO TEXTURE ANALYSIS
cases, pattern indexing is assisted by comparison with standard patterns, and then the crystallographic orientation of the sample volume can be derived as described in Section 6.3.1. Inter-clcti~se,semi-clutornateu'. For an interactive orientation evaluation of a Kikucl~ipattern. the coordinates of a number of bands or axes in the pattern are marked by the user and transferred to a computer, which auton~aticallyindexes the pattern and computes the corresponding orientation. In Section 6.3.2 some approaches for semi-automated pattern indexing and orientation determination will be described. Fullj uutonzated. Con~pleteautomation means that the crystallographic features in the Kikuchi patterns are automatically detected by means of appropriate pattern recognition codes, thus avoiding the time-consuming manual marking of zones or axes. Such fully automated codes are an essential prerequisite for performing orientation microscopy (Chapter 10).
6.3.1 Principle of orientation determination Interpretation of Kikuchi patterns that are obtained in the TEM, or by SAC in the SEM, is still performed in many cases by evaluation of photographic films. Both TEM and SAC Kikuchi patterns can mostly be indexed by visual appraisal, particularly for cubic crystals, or by comparison with standard patterns (Appendix 6). In cubic crystal structures, when the indices of two intersecting bands are ( h l ,l<,,11) and (h2,k?, I?), the corresponding zone axis [u~wv]can be derived from the crossk2,l?), and vice versa. For example in Figure 6.5 the product, i.e. ( h l ,k l , 11) X (h, cross product of band 1 and band 2, (220) and (31 1): gives the zone axis labelled axis 12: [l 121. In non-cubic crystal structures, the crystallographic indices have first to be transformed into an orthonormal coordinate system via multiplication by the inverse crystal matrix (Section 2.2). For both TEM and SAC the photographic plate is arranged parallel to the specimen surface itself, so that the primary electron beam direction, usually referred to as the beam r?onml, BN, corresponds to the specimen normal direction, ND. Thus, to derive the crystallograpl~icorientation, the angles between the indexed bands or poles and the sample normal can readily be determined by measuring their distances to the beam normal, a,, 02,013, on Figure 6.5b. The scaling factor to transform these distances into angles is derived from the distance between two indexed poles or from the band width of a Kikuchi band, which is related through 28 to its indices. (Although the diffraction pattern is a gnonlonic projection (Section 6.2.3 and Appendix 4); distances near the pattern centre can be considered to scale linearly with projection angle). With the information about the crystallographic indices q" (qiqlq;) with i = 1, 2 . . . I ? (where rz is the number of identified bands or poles) and the angles ai between each band or pole i and the beam normal, the orientation of the sample normal in terms of its crystallographic indices (hkl) can be expressed as:
THE KIKUCHI DIFFRACTION PATTERN
The root term in equation 6.1 normalises the indices of the bands or axes, which are typically given as integer numbers, e.g. (022) or [l 121: to unity. Hence the resulting vector (hltl) is directly normalised to unity. It is seen that 3 equations are necessary to determine the vector (hkl), which means that 3 bands or poles have to be indexed. In the example given in Figure 6.5b, the angles a ; were determined to be cq = 8.5", wz = 11.6", a3 = 7.2". Solving equation 6.1 for the corresponding three poles [l 1l], [l 121 and [233], we find (lzkl) = (0.486, 0.541, 0.686), which means that the beam normal is about 2" away from (567). The method outlined above yields the axis of the crystal under investigation which is parallel to the beam normal, i.e. the normal direction ND of the sample. In order to derive the full orientation matrix with respect to the specimen frame, the indices of one further reference axis have to be determined as well. This is more complicated, and typically less accurate, and therefore such procedures are omitted in many textbooks. Nevertheless, although determination of the beam direction may be sufficient for some applications, it must be emphasised that in many cases the absolute orientation with respect to the macroscopic specimen geometry is relevant. (Note that for computation of orientation relationships or misorientations specification of a particular axis in the specimen is not necessary; only the sat~ze reference direction in a pair of crystals in the same specimen. For this purpose a direction such as that defined by the edge of the photographic plate, provided that there is no specimen tilt or rotation between measurements from each of the crystal pair, suffices as the reference axis.) To derive the complete or absolrtte o~+ntatiorz matrix (Section 2.2), specific reference axes must be chosen in the specimen. One of these is the specimen normal direction ND and a second is, for example, the rolling direction RD or the transverse direction TD (Figure 6.5). Determination of any reference axis in the specimen other than its normal requires that this axis can still be identified after specimen preparation and, in particular, after inserting the sample into the TEM. Typically the sample is positioned with its reference direction parallel to the edge of the specimen holder. Image rotations can be taken into account by calibration of the reference direction with the help of a copper net that is inserted in the specimen holder in a similar manner. It is clear that these difficulties of identifying the reference direction are responsible for the inferior accuracy in the TEM of orientation determination of specimen reference axes other than the beam normal. Provided that the direction of the reference axis in the specimen surface, e.g. the edge of the photographic plate or the rolling direction, is known in the final Kikuchi pattern, its crystallographic indices [zrvlt.] can be determined by tilting the sample around this axis and determination of the new beam normal (h'k'l') as described above. [uvw] is then given by (Izkl) X (h'k'l'). In a more rigorous approach, the information necessary to derive the complete orientation matrix can also be obtained from the angle between one indexed Kikuchi
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band and the reference axis in the specimen surface (Figure 6.5b). Following the method proposed by Young et al. (1973) and Heilmann et al. (1982). we define an intermediate pattern frame with the z-axis parallel to the beam normal BN and the y-axis parallel to any of the (indexed) Kikuchi bands with the normal likl, which means y is given by BN X hkl. Thus, the rotation matrix RCpfrom the crystal frame to the pattern frame becomes: Rcp = ( ( B N X h k l ) X BN
BN
X
hkl
BN)
(6.2)
This intermediate pattern frame contains the information of the indexed beam normal BN and differs from the outer reference frame, i.e. the sample frame, only by a rotation of angle 3 around BN = z. 3 is the angle between the Kikuchi band and the sample y-direction, which is equivalent to the angle between the uornzal to that Kikuchi band and the sample X-direction (Figure 6.5b). The corresponding matrix RPS for a rotation by angle 7 about BN = z is: cosy .ps=
i
-
:)
siny 0
Combination of the two rotation matrices RCPand RPSyields the rotation between the crystal and the sample frame Rcs, i.e. the desired orientation matrix g:
Tn the present example, the angle 7 between the (022) band normal and the rolling direction RD is 41.5". Substituting this information and the indices of BN into equations 6.2 to 6.4, we get a rolling direction [uviv] = 1-0.616, -0.345.0.7071, about 4" away from [212], and a transverse direction [qis] = [0.620, -0.766,0.165], about 0.7" away from [431]. Note that the Miller indices rounded to integer values are not perpendicular, which demonstrates one of the main disadvantages of this kind of orientation representation (Section 2.3). Although such methods to determine the orientation from a Kikuchi pattern are theoretically straightforward, they are quite challenging to apply and very tedious when analysing a large number of orientations, if no dedicated computer software is available. In the simplest case. the position of three Kikuchi bands or poles together with their crystallographic indices as well as the angle between one of these Kikuchi bands and the reference direction are input into a computer, which then calculates the three-dimensional orientation matrix g. From this orientation matrix, eventually the Miller indices, Euler angles, Rodrigues vectors or misorientation parameters (Chapter 2) can readily be derived according to standard algorithms and the results can be stored for further evaluation and representation of the data (Chapter 9). In the case of EBSD, pattern indexing is usually easier which is due to the larger steric angle which in turn allows a much larger portion of the gnomonic projection to be surveyed in one pattern (Section 6.2.2 and Figure 6.2b). However subsequent determination of the orientation is more complicated, which is a consequence of the
THE KIKUCHI DIFFRACTION PATTERN
incident electron beam
phosphor screen Figure 6.6 Diagram Illustrating the evaluation of an EBSD pattern (see text for details).
different geometry of experimental set-ups to obtain EBSD patterns. As illustrated on Figures 6.6 and 7.7, for EBSD the specimen is tilted to about 70" in the SEM chamber. Moreover, the microscope hardware may dictate that the camera which records the diffraction pattern cannot always be positioned in a simple geometrical relationship with respect to the specin~en only rarely is the camera parallel to the specimen surfxe as for TEM or SAC. These additional considerations for EBSD mean that distances in the pattern cannot be transformed into angles by a simple relationship. Furthermore, the beam normal typically lies out of the plane of the recorded pattern itself. Therefore, EBSD patterns are usually not evaluated manually, but dedicated computer software was developed already at an early stage in EBSD evolution (Section 7.4) (Dingley et al., 1987). -
indexing and orientation
In Section 6.3.1 the principles of orientation determination from Kikuchi patterns were introduced. For high symmetry materials, and where the geometrical relationship between the microscope and the EBSD camera is simple, patterns can usually be indexed by visual appraisal or by comparison with standard Kikuchi maps. Even so, consistent indexing and subsequent calculation of the orientation is a tedious procedure. In particular the problems caused by the geometrical distortions of EBSD patterns led to the development of a number of different approaches to automatise pattern indexing and orientation determination. One of the prerequisites for such an automatised evaluation of Kikuchi patterns is that the coordinates of the bands or zone axes in the pattern must be entered directly
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into a computer, thus avoiding the time-consuming preparation and evaluation of photographic films. As mentioned in Section 6.3.1, three bands or zone axes are required for orientation determination: but we will see later in this Section that more information may be necessary for unambiguous indexing, especially for non-cubic systems. As described in Section 7.6: EBSD patterns are usually captured with a TV-camera and displayed on a monitor. To transfer the necessary information into a computer, in earlier set-ups the computer was interfaced to the camera such that a mousecontrolled cursor could be superimposed on the live EBSD patterns (Dingley et al., 1987; Hjelen et al., 1993) or, vice versa, the coordinates of a cursor generated by the image processor were automatically fed into the computer through a serial interface (Engler and Gottstein, 1992). Thus, the coordinates of several Kikuchi-bands or zone axes were entered directly into the computer and no video grabber card was required. By contrast, in most current set-ups the EBSD pattern captured by the TV-camera is entirely transferred pixel by pixel to the computer by means of a video grabber card. There, the bands or poles can directly be marked by the user with the computer mouse and then used to determine the orientation of the sampled volume. The basic principles of codes for automated pattern indexing and orientation determination will now be outlined following the approach which was implemented at the Institut fur Metallkunde und Metallphysik in Aachen for evaluation of EBSD patterns (Engler et al., 1996b). Algorithms for automatic pattern recognition will be described in Section 6.3.3, and the procedure for calibrating the specimen-to-screen distance and origin of the gnomonic projection is given in Section 7.7. To relate the two-dimensional screen coordinates (x;,?;T) to the three-dimensional sample coordinate system (denoted by the superscript s in the following), the screen coordinates are transferred into three-dimensional vectors ri by:
With an appropriate calibration of the hardware set-up (Section 7.7), the coordinates of the pattern centre (.xpC,y p C )and the specimen-to-screen distance ZssD are known (Figure 6.6). The rotation matrix A contains all necessary geometrical corrections, including the tilting angle of the sample (usually 60'-75") as well as additional angles between sample and phosphor screen which may occur in the various EBSD set-ups. Now, the angles between the bands or zone axes can be calculated from the dot-product between the corresponding vectors r, and compared to a look-up table that contains all angle combinations possible in the given crystal structure. Some small measuring uncertainty (typically l" for zone axes or 2" for bands) must be tolerated, of course. Besides the interplanar and/or interzonal angles, the widths of the Kikuchi bands can be considered as well. Since the band width is related to the Bragg angle of the corresponding lattice plane (Section 6.2. l), it can be exploited to index the bands or to check the indexing by identifying wrong solutions.
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141
For EBSD, however, the measurement of the band width is not very accurate, which is due to the low signal-to-noise ratio of the pattern. On the other hand, in the case of high symmetry materials (cubic, hexagonal) virtually all EBSD patterns can be unequivocally indexed, so that the band widths are usually not considered for orientation determination. In contrast, for materials with lower symmetry the band widths yield valuable additional information to ensure unequivocal indexing which is particularly important for TEM or SAC where a much smaller portion of the gnomonic projection is visible. After indexing the pattern, the orientation of the sampled crystal volume with respect to the external specimen coordinate system can be determined (Schwarzer and Weiland, 1984; Dingley et al., 1987; Schmidt et ul., 1991; Wright and Adams, 1992; Engler and Gottstein, 1992). The crystallographic orientation is usually given in terms of its orientation matrix g (Section 2.2) which transforms the specimen coordinate system into the coordinate system of the crystal (indexed by the superscript c in the following):
Thus, for each vector r , (axis or band) its vector in the sample frame r.: = (.xi, yg, z:) (equation 6.5) and, after the indexing, also its crystallographic direction r.; = (X;, yf, z:) is known. For non-cubic crystal structures the crystallographic vectors r.: have to be transformed into a orthonor~nalcoordinate system by multiplication with the inverse crystal matrix (Section 2.2): but then they can equally well be examined. It follows from the definition of g (with i E 1, 2, 3):
For computation of the orientation matrix g the second matrix in equation 6.7 must be inverted. which requires its determinant to be different from zero (i.e. the matrix must be regular). Matrix inversion is much easier for orthogonal matrices, where A-' = A ~ SO , that only the rows and columns of the matrix must be exchanged. In order to obtain an orthogonal matrix, 3 new vectors in the sample frame and 7: in the crystal frame are formed according to:
?B
and, correspondingly:
An easier orthogonal arrangement can be gained just by setting = and correspondingly combining the other vectors, as proposed by Wright et al. (1991). I n that case two bands or axes would theoretically be sufficient to compute g. However, with only two vectors indexing is never unequivocal. Even if no tolerance needed to
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INTRODUCTION TO TEXTURE ANALYSIS
be considered, the corresponding two solutions could equally well be exchanged and, moreover, the use of three vectors introduces some averaging during the computation of g. In order to detect wrong solutions and to distinguish between [z/vu,] and [m] directions, a pattern is recalculated from the orientation matrix g and overlaid on the original pattern. If the solution is judged by the operator to be correct. the orientation matrix g is stored in a computer file for further evaluation and representation of the data (e.g. Chapter 9). 6.3.3 Automated evaluation of electron ack-scatter diffraction (E With an interactive procedure as described in the Section 6.3.2 about 50-100 orientations can be evaluated in an hour. For improvement of perforn~ancethe timeconsuming manual marking of zones or axes has to be replaced by an automatic indexing code. In particular for fully automated systems, which are necessary for performing orientation microscopy (Chapter lO), automatic pattern recognition is essential. This requires that the position of the bands or poles is detected automatically by a s~litablepattern recognition code. Subsequently, the pattern can be indexed and the crystallographic orientation can be computed as described in Section 6.3.2. The main problem of automatic pattern indexing codes is the recognition of the low contrast bands in the patterns. In the first attempt of automatic pattern recognition direct detection of zone axis positions was tried (Wright et al., 1991). This was carried out by convolution of the pattern with a filter in the shape of a 'Mexican hat' to detect regions with a high intensity compared to their neighbourhood, i.e. the zone axes. However, it turned out that this method is too sensitive to calibration errors and camera distortions. Therefore, algorithms have been developed to detect the positions of the bands rather than the zone axes: Juul Jensen and Schrnidt (1990) scanned the patterns line by line. searching for local maxima which then yielded the band positions. Wright and Adains (1992) tried to detect the band edges from the local intensity gradients in the pattern by means of the Burns algorithm (Burns et al., 1986). Krieger Lassen et al. (1992) and, later, Kunze et al. (1993) transformed the E pattern by means of the Hough t r ~ ~ ~ z , Y f(or o malternatively the Radon transform, see below). as applied to the automatic evaluation of Laue patterns already by Gottstein (1988). By this procedure. which has been reviewed by Krieger Lassen (1996a). lines in the original image are transformed into points, which can be detected much more easily by computer codes. A careful comparison of these two pattern indexing routines has shown that both the Burns and the Hough algorithms seem to be equally reliable, although both algorithms have their advantages (Kunze et d., 1993). In general, for poorer pattern quality the Hough transform yields slightly more accurate results, whereas the Burns algorithm is advantageous for high quality patterns. Furthermore, the velocity of the Hough transform does not depend on the pattern quality. which favours it for automated systems. In the following, the automatic interpretation of EBSD patterns will be outlined, again by the approach which was implemented by one of the authors at the Institut fiir Metallkunde und Metallphysik in Aachen. Codes for the automatic indexing of TEM Kikuchi patterns by means of the Hough transform have been developed as well, and will be addressed in Section 6.3.4.
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143
After recording the pattern from the video grabber card, first several steps of preprocessing of the original pattern are taken to obtain optimum conditions for the subsequent Hough transform. The original EBSD pattern is substantially reduced by averaging over 5 X 5 pixel blocks to 101 X 101 pixels, which represents a good compromise between sufficient resolution of the EBSD pattern and cutting down on computation time. Figure 6.7a shows such a size reduced EBSD pattern which was taken from a Ni3AI sample. Subsequently, contrast enhancement of the pattern is performed by linear scaling of the grey values I(x, y ) to the entire spectrum 0 < I ( s , j ) < 255. This means that grey contrast differences from pattern to pattern do not affect the subsequent image transformation, which appears to be important with a view to full automation of systems for orientation microscopy. The EBSD patterns contain a high background with variations in the level of background intensity (Figure 6.7a). Hence the next step of image processing is to correct the pattern with respect to both the background intensity and shading differences. The pattern background is determined by convoluting the image with a low pass filter, whose size ( l 0 X 10 pixels) is chosen so as to reduce shading differences most effectively without losing pattern information. After bilinear extrapolation to the original size the background image B(x,J.) is obtained and subtracted from the original image so that I ( x , J,) - B(x, 2.) = T ( s ,y). This procedure is repeated several times so as to yield the smooth background that is necessary for the subsequent image transformation. Figure 6.7b shows the pattern of Figure 6.7a after the pre-processing. The next step of the procedure is the transformation of the pattern and, as already stated earlier in this section, nowadays usually the Hough transform is favoured (e.g. Krieger Lassen, 1996a). During the Hough transform (Hough, 1962), each point (.xi,j>;)of the original image is transformed into a sinusoidal curve in an accumulation space. co~nmonlycalled Hough .space, which is characterised by the coordinates p and p according to the relation:
Figure 6.7 Steps of image processing during automatic evaluation of an EBSD pattern taken from Ni3AI (accelerating voltage 15 kV). (a) Original pattern after size reduction: (b) Pattern after pre-processing; (c) Pattern after the Radon transform.
144
(a) original image
INTRODUCTION TO TEXTURE ANALYSIS
(b) Hough-space
Figure 6.8 Schematic representation of the Hough transform. (a) Two bands in the original image; (b) bands--from (a) in the Hough space.
This relation specifies a line by the angle p between its normal and the pattern x-axis as well as its distance p from the origin - here, the centre of the image with radius R . The principle of the Hough transformation is schematically illustrated in Figure 6.8 for two bands which each are defined by two points. Points which lie on a line in the original image (Figure 6.8a) have the same distance p and angle p with respect to the origin and. hence, they all accumulate in one and the same point in Hough space (Figure 6.8b). For a transforsnation of a grey-tone image such as an EBSD pattern, each point yi) is transformed into the Hough space (p, p) under of the original image (xi, consideration of its intensity I*(xi,yi), a procedure which is referred to as grey-tone weighted Hough transform or, more precisely, as a Radon transfirm (Radon, 1917). For obtaining a homogeneously transformed image without shading effects, only a circle with a diameter of 101 pixels is considered during the transform rather than the entire square pattern. In Figure 6 . 7 ~the . Radon transformed image of Figure 6.7b is represented. The next step is the identification of the peaks in Radon space, i.e. the bands in the original image. During the Radon transform, the bands in the original EBSD pattern lead to peaks with a characteristic butterfly-like shape in the transformed image (Krieger Lassen et al., 1992). Hence, the peaks can be detected by convolution of the Radon space with a filter which exactly has such a butterfly shape. Application of an appropriate filter strongly enhances the centre of the peaks in p-direction and so the angle 9 of the bands can be detected with high accuracy. For detection of the p-value of the peaks - which defines the exact position of the mid of the EBSD bands a multitude of different peaks has been analysed to yield an average 'standard profile' of the peaks in the p-direction, and so, the p-coordinate of the bands in the Radon space can be reliably determined. -
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145
The detected bands are listed according to their relative intensity, and the five bands with the highest intensities are used for further orientation computation as described in Section 6.3.2. Occasionally wrong lines are detected and, consequently, no solution is obtained so the operator has to reject manually the wrong lines. In other cases ambiguous solutions are found, particularly in the presence of symmetrical poles. In that instance, all possible solutions are listed and the user has to select the right one, e.g. by input of additional lines. For a fully automated system, i.e. for orientation microscopy (Chapter 10), interaction by the user is not possible, of course, although in some EBSD packages wrong solutions can be remedied in a post-processing step. Therefore, the detection algorithm had to be improved to yield unique solutions in as many cases as possible. An automatic permutation of the detected bands by consideration of less intense bands has yielded very promising results by strongly reducing the number of nonuniquely indexed patterns. During a critical evaluation of this algorithm it appeared that more than 95% of the patterns, of course strongly depending on the pattern quality, are unambiguously and accurately indexed. With regard to the accuracy of the automatic indexing routine, about 200 orientations were evaluated both manually and by the automatic indexing routine. Comparison of the obtained data yielded an average misorientation of -l0, i.e. no noticeable difference to the relative error for manual EBSD measurements (Section 7.6.3; see Engler et al. (1996b); for reliability data of other systems see Wright and Adains (1992) and Krieger Lassen (1996b)). To sum up, with computation times for the entire automatic pattern solve algorithm below one second on standard personal computers, such automatic indexing routines work very conveniently, with high velocity and, particularly, with high reliability. 6.3.4 Automated evaluation of transmission electron microscopy (TEM) Kikuchi patterns As discussed in Section 6.2.1, TEM Kikuchi patterns rely on the same physical and crystallographic principles as the EBSD patterns in the SEM and therefore they can be evaluated according to analogous methods. In fact, we have introduced the principles of interpretation of Kikuchi patterns in Section 6.3.1 for TEM geometry, as this case is easiest to illustrate. Schwarzer and Weiland (1984) developed a method to interpret Kikuchi patterns in a TEM on-line at the viewing screen of the TEM. For this interactive analysis the pattern is shifted across the viewing screen using the TEM deflection coils until the points to be measured - which may be points in a selected area diffraction (SAD) pattern (Section 8.3.1) or three poles with known indices in a Kikuchi pattern - successively coincide with a reference mark on the screen, e.g., the primary beam stop (Figure 6.9). The voltages needed to shift the pattern are a measure for the position of the respective points on the screen. They can be transferred to a computer through an analog-to-digital converter board and can there be evaluated to yield the crystallographic orientation of the crystal under investigation. In the approach by Schwarzer and Weiland (1984), the positions of minimum
INTRODUCTION TO TEXTURE ANALYSIS
incident electron beam
Figure 6.9 On-line acquisition of Kikuchi patterns in a TEM (Adapted from Schwarzer and Weiland. 1984).
3 Kikuchi bands are characterised by marking 4 points for each band. and this information is used in an automated code for pattern indexing and orientation determination, similar to that described in Section 6.3.2. More recently, based on the success of the automatic indexing of EBSD patterns (Section 6.3.3). there have been several attempts to adapt the respective computer codes for evaluation of Kikuchi patterns in the TEM (Zaefferer and Schwarzer, 1994a; Weiland and Field, 1994; Krieger Eassen. 1995; Engler et crl., 1996b). In these developments: likewise the entire TV image is transferred to the computer through a video grabber card (Section 8.4). However, Kikuchi patterns more serious problenls had to be overcome and the EBS ad to be substantially modified to yield satisfactory results. These problems include: e
TEM Kikuchi patterns have a much higher (Zj~izamicrange due to variations in background intensity as well as the superposition of SAD diffraction spots and the primary beam.
e
TEM Kikuchi patterns usually consist of pairs of bright and dark Kikucl~ilines as opposed to the bands with rather homogeneous intensity in EBSD.
e
Urzctnzhigltous indexing is more complicated because of both the small steric angle and the large number of visible high-index bands.
THE KIKUCHI DIFFRACTION PATTERN
147
The contrast range of TEM Kikuchi patterns is much larger than that of EBSD patterns. Furthermore. the Kikuchi patterns are frequently superimposed by high intensity SAD diffraction spots and, particularly, by the primary beam, whose intensity may exceed that of the Kikuchi lines by several orders of magnitude. As addressed in Section 8.4, cameras and frame-grabber cards with higher resolution (e.g. 12 or 14 bit) are advantageous to handle this large dynamic range of TEM Kikuchi patterns. Alternatively, a continuous gradation filter inserted in front of the camera has been used to level out the uneven background intensity (Zaefferer and Schwarzer, 1994a). Because of the high dynamic range and the stronger variations in background intensity in TEM Kikuchi patterns, additional steps of image processing are required in comparison to EBSD patterns. For instance, strongly localised high intensities can be searched for and replaced by the average value of their neighbourhood, since such regions are likely to correspond to the SAD spots. The area affected by the primary beam spot can similarly be smoothed, or it is simply neglected during the Hough transform. Bright and c/ulcll.lr Kikciclzi lines TEM Kikuchi patterns usually consist of pairs of bright and dark lines (Section 6.2.l), which means that an offset has to be introduced during determination of the background intensity to ensure that the defect Kikuchi lines with intensities below that of the background are not removed. In contrast, EBSD patterns mainly comprise bands with a higher intensity than their surroundings. and only occasionally sharp bright and dark lines are observed at the band edges (Section 6.2.2). Furthermore. low-index lines in the TEM are frequently associated with parallel high-order diffraction lines. For these reasons. the routines for line detection via the Hough transformed image had to be modified as well (see Zaefferer and Schwarzer, 1994a; Krieger Lassen, 1995; Engler et al., 1996b).
Owing to the smaller steric angle, but larger number of high-index bands, unambiguous indexing is more complicated than for EBSD. For instance, whereas in EBSD patterns of fcc materials usually only bands up to (31 1) or maximum (331) are obtained, in TEM Kikuchi patterns bands like (244) are readily visible and must hence be considered during the indexing procedure. On the other hand, the widths of the Kikuchi bands can be determined much more accurately and hence they can also be used to index the bands or at least to identif57 wrong solutions (Section 6.3.2). The large number of different, widespread data that is needed to index the patterns results in the necessity of an appropriate handling of erroneous data. In many cases, still several solutions are obtained which require interaction bp the user. To sum up, automated indexing of TEM Kikuchi patterns is by far more complicated and, to date, less developed than for EBSD patterns. However. since typically a much smaller number of orientations will be analysed in the TEM than by EBSD
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INTRODUCTION TO TEXTURE ANALYSIS
(Chapter 8), an appropriate computer code for an interactive, on-line analysis as described in Section 6.3.2, rather than full automation, is at present the optimum way to facilitate microtexture analysis in the TEM. Recently, another simple method has been introduced which allows a fast and accurate interactive interpretation of TEM Kikuchi patterns using a double-tilt specimen holder (Liu, 1995). To determine the orientation of the sampled volume, the sample is tilted so that a low-index zone axis is close to the beam direction. The tilting angles as well as three features (zone axes or points on the Kikuchi lines) that clearly define the Kikuchi pattern in this position are marked using the postspecimen deflection coils and can be evaluated to yield the crystallographic orientation. This method is particularly well suited to the determination of misorientations in a given sample region with similarly oriented crystals, e.g. in a deformed and recovered subgrain structure (e.g. Wert et al., 1995). If the misorientation angle is less than, say, 15" the low-index zone axis is mostly still visible, so that only the relative position of the actual Kikuchi pattern with regard to the reference pattern has to be determined. For the determination of misorientations in terms of misorientation angle and axis the accuracy is stated to be 0.3" and 3", respectively.
6.3.5 Automated evaluation of selected area channeling (SAC) patterns
It was stated in Section 6.2.1 that the geometry of the diffraction patterns as obtained by SAC is akin to the TEM Kikuchi patterns, although their formation mechanisms are slightly different. Since furthermore both the geometrical arrangement and the capture angle of both techniques are comparable, SAC patterns can generally be evaluated according to the same principles as TEM Kikuchi patterns (e.g. Joy et al., 1982; Lorenz and Hougardy, 1988; Schmidt and Olesen, 1989). However, due to the advantages of the competitive EBSD technique (Sections 7.4-7.8), SAC is increasingly less used for the determination of individual orientations in the SEM and, accordingly, there is nowadays little effort put into the automatisation of interpreting SAC patterns (Section 7.3). SAC patterns form on the viewing screen of the SEM and are usually recorded on photographic films as conventional SEM micrographs. Hence there is again the problem of transferring the pattern information into a computer for the automatic analysis. Zaefferer and Schwarzer (1994b) used the SEM image cursor to mark the bands and to transfer the data through a serial interface to a computer. Alternatively, in modern digital SEMs the entire screen image - thus also the SAC pattern can be stored and used by an external computer to derive the crystallographic orientation of the sampled volume. -
6.4 PATTERN QUALITY Besides the crystallographic orientation (Section 6.3), there is another parameter which can be derived from the patterns by means of image analysis, namely the sharpness or quality of the patterns. Since the Kikuchi line broadening, and in turn
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the whole pattern quality, is related to the number of lattice defects in the sampled volume, it yields information about the state of deformation and internal stresses. However, the pattern quality is also affected by the experimental set-up and specimen preparation, so that only qualitative, or at best semi-quantitative, information can be achieved. Quested et al. (1988) determined the magnification at which the pattern quality in both spot mode and image mode of the SEM were equal. The larger this magnification is, the smaller is the sample area in the image mode that contributes to the pattern, where the pattern is not yet affected by lattice distortions. This information yields a brief estimate of the dislocation density and, consequently on internal stresses. In detailed studies on differently deformed aluminium samples (AA6061), Wilkinson and Dingley (1991) developed a more meaningful method for a quantitative evaluation of the pattern quality. By means of a discrete Fourier analysis they showed that with increasing deformation the sharpness of the edges of the bands in the patterns decreased, which resulted in a continuous reduction of the high frequency parts of the bands in the Fourier spectrum. The pattern quality could be derived in a quantitative manner from the first moment of the Fourier-transformed pattern. However, this method is based on the analysis of patterns which were recorded photographically and is restricted to one particular zone axis ( < l 12>): so that it is not suitable for routine applications. For a quantitative characterisation of the pattern quality, which is routinely applicable during large scale microtexture measurements, a quality parameter IQ, 'Image Quality', has to be derived from the pattern. Kunze et al. (1993) defined a quality parameter IQ as the sum over the peak sharpness of all detected peaks. For a homogeneous, grey pattern IQ becomes zero. The upper limit of IQ is given for each pattern by the minimum peak sharpness and the maximum possible number of peaks, so that this definition yields data whose absolute values cannot be compared quantitatively. Krieger Lassen et al. (1994) adapted the Fourier transform for use in fully automated evaluation codes. As this method appears to be the most general one, it will be described here in more detail. First, a Fourier transform of the entire pattern I(x, y) is performed. To achieve minimum computation time, the Fast Fourier Transform (FFT) is used: 1
n-l
n-i
n (where i is the imaginary number). For the FFT, n must be a multiple of 2 so that only a part of e.g. 64 X 64 pixels (64 = 26) of the pattern is considered. A pattern of poor quality has a more homogeneous spectrum than a pattern of higher quality (Figure 6.10a, b). This means that a sharper pattern consists of a larger part of low frequencies, whereas a worse pattern becomes close to the frequency spectrum of white noise (Figure 6.10c, d). (Note that Wilkinson and Dingley (1991) analysed the Fourier spectrum of the edges of the bands, not the entire pattern.) A measure for the uniformity of the frequency spectrum is the inertia I of the Fourier spectrum
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INTRODUCTION TO TEXTURE ANALYSIS
(a) original EBSD-pattern - poor quality
(b) original EBSD-pattern- high quality
(c) Fourier spectrum of (a) (IQ =0.29)
(d) Fourier spectrum of (b) (ICk0.43)
Figure 6.10 Original and Fourier transformed EBSD patterns with different quality to d e r i ~ ethe image quality IQ.(a) Original poor quality EBSD pattern: (b) Original high quality EBSD pattern: (c) Fourier spectrum of (a) (IQ= 0.29): (d) Fourier spectrum of (b) (IQ = 0.43).
norinalised to the total energy of ihe Fourier spectrum. ith increasing concentration of the Fourier spectrum in the lower frequencies; i.e. with better patterns, I becomes smaller. To obtain values which are quantitatively comparable I is related 10 the value I,,, for a completely homogeneous spectrum. The image quality is then given by:
This definition of the image quality I Q yields nornialised values between 0 and I . For white noise I = I,,,,, thus IQ = 0.The better the pattern, the larger the part of
THE KIKUCHI DIFFRACTION PATTERN
151
low frequencies will be. Consequently. I decreases and IQ increases up to IQ = 1: but in practice, usually patterns with IQ < 0.6 are obtained. The completely automated procedure to derive IQ combined with very short computation times favour incorporating this or a similar method into codes for automated evaluation of Kikuchi patterns. In Section 10.4.5 we will see how the image quality parameter can be ~lsed to produce orientation microscopy images that strongly resemble conventional SEM n~icrographs(Figure 10.7b). It was already mentioned above that the pattern quality is strongly affected by a variety of factors: so that the absolute values of IQ cannot be compared unan~biguously.For instance, changes in experimental parameters like accelerating voltage and beam current as well as other factors like vacuum or quality and alignment of the filament will strongly influence degradation of the pattern. However, during one scan the experimental parameters should remain more or less constant. so that at least a semi-quantitative evaluation of the pattern quality can be achieved. For instance, with the help of the image quality parameter Krieger Lassen et al. (1994) could distinguish between deformed and recrystallised grains in a sample of partially recrystallised aluminium with a very high reliability. It should be noted that the pattern quality IQ may also depend on the orientation of the sampled volume itself, although this has not been quantified to date. Despite the large steric angle covered by EBSD, patterns representing different orientations will depict different regions of the stereogram. Therefore, irrespective of the pattern quality, the Fourier spectra and, hence, the numerical value of IQ will vary for different orientations.
Almost all microtexture work relies on the interpretation of Kikuchi patterns. the Kikuchi patterns in the TEM and the EBSD patterns in the SEM form by Bragg reflection of inelastically scattered electrons at the lattice planes of the illuminated crystal volume. Accordingly, the Kikuchi pattern is directly related to the orientation of the reflecting planes in the specimen and is hence perfectly suited to crystal orientation determination. Although the formation mechanisms of SAC patterns in the SEM are slightly different (pseudo-Kikuchi lines. Section 7.3), all these patterns embody the same geometry and crystallographie information and, consequently, they can all be evaluated according to the same principles. The spatial resolution and angular accuracy is compared for these techniques in Table 12.1. Because of the simple, direct relation between the reflecting planes and the Kikuchi patterns, the interpretation of Kikuchi patterns is in principle straightforward. However. the necessary mathematical operations make it a cumbersome procedure to apply, especially for analysis of a larger number of orientations. Appropriate computer codes may strongly facilitate the procedure. and various levels from computer support with matrix handling during manual, off-line microtexture analysis up to fully automated systems for pattern indexing and orientation determination in on-line applications have been developed.
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The large steric angle covered by EBSD means that the resulting patterns can usually readily be indexed in an unequivocal manner. As the EBSD patterns furthermore comprise a rather homogeneous contrast range, they are very well suited to automatic pattern recognition codes (Hough transform) which automatically detect the Kikuch~bands. Accordingly, fully automated systems to index the patterns and to evaluate the corresponding crystallographic orientation have been developed with great success and are widely available. In contrast, for TEM Kikuchi patterns and SAC patterns much smaller regions of the stereogram, but more highindex bands, are visible so that unequivocal indexing is much more complicated. Here, the additional information provided by the band width must be considered to attain unequivocal solutions. Furthermore, the much higher dynamic range of the TEM Kikuchi patterns renders automated indexing of TEM Kikuchi patterns more difficult. So, despite some success that has been achieved in developing codes for fully automated interpretation of TEM Kikuchi patterns, today they still will probably equally well be evaluated by a reliable, interactive on-line method. Full automation of EBSD has been a vital step forward in the development of microtexture analysis, since the unreasonable time burden on operators and microscopes which were inherent in manual EBSD is reduced or eliminated. Furthermore, automation has allowed the full potential of EBSD to be realised through the subsequent development of orientation microscopy and orientation mapping (Chapter 10).
7. SCANNING ELECTRON MICROSCOPY (SEMI-BASED TECHNIQUES
7.1 INTRODUCTION The SEM is a powerful instrument for acquiring microtexture data and characterismg the local crystallography of materials (Wilkinson and Hirsch. 1997). Electron backscatter diffraction (EBSD) is the technique which is becoming almost universally used to obtain such data, and so most of this Chapter is devoted to a description of the principles and practice of EBSD. Although EBSD (which is also known as Backscatter Kikuchi Diffraction, BKD, or Electron Back-Scatter Patterns, EBSP) is mainly associated with microtexture measurement, it can also be used to assist in phase identification via the symmetry elements in the diffraction pattern (Dingley et al., 1995; Michael, 1997) and to assess lattice strain (Sections 6.4 and 10.4.5). Two other SEM-based techniques, micro Kossel and electron channeling diffraction or selected area channeling (SAC) were forerunners of EBSD and, although they are comparatively little used nowadays, they are included here for completeness.
7.2 MICRO KOSSEL TECHNIQUE The basis of the Kossel technique is the diffraction of X-rays at the crystal lattice (Section 3.2), where the X-ray source is situated \vitlzin the sample under investigation. For crystallographic studies including orientation determination - by means of this technique an electron beam is focused on the sample. Through interaction of the incident electron beam with the electron shells of the atoms characteristic X-rays are emitted which then interfere with the surrounding crystal lattice. Kossel and coworkers (1935) first achieved diffraction patterns according to this method from macroscopically large san~ples.In order to investigate local orientations in microscopic regions (Micro Kossel X-ray Dijfiaction, MKXD), Kossel measurements are usually performed in a micro-probe (Peters and Ogilvie, 1965; Bevis and Swindells, 1967) or an SEM (Dingley and Steeds, 1974; Dingley, 1978). -
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ctron beam
(a)
lattice lane normal
(b)
Figure 7.1 (a) Diagram illustrating the formation of Kossel patterns in reflection: (b) Kossel pattern from titanium (Courtesy of F. Friedel).
Part of the X-radiation which forms through the interaction of the electron beam with the sample material is subject to subsequent reflection at the lattice planes of the sample in accordance with Bragg's law (equation 3.4). Because the lattice planes are irradiated with X-rays coming virtually from all directions, the reflected X-rays form cone shapes, the Kossel cones, with a half apex angle 90" -Q. The reflected X-ray intensity along these cones can be recorded on an X-ray sensitive film which is usually placed several centimetres before (reflection geometry) or behind (transmission geometry) the sample within the microscope chamber (Figure 7.la). In comparison to Mikuchi electron diffraction (Section 6.2.1), the wavelength of the X-rays analysed by the Kossel technique is much larger. Therefore, all Bragg angles Q between 0" and 90" may occur, so that the resulting projection lines on the film plane are strongly curved (Figure 7. l b). To produce Kossel patterns the accelerating voltage of the electrons must be about 2-3 times larger than the critical voltage to produce the characteristic X-ray Karadiation, which is not possible for all materials. By contrast in the case of light elements (e.g. aluminium), the characteristic wavelength of the Kn-radiation of the sample material is too large to yield reflection according to Bragg's law. In both cases X-radiation of a suitable wavelength can be produced by an external X-ray source which is placed as close as possible to the surface of the sample (pseudoKossel Technique). As example. a thin layer of a different material can be deposited on the sample surface (Peters and Ogilvie, 1965; Ferran et al., 1971) or the sample is covered with fine particles of a suitable metal (Ullrich and Schulze, 1972). Although Kossel patterns have been achieved from volumes of only a few microns (Dingley and Steeds, 1974), the spatial resolution is in general cases limited to 5-10 pm, SO that only rather coarse grained samples can be analysed. In the case of the transmission technique the spatial resolution is correlated with the transmitted sample thickness which generally results in poorer spatial resolution (Ferran et al.,
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
l55
1971). Only with extremely thin samples can a spatial resolution of 10-20 pm be achieved (Bellier and Doherty, 1977). Furthermore, Kossel patterns have a very high background and, consequently, a low signal-to-noise ratio. This fact, which is caused by the non-reflected X-rays as well as the Bre~nsstrahlung(contmuous radiation), complicates evaluation of the patterns. Also, Kossel patterns cannot yet be digitally recorded, so that an automated evaluation as for example in the case of Kikuchi patterns (Section 6.3) has turned out to be impossible so far. For these reasons, and with the advent of EBSD, the Kossel technique is no longer competitive for orientation determination. Rather, nowadays Kossel patterns are mainly used for analysis of unknown crystal structures (Tixier and Wache, 1970; Ullrich and Schulze, 1972; Ullrich et nl., 1992) and high precision internal stress measurements (Ellis et al., 1964), as lattice constants a can be derived with an accuracy of better than aala= i o P .
7.3 ELECTRON CHANNELING DIFFRACTION AND SELECTED AREA CHANNELING (SAC) Electron channeling diffraction has been a more common method for local orientation determination in the SEM than has Kossel diffraction. It is based on the diffraction of the electron beam in an SEM at the crystal lattice. The intensity of the reflected electrons depends not only on the site in the sample, but also on the angular direction of the incident electron beam with respect to the lattice planes. Thus if one changes the direction of the incident beam at a constant sample site e.g. in a given grain the anisotropy of back-scattered electrons gives rise to E l ~ c t r o n Cl~annelingPatterns (EC ) (Coates, 1967). These ECP patterns, which are composed of bright bands of a pi n thickness representing distinct crystallographic planes (Figure 7.2a) are akin to TEM Kikuchi patterns and EBSD patterns in both their geometry and the crystallographic information which they contain (Section 6.2) and, consequently, they can be evaluated according to the same principles (Section 6.3). The exact forniatiol anisms of ECP bands. however, differ slightly from that of Kikuchi lines in TE a comprehensive treatment see Joy et al.; 1982), therefore ECP bands are often referred to as 'pseudo Kikuchi bands' (Figure 7.2b). To use ECP for orientation determination the electron beam in the SEM is focused on the sample site to be analysed. Then, the beam is tilted at this position by an angle a either in two perpendicular directions or on concentric circles with angles < a 5 a,,,,,(Figure 7.3a), without leaving the corresponding sample site. If the electron beam fulfils Bragg's condition for reflection at a given set of lattice planes, i.e. o = 8:it will be reflected and contributes to a band in the ECP pattern (Figure 7.3b). In practical applications the angle a,n,, is limited to 5-10'. which means that the angular diameter of the ECP pattern is only 20" maximum. Hence, only a small region of the gnomonic projection can be recorded which makes pattern indexing more complicated. To achieve optimum spatial resolution, minimum probe diameter is required. For that purpose, the beam diameter is usually reduced by suitable apertures to a size of a few microns (Selected Area Electron Clzannelling DiJjinction, SAC or SAECD; -
-
QC
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INTRODUCTION TO TEXTURE ANALYSIS
incident electron beam
sample
lance planes
Figure 7.2 (a) SAC pattern of a recrystallised grain in a partially recrystallised AI-Fe-Si sample (Courtesy of H.W. Erbsloh); (b) formation of pseudo-Kikuchi bands in SAC.
Figure 7.3 (a) Crystal with three lattice planes A. B and C irradiated by an electron beam at angles between u = 0" and cr = N , so ~as to ~ produce ~ ~ an SAC pattern. (b) Resulting SAC pattern if only the three planes in (a) fulfil the Bragg condition (Adapted from Lorenz and Hougardy, 1988).
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van Essen et al., 1970, 1971). In general the spatial resolution is still limited to 10 pm, which is caused by the necessity to tilt the electron beam without leaving the corresponding measuring site. Only with the application of dynamic corrections of the electromagnetic lenses can the spatial resolution be improved to 1-2 pm (van Essen, 1971; Joy and Newbury, 1972). Another drawback of the SAC technique is its high sensitivity to lattice defects, so that only recrystallised or strongly recovered microstructural regions can be analysed. In principle, ECP can be obtained in a TEM as well. when a scanning equipment is available (Section 8.4). In that instance, the spatial resolution is about lpm (Joy et al., 1982; Humphreys, 1984). A very detailed survey on ECP and SAC including a list of literature which appeared to that date can be found in Joy et al. (1982).
-
7.4 EVOLUTlON O F ELECTRON BACK-SCATTER DIFFRACTION (EBSD)
Electrorz back-scatter- diffrnctiorl (EBSD) in an SEM has become the most widelyused technique for the determination of inicrotexture (Randle, 1992). As mentioned in Section 1.2.2. the steps necessary to produce an EBSD pattern in an SEM are: e
Tilt the specimen so that its surface makes an angle >60° with the horizontal
e
Turn off the scan coils to obtain a stationary beam
e
Place a recording medium or device in front of the tilted specimen to capture the diffraction pattern
The main effect of tilting the specimen is to reduce the path length of electrons which have been backscattered by lattice planes as they enter the specimen. thus allowing a far greater proportion of these electrons to undergo diffraction and escape from the specimen (having lost virtually none of their energy) before being absorbed. When the specimen is flat, which is the case for conventional SEM, the path length and hence absorption of the backscattered electrons is too great to produce detectable diffraction. Figure 7.4 shows how the electron/specimen interaction volume is modified from the well-documented 'tear drop' shape to a volume having a very small depth when the specimen is tilted (Schwarzer, 1993). The total effect on the backscatter process of tilting the specimen is more complex than described here, but a simple explanation based on electron path length is adequate for the present pmposes. More details on the physics of EBSD can be found elsewhere (Venables and Bin-Jaya, 1977; Dingley, 1981; Reimer, 1998; Dingley et al., 1987). The observation of high-angle Kikuchi patterns from 'reflected' electrons, which was the forerunner of the EBSD technique, was reported as early as 1933 (Meibom and Rupp, 1933) and 1937 (Boersch, 1937). A report on systematic analysis of highangle Kikuchi patterns was published in 1954 (Alam et al., 1954). Here, a special apparatus consisting of an electron source, a specimen chamber and a camera was used and specimens of cleaved LiF, KI, NaCl and PbS2 were examined. This arrangement captured very wide-angle Kikuchi patterns (up to 164") which were found to have lost almost no energy in the scattering process and to be in agreement with Bragg's law.
INTRODUCTION TO TEXTURE ANALYSIS
bulk sample (SEMI Figure 7.4 Diagram showing the specimen-beam interaction volume in a specimen tilted for EBSD (Courtesy of R. Schwarzer).
It was nearly twenty years later before the discovery of wide-angle Kikuchi patterns was pursued further. At this stage an §EM was used to provide the electron source, and the diffraction patterns from the tilted specimen were captured by a fluorescent screen interfaced to a closed circuit TV camera. At this time the patterns were named 'electron back-scattering patterns' since they comprised the angular distribution of the back-scattered electrons after diffraction. Several advantages of the technique over SAC for obtaining crystallographic information in the SEM were cited: the smaller sampled volume, much larger angular view of the diffraction pattern and simpler modifications to the microscope which were involved (Venables and Harland, 1973). EBSD patterns were obtained from materials such as tungsten, silicon and aluminium. In order to make orientation measurements from EBSD patterns it is essential to identify the following parameters (Venables and Bin-Jaya, 1977), which are illustrated on Figure 7.5 (Randle, 1992): e
The coordinates of the pattern centre, PC, defined in Section 6.2.3 and 6.3.2 The specimen-to-screen distance ZssD or L on Figure 6.6 and 7.5
e
The relationship between reference directions in the microscope X, Y,,,Z,,, specimen X , Y , Z , and screenlpattern xyz
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
159
Figure 7.5 Parameters required for EBSD orlentatlons measurements pattern source pomt on the speclmen, SP, pattern centre on the recording scieen, PC, spec~men-to-screendlstance L (or Zsso), and three sets of orthogonal axes, ' c ~ (scleenlpattern z axes), X, Y,Z, (speclmen axes) and X, Y,,Z,,, (mlcroscope axes)
These parameters depend on the relative positions of the camera and specimen within the microscope, and are discussed in Sections 6.3 (pattern evaluation) and 7.7 (EBSD calibration). The first method to obtain the coordinates of the pattern centre and the specimen-to-screen distance involved measuring the axes of elliptical shadows cast onto the pattern by three fixed spheres (Venables and Bin-Jaya, 1977). Although this method was cumbersome and slow, it was an accurate means of calibration such that individual orientations could be obtained by photographing the diffraction pattern and spheres assembly and subsequently analysing it off-line. The next advances in the evolution of EBSD occurred in the early 1980s. These were: e
A more convenient method of calibration
s On-line interaction with the diffraction pattern 0
Software which gave a rapid evaluation of the orientation
0
Hardware improvements
The calibration method involved use of a silicon crystal prepared so as to reveal {001} and mounted at a predetermined inclination angle so that the known geometry allowed the pattern centre and specimen-to-screen distance to be evaluated. The diffraction pattern could be interrogated in real time on a monitor screen and the
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INTRODUCTION TO TEXTURE ANALYSIS
first versions of software to accomplish this were for cubic materials only (Dingley et al., 1987). The operator was required to move the computer cursor to identify certain zones axes - < l 12> and either < l 14> or < l 1l > from which information the orientation was calculated, the indices of all major poles were superimposed on the pattern and, if the indexing was correct. the orientation was stored in a datafile. Performance of these steps required the operator to have an adequate knowledge of crystallography. The hardware improvements included substituting a Silicon Intensified Target (SIT) low-light camera for the earlier TV camera. This required a lower probe current in the SEM (down to 0.5 nA compared to 20nA) and so provided greater sensitivity (Hjelen et d.,1994). An image processing system was also employed which allowed the live image to be frozen and for a background subtraction/averaging routine to be implemented to improve the quality of the raw EBSD pattern (Section 7.6.4). From the early 1980s onwards EBSD was also being developed as a tool for crystallographic phase identification via measurement of the symmetry elenlents in the diffraction pattern (Dingley et al., 1995). These measurements require as much detail in the pattern and as wide an angular view as possible. and so patterns were recorded on film introduced into the microscope chamber in front of the tilted specimen. Recording the pattern directly onto film is a very sensitive means of producing high contrast, detailed hard copies of diffraction patterns. However, the recording process is tedious and it is not required for orientation determination. Nowadays, phase identification can often be performed on digitally stored EBSD patterns (Michael, 1997). The progression of EBSD described so far, as shown in Figure 7.6, has involved step changes in the technology. Subsequent evolution which occurred from the early 1980s to the early 1990s was less dramatic, involving instead refinements of the fundamental components and routines, and included: -
Further improvements to EBSD hardware, particularly better cameras Intensified Silicon Intensified Target (ISIT) offering even lower light sensitivity and Charge-Coupled Device (CCD) cameras which offer less distortion. Diffraction pattern processing also improved, and a forward-mounted back-scatter detector became available for convenient viewing of the image when it is highly tilted (Section 7.6)
-
Extension of the calibration options for greater accuracy and convenience (Section 7.7) Continuing improvements to the EBSD software, including indexing routines for any crystal system; mouse-driven interrogation of diffraction patterns no longer involving identification of zones; more sophisticated and faster data handling; a choice of options for data output (Section 7.8) Commercial EBSD systems became available Recognition and awareness in the scientific and engineering community of the benefits of EBSD for microtexture measurement were engendered through early
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
161
Electron back-scattering patterns recorded in the SEM Early 1980s....
t
1
Computer routinesfor
Late 1980s.... Automated EBSD pattern
S..,.
Orientation
Figure 7.6 Tlmeline of EBSD development
publication of EBSD-generated data (e.g. Randle and Ralph, 1988; Hjelen and Nes, 1988: Juul Jensen and Randle, 1989) In the early 1990s two more major step changes occurred in EBSD technology. These were the implementation of computer interpretation and indexing of diffraction patterns without operator input (Section 6.3), followed by coupling this facility with beam andior stage control to automate totally the EBSD operation which finally led to orientation mapping (Section 10.4). Nowadays most crystal structures can be successfully indexed, although for some lower symmetries this is still a challenging task. A state-of-the-art EBSD system operated auton~aticallyon a tungsten-filament SEM delivers
INTRODUCTION TO TEXTURE ANALYSIS
162
beam control
objective lens'
I I
phosphor screen
1 I ,lead
glass window
7
sample sample holder
SEM vacuum :hamber stage control
Figure 7.7
Components of an EBSD system
s A pattern solving algorithm which takes a fraction of a second s An accuracy of
< 1"
s An angular resolution of -200-500
nm
Figure 7.7 illustrates the essential components of a typical EBSD system.
7.5 EBSD SPECIMEN PREPARATION One of the attractions of EBSD is that specimen preparation is straightforward, often similar to that for optical microscopy. This is in contrast to the requirements for other electron diffraction-based techniques. in particular TEM and also SAC in the SEM, both of which require complex preparation procedures. The specimen preparation objective for EBSD can be stated very simply: the top 10-50nm of the specimen must be representative of the region from which crystallographic information is sought, since it is only from this surface region that diffraction occurs. The shallowness of the diffraction zone results from the highly tilted specimen position in the SEM chamber and the consequent reduction of the mean free path length for electrons (Section 7.4 and Figure 7.4). The practical upshot
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of this, and indeed the only crucial aspect of specimen preparation for EBSD, is that the specimen surface must not be obscured in any way - by mechanical damage (e.g. grinding), surface layers (e.g. oxides, thick coatings). or contamination. Mounting, grinding and polishing is the standard metallographic preparation route for most specimens, especially metals and alloys. The adaptation of this route for EBSD will be discussed first, since metallic materials are currently the most common group to which EBSD is applied. Mounting the specimens in a conducting medium is clearly advantageous for SEM work; otherwise electrical contact to the specimen can be established by using silver paint, carbon paint or conductive tape, or simply by cutting the specimen from the mount after the preliminary preparation stages. It is the final preparation step which ensures suitability for EBSD. Diamond polishing is not an appropriate final stage because of the mechanical damage entailed. However, many common electropolishes or etchants which are used to reveal the microstructure for optical microscopy do so by attacking the specimen surface, which has the concurrent effect of removing polishing damage. Hence, for many metals and alloys the route used for optical metallography, with minimised diamond polishing to avoid surface damage, is all that is needed for EBSD. A useful guide is that the same electrolytes which are used to prepare TEM thin foils can often also be used to prepare the surfaces of bulk specimens for EBSD. Brittle specimens (typically minerals, ceramics. semiconductors) may be fractured, cleaved, or mounted and polished for EBSD. Specimens which contaminate or oxidise rapidly (e.g. white tin) may need to be prepared immediately prior to SEM examination or alternatively ultrasonically cleaned and/or stored under solvent. Particularly difficult specimens can be ion milled for a few hours. A highly recoininended method for preparing a variety of specimens for EBSD is final polishing in colloidal silica, since this medium does not introduce the harsh mechanical damage associated with diamond polishing. In general, it is not necessary to prepare simple metals and alloys in this way because they respond readily to conventional polishing and etching. However, large particles (e.g. carbides, nitrides) in metallic matrices, low melting point metals and alloys, ceramics, con~positesand minerals are all amenable to polishing for up to several hours in colloidal silica (Case Study No. 5). Inadequate or inappropriate specimen preparation could give rise to misleading data interpretation. For example the diffuseness of an EBSD pattern is a guide to the amount of plastic strain in the lattice (Section 6.4); however if a specimen contained diamond polish damage prior to etching, the pattern might be interpreted erroneously as arising from a deformed specimen. Furthermore, the absence of an EBSD pattern in a representatively sampled region is evidence either that it is noncrystalline, or highly strained, or that the grain size is smaller than the probe size. However, these interpretations are only valid after correct specimen preparation. Although the primary aim of EBSD specimen preparation is to obtain diffraction patterns, a secondary objective may be to reveal the microstructure. This can be done by orientation microscopy (Section 10.3), electropolishing or etching. Often, orientation inicroscopy is carried out in conjunction with either of the other two methods used to reveal microstructure although etching time should be kept short to minimise
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INTRODUCTION TO TEXTURE ANALYSIS
topographical effects which may affect collection of diffraction patterns, particularly close to grain boundaries. The actual depth of the surface layer which gives rise to EBSD patterns is a function of both the accelerating voltage and the atomic number of the specimen. Higher accelerating voltages allow deeper penetration of the electron beam whereas higher atomic numbers are associated with lower elastic mean free paths of electrons, and thus shallower penetration (Section 7.6.3). The sensitivity of EBSD patterns to coating thickness has been explored by coating single crystals of silicon with various thicknesses (from 5 nm to 40nm) of aluminium, nickel or gold (Michael and Goehner, 1994). Figure 7.8 illustrates the effect of coating silicon with 5 nm of nickel. An EBSD pattern from the underlying silicon is easily observed when an accelerating voltage of 40 kV is used, yet for an accelerating voltage of 10 kV Figure 7.8 shows that the beam has hardly penetrated through 5 nm of nickel. It was confirmed that the interaction thickness is approximately twice the elastic mean free path (Joy, 1994). This gives critical interaction depths. i.e. the maximum thickness of surface material that can be penetrated by the electron beam at 40 kV accelerating voltage, of 100 nm, 20 nin and 10 nm for aluminium, nickel and gold respectively.
Figure 7.8 Illustration of the penetration depth of the electron beam in a silicon EBSD specimen. (a) N o coating, 40 kV accelerating voltage: (b) no coating, 10 kV accelerating voltage; (c) coating with 5 nm of nickel. 40 kV accelerating voltage: (d) coating with 5 nm nickel, 10 kV accelerating voltage. There is less beam penetration at l 0 kV since the underlying silicon pattern is indistinct (Courtesy of J.R. Michael).
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
l65
For non-conducting specimens, which includes minerals, diffraction patterns can usually be obtained if very small specimens are embedded in conducting paint. Low accelerating voltages may be needed to reduce charging or beam damage. Otherwise, a conductive coating which is only a few nanometres thick, such that the electron beam can penetrate to the specimen beneath, could be deposited prior to EBSD. If a coating has been deposited it may be helpful to increase the accelerating voltage of the SEM which in turn increases beam penetration as described above. A few examples of preparation methods which are used routinely for EBSD are given belon. This list is of course by no means exhaustive, and intended only to illustrate some of the points made in this Section: e
Comnzerciallj~pure alunziniunz, titanium a l l o ~ s- Electropollsh in 5 % perchloric acid in ethanol at -25°C. (Etchants which deposit a film on aluminium are unsuitable for EBSD).
e
A1ztnziniu~~1-litl~izlnl allojs slightly warmed.
e
Mild steel
e
Marzj. rocks and minerals Diamond polish block specimens followed by colloidal silica polishing for several hours.
e
Polj~silicon- Wash in detergent, immerse for 1 minute in 10% hydrofluoric acid in a plastic beaker.
-
-
Im~nersefor several seconds in Keller's reagent,
Swab with 2% nital for several seconds. -
In summary, specimen preparation for EBSD is generally uncomplicated, and for most cases is based on standard metallographic routes. Diffraction patterns arise from a surface region which is of the order of tens of nanometers thick. the exact thickness varying with microscope conditions such as accelerating voltage and atomic number.
7.6 EXPERIMENTAL CONSIDERATIONS FOR EBSD Although Initial acquisition of EBSD hardware tends to be a 'one-off' consideration, there are then many cholces for the methodology used in an investigation. reflecting the versatility of EBSD. Furthermore appropriate choice of microscope operating conditions is necessary to achieve the optimum balance of convenience and resolution. These points are discussed in this Section.
7.6.1 Hardware An EBSD system can be added as an attachment to practically any SEM, including a Geld emission gun (FEG) SEM. Whereas in the earlier standard set-ups the diffraction camera was mounted horizontally in the rear of the microscope, other geometries with either a tilted camera (in a positive or negative sense with respect to the horizontal) andlor entry through a side port of the microscope are feasible. Since most of the electrons are diffracted in the forward direction from the tilted specimen, the most
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favourable capture geometry is to have the camera also tilted so as to be below the horizontal. However, this poses additional problems for maintaining the camera in a stable position, and so is not generally adopted. The most common reason for mounting the camera in a side port of the microscope is to take advantage of the resident tilt of the eucentric stage, rather than utilising a purpose-built pre-inclined holder. Which of these options is the more convenient will depend on the microscope column layout and the size of specimens that are to be investigated. The most important item of EBSD hardware is of course the camera. The basic choice is between two types of camera the tube type, i.e. SIT or ISIT, incorporating such refinements as fibre optic coupling, or a CCD as mentioned in Section 7.4. The CCD camera, which is now tending to supersede the other models, is usually cooled to reduce the build up of thermally generated charge during operation. The advantages of the CCD camera are (Hjelen et al., 1994; Drake and Vale, 1995): -
0
The patterns are of a consistent quality over a wide range of microscope operating conditions, unlike tube cameras where the diffraction pattern becomes noisy if the accelerating voltage andlor probe current is reduced
e
There is no geometric distortion of the diffraction pattern, whereas tube cameras may introduce considerable distortion
0
They are more robust, since they are not damaged by normal illumination, whereas low-light tube cameras are
e
The lifetime of a CCD cainera is longer than that of a tube camera, and furthermore the CCD camera can be smaller and lighter
There are two options for accommodating the camera/'screen assembly. The camera may view the specimen in the microscope through a lead glass window. with a port in the microscope chamber purpose-modified to house the window. The phosphor screen then resides inside the inicroscope column, independent of the camera which is outside the vacuum system. A more advanced approach, which takes advantage of the smaller and lighter CCD camera, has the phosphor screen mounted on the front ofthe camera and the whole camera/screen assembly is inserted or retracted through a vacuum seal into the SEM chamber. The phosphor screen itself is fragile and touching. cleaning or collision with any objects should be avoided. It may be possible to tolerate some screen blemishes because pattern enhancenlent routines can minimise their effect (Section 7.6.4). Damaged or degraded screens can also be easily replaced. The position of the cainera screen in the coluinn is generally dictated in the first instance by the space available in the inicroscope specimen chamber. and is usually approximately 30 mm from the specimen. The camera screen may not be maintained at a fixed position for several reasons: e
If large specimens, for non-EBSD apphcations. are accommodated in the microscope chamber the camera screen m111 need to be retracted to prevent collision damage to it
e
One of the EBSD calibration routines requires the cainera screen to be operated in two positions (Section 7.7.2)
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES e
l67
If the camera screen is placed further away from the specimen than is usual, the capture angle for the diffraction pattern will be reduced. Figure 7.9 shows that the capture angle is reduced to 14" if the screen is placed 140 mm from the specimen. This configuration gives an increased accuracy and has been used to measure small orientation changes (Wilkinson, 1996). However, the concomitant reduction in light level captured by the camera (approximately 1,120 of that usually encountered for the 30 mm specimen-to-screen distance) means that such a setup is only feasible with a CCD camera using long integration times.
The other main components of an EBSD system are the camera control unit and a suitably configured personal computer. A fully automated system will have a mechanism for controlling the position of the beam andior a stage motor to position the specimen. as shown on Figure 7.7. Finally a forward mounted back-scatter detector (FMBSD), which is also referred to as a forward scattered electron detector or a forescatter detector, may be included in the EBSD system. This detector fulfils the same function as a back-scattered electron detector in conventional imaging, namely uses the primary (back-scattered) electrons and a form of amplification to image the microstructure on the basis of changes in topography, conlposition or orientation. The modification which is required for EBSD is to mount the detector in a , f o r n w d position, either on the camera or phosphor screen itself or on the specimen holder, so as to be in the optinlunl position to detect the back-scattered electrons and use them for imaging. Since the inclination of the specimen gives a much higher yield of backscattered electrons than if the specimen were flat, back-scattered imaging is improved by tilting the specimen, particularly for predominantly low atomic number specimens such as minerals. rocks or aluminium alloys.
specimen
2
usual screen position
screen position for strain measurement
Figure 7.9 Effect of retracting the phosphor screen in an EBSD system to g i ~ ea smaller convergence angle o: and hence belter resolution in the diffraction pattern (Courtesy of A. Wilkiason).
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The response of the forescatter detector to crystallographic variations is shown in Figure 7.10 for a specimen of granite. Figure 7.10a is a normal backscatter image showing atomic number contrast between the phases whereas Figure 7.10b is a forescatter image of the same area which shows, via orientation contrast, subgrains and twins but no atomic number contrast. Hence, the forescatter detector is a useful tool with which to obtain qualitat~veorientation inforn~ation.
7.6.2 Experiment design philosophy The first point to consider when designing an EBSD investigation is the actual choice of analysis technique itself. EBSD may not the best choice if: e
There is too much deformation, e.g. a heavily cold worked material
e
The grainlsubgrain size is too small, e.g. a nanocrystalline material
e
An overview of the average texture is required
For the first two cases TEM would be a better choice and for the last case a macrotexture analysis would be more suitable, although the range of EBSD applicability is increasing to encompass more investigations which have traditionally been the province of other techniques (Sections 7.9 and 9.10).
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Figure 7.10 SEM micrographs of granite from Peru (accelerating voltage 20 kV). (a) Normal back-scatter electron image showing atomic number contrast between plagioclase, quartz and haematite plus ilmenite: (b) Forescatter image of the same area as in (a). As a result of orientation contrast twins and subgrains are visible 111 the quartz and subgrains are visible in the plagioclase (Courtesy of D. Prior).
Having established that EBSD is to be used, the material needs to be sectioned so that an appropriate surface can be prepared. In most cases there is an obvious choice for this surface, although its selection may need some thought. For example a specimen which has been cut from a rolled product may be mounted so as to sample microtexture from the 'side' of the sheet. the RD-ND plane (Section 2.2.1) rather than the more obvious RD-TD plane as shown in Figure 7.11 (Davies and Randle, 1996; Huh et al.. 1998). The orientation sampling schedule can be performed either manually or automatically. In general some preliminary exploration in manual mode to establish the scale of the orientations and how they relate to the microstructure, e.g. the grain size distribution, will clarify the requirements. For example, in order to obtain nearestneighbour misorientations, the orientations must be sampled with a step size which is less than the grain size. In general EBSD is not used only to obtain an overall average texture; because this can normally be done more efficiently by a macrotexture method. However, if EBSD is used to gain a representation of the texture, then consideration must be given to the location and number of the constituent orientations (Section 9.3.3).
INTRODUCTION TO TEXTURE ANALYSIS
(4 Electron beam
\i/
Longitudinal
Rolling plane
I
Electron beam Longitudinal transverse
m Figure 7.11 Examples of choices for mounting an EBSD specimen to sample either (a) the rolling plane or (b) the longitudinal-transverse section.
Once the parameters for the location of the region of interest, i.e. its size plus the step size or indi~iduulcoorditzates needed for orientation measurements have been determined by trial runs, many investigations can then proceed automatically (Section 10.3). Sometimes further manual interaction is required, such as where there are very large differences in the spatial scale of the orientation distribution, for
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example the sampling of fine twins in polysilicon which has a grain size of many millimetres (Wagner et al., 1997). It may also be necessary to establish what phases are present in the specimen. Multi-phase materials can be analysed using EBSD (e.g. Randle and Laird, 1993; Engler et al., 1995b; Powell and Randle, 1997). The need to distinguish between diff'raction patterns from different phases may mean than such work is often carried out interactively, especially at preliminary stages. Progress has been made in automated operation and mapping of multi-phase materials (Section 10.4, Figure 10.7 and Case Study No. 6). Many EBSD packages allow orientations to be analysed concurrently from more than one phase. 7.6.3 Resolution and operational parameters Whereas some EBSD investigations, e.g. those involving very small (sub)grains and small misorientation changes will approach the resolution and accuracy limits of the technique. the majority of studies are performed well within the capabilities of the average EBSD system and SEM. However, it is still instructive to have some insight into the effect that material type and the various microscope operating parameters have on EBSD so that values can be chosen advisedly. The average spatial resolution and accuracy of EBSD are -200-500 nm and l" respectively, and these are influenced by e
Material
e
Specimen/microscope geometry
o
Accelerating voltage
s Probe current s Pattern clarity
The fundamental factor which governs the absolute spatial ~ . e s o l u t i ~ofn EBSD. and indeed of SEM in general, is the interaction volume of the electron beam in the specimen. Because of the very thin surface layer contributing to the EBSD pattern. the interaction volume approximates to the area of the incident beam, i.e. the spot size. Because the large angle of specimen tilt means that the interaction volume is anisotropic (Figure 7.4), the interaction volume is generally quoted as the product of the resolution parallel and perpendicular to the tilt axis, or an average of these two values. However, this implies that the specimen orientation is invariant within the interaction volume which may not be the case for deformed or very fine grained materials. Where overlapping diffraction patterns are sampled in a single volume it may be possible for the software to solve the stronger of the two patterns, giving rise to an effective spatial resolzition which is smaller than the absolute resolution (Humphreys et al., 1999). Material The amount of backscattered signal increases with atomic number, hence often there is more detail and greater clarity in patterns from high atomic number elements than from those with low atomic numbers.
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The three main parameters which can be altered in the physical EBSD set-up are the specimen-to-screen distance, the specimen tilt and the specimen height ('working distance') in the microscope. Values for all these parameters are necessary to calibrate the system (Section 7.7) and hence to index the diffraction pattern (Section 6.3). For general EBSD data collection the specimen-to-screen distarzce, i.e. the camera position, remains fixed (Sections 7.4 and 7.6.1). With regard to specinzerz tilt, EBSD patterns have been observed for specimen tilt angles >45". However, as mentioned in Section 7.1, the path length of the backscattered electrons in the specimen decreases with increased specimen tilt which leads to better contrast in the EBSD pattern with increased tilt angle (Venables and Harland. 1973). Tilt angles >80" are impracticable because of the excessive anisotropy of the sampled volume and distortion in the uncorrected image. A tilt angle of 70" is obligatory for one calibration routine (Section 7.7) and, since this value represents a good compromise with regard to convenience and pattern contrast. it is often used as a standard angle for EBSD. For this tilt angle the anisotropy in the sample volume (which also feeds through as an image distortion) is 3:l in the direction perpendicular to the tilt axis compared to that parallel to it. Although for standard SEM work best resolution and minimised focussing distortions are obtained from a short uvol-king distance, for EBSD the major consideration when choosing the working distance is to locate the specimen so that electrons are backscattered from the specimen towards the camera. Often a limiting factor is also the risk of collision with microscope hardware, particularly the pole piece, at sinall working distances. The optimum specimen position will depend therefore on the geometry of the microscope. It is also convenient, if possible. to choose a working distance which locates the PC near the centre of the phosphor screen. Taking all these factors into consideration. an optimum working distance is in the range 15-25 min. Some calibration routines or eucentric sample stages dictate a fixed working distance, e.g. 19.5 mm.
Acceleratiizg voltage
Essentially, there is a linear relationship between accelerating voltage and interaction volume for a specific element. Hence a low accelerating voltage is chosen if good spatial resolution is required. Since the size of the interaction volume also correlates with the atomic number of the species, in general a higher accelerating voltage can be chosen for high atomic number specimens. Figure 7.12 shows the relationship between accelerating voltage and average lateral spatial resolution in nickel as determined in a standard tungsten filament SEM (Drake and Vale, 1995). It is clear that where good resolution is required: e.g. for fine grain sized or deformed material, significant improvements in resolution are attained by reducing the accelerating voltage. In addition to resolution there are several other factors to consider when selecting an accelerating voltage for a particular inquiry. The advantages of using a high accelerating voltage, i.e. 3 0 4 0 k V , are (Drake and Vale, 1995; Schwarzer et al., 1997a):
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
0
5
10
15
20
25
173
30
Accelerating voltage (kV)
Figure 7.12 Spatial resolution of EBSD in nickel as a function of accelerating voltage (Adapted from Drake and Vale, 1995).
e
The efficiency of the phosphor screen increases with electron energy (i.e. higher accelerating voltage) which results in a brighter diffraction pattern
e
There is less interference from stray electromagnetic fields
e
The electron beam penetrates further, and thus the diffraction pattern originates from a region below the surface such that surface contamination or surface damage effects are minimised
The disadvantages of using a high accelerating voltage are: e
The beam-specimen interaction volume increases. thus strongly reducing spatial resolution
e
Specimens which are poor conductors or are susceptible to beam damage cannot be examined using a high accelerating voltage, unless the specimen is lightly coated with a conductor (Section 7.5)
A change in accelerating voltage changes the electron wavelength, which in turn changes the spacing of the Kikuchi bands according to Bragg's law (equation 3.4). This effect can be seen in Figure 7.8 where the accelerating voltage is varied from 10 kV to 40 kV. However neither the interplanar nor interzonal angles are affected
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and so the principles of orientation measurement (Section 6.3) are independent of accelerating voltage. Probe current
The probe current is selected in accordance with the light sensitivity of the camera and the requirement of imaging the specimen surface, and 5 nA is a good choice for a modern camera. Although the spatial resolution is less sensitive to probe current than to accelerating voltage, if it is convenient to work at a particular accelerating voltage then the probe current can be used to manipulate the resolution. Lateral resolutions (i.e. ignormg the depth penetration because it is an order of magnitude smaller than the other dimensions, Figure 7.4) in aluminium for a standard EBSD set-up have been quoted as 250 X 700nm2 (Hjelen and Nes, 1990) and 200-500 nm (Humphreys et al.. 1999). Although the best absolute resolution corresponds to the smallest interaction volume, small interaction volumes affect adversely the pattern clarity and hence the ability of the software to locate the Kikuchi lines and index the pattern (see also below in this Section). Hence the best working resolution is not achieved with the smallest probe size, as illustrated on Figure 7.13. Here there are minima in the curves
Probe current, A Figure 7.13 The effect of probe current on effective resolution for several aluminium specimens. The minima in the plots are caused by the reduced pattern solving accuracy at 10s probe currents (Courtesy of F.J. Humphreys).
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relating effective resolution to probe current, indicating that there is a compron~ise situation between interaction volume and acceptable pattern clarity. The best resolutions are obtained with a field emission gun (FEG) SEM rather than a conventional tungsten filament SEM due to the small size of the electron beam. Using an accelerating voltage and probe current of 20 kV and 1 nA respectively, a lateral resolution of 20 X 80nm2 and 10 nin depth has been achieved in nickel (Harland et al., 1981). Simulations have shown that lOnm lateral resolution is feasible if 5 kV accelerating voltage is used in a FEG SEM and experiments with modified EBSD hardware are in progress (Troost and Slangen. 1994; Troost and Karnminga, 1994). Pattern c/ar.itjl
Pattern clarity is influenced not only by the innate quality of the pattern arising from the sampled volume of specimen itself, caused by the presence of lattice defects, but also by 'noise' introduced during pattern capture and processing. Degradation of this type will result when the pattern is averaged over too few frames, or digitised to too few points, or when the back-scattered electron signal is reduced. Since the back-scattered signal is proportional to the atomic number, higher atomic number materials give inherently clearer patterns than their low atomic number counterparts. Use of large probe currents increases significantly the amount of back-scattered signal generated. as shown in Figure 7.13, because of the larger sample volume entailed, thus improving the pattern quality. This effect is further illustrated and quantified in Figure 7.14 for single crystal silicon. For the largest probe current used 87% of the
Probe current
- - .-..,.,-
: : a . . . .
o"
0.5'
:.:
F
--
1.OO
Misorientation Figure 7.14 Misorientation measurements between adjacent points on a single crystal silicon specimen for four different probe currents (in Amps). The highest probe current provides the most accurate result (Courtesy of F.J. Humphreys).
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INTRODUCTION TO TEXTURE ANALYSIS
misorientations measured to assess the accuracy were
Specimen tilt angle 70"
e
Working distance 15-25 mm
e
20 kV accelerating voltage
e
5 nA probe current
These apply to a standard SEM using a tungsten-filament and a modern camera. Where high spatial resolution is not an overriding consideration. i.e. when a resolution - ~ l p mis adequate, operational parameters are chosen for convenience (short averaging time, acceptable imaging, etc.). 7.6.4 Diffraction pattern enhancement The real-time or 'raw' diffraction pattern, i.e. that obtained straight from the specimen, is typically rather noisy, blemished, uneven, blurred or dim (e.g. Figure 7.15a,b). Therefore, all modern EBSD systems include forms of diffraction pattern enhancement to improve the quality and so facilitate indexing on the processed
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Figure 7.15 EBSD pattern quality enhancement in a nickel based alloy (a) raw image; (b) raw image stretched to show non-uniform background; (c) Parabolic background correction and stretching. Screen imperfections are visible as black spots; (d) Background subtraction using an averaged EBSD pattern (2 TV rate frames); (e) as (d) but averaged over 8 TV rate frames; (f) as (d) but averaged over 128 TV frames (Courtesy of A. Day).
pattern, which is usually accon~plishedby an image processor like ARGUS 10 (Hamamatsu) or DSP2000 (MTI). Figure 7.15 shows a typical sequence and illustrates the improvement in diffraction pattern quality which can be obtained compared to the raw pattern. The diffuse background, plus minor screen blemishes, are stripped from a pattern by employing a background correction routine. This routine involves obtaining an image of the background intensity on the phosphor by integrating over the patterns obtained from a number of differently oriented grains such that the various patterns 'cancel out' (Figure 7.15~). In a typical as-deformed or finely recrystallised microstructure this can readily be accomplished by switching from spot- to image mode - if necessary at a lower magnification on the specimen - so that the orientations of a multitude of grains contribute to the background image. If the number of grains is too small to yield a uniform background, in particular in single crystal materials, a background can be obtained either by defocusing the objective lens extensively or by rotating the sample during background acquisition. The background image is then electronically subtracted from the diffraction pattern (Figure 7.15d). Alternatively, the diffraction pattern can also be divided by the background image which results in a better correction of shading differences of the processed pattern. A new image of the background intensity must be collected every
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178
time a new specimen is placed into the microscope, or when the operating conditions are changed. Furthermore, the diffraction pattern can be nvemged over a large number of frames to reduce the amount of noise present (Figure 7.15e). Averaging up to about 8 frames per second still retains the 'live' character of the patterns, which is important for the localisation of specific features, e.g. subgrains, grain boundaries, etc., during manual EBSD operation. A longer averaging time improves the pattern quality (Figure 7.15f), but it may noticeably slow down the operation of automated systems. Besides these means of pattern enhancement in analog image processing devices, most modern EBSD softwares include more sophisticated software routines for further image processing, some of which have been addressed in Section 6.3.3.
7.7 CALIBRATION O F A N EBSD SYSTEM EBSD calibration involves a transformation between the two-dimensional distances given in pixels in the diffraction pattern and the corresponding three-dimensional crystallographic angles in the specimen. Careful calibration is most important if accurate orientatio~lsare to be measured. Although standard commercial EBSD packages have calibration routines which include step-by-step instructions, it is useful to understand the principles upon which the calibration is based. and to have an appreciation of the various methods which can be used to obtain the calibration parameters, both of which will be discussed below. The calibration methods are: s
Shadow casting
e~ Known crystal orientation
e
Pattern magnification
s
Iterative pattern fitting
7.7.1 Calibration principles The objective of EBSD calibration is to provide a transformation between angles in the diffraction pattern expressed as screen pixels and the same angles in the specimen. Only two parameters are required for this transformation, and both of them can be described in terms of the gnomonic projection (Appendix 4): e
The origin of the gnomonic projection, labelled N in Figure 6.4 and referred to as the pattern centre. PC
e
The radius of the reference sphere, labelled ON in Figure 6.4 and referred to as the specimen-to-screen distance. ZssD
The specimen-to-screen distance is the line which connects the PC with the point of incidence of the electron beam on the specimen, known as the pattern source point, S P (although strictly S P is a volume, not a point. as shown in Figure 7.4). PC, S P and ZssD are illustrated in Figure 7.5 (Randle, 1992). It is worth mentioning that
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although in Figure 7.5 the PC is shown coinciding with the physical centre of the EBSD pattern, this is not usually the case in reality for two reasons. Firstly, the y (vertical) coordinate of the PC depends on the height of the SP on the specimen, which in turn depends on the height at which the specimen is positioned in the microscope. Secondly. the x (horizontal) coordinate of the PC depends on the location of the optic axis of the microscope, which for some models is displaced from a central position. (These considerations are not valid for TEM or SAC diffraction patterns, since for those techniques the electron beam is concentric with the PC). The orthogonal axes xyz: Xs Y,Z, and X,, Y,,Z, on Figure 7.5 refer to the reference axes of the diffraction pattern, specimen and microscope respectively. Calibration of the system presupposes that X, and X,,, are parallel, which implies selection of a reference direction in the specimen, usually a specimen edge, which in turn may be important for the particular investigation, e.g. a cleaved edge, a facet or a direction of processing. Where X, does correspond to a significant direction in the specimen, it is clearly important to align it carefully parallel to X,,,. Any alteration in the microscope to the diffraction pattern collection geometry will change the calibration parameters. Possible alterations are: s The tilt angle of the specimen stage e
The position of the detecting cameraiscreen
e
The position of the specimen (more specifically the S P on the specimen)
In practice the first two of these do not usually apply because a fixed specimen tilt is chosen to conserve the calibration geometry and, once calibration is completed, the camera,screen assembly remains in a fixed position during data collection. On the other hand the position of the S P on the specimen will move if the focussing controls are used because essentially this operation changes the working distance (objective lens setting). Once the system is calibrated, if refocusing of the image is necessary it is achieved by changing the height of the specimen using the microscope '2'stage control - hence the specimen moves but the S P remains fixed (Dingley et al., 1987). Another way in which the position of S P can move between calibration and data collection is through beam deflection at low magnifications (Day. 1994) which may become an issue in orientation microscopy (Section 10.3). At a magnification of x75, the total beam deflection across the screen is approximately l mm in the xdirection and 3 mm in the y-direction if the specimen is tilted to 70". This means that if sampling points are chosen by moving the beam away from the original SP. large inaccuracies in the PC will result. At higher magnifications, e.g. X 1000, the scanning area on the specimen is very small and discrepancies between the S P as defined in the calibration routine and sampled volumes defined by moving the beam become negligible. If it is necessary to work at low magnification, e.g. if a specimen has a very large grain size. then the beam position should remain at the calibration position and individual grains sampled by translating the specimen. In this way the coordinates of the PC and the value of Zssu are maintained between calibration and data collection. On the whole, the accuracy of calibration feeds through to give an error of approximately 0.5" on a orientation measurement relative only to the calibration parameters and not to the external specimen geometry. Where it is necessary to
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INTRODUCTION TO TEXTURE ANALYSIS
quote an orientation relative to the external specimen geometry, sometimes called the 'absolute orientation', the accuracy degrades to approximately 2" because of specimen location errors. Individual EBSD packages may claim slightly different estimated calibration accuracies. An improved calibration accuracy results if ZssD is increased since the angular resolution is improved (Figure 7.9). The angle subtended by 1 mm on the screen at the PC is reduced from approximately 2" to 0.4" by increasing ZssD from 30 mm to 140 mm. However, as discussed in Section 7.6.1 this camera configuration is not usually convenient because of the reduction in signal which ensues and also because the small capture angle of the diffraction pattern may hinder automatic indexing. 7.7.2 Calibration procedures
Shadow castirzg Objects placed between the specimen and the camera will cast a shadow on the recording screen. If the objects are regular in shape, the shadows cast can be used to find the PC and ZssD As alluded to in Section 7.4, this was the principle of the first calibration used for EBSD, when the patterns were recorded onto film. Three spheres mounted in front of the specimen cast elliptical shadows onto the screen. The major axes of the ellipses, when extended, intercepted at the PC. Another, more recent, shadow casting method uses a fine square mesh attached to the phosphor screen (Schwarzer et al., 1997a). The shadow images of the mesh are trapezoids having their long axes pointing towards the PC. A variant of the shadow casting method is to have two shadow-casting objects, one behind the other, between the phosphor screen and the specimen. These can then be lined up in a similar manner to gun sights to define the PC and SP. Figure 7.16 demonstrates that when the shadows of both sighting wires are superimposed on the image of the diffraction pattern, the PC and SP are defined (Day, 1994). The advantage of the shadow casting method is that it is accurate and repeatable, and may offer some protection from collision between the specimen and screen; the disadvantages are that it requires additional, precision-made hardware and that, unless the shadow casting devices can be retracted, it interferes with automatic pattern recognition.
Kno~cncrystal o~.ierztation The 'known crystal orientation' method was inaugurated as part of the original computer software which analysed EBSD patterns on-line (Dingley et al., 1987). It survived for several years as the main method for EBSD calibration. Essentially a single crystal of silicon, prepared so as to reveal {001), was mounted in the microscope and tilted so as to make an angle of 19.5" with respect to the electron beam. This specimen geometry means that the PC must lie 19.5" vertically below the specimen normal direction, [001], which is [l 141. Hence for an [001] silicon standard, tilted at 70.5" (90-19.5"), the PC is given by the position of the [l 141 zone axis. ZssD is then obtained from measuring the distance in pixels on screen between two zone
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18 1
l Specimen
Figure 7.16 Locat~onof the pattern centre, PC, on the phosphor screen by alignment of two sets of crosswlres located between the specimen and the screen (Courtesy of A Day)
axes, e.g. 101l ] and [001], and relating it to the known angle between them, which is 45" in this case, via the relationship
where 4 is the angle between the two chosen zone axes, X I , x2, y1, y2 are the coordinates of both of them respectively and X ~ C y, p are ~ the coordinates of the PC. ZssD is then the only unknown in the equation. Figure 7.17 shows an EBSD pattern of (001) silicon with the pattern centre marked. Note that if the edge of the silicon is cleaved along < l 10>, and mounted with this edge parallel to the X,,, then the x-axis of the system is defined by the [i10] direction in the diffraction pattern. In other words, the Kikuchi band (zone) which
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INTRODUCTION TO TEXTURE ANALYSIS
Figure 7.17 EBSD pattern from a calibration crystal of single crystal silicon cleaved on jOOlj and along <110>. The pattern centre PC (at <114>) and the crystal normal ND <001> are marked.
contains [l 141 and [001] should be vertical. The fortuitous features of this method are that a specimen tilt angle of approximately 70" is optimal for a compromise between back-scattered contrast and tilt of the image, and the [l141 zone axis is easy to recognise as the PC. The operator is required to input the position of [l 141 and the location of two identified zone axes plus the angle between them. The specimen must be mounted with exactly the same geometry as the calibration crystal. and usually this is achieved by mounting them both together. (001) single crystal silicon is widely used for the known crystal calibration method since the material is readily available, stores well and gives clear diffraction patterns. Germanium or gallium arsenide are also used, or lead sulphide which has the additional advantage of exhibiting natural (001) facets. Errors are introduced into the method if the single crystal is not exactly (001), which is undetectable in the SEM and would necessitate checking by an independent means, such as X-ray methods. Since silicon calibration specimens are usually made by mechanical cutting along (001). which is not a natural cleavage plane in silicon, a slightiy misaligned calibration crystal is a real possibility. The other main error which can occur relates to rotation of the calibration crystal, which can be checked from the ratios of [l 14]:[112], [l 12]:[11l] and [100]:[101] measured in pixels in the diffraction pattern.
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQC'ES
l83
The advantage of the known orientation method is its simplicity. The main disadvantages are that care has to be taken to mount the calibration crystal carefully, and even then there may be 'hidden' errors which are difficult to take into account, and some knowledge of crystallography or ability to recognise elements of the diffraction pattern are required. Pattern nzugnifiication
The effect of increasing ZssD (ON in the gnomonic projection, Figure 6.4) is to increase the distance of a projected point in the diffraction pattern from the pattern origin or PC. In other words, the pattern is more magnified. However. increasing ZssD and magnifjing the pattern does not change the position of the origin of the gnomonic projection, the PC. Thus the P C is identified as the only point which does not shift when the diffraction pattern is magnified, and hence 'zooming' the pattern offers another method for calibrating the PC. Practically. this means that the camera/screen assembly must be capable of retraction to increase ZssD so that the pattern can be viewed at a minimum of two magnifications, which can be called the 'in' and 'out' positions. The easiest way to locate the pattern centre is to outline three Kikuchi bands which form a triangle in the 'in' camera position, and then outline the same three bands in the 'out' position on the magnified diffraction pattern, as shown on Figure 7.18a. The coordinates of each triangle corner for the 'in' and 'out' positions lie on a line upon which the P C also lies. The intersection of the three lines from each triangle corner therefore defines the PC (Figure 7.18b; Hjelen et al., 1993). The pattern magnification method does not require a standard crystal and can be applied to any well-defined diffraction pattern. The specimen-to-screen distance ZssD can be found by the same method as that described above w h e n the camera is in the 'in' position ready for data collection. the distance in screen pixels between two known zone axes is measured and the angle between them input. However, this way of determining ZssD requires recognition of two zone axes, and it may be convenient to keep a specimen with an easily identified EBSD pattern to facilitate this procedure. The sample-to-screen distance can be obtained also directly fi-on1 the position of the zone axes (Hjelen et al., 1993). After calculating the distances Li, and between two zone axes, ZssD can be derived from the relation LiniZSSD= Lout/ (ZssD AZLssD), where AZssD is the distance between the 'out' and the normal 'in' screen position. Greater accuracy is achieved if the third zone axis in introduced in the computations. The error in the absolute orientation, Ag, is related to the error in the positioning of the P C , A P C , and ZssD as follows
+
The advantage of the pattern magnification method is that it requires no special equipment beyond a 'stop' to prevent the camera being wound in too far when locating the 'in' position. There are no great disadvantages, except that, as for the previous case. some pattern recognition capability is needed to input the parameters to find ZssD
INTRODUCTION TO TEXTURE ANALYSIS
Camera 'out' Position
Camera 'in' Position
Figure 7.18 Principles of the 'pattern magnification' method of EBSD calibration (Courtesy of K. Dicks).
Iterative pattern fitting Calibration via iterative pattern fitting requires only an EBSD pattern of reasonable quality in which the operator can identify at least four zone axes. Equation 7.1 can then be formulated n(n-1)/2 times, where n is the number of identified zone axes, substituting in the known indices of two of the poles in turn. These equations have to be solved numerically to obtain X ~ C Y, P C and ZssD (Krieger Lassen et al., 1992; Krieger Lassen and Bilde-Sorensen, 1993). The numerical method needs a 'first guess' for the values of the PC coordinates and Zsso If the first guess is reasonably good, less iterations will be required to converge towards the right solution. Another iterative fitting model uses a comparison between a diffraction pattern from a known material and a simulated pattern. About ten zones (band centres) are required to be input by the operator, and the method will calculate the rotation of the simulated pattern required to bring it into alignment with the real pattern (Day, 1994). Again, a reasonably good 'first guess' is required. The advantage of the iterative pattern fitting method is that the need for any special attachments or standards is eliminated, and it is insensitive to the geometry of the set-up. Hence it is very suitable for non-standard equipment. A disadvantage of this method is that it requires sophisticated programming (although this it not really a disadvantage for the user of a comn~ercialpackage) and a 'first guess' solution.
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
185
Table 7.1 Summary of EBSD calibration options P
P
Method
Advantages
Disadbantages
Shadow casting
Accurate and repeatable. Some protection against screen collision. Simplicity
Requires precision hardware which interferes with automatic pattern recognition
Known crystal orientation Pattern Magnification Iterative pattern fitting
N o special equipment or standards required No special equipment or standards required
Errors associated with crystal and mounting; some pattern recognition required Some pattern recognition required Requires sophisticated programming and a 'first guess' solution
However in most cases unless the experimental set-up has been completely changed, the 'old' calibration parameters can be used as the first guess. Table 7.1 summarises the pertinent features of the calibration routines described above. The optimum approach to calibration is to exploit the best features from the different methods and form a combination procedure. For example, most commonly iterative calibration is used to refine the calibration parameters after they have been obtained by another means.
7.8 OPERATION O F A N EBSD SYSTEM AND PRIMARY DATA OUTPUT Most modern EBSD packages evaluate the diffraction pattern automatically. as described in Section 6.3.2. The parameters which need to be passed to the software are: e
The unit cell dimensions, interplanar angles, crystal system, lattice type, atom types and positions
e
The calibration parameters as described in Section 7.7. The microscope operating conditions, accelerating voltage and probe current may be selected dming calibration
e
The specimen geometry with respect to the camera, i.e. the specimen tilt and rotation angles and the directions of the specimen reference axes
e
The pattern acquisition conditions, i.e. suitable background correction conditions
Patterns are normally solved and stored in a fractlon of a second. Sampling points are selected either manually in real time. or by automatically moving the beam or stage (Section 10.3.1). The automatic route samples either at pre-set intervals so as to form a grid of sampling points on the specimen, or by moving the stagelbeam to preprogrammed individual coordinates on the specimen. Many EBSD packages offer a numerical assessment indxating the goodness of fit between the experimental pattern and the solution. Examples are a 'Mean Angular Deviation' or a 'Confidence Index'. Once a population of orientations has been evaluated from the diffraction patterns they can be displayed and output in the following formats (Chapter 9). which are offered as standard in most commercially available EBSD packages:
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INTRODUCTION TO TEXTURE ANALYSIS
Orientation matrices. These can be exported for further mathematical manipulation (Section 2.2.2) Pole figures. This is the most popular display method for EBSD data (Sections 2.4.1, 9.2, Case Studies No. 1, 3, 5, 11) I m w s e pole jgures (Sections 2.4.2 and 9.2, Case Study No. 6) Ideal orientation for each individual orientation. i.e. {hkl}<~mw>(Section 2.3, Case Study No. 1) E d e r angles plotted in Euler space (Sections 2.5, 9.3, 9.6.2 and Case Study No. 9) Roclrigues vectors for orientations or inisorientations plotted in Rodrigues-Frank space (Sections 2.7, 9.4, 9.6.3 and Case Study No. 3) Mi~orientationpurumetc.rs, typically the anglelaxis of misorientation (Sections 2.6 and 9.6.1) and coincidence site lattice designations (Section 11.2, Case Studies No. 3, 5) Cryrtul orientution mups and their derivatives such as grain boundary maps and image quality maps (Section 10.4, Case Study No. 3)
The descriptors of orientation and representation methods listed above are described in detail in Chapters 2 and 9 respectively. Orientation mapping differs from the other output methods in that it conveys spatial rather than statistical information, which is discussed in detail in Chapter 10. The information contained in the orientation matrix can be further manipulated to determine specific crystallographic directions in the specimen, such as in the analysis of facets, cracks or interfaces. These topics are discussed in Chapter 1 1.
7.9 SUMMATION
A state-of-the-art commercial EBSD system is designed to be operated with minimal knowledge of the science which underpins diffraction pattern evaluation and orientation determination. However this Chapter, which has concentrated mainly on EBSD rather than other less versatile §EM techniques, has aimed to provide sufficient depth of information for an EBSD user to have a basic appreciation of the theoretical principles of the technique. Procedures for automatic diffraction pattern evaluation were described in Chapter 6. Although there is a great diversity in application of EBSD for microtexture-related investigations. the basic principles are generic and are summarised on Figure 7.19. This 1s a scheme for design of a typical EBSD experiment, going through preliminary steps of setting up the experimental parameters to collecting and outputting data. The majority of data relates straightforwardly to grain orientations or misorientations (interfaces). Additionally, as shown on Figure 7.19, intra-grain orientations (Section 9.5.2) and three-dimensional information (Section 11.3) can also be accessed. Aspects of EBSD form all or part of Case Studies Nos. 1-6, 9 and 11.
SCANNING ELECTRON MICROSCOPY (SEM)-BASED TECHNIQUES
157
I TYPICAL EBSD INVESTIGATION DESIGK /
6 Select material
I
Check diffraction patterns available
quality, stabihty, grain size etc. I
Preliminary overview of orientation pararnetersglobal andor contiguous
Figure 7.19 Flow chart showing an overview of the design philosophy for EBSD experiments
The range of applications for EBSD is becoming wider. allowing it to begin to challenge some areas where other techniques. notably TEM and X-rays, have traditionally been used to obtain orientation information. At the time of writing it is likely that, whereas the spatial resolution limit of EBSD in a conventional tungsten filament SEM (< lpm) has been reached, use of an FEG SEM will allow a inuch increased signal-to-noise ratio in the pattern. This will in turn lead to a better spatial resolution and, incorporating software improvements, better accuracy. Analysis in
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INTRODUCTION TO TEXTURE ANALYSIS
the range of one-tenth of a micron then becomes feasible. At the other end of the scale, judicious use of sampling and improved speed in automated measurements will render EBSD suitable to replace X-ray macrotexture determinations in some situations (Section 9.10).
8.1 INTRODUCTION In Chapter 7 we discussed the current techniques to determine local orientations in an SEM. Compared with those methods, in particular EBSD, it turns out that TEMbased techniques can contribute most valuable information in cases where their much higher spatial resolution is a significant factor, such as where the diffraction volume elements are less than about 200nm in size (Case Studies No. S and 9). Furthermore, the TEM yields a wealth of concurrent information on the microstructure of the sample under investigation. e.g. on small precipitates, dislocation substructures and associated measurements, that cannot be obtained in the SE (Case Study No. 7). The present Chapter gives an overview of the techniques that can be used to determine orientations in a TEM. These techniques are: e
High resolution electron inicroscopy (HRE
o
Selected area diffraction (SAD) (Section 8.3)
o
SAD pole figures (Section 8.3.2) Microdiffraction and convergent beam electron diffraction (CBED) (Section 8.4)
Whereas to date high resolution electron microscopy (HREM) is certainly only of peripheral interest for orientation determination, selected area electron diffraction (SAD) has historically been widely used to analyse orientations in a TEM. In contrast to other techniques which yield only orientations of individual crystals, SAD offers the possibility to measure directly pole figures of small volumes in the TEM. However, for the determination of individual orientations the evaluation of Kikuchi patterns (which are obtained by microdiffraction or, if highest spatial resolution is required, by convergent beam electron diffraction (CBED)) is much better suited, as this method combines highest spatial and angular resolution and is easy to perform. For details on the principles of electron diffraction, dynamic diffraction theory and operation of the TEM and also the scanning-transmission electron microscope
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INTRODUCTION TO TEXTURE ANALYSIS
(STEM) (Section 8.4). the reader is referred to standard text books (e.g. Edington 1975, 1976; Wenk, 1976; Hirsch et d.,1977; Heimendahl, 1980; Thomas and Goringe. 1979; Goodhew and Humphreys, 1988; Reimer, 1997; Loretto. 1994; Williams and Carter, 1996). Preparation of specimens for TEM exaniinatio~l involves electropolishing for metallic materials and other procedures such as ion beam milling for non-nietallics. The standard methods used to prepare electron-transparent (i.e. less than about 200 nm thick) specimens which are representative of the bulk material are quite exacting, but they are well-established and documented in several texts (e.g. Goodhew and Humphreys. 1988). Almost always the preparation of TEM specimens is more difficult than preparation of the same materials for EBSD analysis in the SEM.
8.2 HIGH RESOLUTION ELECTRON MICROSCOPY (HREM)
A fascinating technique for determination of local orientations in the very smallest volun~esis the use of high resolution transmission electron niicroscopy (HREM). In this technique the positions of atoms (more precisely. columns of atoms) are imaged by means of an interference method, which enables one to draw directly conclusions on the crystallographic features. HREM is best suited for determination of mis-
Figure 8.1 HREM photograph of a 17", < l o o > grain boundary in gold in which the interfacial structure can be resolved (Courtesy of W. Wunderlich).
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 191
orientations as well as for investigation of grain or phase boundaries. As an example, Figure 8.1 shows the HREM image of a 17"/<100> grain boundary in gold, which depicts subtle details of the grain boundary structure. For routine orientation measurement. however. HKEM is not appropriate (Schwarzer. 1989, 1993) for the following reasons: e
Complicated handling of the microscope to achieve HREM
e
At present the spatial resolution of TEMs is restricted to -0.15 nm, so that only low-index lattice planes can be imaged and analysed
e
Sample preparation is very difficult as HREM requires extremely thin samples (-20 n n ~ )
e
Using so thin samples raises the question of whether the orientations determined are actually representative for the sample volume of interest
e
Interpretation of the results is complicated and requires the use of dedicated software
Although the HREM technique yields impressive images of the ato~nisticstructures, particularly of interfaces. use of this technique for mainstream microtexture purposes is clearly inappropriate.
8.3 SELECTED AREA ELECTRON DIFFRACTION (SAD) The traditional method for orientation determination in the TEM is the use of selected area ielectron) difJkaction (SAD). Although electron microscopes have been in use since 1930, only with the appearance of powerful TEMs with accelerating voltages of l00 kV in about 1950 could thin metal foils be irradiated and their crystallographic orientations analysed (Hirsch et al., 1977). For texture analysis this technique was first utilised by Pitsch et al. (1964) and, simultaneously. by H a e h e r et al. (1964). Nowadays, although SAD is still being used for analysis of local textures, as TEMs with microdiffraction facilities become more prevalent (Section 8.4), SAD has become superseded by the more accurate modern techniques. IHowever, the principles of SAD pattern generation are included here (Section 8.3.1) for completeness and because they form the basis for -FE pole figure measurement (Section 8.3.2). A complete treatise on SAD, as well as its advantages and disadvantages, is given elsewhere (e.g. Edington, 1975; Duggan and Jones, 1977; Humphreys, 1984: Schwarzer: 1989, 1993).
To utilise diffraction of electrons at the crystal lattice for orientation determination, the volume of interest is irradiated with a parallel electron beam in the TEM. In order to achieve a high resolution. the transmitted volume must be very thin (of the order of the mean Gee path of the electrons in the material), so that multiple diffraction effects are avoided. For analysis of small sampled regions the area of view
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INTRODUCTION TO TEXTURE ANALYSIS
is reduced by inserting an appropriate aperture in the plane of the first magnified image which has a magnification of typically x25. Thus, a 50 pm aperture will select an area of size 2 pm in the sample. The diffraction pattern which appears in the back focal plane of the microscope is then focused onto the main viewing screen or a photographic plate. Investigation of diffraction from single crystal volumes yields characteristic patterns which are composed of a regular arrangement of individual diffraction spots. as shown on Figure 8.2, which can be evaluated for orientation determination. The diffraction spots form by coherent elastic scatter of the electrons at the crystal lattice (Figure 8.3a). Because of the very short wavelength of the electron radiation, the diffraction angles are very small as well. As shown in Section 3.6.3, the diffraction angle between the reflecting lattice planes and the primary beam is at the most about 2".This means that the reflecting planes all are situated almost parallel to the primary beam or, in other words, the primary beam is a zone axis of the reflecting planes (Figure 8.3b). Owing to the very small Bragg angles sin H H, Bragg's law (equation 3.4) can be written as:
-
where L is the distance between sample and photographic plate and Rhklis the distance between primary beam and diffraction spot as shown on Figure 8.3a. With 12 = 1 it follows:
Figure 8.2 SAD pattern obtained i n a n A1-1.3%Mn single crystal with a zone axis near <110>
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 193
incident electron beam
primary beam (zone axis) \ V
sample
reflecting lattice planes planes of a
Figure 8.3 Schematic diagrams showing the formation of SAD patterns. (a) Formation of a diffraction spot. (b) Primary beam as a zone axis for several diffracting planes. (c) Indexed diffraction pattern (as in Figure 8.2) of a [l011 zone axis pattern.
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INTRODUCTION TO TEXTURE ANALYSIS
The expression XL which describes the magnification of the diffraction patterns is constant for monochromatic radiation which is approximately the case in the TEM and is called the cunlera constant. As an example, for a TEM operating at 200 kV (which results in a wavelength X = 0.00251 nm, equation 3.12) and a camera length of 250 mm, the camera constant is XL = 0.6275 mm X nm. For a general description of the diffraction effects during SAD the construction of the reciprocal Iuttice is used. Briefly, the distance between a diffraction spot in the diffraction pattern and the primary beam, i.e. the length of the vector is inversely proportional to the lattice spacing d (equation 8.2). Rhklis perpendicular to the reflecting lattice planes, the position of which can be assumed to lie in the origin of the diffraction pattern. These two conditions correspond to the definition of a point in the reciprocal lattice (except for a scaling factor). It follows that each diffraction spot represents the reciprocal lattice point of the corresponding set of lattice planes {lzkl).Thus the diffraction patterns can be interpreted as intersections between the points in the reciprocal lattice and the Ewald sphere or rejlection splzere (Figure 8.3a). As samples which can be trailsmitted by electrons in the TEM must be very thin, the points in the reciprocal lattice are elongated parallel to the foil normal and get an ellipsoidal shape in reciprocal space. Furthermore, as the radius of the reflection sphere, 1/X, is by about 2 orders of magnitude larger than the inverse lattice constant lid, the plane of intersection with the reflection sphere is almost flat. Thus, the ellipsoids or 'spikes' around the points in the reciprocal lattice can intersect with the reflection sphere even when the Bragg condition for diffraction is not exactly met. Therefore, several planes that belong to a given zone axis can contribute simultaneously to the diffraction pattern. as illustrated in Figure 8.3b, c, whereas under the strict application of Bragg's law only one individual diffraction spot would be expected. The concept of the reflection sphere and reciprocal lattice are explained in detail in the standard texts listed in Section 8.1. Unfortunately, the spatial resolution of SAD cannot be improved to any extent just by using a smaller aperture, as with a further reduction of the aperture size defocusing and lens errors cause the problem that the TEM image formed by the primary beam and the SAD pattern formed by the diffracted beam do not coincide in the image plane (Riecke, 1961; Ryder and Pitsch, 1967, 1968; Edington, 1975). In older TEMs this displacement error limits the application of SAD to an area of about l pm. In modern n~icroscopeslens and defocusing errors are smaller. and in particular the higher accelerating voltage (200 kV and more) results in smaller diffraction angles, so that the displacement error is smaller. Nowadays, the availability of sufficiently small apertures (e.g. 5-10 pm) limits the spatial resolution to about -0.2pm (e.g. Schwarzer, 1989, 1993; Willianx and Carter. 1996). Indexing SAD patterns, which is often carried out by comparison with known patterns, is a standard TEM technique and is described elsewhere (e.g. Edington, 1975). Analysis of more complicated patterns, e.g. with overlapping zone axes or of unknown crystal structures, is more difficult, but can nowadays be performed with the help of dedicated computer software (e.g. Stadelmann. 1987). However, SAD patterns comprise a two-fold symmetry, so that an orientation cannot be distinguished from the equivalent one formed by a rotation of 180" around the primary beam. Hence if exact orientation determination is required (e.g. for orientation -
-
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 195
relations) diffraction spots of more than one zone axis must be analysed (Ryder and Pitsch, 1967) or. preferably, a microdiffraction technique should be used (Section 8.4). Whereas rotations around the direction of the primary beam can be detected quite accurately, the diffraction patterns are much less sensitive to rotations around other, non-parallel rotation axes (Laird et al., 1966; Ryder and Pitsch, 1968). This means that the accuracy of orientation determination strongly depends on the given orientation itself, which in turn feeds through as an error for orientation determination by SAD of 5"-10". This value can substantially be improved by considering the relative intensity of the diffraction spots in addition to their position in the diffraction patterns (Laird et ul.. 1966). Analysis of several zone axes also improves the accuracy (Ryder and Pitsch, 1967. 1968). These methods, however. are time consuming and not well suited for routine applications. Therefore, SAD patterns should not nowadays be used for microtexture analysis, since interpretation of Kikuchi patterns (Sections 6.3, 8.4) offers analysis of individual orientations with much higher spatial and angular resolution. 8.3.2 Selected area diffraction pole figures When recording diffraction patterns of polycrystalline sample regions with SAD, concentric ring patterns arise which resemble the Debye-Scherrer diagrams known from X-ray analysis (Section 3.4). Each of these rings f o r m by accumulating the hkl reflections from a multitude of differently oriented (hkl) planes in the sample. and the ring diameters are related through 48 to the Bragg angle of the corresponding lzkl reflection (Figure 8.4a). As an example. Figmes 8.4b. c show the SAD diffraction patterns obtained in two aluminium samples, depending on the texture. Whereas a random texture yields uniform intensities along the Debye-Scherrer rings (Figure 8.4b); in the case of a pronounced texture as in the case of a sample deformed by rolling the intensities are very irregularly distributed (Figure 8 . 4 ~ ) Thus, . the intensity distribution along the individual Debye-Scherrer rings is a measure for the alignment of the corresponding lattice planes {Izkl): which can be exploited for determination of the local texture of the irradiated volume. Such techniques for measuring local pole figures have been developed independently by Schwarzer (1 983) and Humphreys (1983). For experimental details see also (Schwarzer and Weiland, 1986; Weiland and Panchanadeeswaran, 1993). The diffraction rings only reveal the spread of orientations around the incident beam, and hence one diffraction pattern does not comprise the inforn~ationon the entire texture of the sampled volume. Since the Bragg angle of electron diffraction is usually below l". under perpendicular irradiation of the sample the reflecting planes are all arranged approximately parallel to the incident beam (see Figure 8.3b). so that the measured points fall on the outer circle (a = 90") of the pole figure (Figure 8.5a). Thus, to obtain more coverage of the pole figure, the sample has to be tilted similarly to conventional X-ray texture analysis (Section 4.3.3). Tilting of the sample through angle Aa in both directions around an axis perpendicular to the incident beam allows one to record great circles that are situated in the interior of the pole figure as shown on Figure 8.5b and c. -
-
INTRODUCTION TO TEXTURE ANALYSIS incident electron beam
Figure 8.4 (a) Schematic illustration of formation of SAD ring patterns in polycrpstalline assemblies: (b) SAD diffraction pattern of evaporated aluminium with random texture; (c) SAD diffraction pattern of cold rolled aluminium with strong texture (Courtes~of H. Weiland).
I electron beam
Figure 8.5 (a) Formation of SAD pole figures in a TEM; (b and c) coverage of the pole figure as the angle a is gradually increased.
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 197
In order to derive the orientation distribution, for each pole figure point the intensity along the corresponding Debye-Scherrer ring has to be recorded. For that purpose, the diffraction pattern is deflected by the post-specimen deflection coils on a circle such that the selected Debye-Scherrer ring passes across a suitable detector, e.g. a Faraday cage. in the final image plane. Alternatively. the primary beam is deflected along a cone of apex angle 20 pointing at the sample volume. In that case, the diffracted beam whose intensity has to be recorded is parallel to the optic axis of the microscope, which increases the lateral resolution of the technique. In the latter mode the reflected intensities along the Debye-Scherrer rings can also be obtained from the current of the dark-field image on the final image screen. With an appropriate computer control of the TEM for sample tilting, scanning of the DebpeScherrer rings and intensity recording, it is possible to determine the local pole figures of very small regions of the microstructures. A major drawback of the quantitative evaluation of these SAD pole figures is caused by the fact that the intensity distribution along the Debye-Scherrer rings is not directly proportional to the pole density in the pole figures, which requires relatively complicated additional experimental or analytical corrections. including background intensity and increase in diffracting volume and in absorption with increasing sample tilt (Schwarzer, 1983; Humphreys, 1984; Schwarzer, 1989, 1993). With an appropriate computer control, a pole figure scan can typically be performed within about half an hour, and several pole figures can be acquired simultaneously. However, this time exceeds the long-term stability of a TEM. Hence, to reduce the recording time: it has been proposed to record simultaneously several complete hkl rings for a given sample tilt a by using a CCD camera as an area detector (Schwarzer et al., 1997b). The diffraction patterns are stored in a computer and, after the scan, the intensity distributions along the rings can be retrieved either interactively or automatically with the help of suitable image processing algorithms. This parallel recording technique drastically reduces the measuring time in the TEM to about 1 minute. In most TEMs the tilting angle is limited to a range of 0" to approximately 160" so that only incomplete pole figures with a large non-measured 'white' region in the pole figure centre are obtained (Figure 8 . 5 ~ )This . white region can be reduced by a second scan where the sample is tilted around an axis perpendicular to the first one. To receive complete information about the texture, several incomplete pole figures can be determined and, similarly to X-ray texture analysis (Section 5.3), from these an O D F is computed (e.g. Weiland and Panchanadeeswaran, 1993; Helming and Schwarzer, 1994). If required, complete pole figures can then be recalculated. The minimum measured volume that can be analysed by this technique is given by the spatial resolution of the SAD technique, i.e. SAD pole figures can be derived from regions of about 1 pm in diameter. In practice, however, mostly larger regions of several microns up to say 20 pm will be chosen to determine local textures of characteristic microstructural regions. Applications include the microtextures in the deformation zones around particles (Humphreys, 1977) or within deformation or shear bands (Schwarzer, 1988).
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INTRODUCTION TO TEXTURE ANALYSIS
8.4 MICRODIFFRACTION A N D CONVERGENT BEAM ELECTRON DIFFRACTION (CBED) When the diffraction effects in a TEM are analysed in a sample with increasing thickness. one observes that the SAD diffraction spots increasingly attenuate, and instead a system of bright and dark lines, the Kikuchi lines (Kikuchi, 1928), will form on the main viewing screen (Figure 8.6a. b). Since similar patterns also form in the SEM (SAC and EBSD, Chapter 7) the formation principles as well as the crystallographic information embodied in the Kikuchi patterns have already been described generically in Chapter 6. To recapitulate briefly. an electron beam entering a crystalline solid is subject to elastic andlor diffuse scattering. The elastic scattering gives rise to distinct diffraction spots in the back focal plane, which can be utilised for SAD (Section 8.3.1). In case of diffilse scattering the atomic planes of a crystalline specimen are showered by electrons arriving from all directions, and these electrons can then undergo elastic (Bragg) scattering to give a strong, reinforced beam which eventually gives rise to the Kikuchi lines (Figure 8.6 and Figure 6.la). Thus, the Kikuchi pattern is directly related to the orientation of the reflecting planes in the specimen. Accordingly, when the crystal is tilted by a small angle, the entire Kikuchi pattern rotates in a manner which is linked to the crystal rotation and, therefore, Kikuchi patterns are perfectly suited for crystal orientation determination (Heimendahl et al., 1964). To illustrate the strong sensitivity of the Kikuchi patterns to the crystal orientation, the diffraction pattern of the same crystal as Figure 8.6a is shown in Figure 8.6b after rotation by only 2". SAD diffraction spots, in contrast, will not change their position and their intensity will alter only slightly before falling to zero intensity as the crystal is tilted so that the reflection sphere no longer cuts the corresponding spikes around the reciprocal lattice points (Section 8.3.1). For this reason, for the determination in the TEM of the local orientation of small single
(a) untilted
(b) tilted by 2"
Figure 8.6 Diffraction patterns obtained from a oriented copper crystal: (a) untilted. i.e. exact < 100> orientation; (b) tilted by 2' to show the much greater sensitivity of Kikuchi lines than SAD spots to crystal orientation (Courtesy of M. Verdier).
TRANSMISSION ELECTRON MTCROSCOPY (TEM)-BASED TECHNIQUES 199
crystalline regions the interpretation of Kikuchi patterns is superior to SAD analysis, as the former technique combines high accuracy. highest spatial resolution and ease of performance. It was mentioned above that the angular distribution of electrons. which are eventually scattered elastically according to Bragg's law to form the Kikuchi patterns, are provided bp inelastic scattering (e.g. Figure 8.7a). Alternatively, electrons that encounter the lattice planes at angles less than the Bragg angle can also be scattered by tilting the incident electron beam until it hits the Bragg angle (Figure 8.7b). Although in such cases (rocking beam microdiffraction or convergent beam electron diffraction: see below) the exact mechanisms of formation of the diffraction patterns are slightly different. they yield the same crystallographic information and, accordingly, they can be interpreted by the same principles as those described in Chapter 6. When Kikuchi patterns are created under parallel irradiation of the sample as in the SAD operation mode, the spatial resolution is limited to about 1 pm as discussed for the SAD spot patterns (Section 8.3.1). This value is surely sufficient to analyse the orientations of individual grains in most metallic samples. but such inquiries can be addressed much more easily by EBSD in a SEM (Sections 7.4-7.8). For other applications. e.g. for analysis of the orientation or orientation relationships of small precipitates. or orientation gradients within a grain. much finer resolution is required. In particular for the analysis of deformed samples the spatial resolution must be sufficiently small so as to be able to access the undistorted volume elements necessary for formation of Kikuchi patterns. In order to overcome these limitations several techniques have been developed to achieve Kikuchi patterns of small regions of the order of nanoinetres. These techniques are con~monlysummarised as parallel electron beam
I
(a) parallel illumination, thick sample
(b) convergent illumination, thin sample
Figure 8.7 Formation of Kikuchi patterns in a TEM for (a) parallel beam, thick sample; (b) convergent beam. thin sample.
200
INTRODUCTION TO TEXTURE ANALYSIS
n~icrodilffmtion (Warren, 1979; Williams and Carter, 1996) and will briefly be introduced in the following. Ehc14sed aperture microdiffraction uses the same objective lens as that used for scanning-trans~nissio~l mode (STEM), but images the sample in the same manner as in the conventional SAD mode. For the former, however, a strongly demagnifyiilg upper objective lens allows a small (< 10 pm) condenser aperture to be imaged on the sample plane itself, such that only a small portion of the sample is illuminated and it is this small area alone which forms the diffraction pattern (Riecke, 1962). Thus. the role of spherical aberration which limits the spatial resolution in SAD is confined to a slight distortion of the diffraction pattern. This technique permits a quick and effective way to produce microdiffraction patterns froni areas down to 50 nm in diameter. In rocking beam microdiffraction the electron beam is rocked over the sample by the STEM scanning coils (van Oostrum et al., 1973; Geiss, 1975). The diffraction pattern is created by the varying intensity that falls in the STEM detector at the base of the microscope as the beam changes its angle of incidence to the sample. and is recorded on the STEM display screen. With this technique diffraction patterns from crystals smaller than 1Onm have been obtained. It should be mentioned that the rocking beam technique can also be utilised to produce transmission channeling patterns (e.g. Fujimoto et al., 1972), similarly as described for the SEM (Section 7.3). In this case, however, the spatial resolution is again controlled by the spherical aberration of the objective lens, i.e. it is limited to about 1 pm (Warren, 1979). By far the most widely used technique to produce Kikuchi patterns from small volumes is,fijcused beam microdiffraction. Best spatial and angular resolution can be obtained not by using the parallel irradiation that is necessary for SAD, but by using a convergent beam. For this purpose, the electron beam in the TEM is focused by additional coils or by a scanning equipment (STEM) to converge at the region of interest. Since the diffraction pattern arises only from the illuminated volume: the spherical aberration which results froni the aperture selecting only a portion of an illuminated area - does not interfere here. Hence, the spatial resolution is only limited by the spot size of the probe, which can readily be focused down to a few nanometres (Ralph and Ecob, 1984), and even subnanometre analysis is possible (Williams and Carter, 1996). With increasing convergence of the beam, the diffraction spots widen to discs with a diameter that is controlled by the convergence angle. In the resulting diffraction patterns sharp Kikuchi lines are almost invariably seen (Figure 8.8), which is mainly caused by the very small sampled volume. So, in the volume contributing to the diffraction pattern there is usually little or no strain, even when analysing highly deformed materials (e.g. Heilmann et al., 1983). Furthermore, owing to the increased intensity, and to the convergence of the beam itself, Kikuchi patterns form even in very thin sample regions. Within the discs, additional diffraction features, the higher-order Laue zones (HOLZ) patterns, may arise. These HOLZ patterns are enhanced by further increasing the beam convergence, typically to about 0.5", which gives rise to the term convergent beam electron diffraction (CBED) (Steeds, 1979). Figure 8.8 illustrates the effect of exciting HOLZ reflections (which strongly resemble the Kikuchi lines though their formation mechanism is slightly different). Although the HOLZ lines are not -
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 201
Figure 8.8 CBED microciiffraction pattern obtained from a < l l l > oriented cryrtal in Nimonic P E 1 6 ,horn-ing Kikuchi lines and HOLZ.
directly used in microtexture measurement; they give a wealth of allied microstructural or crystallographic information including specimen thickness (Allen and Hall. 1982). unit cell and precise lattice parameters (Ayer, 1989; Randle et ai., 1989), symmetry changes (Ecob et al.,1981). and point grouplspace groups (Steeds, 1979; Tanaka et al.,1983; Steeds and Vincent, 1983). In standard experimental set-ups, the Kikuchi pattern. recorded on a photographic film. can finally be indexed to determine the crystallographic orientation of the corresponding crystal (Section 6.3). Of course this method is very time consuming and not very well suited to on-line measurements. Rather, similarly as described for the EBSD patterns in Section 7.6.1, for on-line orientation determination the Kikuchi patterns can be captured with a low-light CCD camera and then transferred through a video grabber card to a computer for storage and orientation determination. The camera can be attached outside the TEM and can
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INTRODUCTION TO TEXTURE ANALYSIS
record the pattern either from the (tilted) main viewing screen (Hoier et al., 1994) or from an additional screen that has to be moved into the electron beam (Zaefferer and Schwarzer, 1994a). In another set-up. the chip of a Peltier-cooled slow scan CCD camera is inserted into the electron beam slightly above the main viewing screen of the TEM (Krieger Lassen, 1995; Eilgler el al., 1996b), which has the advantage that the recorded Kikuchi pattern is undistorted. Alternatively, the camera is mounted inside the TEM below the main viewing screen (Weiland and Field. 1994). Whereas SAC and EBSD Kikuchi patterns normally have quite a uniform contrast with a rather smooth, low background, the contrast range of TEM Kikuchi patterns is much larger. Furthermore, the Kikuchi patterns are frequently superimposed by high intensity SAD diffraction spots and, particularly, by the primary beam. whose intensity may exceed that of the pattern by several orders of magnitude. In order to handle the enlarged dynamic range of TEM Kikuchi patterns, cameras and frame grabber cards with a finer digitisation are advantageous (Krieger Lassen, 1995). For instance. 12 or even 14 bit digitisation respectively yields more than 4000 or even 16,000 grey values as opposed to the 256 grey values of a standard 8-bit camera, but such high performance systems are still very expensive today. As an alternative. it has been proposed to insert a circular gradation density filter in front of the camera lens to level out the uneven background intensity (Zaefferer and Schwarzer, 1994a). This filter which is a photograph of an 'empty' pattern containing only background intensity, has been reported to reduce the intensity of the primary beam by a factor of 100 (Schwarzer et al., 1997b).
11 this Chapter, the technques to determine local orientations in a TE described. The main advantages of TE -based microtexture techniques are:
e
Excellent spatial resolution. down to the subnanometre range
e
Fine detail in the Kikuchi pattern facilitates analysis from complex crystal structures
e
Wealth of additional microstructural and crystallographic information available, e.g. on precipitates or dislocations
e
No requirement for complex calibration routines
The main clisadvantages of TEM-based microtexture techniques are: e~
Demanding and laborious specimen preparation
e
Very small area of view in a single specimen, therefore low statistics Difficult to relate viewing area to the macroscopic specimen geometry
e
Less potential for automation (Section 6.3.4) and orientation microscopy (Section 10.1)
TRANSMISSION ELECTRON MICROSCOPY (TEM)-BASED TECHNIQUES 203
It is fortuitous that microtexture analysis in the TEM has its most important applications in cases where EBSD fails, which include heavily deformed materials, nanocrystals or complex crystal structures. Considering the more common techniques for orientation determination in the TEM. best results in terms of spatial resolution and accuracy are achieved by the interpretation of Kikuchi patterns obtained by microdiffraction or, if highest spatial resolution is required, by convergent beam electron diffraction (CBED). In contrast, selected area electron diffraction (SAD) should not nowadays be used for orientation determination, as this technique suffers from much poorer spatial resolution (N 1 pm) and, particularly, since errors of as much as 5-10" may frequently occur. However, SAD is still of great interest as this technique offers the unique possibility to directly measure pole figures of small polycrystalline volumes, as e.g. shear bands or deformation zones. There are a vast number of examples in the literature spanning several decades of applications of TEM techniques to orientation determination. Some examples of recent applications, which highlight the exploitation of TEM capabilities and how these can be supplemented by other techniques, are shown in Case Studies 7 , 8 , and 10.
9. EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
9.1
INTRODUCTION
A microtexture is, essentially, a population of individual orientations which can be linked to their location in a specimen and are obtained usually by an electron diffraction technique as described in Chapters 7 and 8. These two factors - the discrete nature of the orientation data and their direct link to the microstructure introduce a much broader and more complex scenario for data evaluation and representation than that for macrotextures, since in the latter case the orientation data evaluation usually involves only direct representation in the form of pole figures and subsequently calculation of an ODF (Chapter 5). The large variety of situations which can be addressed by means of single grain orientation measurements requires many more formats for the representation and interpretation of the measurements. The nature of microtexture gives rise to two fundamental lines of inquiry: Statistical distribution of orientation and misorientation data Orientation and inisorientation data related to the microstructure For most investigations both of these analysis steps will be required, and the data will be selected from relatively small regions of interest. There are two ways in which orientation and misorientation microtexture data can be displayed in a space: as discrete data points or as a continuous density distribution. For simplicity in the case of the latter we will use the terminology ‘orientation (or misorientation) distribution ,fiiizctiorz’ for a11 continuous distributions, without distinguishing if the function applied involves the series expansion method (as for macrotexture data) or the more simple case of a direct contouring function. This has the advantage of allowing generic terminology to be used for both macrotexture and all continuous microtexture data. Some texts, however, reserve use of the term ‘(mis)orientation distribution function, (M)ODF, in microtexture for series expansion methods only, and refer to direct contouring as a ‘(1nis)orientation distribution’. In this Chapter it is helpful to consider data processing for orientations and misorientations separately, since the different physical applications of the measurements 205
INTRODUCTION TO TEXTURE ANALYSIS
in both cases leads to a dichotomy in approach. For example, the anglejaxis descriptor (Section 2.6) has direct relevance to misorientations at interfaces whereas it is inconvenient for orientation descriptions. Hence, Sections 9.2-9.4 and 9.5-9.8 deal with orientation and misorientation representation respectively. For further literature the reader is referred to Randle (1992), Gerth and Schwarzer (1993) and Engler et al. (1994~).
9.1.1 Statistical distribution of orientation and misorientation data For this type of microtexture representation the chief aim is only to represent and quantify the orientation distribution in a region of interest in the microstructure, either by sampling every orientation present or by statistical sampling throughout a large region of the specimen. If only a single orientation is sampled at each, noncontiguous, region then there is no access to misorientation or microstructure information in fact this is equivalent to an 'average' texture, i.e. the microtexture counterpart to a macrotexture. Sometimes, since automated EBSD is very rapid and is able to measure directly the 'true' grain orientation, the statistical analysis mode is used advantageously in place of a traditional macrotexture technique to generate a measure of the overall texture. However it is of prime importance to consider both the size and the distr.iDotion of the sample population in order to obtain a statistically reliable result. The number of grains encountered is usually rnuch srnaller than the number of grains contributing to X-ray or neutron pole figure measurements. Assuming a sampling area of 10 X 10mm2 and an average penetration depth of about 0.05mm (Section 4.3), an X-ray macrotexture of a sample with an average grain size of 25 pm represents the orientation information of about 300,000 grains. In the case of neutron diffraction this number is even higher by several orders of magnitude, which is due to the much higher penetration depth (Sections 3.5, 4.4). In contrast; the number of grains sampled by EBSD typically ranges between 100 and 1000 for semiautomated EBSD and even in the case of fully automated EBSD scans usually less than 100,000 grains are recorded (e.g. Table 10.1). The success of using EBSD for overall texture estimation; therefore, depends on -
s The ability to assess how inany orientation measurements are required 0
An appropriate sampling schedule
Turning now to general evaluation of statistical microtexture data in the form of either orientations or misorientations, it is usual to display first the population using one of the methods described in Chapter 2. Secondary processing steps can then be applied to extract quantified information appropriate to the investigation, e.g. what proportion of specified orientations within a certain angular tolerance are present, either by number or by projected grain area (Section 9.2.2). Many of these processing steps are available as part of commercial software packages or alternatively the user can design customised options. Some of the representation options are those traditionally used for macrotexture, e.g. the pole figure/inverse pole figure (Section 2.4) and Euler space (Section 2.5), whereas other options such as Rodrigues space
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
207
(Section 2.7) and misorientation evaluation (Section 2.6) derive directly from microtexture data. However, the fundamental differences between data collection for microtexture (Chapters 7, 8 and 10) and macrotexture (Chapters 4 and 5 ) feed through to necessitate different approaches for data handling in both cases. Although a 'raw' microtexture population comprises discrete individual orientatlons, which are usually displayed as such. when the number of data points is very large (which is often the case for mapping, Chapter 10) andior the texture is very sharp. it becomes inconvenient to display the data - in any space individually because the representation will soon become unacceptably crowded and lucidity is lost. Instead the data can be shown as density distribution contours (Sections 9.2.2. and 9.3.2). -
9.1.2 Orientation and misorientation data related to the microstructure The most valuable aspect of microtexture is the direct link that it provides between orientation and microstructure, as shown in the model example in Chapter 1 (Figure 1.1) and in real examples in the Case Studies. Hence consideration of the statistical orientation distribution of the sampled region(s) (or subsets of them) as described in Section 9.1.1 is usually only part of the total microtexture investigation. Typically, orientation data collection is accompanied by a record of sampling coordinates in the microstructure, and the location of these individual data points can be marked on the representation, which is usually a pole figure. A similar strategy has traditionally been common practice in TEM. Clearly. this method is only viable for relatively small data subsets since the correlation between measuring site and orientation is lost if the distribution has to be expressed as density contours. The ultimate refinement of orientation/location correlation is the orientation map where each pixel in the map has both orientation and location associated with it. Another asset of microtexture is the access it provides to interfacial parameters, orientation relationships and orientation connectivity in the microstructure. This area gives rise to a further range of data processing and representation options (Section 1.2.1). Even though the representation spaces are the same for orientations and misorientations: they are utilised differently in both cases in order to highlight the required physical features in the data. Further processing of the crystallography of interfaces, and also aspects of microtexture which encompass the way in which the orientation components are distributed in space, i.e. orientation connectivity, are the subjects of Chapter l l .
9.2 REPRESENTATION O F ORIENTATIONS I N A POLE FIGURE O R INVERSE POLE FIGURE The most obvious method of representing a nlicrotexture is in a pole figure (Section 2.4). This is convenient since oilly a single graphic is required to embody the texture information. Local orientation data can be marked in the pole figure (or inverse pole figure) with different symbols or numbers with respect to their location in the microstructure, and the corresponding symbols/nuinbers are marked on a
208
INTRODUCTION TO TEXTURE ANALYSIS
depiction of the microstructure (Section 9.2.1). This method allows a direct correlation of orientation and site in the microstructure but, as mentioned in Section 9.1.2, its application is limited to a quite small number of individual points. For the representation of larger numbers of orientations with statistical relevance in a pole figure, a continuous density distribution is superior (Section 9.2.2), though in this case the direct correlation of orientation and site in the microstructure is lost.
9.2.1 Individual orientations An example of pole figure/microstructure correspondence for individual orientations is illustrated in Figure 9.1, which shows a micrograph of several deformed grains in cold rolled binary A1-3%Mg (Figure 9.la) and an accompanying pole figure (Figure 9.lb), where the orientations of these grains were determined by EBSD. In Figure 9.lb the orientations are further distinguished whether or not the corresponding grains developed shear bands during deformation. Orientations of grains which developed shear bands during deformation are marked with filled symbols and those of homogeneously deforming grains with open symbols (Engler et al., 1992). Inverse pole figures are often used for axial symmetric samples, where only one of the sample axes is defined (Section 2.4.2). For instance; for compression or tensile samples the orientation changes of the compression/tensile axis are plotted in the crystal coordinate system. According to the crystal symmetry it is not necessary to show the entire inverse pole figure, but one stereographic triangle usually suffices. In the case of cubic crystal symmetry the well known stereographic triangle . < l 10> and is shown (Case Study No. 6). Figure 9.2 is an example, showing the compression axis inverse pole figure of the intermetallic compound NiAl which has been deformed at 900°C (Figure 9.2b). EBSD single grain orientation measurements close to a grain boundary depict reinarkable orientation changes as
(a) microstructure
(b) orientations of a)
Figure 9.1 Example of the presentation of microtexture data in a pole figure. showing the influence of the matrix orlentation on shear band formation in polycrystalline A1&3%Mg, 90% cold rolled. Filled symbols grains with shear bands; open symbols - grains without shear bands.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
209
Figure 9.2 (a) Microstructure of NiAl deformed at 900°C by 70% in compression, showing a subgrain structure in the vicinity of a grain bouudary marked GB (CD: compression direction). (b) Inverse pole figure of the subgrain orientatious 1-7 shown in (a). The X-ray macrotexture of the sample mainly showed intensities close to < I l l > . as indicated by the iso-intensity lines. EBSD single grain orientation measurements of subgrains 1-7 close to the GB depicted remarkable orientation changes. (c) Pole figure of the subgrain orientations 1-7 showing that the orientations of subgrains 6 and 7 were quite different (Courtesy of L. Lochte aud J. Fischer-Biihner).
shown from the position of the subgrain orientations 1-6 in the inverse pole figure, whereas the macrotexture of that sample showed a pronounced
210
INTRODUCTION TO TEXTURE ANALYSIS
systems (Raabe et d.,1997a). According to Figure 9.2b the orientations of the compression axes of subgrains 6 and 7 are very similar, but note that this cannot lead to the assumption that these subgrains have nearly the same orientation. This is because the inverse pole figure only shows the orientation of one reference axis h e r e the compression axis - but rotations about this axis are not considered. In fact the full grain orientation of subgrains 6 and 7 were quite different as becomes obvious when plotting the same orientations in a pole figure (Figure 9 . 2 ~ ) .Hence. for a complete representation of the three-dimensional orientation information two inverse pole figures showing two of the three different reference axes are required. (The third reference axis is redundant because it is defined by a cross-product of the first two.)
9.2.2 Density distributions In order either to get more quantitative information on the relative density of microtexture data andior to relieve a congestion of individual data points, it is beneficial to represent the density distribution from single grain orientation data as density levels in the pole figure. Then: however, the direct correlation of orientation and site is lost. Density distributions resulting from a given set of single orientations can readily be determined by subdividing the pole figure into a grid with distinct angular cells, e.g. a X ,3 = 5" X 5". For each orientation g (where g denotes the rotation matrix between the sample and the crystal coordinate system, Section 2.2.2) the pole figure angles ai and ,ai(Section 2.4) for each pole (h,, ki. li) (e.g. (1 1 l), ( l i l ) , etc.) are given by:
in equation In the case of an inverse pole figure the inverse orientation matrix 9.1 is replaced by the orientation matrix g , and the vector (Izikili) is replaced by a vector (XsYsZs) of the sample coordinate system, e.g. RD = (100). = (001) (Section 2.2.1). If only one inverse pole figure is presented this means that just one axis is considered, i.e. i = 1. The angles a and 3 now define the position of the orientation g in the inverse pole figure. After calculation of the angles ai and D,, the value of the corresponding cell (oi, Oi) is incremented by one. Finally, the data are normalised with the total number of orientations 1V and the number of poles i and can then be plotted in a pole figure or inverse pole figure. As an example, Figure 9.3 shows micro- and macrotexture results of a partially recrystallised Al-1.8wt%Cu single crystal (Engler et al., 1993b). In Figure 9.3a some orientations as measured by EBSD in the deformed and in the recrystallised regions are plotted in the stereographic projection. Figure 9.3b shows the density distribution calculated from about 500 individual orientations as described before. For comparison, in Figure 9.3c the X-ray macrotexture of the sample is presented.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
21 1
A con~parisonof microtexture (Figure 9.3b) and macrotexture (Figure 9 . 3 ~ shows ) very good correspondence in the positions of the main peaks. A quantitative comparison of the orientation density, however, is not possible. as already evident from the different heights of the texture maxima (32.6 in Figure 9.3b compared to 10.9 in Figure 9 . 3 ~ ) This . difference in intensity distribution is caused by the different measuring procedures applied: X-ray pole figures are textzwes b j volumc (Bleck and Bunge, 198 1): which means that each orientation contributes to the texture according to its volume. By contrast during the EBSD single grain orientation measmements
Figure 9.3 Comparison of EBSD microtexture and X-ray macrotexture pole figures of a partially recrystallised AI-1.8%Cu single crystal (rolled to 80% reduction, annealed for 300s at 300°C). (a) Examples of EBSD single grain orientation measurements in the as-deformed matrix and the new recrystallised grains: (b) continuous density distribution as computed from about 500 orientations similar as in (a): (c) X-ray macrotexture of the same sample.
212
INTRODUCTION TO TEXTURE ANALYSIS
the orientation of large recrystallised grains was measured only once, yielding a texture by number. Therefore, in comparison to the X-ray texture the intensity of these large grains is underestimated. Of course, it is entirely possible to measure textures by volume by single grain orientation measurements, e.g. by correction of the data with respect to the corresponding grain size. Similarly, orientation mapping as applied during orientation microscopy (Section 10.3) directly yields textures by volume, as in this case the number of measurements per grain is a measure for the volume or, more precisely, the projected area of the respective grain. In most cases, however, it is not the main purpose of single grain orientation measurements to determine overall textures. but single grain orientation measurements are predominantly performed to obtain information on the local arrangement of orientations; in the present case on the orientations of the deformed matrix and the newly forming recrystallised grains. If the number of orientations is smaller or the orientation scatter is larger than in the example in Figure 9.3 the pole figures may look rather unsystematic or 'spiky'. In that case, it might be appropriate to represent only the discrete data or to smooth the pole figure data by convolution of the intensity data in the pole figures with a suitable filter. An appropriate two-dimensional filter with an approximate Gaussshape is shown in Figure 9.4a. Application of this filter would consider the intensity of the current pole figure position cu and 8 with 33%, the intensities of direct neighbouring cells with 12'34, and that of second-next neighbours with 4.5%. These
Figure 9.4 Smoothing of pole figure data. (a) Two-dimensional filter with approximate Gauss-shape t o smooth pole figures; (b) pole figure as derived from pole figure data as in Figure 9 . 3 ~but smoothed with the filter in (a).
EVALUATION AND REPRESENTATION O F MICROTEXTURE DATA
21 3
values do not change the integral intensity (i.e. the normalisation) of the pole figure, but in general they reduce the maximum intensity. Figure 9.4b shows the pole figure of Figure 9 . 3 after ~ smoothing according to this procedure. Another way to derive continuous pole figure or inverse pole figure plots from single grain orientation data which is sometimes applied is to calculate an ODF (Section 9.3.2) and then to recalculate the desired (inverse) pole figure (Section 5.3.1). Provided the series expansion method is applied, the pole figure data are automatically smoothed.
9.3 REPRESENTATION O F ORIENTATIONS I N EULER SPACE Since single grain orientations already contain the entire orientation information in an unequivocal manner, no further information is yielded if a microtexture is plotted in a three-dimensional space such as Euler space or Rodrigues space than if it is plotted in a pole figure. However, representation in a three-dimensional orientation space could be advantageous for complex data sets, and also if direct comparison to X-ray macrotexture data, where the O D F in Euler space is the traditional representation method, is required (Case Study No. 9). This is discussed in Sections 9.3.1 and 9.3.2, and in particular direct and indirect methods to derive a continuous ODF from single grain orientations will be compared. Despite the large number of single grain orientation data which nowadays can be measured by techniques for individual grain orientation measurements, it is still usually small compared to the number of grains that are recorded by X-ray or, particularly, by neutron pole figure measurements. Therefore, in Section 9.3.3 the minimum number of single grain orientation measurements necessary to represent adequately a texture in Euler space will be discussed.
9.3.1 Individual orientations As discussed in Section 2.5, a given orientation can be assigned a point with the coordinates (p1, Q, p2) in the three-dimensional Euler angle space. Thus, a set of orientations determined by single grain orientation measurements can be expressed in terms of the Euler angles and, after appropriate reduction with regard to sample and crystal symmetry (Section 2.5.2), displayed as points or with appropriate symbols in the Euler space. Similarly as discussed for the representation of macrotextures (Section 5.4), microtextures are likewise usually represented as a set of twodimensional sections along cpl or 972 through the Euler space. Orientations which do not fall exactly in one of the sections - i.e. virtually all orientations are plotted into the adjacent section. The resulting error can slightly be reduced by projecting the orientations under consideration of the condition p1 + p 2 = constant (which in this strict form only applies for Q, = 0). Another problem relates to the fact that usually only a subspace of the full Euler space is used for presentation of orientations. For cubic crystals and orthotropic sample symmetry this is 0" 5 (p1, Q, p2) < 90" (1132 of the full Euler space) which still does not take the 3-fold axis of cubic crystal symmetry into account (Section -
214
INTRODUCTION TO TEXTURE ANALYSIS
2.5.2). Hence, each individual orientation has to appear three times in this subspace, but the single grain orientation measurement of course provides only one variant. The two missing variants can readily be generated by successive 12O0<111> rotations from the original orientation. Figure 9.5a shows an example of the orientations of 1000 grains in a recrystallised AI-Mn sample plotted in the Euler angle space. The main textural features strongly resemble the macrotexture of the same sample as obtained by X-ray diffraction (Figure 9.5b), though fine details and smaller intensity variations cannot be resolved. 9.3.2 Continuous distributions The easiest way to determine a continuous orientation distribution in Euler space from single grain orientation data is to apply a direct method (cell method) similar to that described above for the pole figures. As shown in Section 9.3.1. for each orientation a point in Euler space is obtained. If one now subdivides the Euler angle space into snlall cells of an appropriate size, the number of points in each cell can be counted and a continuous distribution function is derived. This method was first applied by Perlwitz el al. (1969) to single orientations determined by SAD in a TEM. In the dlrect methods, the orientations are associated with discrete cells of finite size in the orientation space. so that a direct method provides a discrete ODF. Typically, the
U
(a)
EBSD
,
(D,
=const.
comol. ODF
(b) X-ray
Figure 9.5 Presentation in Euler space of texture data of a recrystallised AlP1.3%Mn sample (rolled to 92% reduction, annealed for 10s at 45O"C). (a) Orientations of 1000 grains as obtained by EBSD: (b) continuous intensity function (ODF) as derived from X-ray pole figures of the same sample, showing good reproducibility of the main texture features.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
215
Euler angle space is subdivided into cells of size A p l = A@ = = 5" or 10". For each individual orientation gS the content of the corresponding cell i with the orientation g, - A g J 2 < gS < gi A g J 2 is incremented. Finally, the value of each cell is normalised to the total number of single grain orientation measurements N and to the cell size Ag;. If necessary, a filter can be applied to smooth the resulting orientation distribution, as discussed above for the pole figures. It is seen that the presentation corresponds to a histogram of orientation densities A n i / ( N . A g i ) defined by the number of orientations A n , falling into the ith cell and normalised by the number N of measured orientations with regard to the cell size Ag;. Although this method appears easy and straightforward, it has several drawbacks. Firstly, the number of measured orientations is often too low for statistical relevance (see Section 9.3.3). As an example, for a cell size of 10" and a statistical reliability of 80% about 25 points per cell must be considered, which for cubic crystals would require about 3000 single grain orientation measurements (Bunge, 1982). Consequently. in the case of a smaller number of orientations the cell size would become very large, which would result in a falsification of the ODF. Therefore. it may be necessary to use the series expansion method to derive an interpolation function as will be discussed below in this Section. A second drawback is that in direct methods it is very complicated to take the strong distortion at small angles properly into account, and therefore, this is in general not done. This means that an ODF as derived by means of a direct method from single grain orientation data could comprise a strong peak at a given orientation having small B, though continuous densities obeying the condition p1 + p2 = constant would be more appropriate. Because of the problems described above, an inch'rect nzetldmight be advantageous (Pospiech and Liicke, 1975; Wagner ct ul.: 1981; Bunge, 1982). Each single grain orientation gi is associated with a Gauss type distribution in Euler angle space with the (half) scatter width '&. The distribution of the orientation density S as function of the angular distance from the exact orientation can be computed according to:
+
For single grain orientation measurements the density So is equal to 1. For a texture consisting of N individual orientations the ODF can be determined by superposition of the corresponding N Gauss-peaks. The main advantage of this method is that it provides an easy computation of the C-coefficients of the series expansion method by:
With regard to the discussion of the ghost error as caused by the lack in odd order C-coefficients in inacrotexture experiments (Section 5.3.1), note that equation 9.3
216
INTRODUCTION TO TEXTURE ANALYSIS
yields both odd and even order C-coefficients so that the microtexture approach does not suffer from this error. For a microtexture ODF calculated according to equation 9.3, each individual orientation i contributes to the final texture with the same weight of 1/N, i.e. it yields a texture by number. To determine textures by volume, the term 1/N has to be replaced by a term that takes the volume of each grain into account. The same would also apply for the scatter widths do if it turned out that the angle $o is different for different orientations or orientation classes, e.g. for the as-deformed matrix and recovered subgrains (e.g. Engler et al., 1996a). As an example for calculation of an O D F from single grain orientation data, Figures 9.6a and b show the ODFs of a recrystallised A1-1.3%Mn sample, which have been derived from 100 and 1000 EBSD measurements, respectively. The latter ODF corresponds to the individual data shown in Figure 9.5a. If the C-coefficients are calculated for microtexture ODFs, then parallel calculations to those derived for macrotexture O D F evaluation can be carried out (Bunge, 1982). For instance plots of individual texture profiles, data of volume fractions and scatter widths, and subordinate functions such as (inverse) pole figures can be derived; furthermore, mechanical properties and anisotropic behaviour can be predicted (Bunge and Esling, 1986). An immediate comparison of ODFs as derived according to the direct and indirect methods is difficult for the following reasons. In the case of the (discrete) cell method the intensities depend on the dimension of the cells and are further locally affected by the distortion of the orientation space. In the continuous representation the density
,
(a) N=100
(4, =const.
GAUSS ODF
I @, @ =conSt. GAUSS ODF
(b) N=1000
Figure 9.6 ODFs of recrystallised AI-1.3%Mn, as in Figure 9.5, calculated from different numbers N of EBSD single grain orientation measurements.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
217
values strongly depend on the chosen scatter width go.Moreover, they are locally influenced by the error introduced by the truncation of the series expansion, which is generally the more severe the higher the local concentration of the measured single grain orientations. Both the size of the cells in the discrete presentation as well as the scatter width in the continuous case should not be chosen arbitrarily but should be defined in connection with the resolution of the measurement as well as the computation procedure. As a lower limit for both cell size and scatter width the absolute accuracy of single grain orientation measurements, e.g. 5", has been proposed (Engler et al., 1994b). Wagner et al. (1981) have discussed that the scatter width of the series expansion method should depend on the maximum series expansion coefficient l,,, which means that for l,,,, = 22 and l,, = 34, yo should not be smaller than 8" and 5" respectively. In any case values exceeding 10" should be avoided as t h ~ swould cause smearing of the peaks and errors in the textures (Engler et al., 1994~). $J"
9.3.3 Statistical relevance of single grain orientation measurements The number of single grain orientation measurements which is necessary to represent adequately a texture strongly depends on the texture sharpness. Whereas in the ideal case of a single crystal one measurement would be sufficient, for weak textures several thousand individual orientations must be measured. In an analytical study by Jura et al. (1996) it was even concluded that for weak textures as much as several tens of thousands of orientations must be considered. However analysis of other experimental results showed that a much smaller number of single grain orientation measurements are sufficient to reproduce a given texture. In one of the first studies on this subject Wright and Adams (1990) determined the texture of a rolled Al-sheet from 100. 200, 300. . . up to 1000 individual orientations and compared the resulting ODFs with the corresponding X-ray texture. The O D F derived from 100 orientations already comprised the main characteristics of the texture, but for a statistically sound representation of the texture 750 orientations were necessary. The authors concluded that for weak textures up to 2400 orientations are required. Baudin and Penelle (1993) performed similar investigations in an Fee3%Si sample and came to very similar conclusions: with 100 orientations the major texture components were already fairly well reproduced, but for a statistically reliable texture representation 800 orientation measurements had to be analysed. Because of the strong influence of the texture sharpness on the accuracy of reproducing an ODF from single grain orientation measurements it is necessary to develop a quantitative measure for the minimum number of single grain orientation measurements :V1 which is required to represent adequately the texture of a given sample. Although in all experimental investigations of this problem so far, the texture as derived from single grain orientation data has been compared with X-ray inacrotextures, such a criterion must as well hold for the reproduction of a microtexture of a given part of the microstructure. As such a criterion, Pospiech et al. (1994) proposed to analyse the convergence of the statistical parameter p, which represents the relative mean square deviation between two textures. To determine the minimum number of single grain orientation
218
INTRODUCTION TO TEXTURE ANALYSIS
measurements Ni necessary to reproduce appropriately the corresponding texture, ODFs are computed for an increasing number N of single grain orientation measurements, e.g. in successive steps of 100 orientations. As an example, Figure 9.6 shows the ODFs of recrystallised A1-1.3%Mn as computed from 100 and 1000 EBSD-measurements. It is obvious that also in this example N = 100 already yields a good qualitative estimate of the X-ray macrotexture given in Figure 9.5b. A parameter po, comparing the intensity valuesJ;~of the O D F as computed from N single grain orientation data with the va1ues.h of a reference ODF, e.g. the macrotexture, is then calculated according to:
In Figure 9.7a the evolution of p o . . ~is shown for the example in Figure 9.6 (AN = 100). The critical number of measurements N I is obtained when the evolution of p with h' approaches saturation. It is seen that in the present example at numbers exceeding about 600 orientations the X-ray texture is reproduced with statistically sufficient accuracy. This procedure can be further refined by a local parameter RMD, which additionally compares density profiles of O D F components (Pospiech et al., 1994). Furthermore, Figure 9.7a shows the evolution of p o , , ~as derived separately for various intensity ranges f (g) of the ODFs. Here, f(g)> 1, 2, . . . etc. means that only O D F regions with f ( g ) > 1,2,. . . were considered for the calculation of p. It appears
Ih
step sue 4pm
tep size
Figure 9.7 Evolution of the statistical parameter p with increasing number of single grain orientation for measurements IN' for different examples (see text for details). (a) Evolution of po,, and recrystallised A1-1.3%Mn (See Figures 9.5 and 9.6), (b) evolution of p,,.,, for recrystallised Ni3AI, obtained by orientation microscopy with two different step sizes.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
219
converges faster for the ODF-ranges with higher f (g)-values. This reflects that the fact that for a more qualitative description of the ODF lower numbers of single grain orientation measurements are sufficient. Single grain orientation measurements are used more frequently to determine local microtextures rather than bulk textures, so that no data of a reference textureh may be available. In such cases. a parameter pAyl,v can be calculated from the ODF-values and J;. for two consecutive stages of the single grain orientation measurement procedure, say for hi' = 300 and N = 400. Equation 9.4 is then modified to:
(plotted with filled symbols) has As shown in Figure 9.7a; the parameter pAv~,;:\. much lower values than p"..\- relating the texture to a reference ODF, as potential measuring errors would show up in all data sets in an equal manner. Qualitatively, however. the two curves are very similar. Thus, if no macrotexture data are available, p\-!, ,\. also provides a reasonable approximation of N I . The texture in the example in Figure 9.7a was fairly sharp (,f'(g),n,,= 10) and comprised only one major texture component on top of a random background. Consequently, 100 orientations already yielded a fairly good reproduction of the macrotexture, and with only 600 orientations a statistically reliable texture could be derived. In Figure 9.7b the evolution of p ~ : . , ,is shown for an example of recrystallised Ni;Al with a much weaker macrotexture (,f(g),,, = 3). It is obvious that in this case many more orientation measurements are required to reproduce the X-ray texture ( N l = 3000). To summarise the discussion on how many orientation measurements are needed to describe an ODF with statistical relevance, with an increasing number of single ~ p2ji,l,p, approach a grain orientation measurements h' both parameters, p ~ . . , and steady-state value. Convergence behaviour of p indicates that a sufficient number of measurements has been acquired from single grain orientation data. The parameter PO. .v which relates the microtextures to a given macrotexture shows slower convergence than ply!. !,and is therefore better suited for determination of the necessary number of measurements Nl, but in cases where no macrotexture data can be derived, e.g. for microtexture analysis, p j v , s also provides a reasonable approximation of N I . (In an alternative procedure, Matthies and Wagner (1996) analyse the evolution of the texture index, F2: as a function of l / N . This parameter, F2(1,'N),approaches linear behaviour when the number of orientations is sufficiently large.) Turning now to selection of an appropriate sampling schedule. the statistical relevance of microtexture results does not depend solely on the number of measurements N, but it may also be affected by the measurement procedure, which becomes particularly important in the context of automated EBSD measurements. To illustrate this: the same Ni3A1 sample was scanned again with a different step size. The first scan was performed with a step size of 80 pm. Since the grain size of the Ni3A1 sample was about 40pm, this means that virtually all measured points belonged to different, mostly not even adjacent, grains. Of course, this procedure is not suited for generation of an orientation map (Chapter 10) or computation of
220
INTRODUCTION TO TEXTURE ANALYSIS
misorientation parameters (Sections 9.5 and 9.6), but it gives a good statistical representation of the orientation distribution of the sample. The second scan was performed with a step width of only 4 pm such that each grain was covered by a quite large number of measurements. Whereas these results gave a satisfactory orientation map, the statistical relevance of the measurements is much poorer than in the first scan resulting in a much slower convergence of the parameter p ( N I 6000; Figure 9.7b; Engler et al., 1999).
9.4 REPRESENTATION O F ORIENTATIONS IN RODRIGUES SPACE Use of Rodrigues space, otherwise known as Rodrigues-Frank space, wherein a population of R-vectors resides (Section 2.71, is a relatively new field compared with the long-established Euler space approach for representation of orientations. The explanation for the late introduction of Rodrigues space is that its application is intimately connected with microtexture, i.e. discrete orientation measurements, whereas representation of texture in Euler space relates traditionally to the representation of continuous orientation distributions as calculated from pole figures (Section 5.4). As already stated above, microtexture data can be represented in either Euler space or Rodrigues space whereas macrotexture data tend to be represented only in Euler space because most commercial systems are configured for this approach. Although Rodrigues space. which is particularly well suited to analysis of misorientations (Section 9.6.31, has not been very widely used to date it has several advantages over Euler space (Section 2.7.2) and so may well be adopted more in the future. Orientations are usually best displayed in the whole of the fundamental zone of Rodrigues space (Section 2.7). Since for orientation representation the axes of the zone are coupled to the axes of the specimen, the zone could be divided according to the symmetry of the specimen itself. For example. a rolled specimen has three axes of symmetry, and so an appropriate reference frame is for the RD, TD, ND to be parallel to the X; Y. Z axes of the fundamental zone respectively. For this case, 118 of the fundamental zone could be used. The advantages of using this reduced volume is that if the sample population is small; textwe trends are more discernible in the smaller volume. However. valuable information can be lost in that the assumed symmetry of the material may not be borne out in practice. For example, a rolling process could bias the textures on one side of the strip or plate more than on the other and this would be evident if the whole fundamental zone were used to represent the data (Section 5.4.2). A further consideration to be taken into account if orthotropic sample symmetry is enforced by collapsing the fundamental zone into a reduced volume is that related texture variants having the same Miller indices families are not distinguished (Becker, 1991). There are in the most general case four different orientations having the same Miller indices families (Section 2.3). As an example the variants of the S-orientation are (123)[634], (123)[634], (123)[6%] and (123)[634] (Table 2.1 and Appendix 3). to quote them in their lowest angle forms. These are distinct orientations even though they feature common families of planes which are only
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
221
symmetrical with regard to orthotropic sample symmetry. The whole of the zone is required to display correctly all these distinct variants, whose R-vectors are:
Figure 9.8 shows texture results from an aluminium-lithium alloy where two different near-S orientations were observed in two different parts of the sheet (Randle and Day, 1993). There are fifty grains in each sample population. In the first set (Figure 9.8a) the microtexture comprises 68% near-S1 orientations, and in the second set (Figure 9.8b) there is 42% near-S2 orientations. (001) <100> and (01 3 ) <100>
Figure 9.8 Orientations from an aluminium-lithium alloy displayed in Rodrigues space as stereo pairs. Data sets are of 50 grains having (a) predominantly S1-orientation and (b) predominantly S2-orientation (Randle and Day, 1993).
222
INTRODUCTION TO TEXTURE ANALYSIS
are present in both as secondary components. We can see that the clusters which represents these two textures are occurring in different regions in Rodrigues space. This information would have been lost if the data had been compressed into a single octant. Distributions of orientations in Rodrigues space are usually generated by a computer program such that the zone can be rotated on-screen in order to gain maximum visual appraisal of orientation clusters. However, like any other three-dimensional space, Rodrigues space is problematic to display in the two-dimensional medium of a printed page. Several options have been trialled. The first method to be used was to view the whole fundamental zone as a stereo pair (Frank, 1988; Ashbee and Sargent, 1990; Randle and Day, 1993). This is not a very convenient method because most people need a stereo viewer to appreciate the three-dimensional depth perspective (Figure 9.8). Viewing the whole zone in perspective is a feasible option for gaining an appraisal of the data spread especially if either different colours/sizes/ shading are attributed to the data points to give pseudo-depth information (Neumann, 1991a, b) or contouring is used (Hughes and Kumar, 1997; Case Study No. 8). A useful perspective of Rodrigues space is to view all the data by looking directly down one of the reference axes. The Z-axis perspective is then equivalent to a standard {100$ pole figure. Figure 9.9a and b show the Z-axis perspective partners to the views in Figures 9.8a and b. It can be informative to study more than one perspective view. For example, two fibre textures, (103)[uv~.]and (T13)[zlv~,],are shown by three different perspectives in Rodrigues space in Figure 9.10 and it can be seen that the X-axis rather than the Z-axis perspective is better for appreciating the pattern of how the textures lie on straight lines. The R-vectors for these, and other, fibre textures are listed in Table 9.1. Note that here one of the orientations, (103) [010], is located outside the fundamental zone in order to preserve the feature that fibre textures map onto a straight line in Rodrigues space. This point could alternatively
......
..:.::..:.:,:,:
*2
;
@
...:.:.. .:.
g";
.
.:+:.:.:..: .<
+
@@ @
@
@@
98
Figure 9.9 Rodrigues space Z-axis perspective view of the microtextures in Figure 9.8a and Figure 9.8b respectively (Randle and Day, 1993).
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
223
Figure 9.10 Fibre tewturea 111 Rodrigues space for {103) (dotted line) and {T13) (dashed line) Both (a) the X-ax~speispective and (b) the Z-axis perspective are shown (Randle and Day. 1993)
Table 9.1 Components of the Rodrigues vector for orientations u w along various fibre axes (Randle and Day, 1993)
001 fibre axis 100 3T0 210 320 l 10 101 fibre axis 101 212 111 121 131 111 fibre axis 112 101
-
111 fibre axis 312 21 1 110 103 fibre axis 010 337 327 317 301
-
113 fibre axis 332 27 1 30 1 110
be re-expressed in its lowest angle form (Section 2.2.3) so that it would then plot within the zone. It has transpired that the most popular method to represent Rodrigues space is by sections through it. as for Euler space. and using the eye to reconstruct the sections into a three-dimensional format (e.g. Engler et al., 1994c; Sarma and Dawson, 1996). Figure 9.11 shows an example of such a representation for cubic symmetry (Becker and Panchanadeeswaran, 1989). The dimensions of Rodrigues space can be represented in absolute R-vectors or normalised so that the length of the major axis is unity, and the R-vectors scaled accordingly. As for any other representation of inicrotexture, contouring of the data may be substituted for discrete data points.
INTRODUCTION TO TEXTURE ANALYSIS
224
TRANSVERSE
Figure 9.11 Example of orientation distributions in Rodrigues space displayed as sections in R3. Copper rolling texture, cubic symmetry (Becker and Panchanadeeswaran, 1989).
9.5
GENERAL REPRESENTATION O F MISORIENTATION DATA
The formulation of a misorientation from pairs of orientations was described in Section 2.5. The following categories of misorientation populations can be distinguished: Misorientations between nearest neighbour grains, i.e. grain boundary misorientation (also sometimes called mesotestz~re).This is by far the most common case. Misorientations between a single, specific grain and other grains, for example a large grain and all sinall grains (Shimizu et al., 1990). Intra-grain misorientations, or misorientations sampled from several sites within neighbouring grains (Section 9.5.2). It has been shown that even in recrystallised materials small orientation changes. i.e. lattice rotation or bending, exist within a grain (Adams et al.. 1993; Davies and Randle, 1997). Misorientation distribution functions (MODFs) and their derivatives (Section 9.7). Orientation relationships between different phases (Section 9.8). Analysis of inisorientations at interfaces also gives rise to a further level of processing and crystallographic analysis, for instance categorising the statistics according to the coincidence site lattice model. These methodologies are discussed in detail in Chapter 11.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
225
9.5.1 Representations based on the angle/axis descriptor The anglelaxis descriptor is the most popular method for representing a misorientation. As described in Section 2.2.3 there is a number of crystallographicallyrelated. mathematically equivalent solutions for a misorientation - for example 24 and 12 for cubic and hexagonal crystal symmetry respectively. In order to avoid confusion it has become common practice to select, from all the solutions, the angle/ axis pair which has the lnini~nummisorientation angle to represent the misorientation. This solution has been named the disorietztatiorz to conveniently distinguish it from the remaining ones (Warrington and Boon, 1975). The angle of misorientation and axis of misorientation provide different physical information: a low-angle or high-angle grain boundary is recognised from the angle alone, and certain misorientation axes are considered to be important in processes such as recrystallisation. Often, then, the angle and axis of misorientation are represented separately, both for convenience on the printed page and in order to focus on either the angle or axis alone. Alternatively, both parameters can be displayed in cylindrical anglelaxis space (Section 9.6.2). A combination of angle and axis is used for representation in Rodrigues space (Section 9.6.3). Considering first the representation of disorientation axes, these are conveniently represented on the stereographic projection or, more compactly, on a single unit triangle of the stereographic projection since the density of axes becomes multiplied by the number of unit triangles, as for an inverse pole figure. The probability density distribution of disorientation axes in a unit triangle has been calculated for cubic crystals (Mackenzie, 1958, 1964) and is shown in Figure 9.12a. The plot consists of density contours centred on both [l001 and [l 1l]. The density rises as the arc AB of the great circle 2 ' 1 2 = ~ K + L: where H, K, L are the Miller indices of the rotation axis and H > K > L, is approached. A more useful form of the probability distribution is shown in Figure 9.12b, which shows the percentages of disorientation axes which are expected to lie in each marked region if the distribution is random. This has come to be known as the 'Mackenzie axis distribution' or 'Mackenzie triangle' and is used to display disorientation axes and to assess the degree of departure from randomness of a sample population. Figure 9 . 1 2 ~shows an example of the use of the Mackenzie triangle (Randle and Caul, 1996). Here, different symbols are used to denote several CSL types (Section 11.2.1). Another option, although not one that is used frequently, is to display misorientation axes relative to some external axes, rather than those of the crystal, for example a major stress axis or a dominant axis associated with the specimen processing, for instance a rolling direction. Such a distribution is analogous to choosing a pole figure representation rather than an inverse pole figure for orientations (Heidelbach et al., 1996). The probability distribution for disorientation angles in cubic polycrystals has also been derived (Warrington and Boon, 1975), and has a maximum at 45' and a cut-off at 62.8". Similarly to the case for the axis distribution. the Mackenzie angle distribution is used to assess the departure from randomness in a sample population of disorientation angles (e.g. Uhbi and Bowen, 1996; Case Study No. 3). Figure 9.13 shows an example taken of two sample populations from directionally solidified nickel: equiaxed grains have a random distribution whereas columnar grains
INTRODUCTION TO TEXTURE ANALYSIS
Figure 9.12 Distribution of disorientation axes for cubic crystals in a single unit triangle of the stereographic projection. (a) probability density plot; (b) percentage of disorientation axes lying in the various regions (Mackenzie, 1964); (c) use of the Mackenzie triangle to display disorientation axes. Different symbols are used to denote several different CSL types (Randle and Caul, 1996).
show considerable bias towards smaller disorientation angles (West and Adams, 1997). Often the angle and axis distribution are shown together, which then requires cross-referencing between particular angles and axes. Case Study No. 5 and Figure 9.14, which is taken from an austenitic steel. illustrate this point (Gertsman and Tangri. 1991): the high proportion of near 60" misorientation angles corresponds to near misorientation axes, and this 60°/<1 1l > combination represents a C3 CSL or twin grain boundary (Section 11.2.1). It is important to bear in mind that, whereas it may sometimes be sound to separate the angle and axis of misorientation in order to concentrate on particular physical aspects of the data, the total misorientation embodies three independent variables (Section 2.6). These can be expressed by the same parameters as orientations, and the anglelaxis pair is only one descriptor (Chapter 2). Hence,
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
227
Misorientation angle (") Figure 9.13 Disorientation angle distributions for two types of grains. equiaxed and columnar, in nickel (Adapted from West and Adams, 1997).
Figure 9.14 Misorientation distribution, comprising both the angle and axis distribution, from stainless steel (Adapted from Gertsman and Tangri, 1991).
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INTRODUCTION TO TEXTURE ANALYSIS
when comparing amounts of misorientation it can be misleading to compare angles of misorientation alone, unless the misorientation axis is constant. Furthermore it may be relevant to assess if the misorientation is in a positive or negative sense, for example in the evaluation of cumulative misorientations (Driver et d., 1996). As stated at the beginning of this Section, it is common practice to select the disorientation, from all the symmetry-related solutions, to describe a misorientation. Although this convention simplifies the data processing, there is a risk that insights into the physics of misorientation-related phenomena may be overlooked if only one angle,'axis pair is examined. It is probably true to say that if the misorientation angle alone is of interest in an investigation, it will almost always be the lowest angle solution that is most meaningful. However: if an investigation concerns the misorientation axis, it may not be that of the disorientation which is of interest (Hutchinson et al., 1996). For example in bcc metals it is reported that high mobility grain boundaries are misoriented on <110>. For such investigations it is clearly better to choose, from the 24 crystallographically-related solutions, the one having a niisorientation axis closest to < l l0>. Figure 9.15 shows the distribution of 10,000 random computer-generated misorientations displayed as those misorientation angle having axes closest to , < l 1l > and <100> (Figures 9.15a, b, c respectively) which can be compared with the distribution for the disorientation angles for a random
Figure 9.15 D~strlbutlonsof computer-generated misorientatlon angles for a landom distribution plotted for misorientation axes closest to (a) < l 10>; (b) < l 11>; (c) ; and (d) disorientation axes (Courtesy of W.B. Hutchinson).
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
229
distribution (Figure 9.15d). Plots for misorientations within 5" and 10" of the given axis are included. It is clear that even for random misorientations there are peaks in the distributions of misorientation axes, which must be taken into account during data interpretation. A second reason for examining misorientations from the standpoint of particular misorientation axes is that where three (or more) grain boundaries adjoin at a grain junction they share a common axis of misorientation (Section 11.4). For example (Randle, 1990. 1993):
refers to the relationship of three grain boundaries at a junction misoriented on <210>, according to the 'addition rule' (Section 11.4). None of these are in the lowest angle form and this relationship could not have been recognised from the disorientation since for that case the axes are all different:
9.5.2 Intra-grain misorientations Measurements of intra-grain misorientations in metals and alloys have revealed the presence of local variations in orientation (e.g. Liu and Chakrabarti, 1996). Whereas this is well-known from TEM observations of deformed materials (Hughes and Kumar, 1997) it is only recently that EBSD has been used to study subgrain misorientation and to show the presence of local 'lattice bending' even in recrystallised materials (Adams et al., 1993; Thomson and Randle, 1997a). As far as data processing of intra-grain misorientations is concerned, it is important to choose a relevant sampling step size in order to reveal orientation changes. A suitable step size will depend on the scale of the microstructure: typically a recrystallised microstructure might require a step size of 2 pm whereas a deformed material might be sampled every 0.1-0.5 pm using EBSD or an order of magnitude less in the TEM. A useful combination of intra-grain spatial and numerical information is to show a linear scan of orientation changes across one or more grains along the specimen normal direction (ND) (Figure 9.16). This display method has the advantage of both focussing attention on the area of interest and providing concurrently quantification of the orientation change. A disadvantage is that a compromise must be made in order to represent compactly the three independent variables which describe the misorientation, in addition to the positional information. The most simple approach is to use only the inisorientation angle, which is the case shown in Figure 9.16. When misorientations between neighbouring sampling points are calculated, different values may be obtained depending on the spacing of the sampling points. Although this methodology of calculating misorientations between nearest neighbour points may be valid for some cases, e.g. to monitor cunlulative misorientations (Hjelen et al., 1991), the ambiguous effect of the sampling step size can be circumvented by plotting an orientation, rather than a misorientation, parameter. Such a parameter
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INTRODUCTION TO TEXTURE ANALYSIS
(Sub)grain number along ND
Figure 9.16 Linear scan of misorientation angles between adjacent subgrains in a superplastically deformed alloy (Adapted from Liu and Chakrabarti, 1996).
can be calculated for each sampling point with reference to a startmg-point or an average orientation. An example is discussed in Chapter 10 with reference to Figure 10.3b and c. Alternatively, a con~pressedorientation parameter can be calculated using either Euler angles or Rodrigues vectors (Randle et al., 1996). The compressed orientation parameter D in Rodrigues space is given by:
R3R are the components of the R-vector at the where R I :R2;R3 and R I R , measuring point and the reference point (e.g. the grain centre) respectively. An inevitable disadvantage of the compressed orientation parameter D is that occasionally different orientations can have the same D value. Finally, fine-scale orientation mapping is the optimal method for visual appreciation of spatial representation of local orientation changes, and this is discussed further in Chapter 10.
9.6 REPRESENTATION O F MISORIENTATIONS IN THREE-DIMENSIONAL SPACES Misorientations can be displayed in Euler space (Section 9.6.1) or spaces based on the angle 'axis pair, namely cylindrical angle 'axis space (Section 9.6.2) and Rodrigues space (Section 9.6.3). It turns out that representation in Rodrigues space is the most advantageous.
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
23 1
9.6.1 Representation of misorientations in Euler space For representation of inisorientations in the Euler angle space, the Euler angles pl, Q, p2 are calculated from the misorientation matrices M according to standard procedures (Section 2.5). As has been described in detail in Section 9.3, the resulting set of Euler angles can be transposed in the Euler space either by direct cell methods or by the series expansion method. As an example, Figure 9.17 shows a misorientation distribution which was derived from EBSD single grain orientation measurements in cyclically deformed nickel. The sample was deformed at 600 "C in 300 cycles with a frequency of 13 X lop' sp' at a strain amplitude of 0.5%. Under such circumstances the grain boundaries tend to rearrange, and particularly C3-twin grain boundaries (Section 11.2.1) were observed to remain stable in the microstructure (Brodesser and Gottstein, 1993). Besides the C3-peak in the distribution, strong intensities close to the origin of the Euler space, i.e. at p1 = Q = p2 = 0". prevail, representing the small-angle grain boundaries (Figure 9.17). The same data will also be shown in other spaces in Figures 9.18 and 9.19b. The main advantage of the representation of misorientation distributions in Euler space is the relatively easy computation procedure, and that the M O D F and its derivatives can be obtained, if required (Section 9.7). However, the technique suffers from several disadvantages: e
The Euler angle representation does not yield direct information on the crystallographic orientation relation between the two considered orientations, which means it is difficult to visualise. Thus, evaluation of MODFs in the Euler MODF
MODF
1 (a) cp2=const.
1
I
LEVELS I3 5
10
30
(b) @=const
Figure 9.17 Representation of the MODF of a cyclically deformed nickel specimen in Euler space. (a) v,-? = constant; (b) G? =constant.
INTRODUCTION TO TEXTURE ANALYSIS
Levels: 1 - 5 - 1 0 - 1 5 - 3 0 Figure 9.18 MODF of a cyclically deformed nickel specimen in cylindrical angle/axis space (see also Figure 9.17).
space needs detailed legends or marking of special misorientations as done in Figure 9.17. e
The orientation space exhibits strong distortions for small angles Q (Section 5.4.1), as becomes most evident from representing the M O D F in @-sections (Figure 9.17b). In order to overcome this shortcoming, Zhao and Adams (1988) proposed to use a subspace with very high @-angles (Q 1 70°), where the distortions are minimal. The borders of this subspace are marked in Figure 9.17b. As the two orientations which define the mutual misorientation can be treated equivalently (Section 2.5.2), this subspace can be further subdivided into two equivalent asymmetric domains, as also indicated in Figure 9.17b. Furthermore, in Figure 9.17b the positions of the special grain boundary orientation relationships with C < 17 are marked. It can be seen that each C-position can be found exactly once in this subspace.
e
The surfaces between the various basic domains of the cubic/cubic symmetry are strongly curved (which is induced by the three-fold axis). causing difficulties in identification. The C3-twin relation whlch is located on the interface between adjacent subspaces is visible three times in the standard Euler angle space. Note that all orientation relationships with low C lie at the border surfaces of the subspaces rather than in their interior (Figure 9.17). This is due to the rotation about symmetrical - or -axes for such orientation relationships.
EVALUATION AND REPRESENTATION O F MICROTEXTURE DATA
233
<
Figure 9.19 Representation of misorientations in Rodrigues space. (a) R-vectors for CSLs having C 45 in a Z-axis perspective of the sub>olurne (Randle, 1990): (b) R-vector density distribution, displayed in sections through a left- and right- handed subvolume, for cyclically deformed nickel (see also Figure 9.17): (c) Z-axis perspective of misorientations in an a~lsteniticsteel, indicating a high proportion of C3s (Randle and Day. 1993).
For all these reasons the representation of rnisorientations in Euler space is very difficult to visualise, which may cause uncertainties in evaluation of misorientations in the Euler space. Rather, misorientations are better represented in anglelaxis spaces. 9.6.2 Representation of misorientations in the cylindrical anglelaxis space Misorientations can conveniently be represented in an orientation space given by the parameters of the ang1e:axis pair description. For that purpose, the rotation axis r is expressed in terms of two angles .3 and @ (equation 2.25), such that the misorientation
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INTRODUCTION TO TEXTURE ANALYSIS
is now given by the three angles d , and 0. All possible orientations and - G < - 360") misorientations are given in the interval (0" 5 Q 5 360°, 0" 5 ~9 5 90°, 0" < and, as shown on Figure 2.12, can be represented in a cylindrically shaped threedimensional space with the cylinder-axis parallel to the 0-axis. In this arrangement, the base plane of the cylinder corresponds to a stereographic projection of the rotation axis v. Figure 9.18 shows the misorientation distribution of a cyclically deformed nickel sample (also shown in Euler angle space, Figure 9.17) in sections parallel to 0 (A0 = 5') through a subsection of the cylindrical orientation space. The basic domain, i.e. the smallest subspace in which each possible misorientation would exist exactly once, is marked with thick lines (for cubic/cubic symmetry). The main advantage of this space is given by the fact that for each 0-section the rotation axis r can readily be derived from its position in the stereographic triangle which is a very fa~niliar way to evaluate misorientation data. Another example of an M O D F represented in the cylindrical anglelaxis space is given in Section 9.8. A major disadvantage of the cylindrical angle/axis space is caused by the relatively large distortion of this space. For decreasing 0-angles the resolution for the position of the rotation axis becomes increasingly smaller and is indefinite for 0 = 0" (Pospiech et al., 1986). Since the Rodrigues space described in Section 9.6.3 offers several advantages, the cylindrical angle/axis space will not be discussed here any further. 9.6.3 Representation of misorientations in Rodrigues space When analysing misorientations, the coordinate frame of Rodrigues space is parallel to the crystal axes. The whole of the fundamental zone does not need to be used to display misorientations if all the misorientation axes are expressed in the same form, i.e. H 2 K > L and all positive. For the cubic system, for instance, the fundamental zone can be divided up along planes of symmetry into 48 identical and equivalent subvolumes (Randle, 1990; Field, 1995). This process is identical to the division of the reference sphere and hence the stereogram into unit triangles (Appendix 4), or division of Euler space into asymmetric domains (Section 9.6.1). The subvolume of the fundamental zone is also called the Mackenzie cell after the Mackenzie triangle which shows the distribution of random misorientations for the cubic system (Section 9.5.1). Table 2.3 included the size of the subvolume as a fraction of the fundamental zone for each crystal system. Figure 2.13 shows how the fundamental zone for cubic lattices is divided along the surfaces equivalent to planes perpendicular to the symmetry axes, i.e. : < l 1l > , < l 10>. The , < l 1l > and < l 10> axes radiate from the origin along edges of the subvolume. All misorientations can be constrained to lie either within the subvolume or on a surface, edge or vertex. Misorientations which lie on the boundary of the subvolume are 'geometrically special', by having a rational, lowindex axis of misorientation. Three types of boundary can be recognised readily in the subvolume of Rodrigues space (Case Study No. 3) low-angle boundaries, boundaries with preferred misorientation axes and coincidence site lattice (CSL) types (Section 11.2). Low-angle boundaries are recognised by their proximity to the origin of Rodrigues space, -
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
235
whereas preferred misorientation axes lie along straight lines in the subvolume, analogous to the case of orientation fibre textures. Often, preferred inisorientation axes are low-index types, <100>, < l 1 l> or <1 10>, and so coincide with the edges of the space. It should be noted that distributions of misorientations in Rodrigues space take account only of the lowest angle solution, and so alternative descriptions, which may show up common inisorientation axes, are not recognised. For example, in CSL notation C3, C5 and C15 are all inisoriented on a <210> axis (equation 9.6) yet only C15 is in its lowest angle form. Rodrigues space is especially apposite for the representation of low-C CSL misorientations in cubic systems since the Rodrigues vectors are rational fractions and therefore easy to recognise when plotted (Randle, 1990). The representation of CSLs in Rodrigues space for crystal systems other than cubic has been addressed little as yet and so will not be discussed here. R-vectors for CSLs up to C = 45 are listed in Table 9.2 and shown as a Z-axis perspective in the subvolume of the fundamental zone in Figure 9.19a. Table 9.2 Rodr~guesvectors for CSLs up to C Ax~slC
RI
R?
Ri
= 45
Axis X
RI
R2
R?
236
INTRODUCTION TO TEXTURE ANALYSIS
The subvolume of the fundamental zone is usually displayed as sections parallel to the XY plane. Either a single subvolume is used, which gives a triangular base, or two subvolumes are displayed which gives a square base and would allow both right- and left-handed misorientations to be displayed, if required. Figure 9.19b shows a sample population of misorientations from the cyclically deformed nickel sample which has already been shown in the Euler angle space (Figure 9.17) and the cylindrical angle/ axis space (Figure 9.18). In Figure 9.19b, the MODF is plotted with contour lines in 118 of the fundamental zone by sectioning Rodrigues space perpendicular to the [001]-axis in equal distances of tan8/2. In comparison to constant sections with A0 = 5" this yields a maximum error of 0.9": which demonstrates the small distortions of this space. This kind of representation of a cubic space comprising 118 of the fundamental zone, which differs from the Mackenzie-cell, was chosen because of its similarity to the well known sections through the Euler space (Engler et al., 1994~).Figure 9 . 1 9 ~shows a distribution of misorientations from austenitic steel, again having a high proportion of C3 boundaries, in a single subvolume. Finally, attempts are being made to superimpose Rodrigues space and real space coordinates in order to provide an orientation/microstructure link (Case Study No. 8; Hughes and Kumar, 1997; Weiland, 1992).
9.7 NORMALISATION AND EVALUATION OF THE MISORIENTATION DISTRIBUTION FUNCTION (MODF) In analogy to the continuous distribution of the orientations, the ODF, a continuous distribution function of misorientations can be defined, that is the misorientation distribution fzmction, MODF or, sometimes, M D F (Bunge, 1982; Pospiech et al., 1986; Adams, 1986). Similarly as described previously for the ODF: continuous misorientation distributions can be derived either by direct methods in an appropriate three-dimensional (mis)orientation space or by the series expansion method in Fourier space. The functions for the series expansion have originally been defined in terms of the Euler angles (Section 5.3.1). whereas comparable functions for the angle/axis and Rodrigues vector parameters have not yet been developed. Therefore, although the anglelaxis and Rodrigues vector description of misorientations are generally considered superior to the Euler angle description, the computations are usually performed in Euler space and then the results are transformed into other spaces (e.g. Pospiech et al., 1986). In the direct methods, the MODF can directly be derived by superposition of the corresponding misorientations in the desired (mis)orientation space (Engler et al., 1994c; Matthies and Vinel, 1994). Under the assumption of a random arrangement of the orientations in the microstructure, the MODF can directly be derived from the ODF, i.e. it can be computed from macrotexture data. The resulting MODF, the so-called texture-derived MODF, statistical MODF or, following the nomenclature introduced by Pospiech et al. (1993), orientation difference distribution function (ODDF) can be used to normalise the MODF, but it does not contain any valuable information on the spatial arrangement of the individual orientations. In real microstructures, the orientations are not randomly arranged such that the 'real' MODF generally differs from the
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
237
ODDF. Thus, determination of the measured, or physical, MODF requires analysis of the orientations of the corresponding grains in the microstructures together with their spatial arrangements, which can only be accomplished by techniques for microtexture analysis. In order to distinguish between the features in the MODF that arise from a given local arrangement of the corresponding orientations in the microstructure and the features which are controlled by the macrotexture of the sample, the measured MODF has to be normalised with respect to the macrotexture, which can be accomplished by dividing the MODF by the O D D F (Plege, 1987; Bunge and Weiland, 1988; Zhao et al.. 1988; Pospiech et al., 1993: Mainprice er al., 1993; Heidelbach et al., 1996). Thus, the resulting texture-reduced MODF discloses features arising from a correlation in nearest-neighbour orientation relationships. which gives rise to the terms correlated MODF or o r i e ~ f n t i o ncor1.elation,finctio7?(OCF). To illustrate the potential applications of MODFs most clearly, in the following the various possibilities of normalisation of the M O D F and the resulting information are discussed using an example with a model distribution of two texture components (Engler et al., 1994~).To compute the model texture two components labelled A and B at the Euler angles (75"; 45", 0") and (54": 29", 78") with a mutual misorientation of 35"/ < l 12> were chosen. Both components were associated with a Gauss type scatter in Euler angle space with volume fraction M = 50% and a half scatter width t10 = 8.5". The corresponding O D F which was calculated as described in Section 9.3.2 is shown in Figure 9.20. Subsequently. this ODF was discretised,
r7-P
(P2
GAUSS ODF
Figure 9.20 Model O D F consisting of two components A and B with a 35'/<117> relationship, used to illustrate the normalisation of MODFs (Figure 9.21).
orientation
238
INTRODUCTION TO TEXTURE ANALYSIS
yielding a set of approximately 500 single grain orientations. With this data set three different calculations were performed. (i) The orientation difference distribution function (ODDF) was computed by determination of the orientation difference between each individual orientation and all other individual orientations of the entire data set. The ODDF yields statistical information on the potential misorientations between all orientations in a specimen without consideration of the spatial arrangement of the corresponding grains in the microstructure, in particular of their neighbourhood. Thus, the ODDF is only influenced by the macrotexture of the given sample and, consequently. it can be computed from macrotexture data (Plege. 1987; Zhao et al., 1988; Pospiech et al., 1993), though it is typically derived from the microtexture data by computing the misorientations between each grain and all other grains irrespective of their local arrangement. The resulting ODDF of the present example is plotted in Figure 9.21a in the Rodrigues space. As could be expected from the model texture, it comprises two maxima at the position of small-angle grain boundaries (near the origin), indicating A-A and B-B relationships, and at the A-B orientation relationship (35') < l l?>). (ii) The MODF was computed by artificially 'arranging' the grains in such a way, that A-B large angle grain boundaries were preferred over small angle grain boundaries A-A and B-B. This means that A-grains are preferentially surrounded by B-grains and vice versa. The corresponding Rodrigues plot is shown in Figure 9.21b. It reveals a strong peak at 35",< l 12>. which is significantly stronger than the corresponding peak in the ODDF (Figure 9.21a). However, there remains a peak near the origin which indicates the presence of still a large number of small angle grain boundaries. In the case of such a strong macrotexture as in the present example, despite the bias towards A-B misorientations still many A-A and B-B grain boundaries will arise. which is simply due to their statistical frequency. Thus, the MODF yields information about the frequency of orientation relations between neighbouring grains in the microstructure. This particularly favours the MODF for problems related to grain boundaries like grain boundary diffusion. grain boundary sliding, etc. (e.g. Pumphrey, 1976). In that context; the special treatment of a material to achieve an optimum grain boundary distribution is referred to as grain bourzdaq. engiuee~.iqor design (Watanabe. 1988; Palurnbo and Aust, 1992; Randle, 1996). (iii) However. for an analysis of the orientation correlation, as e.g. the preferred occurrence of A-B grain boundaries, the MODF is not suited, since it is influenced by the texture of the sample. To yield information on orientation correlations. the MODF is divided by the ODDF. This way one obtains a texture-reduced form of the misorientation function; namely the correlated MODF or orierztutiorz correlntion function (OCF). In contrast to the MODF, the OCF yields information about the texture reduced spatial correlation of misorientations, e.g. whether deformation or recrystallisation leads to a preferred development of distinct neighbourhood relations. Occurrence of a given misorientation with ,f > 1 (or f < 1) in the OCF indicates its stronger (or weaker) occurrence compared to a statistical arrangement of the grains in the microstructure. Hence, the OCF is suitable to allow conclusions to be drawn on underlying mechanisms. In the present example in the OCF no small-
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
Max: 20.4
Max: 9.8
Levels:I - 2 - 4 - 7 - 1 0 - 2 0
Levels: 1 - 2 - 4 - 7
(a) ODDF
(b) MODF
-
-
Levels: 1 2 - 4 6
1
(c) OCF Figure 9.21
(a) ODDF. (b) MODF. and (c) OCF of the model texture in F ~ g u i e9 20
angle grain boundaries arise, only the peak at 3501 <112> (A-B) renmns, which exactly reflects the distribution which is to be expected from the artificial arrangement of the orientations (Figure 9.21~).
INTRODUCTION TO TEXTURE ANALYSIS
9.8 MISORIENTATION DISTRIBUTIONS BETWEEN PHASES Analogous to the misorientation between two different grains, the misorientation between different phases can be derived and evaluated to yield information on phase boundaries, phase orientation relationships, etc. In the case of two cubic structures; e.g. fcc and bcc, a computation procedure analogous to the determination of MODFs in single-phase materials can be carried out. In all other conceivable phase combinations, however, the different crystal syn~metrieshave to be considered which renders evaluation of MODFs a complicated task. Therefore, to the best knowledge of the authors so far only examples on cubic/cubic two-phase systems have been published. Schwarzer and Weiland (1988) studied orientations and orientation relationships in a ferrite,/martensite duplex steel by Kikuchi pattern analysis in a TEM. From the orientations of contiguous ferrite and martensite grains an MODF was computed and represented in Euler space. The MODF showed low-angle boundaries and 4j0/<1 10> misorientations pointing at a Bain orientation relationship between ferrite and martensite (Bain, 1924). Field et al. (1996) determined the M O D F between ferrite and retained austenite by means of automated EBSD and presented it in the Rodrigues space (Section 9.6.3): they likewise observed a Bain-like orientation relationship between the two phases. Figure 9.22 shows an example of the MODF between the tungsten and nickel phases in a W/Ni composite material. The orientations of both phases were determined by means of automated EBSD orientation microscopy; examples of the resulting maps are shown in Figure 10.7. Subsequently, the misorientations between bcc tungsten and fcc nickel across the phase boundaries were determined, expressed in terms of the parameters misorientation axis P and angle B and represented in the form of H-sections through the cylindrical angleiaxis space (Section 9.6.2). As in the previous two examples. the resulting MODF (Figure 9.22) shows a strong maximum of 4j0;<100> misorientations which indicates a Bain-orientation relationship between 1998). the two phases (Sinclair et d.;
9.9 EXTRACTION O F QUANTIFIED DATA Display of orientation or misorientation data in an appropriate space is usually only part of data evaluation. Secondary processing is needed to extract quantified information from the data. As far as statistical information is concerned, one means to do this was mentioned in Section 9.3.2, namely calculate the C-coefficients of the series expansion method and call upon the analysis methods derived for macrotexture. However, this route may be unnecessarily complicated, especially for small data sets. Proportions of grains having specified texture components, within a predetermined tolerance angle, can be extracted directly from the orientation matrices (Case Study No. 1). Indeed, these statistics could be output directly without recourse to any orientation space, but it is convenient to have a depiction of the texture distribution to facilitate qualitative characterisation by eye. Figure 9.23a shows a { l l l ) microtexture pole figure con~prising300 grains from a deformed and partially recrystallised aluminium alloy (Davies and Randle, 1996).
241
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA 111
5"
A
10"
111
15"
111
20"
4
350
111
40"
111
60"
A
1
001 25"
A
- 1
A
001
101 001
45 "
50'
101 001 111
4
55"
A I 101
OC 65"
A A
1
001
111
6.03 3.31 1.82 1 .oo
min 0.02
Figure 9.22 MODF between two different phases in a nickel/tungsten composite. The MODF was computed between the orientations of adjacent grains of fcc nickel and bcc tungsten which were obtained by EBSD (See also Figure 10.7).
The texture components extracted, using an angular tolerance of 15", are shown in Table 9.3 and some of the main extracted components are shown separately on pole figures in Figures 9.23b-d. These are { 123}<634>, (21 1}<111> and a 11" rotated component of {211}<111>, {436}<323>. A comparison of Figures 9 . 2 3 ~and d shows that on the basis of the pole figure these last two components are hard to separate. However, analytically they can be distinguished. A few orientations could be classified as either (21 1}<111> or {436}<323>, based on a 15" spread. For these cases the orientation was classified according to the smaller deviation angle. The analytical procedure involved a search through a look-up table of specified orientations. Many commercial software packages include routines for similai calculations. Note that each crystallographically-related solution was searched for separately and, although all the variants are included together in Table 9.3, it turned
242
INTRODUCTION TO T E X T U R E ANALYSIS c
3
on C .-m ui
2 on
c a
0
e,
-
0
C
d
0
0
zal X
c L
EVALUATION AND REPRESENTATION OF MICROTEXTURE DATA
243
Table 9.3 Proportion ( % ) of grains representative of each n m n texture component from the data in Figure 9.23a (Davies and Randle. 1996) (001)
(00l] <310>
(103; <311>
jl0l)
(211)
j123) <634>
{4363 <323>
Random
8.7
3.3
4.3
4.7
21.0
28.0
19.3
10.7
out that each variant Lvas not represented evenly. For example, for the (436)<323> component only two of the four variants were present. As far as quantification of links between orientation or misorientation and the n~icrostructureis concerned: the range of possible applications is so diverse that it is difficult to be prescriptive with regard to methodology. Usually. the required information is obvious in the context of the inquiry. and this is illustrated by the Case Studies. The application of microtexture analysis to orientation connectivity topics is discussed in Section 1 1.4.
9.10 SUMMATION The aim of this Chapter has been to provide a guide to the broad array of choices which exist for evaluation and representation of microtexture data. by giving a concise exposition of the principles and ~nethodologyof each. Formalism has been deliberately kept to a minimum. and the reader can obtain fuller information elsewhere (see Bibliography). Selection of a method for data representation and analysis \.\,illdepend on the nature of the inquiry and the user's preference. A key point is that single orientation and therefore their representation (and measurements are directlj. t1zr.e~-dinzc~risior1NI subsequent extraction of statistics) in pole figures. Euler space or Rodrigues space is entirely equivalent. As far as direct representation of misorientations is concerned, axkangle representations, in particular those in Rodrigues space: offer significant advantages over use of Euler space. When analysing microtexture data, it must always be borne in mind that the quantification can be by rolzme (or area for interfaces) or by nuniber. As a consequence of ongoing improvements in the speed and automation aspects of EBSD analysis, this technique is now capable of providing an acceptable alternative to rnacrotexture for the measurement of overall texture. provided that attention is paid to appropriate sampling schemes and population size. The EBSDbased method is free from the errors introduced in the series expansion method (Section 5.3) and the cost of a standard SEM plus EBSD system is considerably less than a full X-ray texture goniometer. Furthermore, EBSD of course gives access to a whole range of local orientation studies, which are usually the main focus. On the other hand, potential disadvantages of EBSD to measure overall texture are that only surface p i n s are sampled (although sequential surface depths could be prepared). all grains must provide indexable patterns, which restricts deformation levels, and X-ray macrotexture determination is much faster. It is probable,
244
INTRODUCTION TO TEXTURE ANALYSIS
therefore. that in the future both techniques will be used to generate overall textures. Neutron diffraction, with its greater depth of penetration (Section 4.4) offers considerable advantages over EBSD, although these are offset by the inconvenience associated with the nature of neutron production. Often, it is expedient to use a multi-scale approach in an investigation and combine more than one texture technique (Section 12.4; Case Studies No. 9 and 10).
10. ORIENTATION MICROSCOPY AND ORIENTATION MAPPING
10.1 INTRODUCTION The concepts of orientation microscopy and orientation mapping were introduced in Section 1.2.3. Orientation microscopv refers to the (usually) automated measurement and storage of orientations according to a pre-defined pattern of coordinates on the sampling plane of the specimen. The pictorial output of these orientations with reference to the sampling coordinates provides a map of the spatial orientation distribution, i.e. derives the 'orientation topography'. Once this basic information, i.e. orientation plus position, is stored it can be processed according to e
The orientations themselves
e
Orientation changes
e
Pattern quality
In the final output various methods of colouring, symbols or shading can be used to convey and depict aspects of the orientation topography. These outputs are known collectively as orientation nzaps. The defining feature of orientation mapping is that it depicts the microstructure in terms of its orientation constituents. These orientation constituents, or subsets of them, are also available for output and analysis in any of the ways described in Chapter 9. Hence. not only does a map have great visual appeal but it also gives access to fully quantified orientation statistics. In this Chapter, after first considering the historical evolution of orientation mapping (Section 10.2) and also experimental features of orientation microscopy using EBSD (Section 10.3), the types of orientation mapping output and the information entailed in them are discussed in Section 10.4. With regard to the potential for orientation microscopy and orientation mapping in the TEM, at present TEM diffraction patterns can be solved automatically on-line, which is discussed in detail in Section 6.3.4 (Zaefferer and Schwarzer, 1994a; Weiland and Field, 1994; Krieger Lassen, 1995; Engler et al., 1996b). Briefly, SEM
246
INTRODUCTION TO TEXTLRE ANALYSIS
methods cannot be adapted directly for TEM, mainly since the background varies across the diffraction pattern. diffraction spots are superimposed on the pattern and excewdefect (bright,'dark) Kikuchi lines have to be considered. At the time of writing automated orientation microscopy in the TEM is still only in the early stages of development and TEM orientation maps shown in the literature are usually obtained by manual positioning of the electron beam (Juul Jensen, 1997; Schwarzer, 1997; Case Study No. 8). 1997a: Zaefferer rt d..
10.2 HISTORICAL EVOLUTION The practice of linking microstructure and orientation in a map has existed ever since individual orientation measurements have been made. although it has not been reported frequently. A typical procedure would be to observe or record the grain structure, measure orientations of individual grains one orientation per grain, usually - then coinpile a map of the orientations (andlor grain boundary misorientations) superimposed on a diagram of the grain structure. Typically orientations were classified into groups so that they could be represented in a simple manner by symbols, letters. patterns or colours on the diagram, the last giving a 'stained glass window' effect. Figure 10.1 is an example showing the distribution of S. brass and rotated cube orientations i11 an alu~l~iniurn-lithium alloy. A more advanced procedure involved using a computer algorithm to assign a colour to a grain depending on its orientation contrast in a forescatter image (Day, 1994. 1998). Although the method described above fulfils the criterion of linking microstructure and orientation. the production of such a map essentially involves a postprocessing step. That is to say. the components of the microstructure and the orientations are first identified separately and then combined to produce the map. The technique therefore has to assume that orientation components can be --
Figure 10.1 'Pre-orientation mapping' diagram representing the distribution of brass (B), S and rotated cube (R) components in an aluminium-lithium allo). Low-angle and high-angle boundaries are depicted with dotted and full lines respecti~ely(Randle, 1995a).
ORIENTATION MICROSCOPY AND ORIENTATION MAPPING
247
recognised by their appearance in the microstructure after etching, or a similar process. or by interactive exploration using EBSD. TEM or SAC. Truc orientation mapping takes a fundamentally different approach: diffraction patterns are collected according to a grid, and the spatial arrangement of the orientations thus obtained constitutes directly the map. To the best of the authors' knowledge, the first time a true orientation map was published was in the early 1970s. where the technique was referred to as orierztation tol~ogruplzy(Gotthardt et d.. 1972). The map was collected from rolled copper using SAD in the TEM. with n square grid spacing of l pm. Figure 10.2a shows the map together with a pole figure (Figure 10.2b) which provides an orientation key to the symbols used. The map, which relied on manual diffraction pattern evaluation: is certainly crude by modern standards but it does embody the filndamental orientation mapping principle of constructing a diagram of the microstructure based on orientation measurements collected on a regular grid .sa~zpli?zgscherlulc~. Furthermore, each measurement encompasses qunnt(fi'catior~of t l v orierztatio~i,which is conveyed in this case by a pole figure. Preparation of a manual orientation map would entail an immense amount of work. For example, a square grid of l pm-step sampling points measuring 100 pm X 100 p requires 104 orientation measurements. This can only be achieved
Figure 10.2 (a) Early orientation map. collected manually in the TEM from rolled copper using a square grid size of l pm in the rolling plane. The symbols on the nlap refer to the pole figure in (b) (Gotthardt et al., 1971).
248
INTRODUCTION TO TEXTURE ANALYSIS
realistically by an automated system, i.e. one in which both the sampling coordinates are located automatically and the diffraction patterns are solved automatically. Location of the sampling grid points is carried out by methods already in use for techniques such as energy dispersive X-ray analysis (EDX). The key stage in the development of orientation microscopy has therefore been provision of a reliable routine to solve diffraction patterns on-line, which was not achieved until the beginning of the 1990s for EBSD patterns (Section 6.3.3) and a few years later for TEM (Section 6.3.4). By the mid-1990s orientation mapping was realised as an extension to EBSD and various systems were commercially available. At the time of writing it takes a fraction of a second to solve and plot an orientation point. Orientation mapping in the TEM is not yet available as in the SEM, due both to the larger problem of automated pattern indexing in the TEM compared to in the SEM and the far fewer applications for mapping on the much smaller scale of TEM.
10.3 ORIENTATION MICROSCOPY The principal issues concerning orientation microscopy by EBSD, i.e. the actual collection of the raw data to produce a map, are e
Mechanisms to locate sampling points
e
Specification of grid step size
0
Data storage, display and retrieval
echanisms to locate sampling points There is a choice of two basic approaches for location of the sampling points on a specimen surface: e
The specimen is moved with respect to the stationary incident electron beam by means of a computer controlled specimen stage, i.e. Jtage confl-ol (e.g. Adams rt al., 1993). The measurement rate is currently up to 1 S-'.
e
The electron beam itself is deflected across the stationary sample surface, similar to conventional SEM, i.e. beam corztl-ol (e.g. Schwarzer et al., 1995). The measurement rate is currently up to 5 S-'.
Stage control can be performed with the help of a high-precision specimen stage that positions the sample relative to the electron beam. However in a modern computercontrolled SEM motorised sample motion is often a standard feature, so that additional hardware is no longer necessary. The main advantage of stage control is its ease of performance. With an eucentric specimen stage, the sample can readily be moved to the desired X,]. coordinates; otherwise the sample height, i.e. its z-position, has to be adapted to correct for sample tilt. Therewith, the sample is always located at the position for which calibration and focussing were derived, which means that large samples can be scanned in one run without the necessity to re-focus or to recalibrate the system.
ORIENTATION MICROSCOPY A N D ORIENTATION MAPPING
249
Bern c.o~?tr.ol is achieved by dedicated software to control the SEM deflection coils. The attraction of beam control is that the electron beam can be positioned much more precisely, and much faster, than by mechanical movement of the microscope stage. As the resolution of an SEM does not depend on the magnification, but only on the spot size, in principle large sampling areas can be scanned at low magnifications. However, scanning at low magnifications necessitates large deflections of the electron beam away from the image centre. Because of the fact that the specimen is highly tilted (e.g. 70"), the electron image of the sampling area may be distorted. Unless some dynamic correction is applied, the foreshortening effect will feed through into the map, as shown on Figure 10.3a. Dynamic focussing can either be accomplished by the SEM hardware or, if this option is not available or is not applicable for the corresponding experimental set-up, by an external software control (Schwarzer, 1997b). Furthermore, for the regions well away from the sample centre the calibration of the EBSD set-up no longer applies. Updating of the current calibration parameters can be accomplished by an automated calibration routine (Section 7.7.2) or, more easily, by interpolation of the parameters obtained for the start and end points of the scan. If such measures are not applied, sampling is restricted to areas of ii few tenths of a millimetre wide, which means that large area maps must be montaged. On the other hand, if the sampling area is limited to SEM images obtained at magnifications exceeding x 500, focussing corrections are largely unnecessary. Considering the advantages and disadvantages of stage versus beam control, it turns out that the optimum solution for large maps is a combination of both. Using stage control the sample is positioned at the region of interest, where the orientations are scanned in the desired grid by beam control, before the sample is mechanically moved to the next position and the process is repeated.
10.3.2 Specification of grid step size
The lower limit of the grid step size is defined by the spatial resolution of EBSD; hence steps of down to 0.1-0.5 pm are meaningful in a standard SEM (Section 7.6.3). The step size chosen will depend very much on the nature of the inquiry and effective use of resources. For example, the spatial distribution of microtexture components in a region can be obtained by using a step size of the order of the smallest grains whereas a more detailed map, in terms of grain boundary positions etc., will require a much smaller step size. Finally, fine detail of subtle orientation shifts within grains requires the smallest step size. Clearly, use of a too small step size is an inefficient use of microscope time since the map could take several hours longer to complete than is necessary and requires excess computer storage and post-processing time. Choice of a pertinent step size usually involves both a certain amount of preliminary experimental work to gauge the scale of the microstructure and thoughtful consideration of the inquiry in order to make an appropriate decision. For example, several maps of an area could be collected using different step sizes. Figure 10.3 contains examples of the effect of step size on the final orientation map. In Figure 10.3a a very coarse step size of 5 pm was used in order to give a rapid appra'isal of the microtexture throughout the entire thickness of a steel sheet. Here a post-processing step was used to combine the basic map with the secondary electron
250
INTRODUCTION TO TEXTURE ANALYSIS
Crystal Axis
‘1 A
A [OOlI Colour Key
ORIEKTATION MICROSCOPY AND ORIENTATION MAPPING
25 1
GB
(C)
1 2 3 4
5
6
7
8
9 1011 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 22232425262728293031
Sampling Distance pm
Figure 10.3 (a) Coarse step size (5 pm) orientation map from the entire through-thickness of a recrystallised interstitial-free steel sheet. superinlposed on the (tilted) secondary electron image. A colour key is i~lcluded.The large step size used is still sufficient to reveal that the < l 1 l > directions (blue) are inhomogeneously distributed throughout the sheet thickness; (b) 0.5 pm step size orientation map, specimen normal ( 2 )direction. in 20% deformed. 99.999% aluminium (colour key as for (a)). Inter- and intra-grain orientation perturbations around the two grain junctions are revealed: (c) Horizontal orientation scan. nine pixels from the top of the map in (b). showing orientation perturbations close to the boundary.
or backscattered image. which has the useful effect of smoothing the map and reducing the nun~bersof points required to give an acceptably 'unblocky' picture, hence saving time. Furthermore, the microstructure is related directly to the orientation map, and reveals at a glance that the microtexture at the sheet surfaces is modified compared to the midplane region. Figure 10.3b, on the other hand, was collected using a step size of 0.5 pm and is able to depict subtle orientation perturbations within grains. Orientation microscopy methodology employs either a square or a hexagonal sampling grid. The hexagonal grid has the advantage that each measuring point has six nearest neighbours whereas in a square grid four nearest neighbours plus four second nearest neighbours must be considered as shown on Figure 10.4. Furthermore. the hexagonal grid facilitates delineation of interfaces. Generally, the sampling area is scanned by a horizontal raster. and the orientation map builds up pixel by pixel as each orientation is measured. Clearly, stable microscope operating conditions should prevail while the orientation map is being collected. 10.3.3 Data storage, display and retrieval Each sampling point is stored with its orientation. spatial coordinates and often a pattern quality index. These data then allow the output of a variety of user-defined map formats, as described in Section 10.4. Moreover, it is important to realise that the orientation data which comprise the map are entirely quantitative and accessible for the various forms of statistical output as described in Chapters 2 and 9 (e.g. Case
INTRODUCTION TO TEXTURE ANALYSIS
Figure 10.4 (a) H e x a g o ~ dgrid and (b) square grid. where orientat~onmicroscopy sampling points have six and eight nearest neighbours respectively; (c) orientation microscopy recognition of a grain boundary as a region bet~veenneighbouring pixcls having an orientation difference greater than a pre-set value.
Study No. 3). For example. the graph in Figure 1 0 . 3 has ~ been formulated from orientation data which comprises the map in Figure 10.3b. It may be appropriate to select all or only part of the data for statistical output. For example. if all the data points are included, the microtexture thus obtained is weighted according to the projected grain area in the section plane, i.e. it represents a 'texture by volume'. If one orientation per grain is selected then the texture is weighted according to numbers of grains, i.e. 'texture by number' (Section 9.2.2). Orientation microscopy and orientation mapping when applied to multi-phase materials require some additional considerations since there are technical challenges involved in the simultaneous automatic pattern recognition of two or more phases (Field et d., 1996). These issues include:
ORIENTATION MICROSCOPY AND ORIENTATION MAPPING
253
e The 'background' diffraction signal may vary for each phase, which can be
addressed by various hardware or software solutions. e The coinputer algorithm needs to recognise correctly each phase. To search
simply for the 'best fit' solution may have an unacceptable time penalty, and so a 'confidence index' approach may be used. This. however. requires some preliminary work to asses the confidence index parameters of each phase in the material (Case Study No. 6). a
Following from the previous point, multl-phase materials where the phases have the same crystal structures clearly cannot be analysed automatically.
e
Specimen preparation for EBSD of multi-phase materials, where the constituent phases polish at different rates, may be more complicated than for a single phase matrix.
10.4 APPLICATIONS O F ORIENTATION MAPPING The applications of orientation mapping can be categorised, depending on the information which is being sought, as follows: e
Spatial distribution of texture components
e
Misorientations and interfaces
e
Orientation perturbations within grains
8
True grain size,'shape distributions
8
Deformation (i.e. pattern quality) maps
Table 10.1 illustrates the applications of orientation mapping by summarising a brief selection of investigations. The information includes: The material and topic studied (column 1) 8
The area scanned and sampling step size used to produce the map (column 2)
e A brief description of the value of orientation mapping to the investigation
(colun~n3) e
Main application(s) of orientation mapping in the particular investigation, categorised as 1 to 5 above (columns 4-8)
e
Publication reference (column 9)
The data in the table highlight the versatility and range of orientation mapping. The materials illustrated are diverse, ranging from metals such as aluminium and nickel, to diamond and minerals. The topics covered reflect the close links between microstructure and history (e.g. twinning), processing (e.g. deformation. recrystallisation) and in-service behaviour (e.g. stress-voiding, fatigue). Orientation mapping is rapidly growing in popularity and there are many more examples which could have
254
INTRODUCTION TO TEXTURE ANALYSIS
Table 10.1 Selected example5 illust~atmgthe scope of o l ~ e n t a t ~ omapping n (see text) M,rteiial & t o p ~ c
Map dsta~ls
Orientat~oninappmg contribution
Proccicmg of aluminium
1600 polnis at 30. 10 and I 5 pm steps
Reconatruct~onof g a i n shapc. o r ~ c n r a t ~ o n m the correlation, ~ h o nsome period~c~t) nncrostriictule, comparison betnesn microtexture and macrotexture.
Deformailon at gram junctions in pure aluminium Recr! stall~sation111 cxtiuded aluminium alloy
Shoa, largci or~entatioiiperturbations near gram junctions than nithm grams; inhomogeneous stram d i i t r ~ b u t ~ obetneen n g r a m at a lunct~on. 25 lim and 40 pm steps
1996
Largest cube grams and non-cube grains are the same size. recr)stalllsation texture from or~entatednuclemon
er a / ,
Defommtion m aluminium alloy
Correlates g r a m in deformed structure a ~ t h original grams: shows an orlentation ipredd up io 25" within grains; probes d~sagrcemcnt between experimental and modclled textures.
Panchanadcesnaran er al.: 1996
Recrystalliiat~onin sxtruded aluminiunl allo)
Grams close to cube texture nucledte faster than non-cube grams: exidencc for oriented i~ucleation.
er al..
Doherty 1992
Samajdar
Intergranular cracking In aluminium alloy rolled plate
not quoted
V~sual~satmn of cracks a\oidiiig grain boundaries ii hich ha\ e onc plane near [ I l l }
Field ei al.. 1995
Stress-voiding failure in aluminium interconnects
0 4 pm steps
Identification of those grain boundarres pronc to electromigrat~on.
Field and Dinglc). l995
Grain boundaries and gram growth In A-Rlg
00225mm',O8pm cteps
Some 1 3 boundanss h~ghl! nlobiic. lowanglc boundaric\ l c s inoblle than randoin boundaries. Tracking of gram break-up durmg I I I situ itramlng. correlation between orientation and strain pattern. also surface roughness
Structure e\.olution during deformation of A-j%\lg Tnin break-up m copper after deformat~on
l mm2.1 pm steps, 50"h dcforniation brcaki a twm 15.000 o r ~ e n t a t ~ o i ~ s boundary into { l 10) and :211) components.
Recrystdllisat~onin homogeneousl) and inhomogeneouslq cold rolled copper
200 x 150 pm', l pm steps. 30.000 orientations
Cube-oriented grams marked irith black colour: the larger the d e l i a t ~ o nfrom thc exact cube orientatioil 1s. the bnghter the grams \+ere shaded.
Creep In nickel
I mm'. 400 grain boundar~es
Crackmg along non-CSL boundaries \~sualisationof affected boundaries' connccti\Iry.
Microstructure m unidirectionally cast nickel
not quoted
Shoms clusters of 'supergrain,' small grains with s~milarorientations ~ndicatinga preferrcd groat11 inechan~sm
Strcsi corrosion crackmg in nickelbased Allov X-750
D i s p l a ) ~connecti\~t).of the grain 10 liin stcps, 10 boundar) nctnork Propens~tyfor serial sections. 90.000 orientatlolls on cracking correlated a i t h gram boundary crystallographq zach
Orientat~onclustering in a nickel alloy Ingot
Many sect~ons. 10,000 orientatlons (1100-2400 grains) on each
In\estlgates clustering of grains, where clusters are separatcd b) h~gh-angle boundarm Strong fibre texture and low-angle boundaries aithin clusters.
Inteqrdnular stresscorrosmn cracking in Inconel
1.5-10 pm step i1zes
D~spla!s 'decision tree' of crack at triple junction, Cracks shonn by poor pattern quality. Cracks a o r d twin boundaries.
1
1 1
Huh cl a1 1998
Lshockey and Palumbo, 1997 Mason and Adams, 1994 Pan er a/ 1996
West and Adams, 1997
INTRODUCTION TO TEXTURE ANALYSIS
256
S
I
-J
Orientation [All E u k ] [Calolte (loo%]
-I :0
Eulrr I
40
20
60
10
80
Eulrr 2: Euler 3:I
0
2b
4b
sb
I!
-
I00
11
180-
lbd
180
li0
Figure 10.5 (a) Euler angle orientation map from deformed and recrystallised limestone. The map covers an area of 1 mm2 and was obtained with a grid step size of 4pm, giving 62,500 orientations altogether; (b) Euler angle colour key; (c) (0001) pole figure, obtained from the orientations in (a) and applying the Euler colour scheme (Courtesy of B. Neumann).
naturally deformed and recrystallised limestone, i.e. calcite, which has a trigonal crystal structure. In addition to the orientation information, the mapping has allowed grains to be defined very clearly, enabling lineation (elongated grains) at approximately 20" to the horizontal to be recognised (Jensen, 1998). The map is accompanied by a colour key for the Euler angles (Figure 10.5b) and a (0001) pole figure (Figure IOSc), applying the Euler colour scheme, which shows that the texture is very weak. User-defined colours are added to maps in a post-processing step. This is demonstrated in Figure 10.6a where cube and R-oriented grains in recrystallised aluminium have been coloured red and blue respectively both in the map and in the accompanying pole figure (Figure 10.6b). The respective colours fade with increasing angular deviation from the ideal cube and R orientation. Note that all four crystallographically equivalent components of the R orientation had to be identified to mark all R-orientated grains (Section 2.2.3). A multi-phase system can be represented in the same way, and Figure 10.7a shows a map obtained in a two-phase nickel-tungsten material. Orientations that could better be indexed for fcc crystal structure, i.e. those
0 I< I E N TAT I0 N M I C It0 SCO I'Y A N I > 0 I< I I I N TAT I0 N M A 1'1'1
N
351
111 2100 prn
-
boundary levels. 3.0' 15.0' 70 slcps Orirntalionr
which are more likely t o belong t o the nickel. are coloiired blue wliercns the bcc o ric n t ;i t i o 11s ( i .e . t i t rigst e n ) ;I re co I o i t red ye1I ow . F 11rt lie rtii o re. 11i g 11 ;I 11 g le bo 1111d ;i r ies arc highlighted (blnck lines. Section 10.4.2). This representation displays ~ ' ~ ' 1 tlie 1 topographic at-rangemetit of tlic two phases. and the respective volume fractions c;in also be estimated. In the present csatiiplc i t turns out t h a t tlie minority component, the tungstcn. is situated prcl'crcntially at tlic nickel grain boundaries. 10.4.2 Misorientations and interfaces Regions o f oricn t ;I t io ti clia ngc ;I re i m mcd ia t cl y obvious i 11 ;I s t ;I nda rd orion t ;I t io n map bccaiisc they arc associated with ;I colour change. The presence o f these orientation changes is enhanced by instructing t h a t ;I line is dra\vn bet\\wn all neighbouring points \\,liere tlic misoricntatioti exceeds ;I certain value. ;IS slio\vn schematically i n Figure 1 0 . 4 ~ . Usually. ;I lienvy blnck line is chosen if tlic misoricntntion csceeds 15' . i.e. t o denote ;I high-angle boundary. and ;I low-angle bounchry can then be depicted by ;I thinner line ;IS s1ion.n in Figitre 10.h.Phase boundaries can be rcprcsented similarly. Alternntivcly. misoricntation axes can be shown r;i t her t h a n misorient ;t t ion ;i ngles ( Schwa I-zeI-. I 99 7a ) . 111t ra-gra i 11 v;i ria t io 11s can also be represented: if small tiiisol.ictitations bet\vccn neighbouring pisels ;ire mapped. typicall!. -2 . i t is possible to represent the locnl dcl'ormcd state of tlie microstriicturc (I
INTRODUCTION T O TEXTURE ANALYSIS
258
m (a) phase
I (b) image quality
Figure 10.7 (a) Map highlighting different phases, fcc (blue) and bcc (yellow), that were automatically detected through scanning a two-phase nickel-tungsten specimen. (b) Image quality version of the map in (a). Darker shading corresponds to poorer image quality. Note that grain and phase boundaries were not specially marked, but show up due to their low image quality.
in a map. Similarly to orientation connectivity, the linking of certain boundary types is then apparent. Figure 10.8 shows a map including a crack, which displays directly as a wide black region in the map since no diffraction patterns could be sampled from the cracked region (Pan et al., 1995). C3, E9 and C27 boundaries are superimposed in red, green and yellow respectively. This is an important aspect of interfacial studies, which orientation mapping facilitates, since the connectivity of ‘poor’ boundaries is a factor contributing to intergranular degradation and failure in service. The statistics of grain boundary populations, both in terms of frequency distributions and projected area distributions of selected boundary types, can be calculated from maps such as that shown in Figure 10.8. This Figure is also discussed with respect to orientation connectivity in Section 1 1.4. 10.4.3 Orientation perturbations within grains Automated EBSD measurements have highlighted that, although a grain is defined as a unit of microstructure having a single orientation, in reality it has an average orientation since even undeformed materials may show some lattice curvature close
ORIENTATION MICROSCOPY AND ORIENTATION MAPPING
259
1 - 100.0 pm
10 steps
bounda
Figure 10.8 Orientation map revealing a crack in a superalloy. The map consists of image quality shading, where the crack plots directly as a black region, and grain boundaries coloured red, green, yellow and black for C3, C9, C27 and random boundaries respectively (Courtesy of D.P. Field).
to grain boundaries, and deformed grains exhibit a whole range of substructures which are accompanied by orientation shifts (Figure 10.3b and Figure 10.9). Orientation mapping provides a means of studying these orientation perturbations when a very small step size is used (Adams et al., 1993). The results of such studies are helping to pinpoint mechanisms of strain accommodation which in turn have a
260
INTRODUCTION TO TEXTURE ANALYSIS
Figure 10.9 Orientation map across a grain boundary area in coarse-grained aluminium cold rolled to 50%. using colour and shading to depict deviations from various texture components. Parallel patterns of high and low deviation can be discerned, particularly in the green (brass, B) orientations (Courtesy of N.C. Krieger Lassen).
large influence over a wide range of thermomechanical processes including both commercial fabrication of materials and geological studies. Two examples of orientation perturbations, i.e. intra-grain orientations (Section 9.5.2) are given here. Figure 10.3b is an orientation map showing the accommodation of strain near a triple junction in an aluminium alloy and Figure 1 0 . 3 is ~ an accompanying horizontal orientation scan across one boundary (nine pixels from the top), where each point is characterised by the orientation relative to the endpoint orientation in each grain, i.e. points 1 and 29. Figure 10.9 shows use of a novel combination of colour and shading to show deformation substructure near a grain boundary in aluminium in terms of deviations from selected texture-components such as brass, Goss, cube etc.
10.4.4 True grain sizelshape distributions Access to a number of morphological parameters is another important strand of orientation mapping. Indeed, a map constructed with an appropriately small step size and focus correction is a replacement for a conventional micrograph (albeit obtained by a time-consuming route) and can be an improvement since aff boundaries are visible plus, if required, the intragranular structure. Those features which are represented in a conventional micrograph, i.e. grain size, shape, number of boundaries per grain, etc. can be computed from an orientation map (Geier et al., 1994; Day, 1998). Furthermore, features such as grain size classes of interest can be highlighted by colour or shading (Case Study No. 3).
ORIENTATION MICROSCOPY AND ORIENTATION MAPPING
26 1
10.4.5 Deformation (i.e. pattern quality) maps Besides the orientations of the individual sampling points, most automated EBSD systems provide a numerical estimate of the pattern blurredness or image quality IQ (Section 6.4). To display this information in a map, the available range of greyscale values (typically 256) is normalised to the range of image quality values in the given map, so that points with minimum IQ are shaded black and increasing brightness indicates increasing IQ. Figure 10.7b show the same map as in Figure 10.7a but plotted with image quality shading rather than colouring the points according to their crystallographic phase. This map displays well the different microstructural features. The bcc tungsten phases generally have different pattern qualities than their surroundings but, as mentioned in Section 6.4, differently oriented grains of the nickel matrix may also depict differences in pattern quality, so that the various grains and phases can readily be distinguished. Furthermore, patterns obtained at (or very close to) grain and phase boundaries are characterised either by poor pattern quality or no difraction pattern at all. and hence appear as dark lines in the map. The representation using pattern quality is of particular interest for deformed materials. As deformation in the microstructure produces degradation in the diffraction pattern quality, this parameter can be used to produce an alternative type of orientation mapping which highlights regions of specific deformation. e.g. strains associated with welds. etc. (Adams et al., 1993; Wright, 1993). Figure 10.8 includes both image quality shading and CSLs. 10.5 SUMMATION Orientation mapping has opened up many new possibilities for investigations of materials. Its central strengths are: aspect of orientation mapping is without doubt its most compelThe ~~isualisation ling and novel feature. The type of map is selected by the operator: orientations, grain boundary parameters, strain (from pattern quality) or combinations of these. A less conspicuous. but equally significant merit of orientation mapping is that it embodies total qua~itifi'cationof the orientation aspects of microstructure, which complements the visual appraisal aspects (Case Study No. 3). Hence texture or grain boundary statistics can be accessed from a map and represented by standard methods (Chapter 9). For example, the quantitative data in Figure 10.3~ has been extracted directly from the map in Figure 10.3b. Orientation mapping offers the ability to compact into one dataset measurements which had previously been obtained by disparate methods (e.g., grain structure, texture distributions and grain boundary types) or indeed had been virtually inaccessible (e.g., maps of local deformation within grains). Hence the need for elaborate correlation techniques between various separately determined parameters is obviated. The n ~ o ~ p h o l o gofy grains is depicted using orientation mapping, which allows a direct measure of the true grain size and distribution on the particular section
INTRODUCTION TO TEXTURE ANALYSIS
through the microstructure. This feature is especially relevant to materials where grain interfaces are difficult to observe in an image, e.g. low-angle or 'special' boundaries, materials such as superconductors which are unresponsive to etchants and materials such as aluminium whose microstructures are best revealed by deposition process such as anodising, which is unsuitable for EBSD. 8
The information contained in an orientation map depends on the scale qf' the nirasuren~entsand the nature of the inquiry. both of which are under operator control. For example the basic microtexture distribution in a specimen can be obtained by using a sampling step size of the order of the smallest grains over a region encompassing many grains, whereas fine details of orientation shifts close to grain boundaries can be studied in the same region of specimen by preparing a map which has a much smaller step size. typically < l pm. Most methodologies will involve preliminary explorations to overview the microstructure and microtexture, followed by the data collection to construct maps from selected regions and,'or using different step sizes.
On the other hand drawbacks of orientation mapping are that it is inefficient in terms of SEM usage and large amounts of redundant data are often generated. This emphasises the need for thoughtful design of an orientation mapping investigation, both to provide data which are pertinent to the inquiry and to make effective use of resources. For exanlple, a coarse-scale map such as that in Figure 10.3a will often provide sufficient microtexture information even though it is not as aesthetically appealing as a fine-scale map. If grain misorientations and grain size information is required. then a one-dimensional 'linear-intercept' type of map can be constructed which will contain the relevant information.
11. CRYSTALLOGRAPHIC ANALYSIS OF INTERFACES, SURFACES AND CONNECTIV
1 1.1 INTRODUCTION A spin-off to the spatial specificity of inicrotexture measurements is that planar surfaces or facets can be analysed in some crystallographic detail. These surfaces include e
Grain boundaries
e
Phase boundaries
e
Slip traces
e
Cracks
e
External surfaces of the specimen such as facets or fracture surfaces
Representation of grain and phase boundaries in terms of misorientation alone is achieved via the usual orientation parameters. namely the Euler angles, anglelaxis pair or Rodrigues vector (Chapter 2). The most popular extension categorisation of misorientation for grain boundaries is the coincidence site lattice model which is explained in Section 11.2.1. However, ,five independent variables are required to specify the crystallographic orientation of a surface (Section 11.2.2), and the misorientation only supplies three. A full analysis of a planar surface requires spatial information, additional to the grain orientation. in order to calculate the crystallographic indices. In this Chapter the methodologies of such investigations: and some applications, are described (Section 11.3). Although grainiphase boundaries and slip traces can be analysed in the TEM, there are considerable disadvantages, mainly in terms of statistical reliability. Furthermore, analysis of cracks or external specimen surfaces is not possible in the TEM because of specimen preparation restrictions. On the other hand SEM is ideal for studying the morphology of surfaces and so it is mainly EBSD that has been adapted to crystallographic studies of surfaces. Such applications of EBSD tend to be customised (e.g. Case Study No. 4) and are not yet mainstream as for studies involving only orientations and misorientations.
264
INTRODUCTION TO TEXTURE ANALYSIS
A inicrotexture derivative is that it is often pertinent to characterise how orientation components are spatially related in the microstructure. Such studies may involve only orientation information: but also may encompass more in-depth investigations, leading ultimately to a full three-dimensional crystallographic characterisation of the microstructure. These extensions of microtexture are discussed in Section 11.4.
11.2 CRYSTALLOGRAPHIC ANALYSIS O F GRAIN BOUNDARIES 11.2.1 The coincidence site lattice The categorisation of grain boundary misorientations as anglelaxis pairs is often supplemented by further classification according to the coincidence site lattice (CSL) model, especially for cubic materials. This is the most commonly used classification tool for grain boundaries in cubic materials. It owes its popularity both to convenience, since the analysis is easy to apply, and to the fact that some CSLs (Randle, 1996) may be so-called 'special' boundaries, i.e. those often having beneficial properties associated with them, such as resistance to intergranular degradation, which is a major cause of component failure in service (Aust, 1994). A 'coincidence plot', i.e. the interpenetration of two lattices of the same crystal structure as shown in Figure 11.1, is the most visual way to illustrate the concept of a CSL. For certain discrete misorientations between interpenetrating lattices a proportion of lattice sites will coincide. forming a periodic sublattice in three dimensions (Grimmer et al., 1974). The parameter C is the volume ratio of the unit cell of the CSL to that of the crystal lattice, or equivalently C is described as the
..
CRYSTAL 1 CRYSTAL 2
OCSL
Figure 11.1 Formation of a coincidence site lattice (CSL) boundary. (a) Notional interpenetration of two neighbouring lattices misoriented by H CVW, where 0 and U V W are the angle and axis of misorientation respectively. (For simplicity, a tilt boundary is represented). ib) Tmo-dimensional representation of a lattice interpenetration showing how a misorientation of 36.9'/<100> leads to the creation of a C 5 CSL, i.e. l in 5 lattice sites from each crystal coincide.
CRYSTALLOGRAPHIC ANALYSIS
265
reciprocal density of coinciding sites. In Figure l l . l b 1 in 5 lattice sites are coinciding, so this is described as a C5 relationship. The actual grain boundary surface or boundarj. plane itself is realised as a plane running through the CSL as shown in Figure l l . la. On average, the plane will intercept 1 in C atoms. Boundaries associated with low CS, generally < C29. are of interest from a data processing point of view (Randle, 1996). C3 boundaries, of which annealing twins are a subset (Case Study No. 2), are of special significance because they are ubiquitous in some materials and they are very strongly associated with special properties (Lin et al., 1995). The requirement for a CSL analysis of a population of misorientations is to identify all rnisorientatio~lswhich are 'close to' low-C CSLs in terms of an angular deviation. The structural analogies between low-angle boundaries and CSLs led to the adoption for the CSL case of the following well-known Read-Shockley relationship which links the dislocation density d in the boundary with the Burgers vector b and the angular misorientation 6: 6 v,,
= b/d =
b/d
(low-angle) (CSL)
Thus v,. the maximum deviation from an exact CSL, corresponds to the highest denslty of dislocations possible in the boundary. The density of dislocations which can be accommodated in a CSL is related to its periodicity, i.e. C. The variation of v,,, with C is usually taken to be as C-'12, called the Brandun criterion, which corresponds to a relationship based on periodicity alone (Brandon. 1966). Hence
where v0 is a proportionality constant based on the angular limit for a low-angle boundary, i.e. typically 15". If C = 1 is substituted into equation (1 1.2), v,,, is 15", which is the low-angle boundary limit. Hence a low-angle boundary can be described as C l . The Brandon criterion has been adopted almost universally for deciding v,,,. Although other criteria exist for v,,, and arguably the Brandon criterion may not be the most accurate descriptor on a physical basis, for reasons mainly of consistency between investigations it is used almost universally as the CSL deviation cut-off (Randle, 1996). Another criterion, whose usage is becoming popular, suggests a narrower C-'l6 dependency (Palumbo and Aust, 1992). As for all other aspects of inicrotexture analysis, both statistical and spatial information regarding CSLs needs to be available for a full interpretation of the data. The statistics of CSLs. either as numbers of boundaries or weighted according to projected grain boundary length, can be conveniently displayed in angle,'axis space or Rodrigues space (Section 9.6 and Figure 9.19) or as frequency distributions according to C-value (Case Study No. 3). Although CSLs can also be displayed in the asymmetric domain of Euler space (Zhao and Adams, 1988): this method is cumbersome compared to other methods and so is little used. Orientation mapping allows connectivity between various CSLs to be evaluated according to numbers, grain boundary projected length and grain boundary type (Section 10.4.2).
INTRODUCTION TO TEXTURE ANALYSIS
266
More information on CSL formalism and application to materials can be found elsewhere (e.g. Brandon, 1966; Grimmer et al., 1974; Randle, 1993; Aust. 1994) as can details of other grain boundary descriptors such as the 0-lattice (Bollmann, 1970). These are not included here because the CSL is by far the most popular method for grain boundary misorientation analysis. The CSL model can be applied directly to cubic materials, and it can be modified according to the 'near' or 'constrained' CSL for application to materials having more complex crystal structures (e.g. Chan. 1994). 11.2.2 The interface-plane scheme A more advanced method of grain boundary analysis, which usually includes CSL classification. focusses on crystallographic analysis of the grain boundary plane. Five degrees of freedom are required to specify the boundary, and these are most familiarly approached from the ang1e;axis of misorientation viewpoint, sometimes including the boundary plane expressed in one grain. An alternative viewpoint is to use four degrees of freedom to specify the Miller indices or direction cosines of the boundary plane in the coordinate systems of both interfacing grains and one degree of freedom to specify the twist angle @ of both plane stacks normal to the boundary plane as illustrated in Figure 11.2. This is known as the interfuce-plune .scheme for describing grain boundary crystallography (Wolf and Lutsko, 1989; Wolf. 1992). The interface-plane scheme is particularly powerful for CSL boundaries where it is especially relevant to identify tilt and twist components, sincc thesc types are thought to have lower free volumes than boundaries having a totally random structure. Four classes of boundaries are identified: e
{ h l k l l l )= { / 1 ~ 1 1 ~and 1 ~ )Q, = 0; a sjwnzetricnl t ~ l boz~ndnrj, t STGB
e
{ h l k l l l )f {/i2k2l2}and Q, = 0; an aLrynzmetrical tilt hozlndury. ATGB
e
{ / z l k l l l )= {h2k212)and @ f 0: a t ~ i s bou~zdaq., t TWGB
e
{ h l l i l l l )f { h 2 k 2 h )and cP
# 0: a general or mndom bourzduq~
where h l k l l l and h2k& are the indices of the boundary plane expressed in the coordinate systems of both interfacing grains respectively. The tilt and twist boundaries are described in more detail as follows:
Tilt boundaries A tilt boundary has the tilt axis parallel to the axis of misorientation. A STGB has planes from the same family (i.e. the same form of Miller indices) on either side of the boundary whereas an ATGB has different plane types interfacing the boundary. Both of these tilts are recognised in the interface plane scheme by Q, = 0. For a particular CSL there are up to two STGBs; for example the C3 system has STGBs on {l 1 l ] and (21 1). There are an unlimited number of ATGBs. although it is practical to consider only those with planes having low Miller indices. If Q, = 0 and { / z l k l l l ) ;{h2k212)are different but commensurate. the boundary is an ATGB. A commensurate boundary arises when the ratio of the planar grain
CRYSTALLOGRAPHIC ANALYSTS
267
boundary unit cells is an integer G. The planar grain boundary unit cell area is proportional to the separation of planes parallel to the boundary. G is therefore given by the square root of the ratio of the sum of the squares of the Miller indices. For example, (7511 and { l 11) are commensurate planes because
i.e. an integer. These two planes constitute an ATGB in the C5 system. Since a CSL can give rise to two STGBs at the most but many ATGBs, this is one reason why ATGBs have been found to be ubiquitous in polycrystals. Twist botozduries
A TWGB has the twist axis perpendicular to the axis of misorientation, and is characterised by the condition {lzlklll) = {h21c2l2)and cD > 0. All the TWGBs for a particular CSL are given by the misorientation axis for each of the 24 crystallographically-related solutions (Sections 2.2.3 and 9.6.1). An asymmetric twist grain boundary is in fact better known as a random or general high angle boundary. Figure 11.2a shows a representation of an ATGB and Figure 11.2b depicts an asymmetric twist grain boundary. There is evidence that boundary planes which are near quite densely packed lattice planes (i.e. those with low Miller indices) are significant in polycrystals, particularly non-cubic ones (Randle, 1997, 1998). Such boundaries may be tilt or twist types. for example the ATGB in the C3 system having {110)1{411), indexed in both grains, or they may not have a periodic; i.e. CSL, structure at all. f i r example (1 1l ) , {100)2. Conversely some misorientations may be periodic in nature. i.e. a CSE, but the boundary planes are irrational. Provided it. is known whether a boundary is a STGB,
Figure 11.2 Notional formation of a grain boundary by the joining of plane stacks from adjacent grains, i.e. the interface-plane scheme (see text). (a) An asymmetrical tilt boundary; (b) an asymmetrical twist or general boundary (Courtesq of D.P. Wolf).
268
INTRODUCTION TO TEXTURE ANALYSIS
ATGB or TWGB, it is generally sufficient to specify the Miller indices of the plane in both grains alone, and omit a. The term 'special' CSL has been adopted to refer to those CSLs which do not have irrational boundary planes (Randle. 1997, 1998). In conclusion to this Section: the central tenet of the interface-plane scheme as applied to grain boundaries in polycrystals is to focus primarily on the crystallography of the plane, with the misorientation a secondary consideration. More details can be found elsewhere (Wolf, 1992; Randle, 1998). The practical aspects of how the indices of boundary planes, and also cracks and facets. are determined experimentally is discussed in the next Section.
11.3 CRYSTALLOGRAPHIC ANALYSIS OF SURFACES The measurement of grain orientation is made from a flat surface. giving the orientation of the crystallographic direction which is normal to this surface and two other directions which are mutually perpendicular to the surface normal. The orientation of other surfaces in the specimen may also be of interest. Such surfaces include: e
Microcracks. either intergranular or transgranular (Liu et al., 1992)
e
Fractures surfaces or facets, i.e. external surfaces of the specimen (Slavik and Gangloff. 1966; Randle and Hoile, 1998)
e
The plane of grain or phase boundaries (Randle, 1995a: Pan et al.. 1996)
Slip traces (Blochwitz et al.. 1996; Lin and Pope. 1996; Raabe et al.. 1997b) It is possible to extend techniques for measuring single orientations, most especially EBSD, to obtain the crystallographic indices of flat surfaces (i.e. planes) associated with the specimen. This strategy is a precursor to a three-dimensional view of microtexture, since some kind of sectioning or photogrammetric (fractographic) technique is required to obtain the spatial coordinates of surfaces which is then coupled with orientation information (Section 11.4). As mentioned in Section 11.1, TEM is not well suited to the crystallographic analysis of surfaces, although it is sometimes used to give detailed information on individual or small numbers of boundaries (Section 11.3.1). Specimen preparation restrictions preclude examination of facets or cracks. and usually only a feu?suitable grain:phase boundaries are found in each foil. By far the most convenient technique for crystallographic analysis of surfaces is EBSD, since all the relevant imaging capabilities are encompassed in the SEM (large depth of field, specimen tilt, stereo irnaging). The remainder of these Sections on crystallographic analysis will therefore pertain mainly to EBSD-based analysis. Surfaces associated with polycrystals are of two types, as far as investigation methodology is concerned: exposed or unexposed. Exposed facets. such as a fracture surface, can be viewed directly in the SEM whereas unexposed facets, such as grain boundaries or slip traces: are firstly revealed as an etched trace on a polished surface, a change of contrast in a backscattered image or by various orientation mapping
CRYSTALLOGRAPHIC ANALYSIS
269
options (Section 10.4). Different experimental approaches are required to deal with each type of facet. 11.3.1 Sectioning techniques
The principle of using sectioning techniques to obtain the orientation of a planar surface (e.g. a grain boundary or an exposed facet) relies on the trace of the plane being evaluated on at least two sections through the specimen. There are three stages in the analysis (Randle. 1993. 1995b): e
Acquisition of the inclination of the plane with respect to reference axes in the specimen
a
Measurement of the orientation of the grain(s) h ~ c habut the plane. relative to the ~u777ereference axes as the inclination measurement
a
Calculation of the direction cosines of the plane from the orientation and mcl~nationparameters
The plane inclination is described by the angles cr and 3. which are shown on Figure 11.3 with respect to the orthogonal reference axes XYZ in the specimen. Clearly. the evaluation applies only to surfaces which actually are planar, and not to curved surfaces such as grain boundaries immediately following recrystallisation. For convenience, X is horizontal in the microscope and EBSD measurements are made on the XY reference surface. Where the plane is exposed, e.g. a fracture surface, the right-hand part of Figure 11.3 is missing. Experimentally. the sectioning process is most usually carried out by the standard metallographic technique of mounting the specimen in a suitable medium, grinding down the exposed surface of the specimen followed by appropriate polishing/ finishing steps. These procedures can be performed manually or automatically using
Figure 11.3 Parameters required to measure the orientation of a surface. XYZ are the specimen axes. (4 and 3 are the angles relating the surface trace to the X-axis and T is the experimental section depth (see text). For an exposed rather than an unexposed facet, the right-hand part of the diagram (dashed lines) is missing.
270
INTRODUCTION TO TEXTLRE ANALYSIS
equipn~entdesigned for that purpose. A potential alternative to the traditional grinding method for sectioning is ultramicrotorny, i.e. sectioning with a diamond knife. This technique is a mainstay of TEM specimen preparation in the life sciences, and has been adapted for many materials applications ( alis and Steele, 1990).
Relatively hard materials can be polished on two mutually perpendicular and adjoining surfaces so that the plane trace is revealed on both surfaces as shown in Figure 11.3. 0 and 3 can easily be measured in the SEM by tilting the specimen about the X-axis so that both angles are visible together. The true angles. n r and 3T, are corrected to account for the microscope tilt angle Q as follows (Randle and Dingley, 1989): n~ = tan-' (tan a cos B) (1 1.4a)
JT
= tan-'(tan
3 ~cos(90 / - H))
(1 1.4b)
where u . ~ and 3.2fare the measured angles after tilting. An example of the application of the two-surface sectioning approach to interface crystallography is a study of fatigue crack initiation in copper (Liu et al., 1992). Fatigue cracks often initiate as a result of impingement of persistent slip bands against grain boundaries, and application of this method allowed the grain boundary crystallography to be determined and related to the slip band interaction.
The second procedure that can be used to obtain plane inclinations is to section the specimen parallel to the E SD measurement surface. i.e. the XY surface in Figure 11.3. If the section depth is known. then the plane inclination can be calculated from accurate measurement of the trace of the plane on the XY surface before and after sectioning. The section depth, where a layer of material is removed by grinding. can be measured by inserting hardness indents into the specimen surface at the start of the procedure. The reduced size of the indent after sectioning, coupled with the knowledge that the angle between opposite faces of the diamond indenter is 136", allows the depth of material removed to be calculated (Figure 11.4). The image of the hardness indents also facilitates accurate alignment between serial sections, which is crucial. The thickness of specimen removed, T, is given by
where Dh and D, are the average distance from the centre to the closest point on the edge of the indent before and after grinding respectively. Then if the offset on the specimen surface of a grain boundary before and after grinding is 0. 3 is given by
or
p = 90'
-
tan-' ( o / T )
(1 1.6b)
C R Y STALLOGRAPHIC ANALYSIS
27 1
Figure 11.4 Calibrated serial sectioning ~neihodologyshowing measurements obtained from the imprint of the hardness indent before and ai'rer sectioning (Courtesy of C. Hoile).
depending on the sense of the plane inclination in the specimen. a is measured directly from the boundary trace on the specimen surface. Case Study No. 2 gives more details of the application of this procedure. The principal disadvantage of the calibrated serial sectioning method is that, for an unexposed surface. planarity within the measurement depth is assumed. Several small depth sections can be made to validate this assumption, although the procedure then becomes very time consunling. In order to minimise measurement errors, the total depth of material removed should be maximised while remaining within the same grain. Hence there is a lower restriction on grain size of approximately 50 pm. The reliability of this method has been estimated to be f43, including the orientation measurement (Randle et al., 1997). Once the grain orientation has been measured near to the plane, its crystallographic indices are calculated as follows. Two vectors which lie in the boundary plane are given by A and B; which are obtained from a: and 3 respectively. The coordinates of A and B are given by
The cross product of A and B gives N , the plane normal, which can then be related to the crystal axes of the grains through the orientation matrix. The whole procedure is illustrated on the stereogram in Figure 11.5. For the case of the crystallographic analysis of slip traces, the slip plane family has been identified by a one-surface trace analysis: which can be adopted for other surfaces. This is carried out by aligning a set of slip traces in an individual grain with the X-axis in the microscope. The first column of the orientation matrix then gives the slip plane. By this method. slip traces in Ni3Al were confirmed using EBSD to be on {l 11) (Lin and Pope, 1996). Alternatively. if the angle between the slip trace and the X-axis is measured the identity of the slip plane assuming that the -
INTRODUCTION TO TEXTURE ANALYSIS
Figure 11.5 Stereographic projection shelving the relationship bet\\een XYZ. n. i (from Figure 11.3); the crystal axes of one grain and the boundary plane normal. N.
slip band is perpendicular to the specimen surface - can be calculated. Such an analysis cannot give full information without this assumption but nonetheless is useful for some applications and has the advantage of not requiring sectioning.
The most straightforward way to obtain a plane norinal in the TEM is to tilt the foil until the plane is upright in it, then obtain the plane nornial from the cross product of the foil norinal and the direction of the boundary trace. However, the projected grain boundary width has a minimum value oves several degrees which degrades the accuracy of this niethod. The most convenient and accurate method to obtain a plane normal in the TEM is to measure the projected width of a highly inclined plane in the foil coupled with measurement of the foil thickness and grain orientations, both using microdiffraction. More details of these methods can be found elsewhere (Randle, 1993).
11 .X Photogrammetric techniques Photogrammetry in the SEM is a well established technique for determining the coordinates of an exposed facet plane relative to a fixed coordinate system in the
CRYSTALLOGRAPHIC ANALYSIS
273
microscope (Hilliard, 1972). These data can then be coupled with diffraction data to give the crystallographic orientation of the facet. The photogrammetric procedure involves producing a series of micrographs of the facet using at least two tilt angles (e.g. 0°, 10°, 20" . . . ) and identifying the projected (X, J.) coordinates of point features in the plane, relative to the reference axes, at the various tilt angles. If the coordinates of one point at tilt angles HI and Q2 are (.vl.?;I) and ( . Y ~ . J .then ~ ) the true coordinates X T ~ T Z T of the point in space are given by (Themelis et ul., 1990): \C7 =
(.v1 sin 132
-
J'T = ?'g = 1'2 =
ZT = (-XI
.YZ
sin HI)/ sin(& - H I )
(yl + ?$/2 xz COS Q l ) /sin(& - H1)
COS 87
+
(1 1.8a) (11.8b) (1 1 . 8 ~ )
The coordinates of three points on the plane can then be used to compute the equation of the plane normal. This brief description of the photogrammetric procedure is amplified elsewhere (Themelis et al., 1990; Slavik et al., 1993). An extension of the photogrammetry technique is computer assisted sfereoplzotogrcmmefry, where a three-dimensional reconstructed image of the facet is coupled with EBSD diffraction information. The crystallographic orientation of the grain containing the facet is then obtained as a separate step, taking care to ensure that the specimen coordinate system is identical for both photogramsnetry and diffraction. A key point is, similarly to the sectioning technique described above, diffraction is performed on a surface adjncerzt to the facet, rather than on the facet itself. This allows the fracture surface to remain undisturbed while a full rnetallographic procedure is used to prepare the adjacent surface for diffraction. Figure 11.6 shows schematically the set-up for obtaining EBSD from a fractured specimen. Once both the equation of the plane normal and the grain orientation have been obtained, the crystallographic orientation of
srde view
Font view
Figure 11.6 Mounting of a specimen conlaining external facets for EBSD analysis (Courtesy of C. Hoile).
274
INTRODUCTION TO TEXTURE ANALYSIS
the plane is obtained by adapting the principle illustrated 011 the stereogram in Figure 11.5. The crystallography of facet planes is an important subset of materials studies, concerning e.g. brittle fracture, fatigue and stress corrosion cracking. Two examples of the photogrammetric approach to these studies are with respect to fracture surfaces in Cu-Bi bicrystals, using Laue back-reflection (Themelis et al., 1990) and fatigue fracture in AI-alloys. using EBSD (Slavik and Gangloff. 1996). It is occasionally possible to align the facet surface so that diffraction information can be obtained from it directly. However. this method is only feasible for large surfaces, and often the deformation processes associated with the formation of the facet interfere with the clarity of the diffraction pattern. This method is not therefore suitable for general analysis.
1 1.4 ORIENTATION CONNECTIVITY AND SPATIAL DISTRIBUTION The topological parameters associated with poiycrystals. i.e. numbers of grains, and grain surfaces. edges and corners, have been studied for many years. Stereological techniques, i.e. the reconstruction of three-dimensional information based on that accrued from two-dimensional sections, have generally been used to investigate the topological connectivity of grains aggregated in a polycrystal (Rhines et d., 1976). The sectioning planes can be random. which gives a statistical result, or parallel to build up an accurate impression of the spatiality of the microstructure. These topological aspects of microstructure have traditionally been studied separately to the crystaiiographic considerations. However more recently the advent of allows large regions of n~icrostructureto be accessed in a single section in te both the crystallography and area1 topology. These studies access the way in which grains having various orientations connect together, i.e. orientation c o m c ~ c t i v i t j ~ . Another aspect of orientation connectivity is the spatial distribution of orientations with respect to some features of the microstructure, for example how grain textures relate to grain size (Vogel and Klimanek. 1996: Engler. 1998) or the clustering of grains which are separated by low-angle boundaries. where a single cluster is enclosed by high-angle boundaries ( est and Adams, 1997). cluster structure analysis is carried out usilig aspects of percolation theory, e.g. fractal analysis. Examples of specific orientation connectivity topics include: B
Grain shape. distribution and orielitation
o Particle distribution with regard to grains of particular orientations s
'Decision trees' for crack paths throughout the rnicfostructure
e
Grain boundary connectivity
e
Clustering of orientation elements
Although any of the a b o ~ examples e can be explored on a single section through the microstructure, if sectioning techniques are employed then three-dimensional aspects of orientation connectivity can be addressed u,hich render the data much more
CRYSTALLOGRAPHIC ANALYSIS
215
powerful. Methods for obtaining the indices of crystallographic planes and facets described earlier (Section 1 1.3) detailed the experimental methodology for calibrated, parallel sectioning. These techniques can be extrapolated to obtain volume information by performing a series of parallel sections which can be reconstructed or otherwise interpreted. Figure 11.7 illustrates this technique by showing part of a volume which measured 220pm x 150pm x 20pm in a recrystallised aluminium alloy. The objective of the study was to ascertain the role of particles in the recrystallisation process. The sample surface was removed in 1.5 pm steps, EBSD measurements were obtained and the planar sections reconstructed to give volume representations. Cube oriented grains, randomly oriented grains and particles are depicted in Figures 1 1.7a, b, c respectively. Analysis of the data showed that the cube grains were unrelated to particles whereas 50% of all grain had a random texture and these were nucleated at particles (Weiland et d., 1994). Another aspect of orientation connectivity relates to grub jirnctions, i.e. where more than two grains meet, often called a 'triple line'. This relates to the geometry of the (usually) three grain boundaries which conjoin to form the junction. An 'addition rule' applies which predicts the geometry of the third boundary from the relationship
M I M z M= ~ I
(11.9)
where M1-3 are the misorientation matrices of three boundaries at a common junction and I is the identity matrix. This rule has the most significant consequences for conjoining CSL boundaries since they share a common misorientation axis, the sum of two of the misorientation angles gives the third, and the product or quotient of two of the E-values gives the third (Ranganathan, 1966). An example is 60"/< 1 1 1 > - 2 1.8"/< 1 1 1 > = 38.2"/< I 1 I > i.e. C3 - C2lcr = C7
(11.10)
The interest in grain junctions from a practical point of view is that any interfacial propagation phenomenon (diffusion, corrosion, cracking, etc.) is controlled by the juxtaposition of boundaries at grain junctions. The phenomenon will only be likely to propagate along non-special boundaries, therefore it is pertinent to categorise grain junctions in terms of the combinations of special/non special types (Garbacz rt d . , 1995; Thomson and Randle, 1997b). Figure 10.8 is an orientation map showing a crack travelling intergranularly in a nickel-chromium-iron superalloy, and then becoming transgranular in the upper part of the map, and serves well to illustrate the concept of parallel study of topology and crystallography in an area of microstructure (Pan et NI., 1995). As the crack arrives at each triple junction, there is a 'decision tree' for its continuation, that is, if it is to continue to propagate intergranularly, there is a choice of two possible boundary paths. The crack is seen to be unable to propagate along C3 boundaries since these are 'special' (Section 11.2.1) and are depicted in red in the orientation map in Figure 10.8. Hence, to prevent the percolation of intergranular degradation, it is desirable to avoid a connected path of high angle boundaries in the microstructure. This example
CRYSTALLOGRAPHIC ANALYSIS
277
- 0.2544
mml
I
- OS6X !15
Figure 11.7 Volume of an aluminium alloy (dimensions in pm) with axes parallel to the specimen axes RD, TD, ND. A combination of EBSD and serial sectioning has allowed reconstruction of the microtexture and microstructure in three dimensions. (a) Cube grains only; (b) randomly oriented grains; (c) particles. Slightly different shading is used to depict where a feature cuts the surface of the volume (Courtesy of H. Weiland).
highlights the need to investigate not only which orientation components exist in the microstructure, but also how they are connected. Although orientation connectivity aspects of the microstructure can be addressed partially via a single section through the microstructure, far more in-depth data can be obtained from incorporating sectioning techniques to gain a three-dimensional viewpoint. This can be done either statistically or specifically. The Intercrystalline Structure Distribution Function (ISDF) and related Interface Damage Function (IDF) (Adams, 1986; Adams et al., 1990) take a statistical approach in order to couple the spatial distribution of grain boundaries with the misorientation distribution function (MODF) (Section 9.7). In one series of experiments, grain misorientations were measured on four section planes e.g. in plate material at O", lo", 30" and 60" to the plane of the plate. From these data MODFs for various interface normal inclinations could be plotted in Euler space, which yielded peaks of preferred boundary misorientation. Note that here the attention focusses on probabilities rather than on the crystallography of individual boundaries, and that the boundary orientation data relate to specimen, rather than crystal, geometry. This methodology has been used to analyse creep damage in copper (Adams et al., 1990).
278
INTRODUCTION TO TEXTURE ANALYSIS
The 'specific' approach to three-dimensional orientation connectivity allows individual orientation components of the microstructure to be investigated by sectioning the specimen and either overlaying accurately subsequent images o r maps of each section to form a volume impression of the microstructure, o r extracting relevant information from each section as shown in Figure 11.7. In another example, examination of crack paths similar to those in Figure 10.8 on sections of different depths show that the crack path changes from transgranular to intergranular as it penetrates the specimen (Pan er NI., 1996).
1 1.5
SUMMATION
This Chapter has shown how microtexture analysis can be extended to obtain crystallographic characterisation of other features, particularly grain and phase boundaries, cracks and facets and the whole connectivity of the microstructure. Such analyses, which have mainly arisen from EBSD, involve non-standard, multi-stage methodologies. No doubt over the next few years there will be a n increase in sophistication and availability of these approaches, which in turn will enhance our understanding of microstructure and properties.
12.1 YNTRODUCTTON In this Chapter we will introduce some methods for texture determination which do not fit into the earlier Sections on X-ray or neutron macrotextures or electron microscope based microtexture techniques. Although X-ray diffraction is usually associated with macrotexture (Section 4.3). such methods can in principle also be used for individual-grain. i.e. microtexture, measurements if it can be arranged that the necessary small volumes in a sample are irradiated with a sufficiently fine X-ray beam. X-rays generated in a sjw4r.otr.on offer this possibility since they are characterised by an intensity several orders of magnitude higher than those generated by conventional X-ray tubes: combined with a high brilliance, i.e. low angular divergence. Accordingly. kaue-patterns from individual cherrer patterns from small polycrystalline volumes can be obtained ereas the underlying methods, h u e and Debye-Scherrer, have been s (see Section 3.4), and have in some cases been used to obtain orientations in very large-grained specimens (e.g. Ferra-xi et al., 1971). the application of synchrotron radiation in texture research is a very new. promising field where at the time of writing first scientific pLi.blications just haIie appeared. AI1 techniques for texture analysis mentioned so far are founded upon the ffraction of radiation electrons. X-rays or neutrons at the crystal lattice. esides those diffraction-based techniques there are methods to derive macrotextures or microtextures which are not based on diffraction of radiation, e.g. ultrasonic or magnetic measurements or optical methods. Since such techniques inap still be of some interest for special applications, a short discussion is warranted (Section 12.3). Finally, in Section 12.4 the various techniques to determine macrotextures and local orientations which have been discussed in this book are summarised and their main advantages and disadvantages are compared. -
-
279
280
INTRODUCTION TO TEXTURE ANALYSIS
12.2 TEXTURE ANALYSIS BY SYNCHROTRON RADIATION Modern electron synchrotrons with energies of 60 keV and more yield a white spectrum of X-rays with intensities of 6 to 7 orders of magnitude higher than conventional X-ray tubes. combined with minimum angular divergence of less than 2 mrad. This unique combination of high intensity, small beam size and free choice of wavelength opens a wide range of possibilities for texture analysis. The strongly focused. high intensity. polychromatic radiation favours synchrotron X-rays for local analysis of individual crystallites by means of the Laue technique (Section 3.4), and this is discussed in Section 12.2.1. Alternatively, inonochroinatic radiation can be produced by using a monochromator crystal, and then synchrotron radiation can be used to generate Debye--Scherrer patterns in polycrystalline volun~es(Section 12.2.2). Synchrotron radiation yields a much greater penetration depth. up to the order of centimetres, compared to values of about 100 pm in conventional X-ray applications. This allows global textures of big samples to be measured, similarly as discussed for neutron diffraction experiments (Section 4.4). However, measurements can also be performed on a local basis in the interior of large bulk specimens, which will also be explained in Section 12.2.2. Thus, interesting applications of synchrotron radiation may include (Garbe et al.. 1997): e
Non-destructive local texture measurements
B
High resolution texture analysis
B
Investigation of texture gradients
@
hz situ texture measurements of texture transformations
e
Determination of misorientation and characterisation of grain boundaries, including boundary plane
B
Internal strain in selected local volumes
Furthermore, because it is possible to tune arbitrarily the synchrotron X-ray wavelength in a wide range, textures of non-centrosymmetric crystals can potentially be studied by means of observations near the absorption edge (Section 3.6.1) with high anomalous scattering where Friedel's law does not apply (Bunge and Esling, 1981). though this has not been tested so far. The main drawback of the synchrotron techniques is caused by the very limited availability of synchrotron sources in large scale research facilities, similarly to the case of neutron diffraction. Furthermore, interpretation of the data is much more complicated than for standard X-ray or electron diffraction techniques. Finally, unlike in electron microscopy, the microstructure of the sample cannot be imaged simultaneously with the orientation measurements, so that the microstructure has to be recorded prior to the synchrotron investigation. Thus, experiments using synchrotron radiation will most likely be limited to special cases that cannot be accomplished by conventional X-ray or electron diffraction techniques.
COMPARISONS BETWEEN METHODS
28 1
12.2.1 Individual orientations from Laue patterns The most common technique to determine the orientation of single crystallites by means of X-ray diffraction is the Laue-method which has been introduced in Section 3.4. In this method the use of white, i.e. polychromatic, radiation ensures that for each set of lattice planes X-rays will exist which fulfil Bragg's condition for diffraction. Upon intersection with a recording medium the diffracted beams generate characteristic point-patterns, e.g. Figure 3.6, which can be evaluated to derive the crystallographic orientation of the sampled volume. However, the beam diameter of conventional X-ray tubes is about 1 mm. As X-rays cannot easily be focused, small beam diameters of the order of microns w h i c h would be necessary to achieve spatial resolution competitive with electron diffraction techniques - can only be obtained by a collimation of the primary beam. This, however. results in a strong reduction in intensity and, consequently. in unacceptably long times to record the diffraction patterns. Even under application of rotating anode X-ray tubes a beam diameter of 10 pm would require illumination times of the order of 10 hours which makes large scale microtexture analysis an unrealistic task (Gottstein, 1988). Therefore. the Laue technique is usually only applied for analysis of single crystals or coarse grained structures with a grain size exceeding l00 pm (e.g. Ferran e t al.; 1971). To overcome these limitations. Gottstein (1986. 1988) suggested use of the enormous intensity of synchrotron X-rays for Laue orientation determination. Even with a collimation of the synchrotron beam to a size of only 5 X 5 pm'; which is sufficiently fine for many applications. illuminating times of the order of l s could be achieved. The sample was mounted in back reflection geometry on a motorised my stage which allowed scanning of the sample surface in a constant grid with steps of e.g. 10 pm. Fairly large numbers of orientations could be determined by recording the diffraction patterns on photographic films or X-ray sensitive image-plates, digitisation of the patterns and a subsequent, automated evaluation similar to that described for Kikuchi patterns (Brodesser e t al.. 1991). With the advent of CCD cameras it is now possible to record and interpret Laue patterns on-line (Wenk et al., 1997). For this purpose, the Laue patterns (Figure 12.1a) are converted to gnomonic projection (Appendix 4) where co-zonal poles ( l k l ) lie on straight lines (Figure 12.1b). Next the Hough transform is applied, converting lines to points (see Section 6.3.3). The computer then detects the points and calculates the interzonal angles which are used to index the patterns, so that finally the orientation can be determined. The sampling area investigated by .this method is given by the selected beam diameter which can be reduced to as little as a few microns. However, considering the large penetration depth of the high energy synchrotron X-rays, this means that highly elongated volumes of e.g. 10 X 10 X 1000 pm3 contribute to the Laue patterns, which prevents high resolution local texture analysis in bulk materials. To overcome this limitation, thin sections of the materials can be analysed in transmission geometry (Wenk e t al., 1997). This results in a spatial resolution of the order of 10 pm which is sufficient for analysis of many geological materials and many metals in the recrystallised state, but is definitely inferior to electron microscopy based
282
INTRODUCTION TO TEXTURE ANALYSIS
Figure 12.1 Laue patterns obtained with synchrotron X-radiation. (a) Digitised L a ~ i epattern from an olivine crlstal (the small cross represents the pattern centre): (b) pattern from (a) transformed into a gnomonic pro~ectionm ~ t ho\eriaid zones that were identified by a Hough transform (Courtesy of H.K. Wenk).
techniques for orientation determination. On the other hand, the signal-to-noise ratio of the synchrotron Laue technique is superior and, furthermore. sample preparation is easy and non-conductive materials can readily be analysed.
For texture analysis in fine-grained polycrystalline samples by means of the DebyeScherrer method with synchrotron radiation, a narrow beam of nlonochromatic X-rays is required. To achieve monochromatic radiation. the beam first passes a cooled copper plate which absorbs radiation with long wavelengths and then a monochromator crystal selects the desired wavelength in the range of 0.01 nmn0.1 nm (Poulsen and Juul Jensen. 1995). Finally, the monochromatic beam is collimated to the desired size and falls onto the sample investigated. where the Debye-Scherrer patterns are generated in transmission geometry (Figure 1 2 . 2 ~ ~ ) . Szpunar and Davies (1984) have demonstrated that it is possible to obtain reliable texture information from the Debye-Scherrer patterns which they recorded on photographic films. Nowadays, the diffraction patterns are either recorded with image plates or with a CCD camera (Section 4.3.4). For on-line analysis GCD cameras are superior as they provide digitised intensities with a high spatial resolution and a wide dynamic range within a few seconds. As discussed in Section 3.4, diffraction from a powder sample with random texture gives rise to a set of homogeneous concentric Debye-Scherrer rings which are related to the various reflecting lattice planes (Izlcl). If the sample has a pronounced texture, the rings display intensity variations which are characteristic of the texture of the sampled volunle (e.g. Wever, 1924). As an example, Figure 12.2b illustrates the
COMPARISONS BETWEEN METHODS
253
CCD-camera or image plate
Figure 12.2 Texture analy& by the Deble-Scherrer method with synchrotron radiation. (a) Schematic view of the experimental set-up. 28 is the scattering angle, a describes the sample rotation and o denotes the angle along the Dehye-Scherrer ring: (b) example from a cold rolled aluminium sheet (38% reducuon. layer 2850 pm from the surface. li = 5", exposure time 45 S. beam current 68 111A (Courtesy of 0. Mishin and D. Juul Jensen).
diffraction pattern recorded with a CCD camera from a 38% cold rolled aluminium sheet. After appropriate intensity corrections, the texture can be derived from analysis of the intensity \-ariations along the Debye-Scherrer rings. If samples with axially symn~etrictextures, e.g. wires, are analysed in transmission with the fibre axis normal to the incident X-ray beam, then a single set of DebyeScherrer rings contains already the necessary texture information. texture variations in small rods of Ni-Fe alloys. prepared by electroforming techniques. have been documented and represented in inverse pole figures (Biick1996). For non-axial textures one set of Debye-Scherrer rings is not strsin et d., sufficient but. in analogy to SAD-pole figures in the TE (Section 8.3.2). the sample needs to be rotated by an angle (Figure 12.2a) and in each setting a pattern has to be recorded (e.g. Kawasaki and Iwasaki, 1995; Poulsen and Juul Jensen, 1995: Wenk and Heidelbach; 1998). Geometric expressions for the transformation of the angles o; Q and i ~ !into the pole figure coordinates cu and /3have been given by Poulsen and Juul Jensen (1995) and Backstrsm et al. (1996). As already stated above. because of the extremely small divergence of the incident beam in synchrotron diffraction experiments, such diffraction patterns can be exploited to yield information on microstructure and texture of small volun~es, tenths of millin~etres.within bulk samples of centinletre size (Black et al., 1991). As illustrated in Figure 12.3, the radial width of the Debye-Scherrer rings is linked to the sample thickness along the beam path. For a Bragg angle of 5" and a sample thickness of 10rnn1 the width of the Debye-Scherrer rings is of the order of 2mn1, and the different parts of the Debye-Scherrer rings disclose the texture information of different layers within the sample volume. Thus, the spatial resolution in terms of the part analysed along the beam path is determined by the radial resolution of the Debye-Scherrer rings. Provided a CCD camera with a resolution of l00 pm is used, small volumes with a length of the order of 500 pm along the beam direction can be resolved. Reeves et d.(1996) obtained diffraction patterns from three different
INTRODUCTION TO TEXTURE ANALYSIS
Debye-Scherrer
sample thickness
A
Debve-Scherrer
l detector
Figure 12.3 Broademng of Debqe-Scherrer rmgs (grains sh'lded g r q conipllse ldent~calor~entations)
through-thickness layers in a 15 mm plate of the aluminium alloy AA2024. Garbe al. (1997) refined this method by introducing a conical Debye-Scherrer slit system in conlbination with a pinhole to define the local volume. With this set-up, pole figures of small volumes with size down to 50 X 50 X 600 p m h o u l d be distinguished.
12.3 TEXTURE ANALYSIS BY NON-DIFFRACTION TECHNIQUES The vast majority of techniques that are in use for texture analysis are based on the diffraction of radiation at the crystal lattice planes (Chapter 3). However. besides those diffraction-based techniques there are some other methods to derive either inacrotextures or microtextures which may still be of some interest for special applications. In general, such techniques make use of a crystal property being anisotropic in different crystallographic directions. For instance, measurements of u l t ~ ~ u s o ~velocity ic or ~ m g n e t i s m yield information on the integral orientation distribution, i.e. the macrotexture, of the sample analysed (Section 12.3.1). The optical reflecti1;itj. of a crystal depends on the crystallographic nature of its surface, which can be used to derive information on the crystallographic orientation of individual crystals (Section 12.3.2). Alternatively. it has long been known that some etclzir~gfechniqzres differently affect grains with different orientations, which can also be used to determine the orientations of individual grains. In the following Sections, some non-diffraction techniques and their main applications will be introduced. 12.3.1 Ultrasonic velocity
Methods to derive the crystallographic texture by means of ultrasonic measurements are based on the anisotropy of the ultrasonic velocity, i.e. on the fact that ultrasonic waves propagate with different velocities in different crystallographic directions. Figure 12.4 schelnatically shows the transit time per unit distance of ultrasonic waves (i.e. the inverse of their velocity or their sloitwess) in different directions of a sheet, represented in a polar plot. Both laser zrltrasorzic methods and, more commonly,
COMPARISONS BETWEEN METHODS
RD l
pronounced
Figure 12.4 Anisotropy of the ve1ocit)- of somld in different directions in a textured sheet (polar plot).
rlectronzugnetic acoustic transducers can be used to analyse the texture of sheet materials (Lu et al., 1997). Thus. ultrasonic measurements are techliiques for application to polycrystals that yield information on the macrotexture of the sheets. With measurements of the ultrasonic velocity only the fourth order C-coefficients can be derived. which give a quite poor, lowresolution description of the ODF. However. the fourth order C-coefficients disclose the elastic as well as plastic behaviour of sheet materials ( unge, 1982) and, hence. ultrasonic methods are of interest in industrial applications for rapid non-destructive testing and on-line detern~inationof sheet quality (Papadakis et 01.. 1993; Kopinek: 1994). For instance, in steel sheets a correlation between ultrasonic velocity and the formability parameters r and Ar has been established (Thompson r f al.. 1993). In aluminium the differences in ultrasonic velocity in the various crystallographic directions are smaller, but. nonetheless, the anisotropy in I--value and elastic modulus as well as earing behaviour could be determined (Thompson et al., 1993; Lu et U / . , 1997; Schneider and Osterlein, 1996). Note that the ultrasonic velocity also depends on internal stresses which renders texture analysis by ultrasonic methods in deformed materials more difficult. In conclusion^ within some limitations. ultrasoilic methods have been developed to yield quantitative texture data and, in particular, information on the elastic and plastic anisotropy of sheet materials.
12.3.2 Optical methods When optically anisotropic crystals are studied with polarised light In an optical microscope, the colour and the intensity of the reflected or transmitted light is dependent on the crystal orientation. To use this effect for orientation determination, thin sections of the sample are mounted on a microscope with a so-called
INTRODUCTION TO TEXTURE ANALYSIS
universal stage (U-stage) which allows rotation of the sample about four or five axes (similar to the texture goniometer). The crystal under investigation is rotated until its optical axis is aligned parallel to the microscope axis. The required rotation angles then yield the crystallographic orientation with regard to the external sample frame. This method finds wide applications in geology where most materials have low crystal symmetry and. hence. are optically anisotropic (Wenk. 1985; Bunge et al., 1994; Case Study No. 5). Owing to their high symmetry most metals are optically isotropic, but in some cases an appropriate chenlical etch can introduce an anisotropy of the optical properties which then can be used to derive the orientation of the individual crystallites (e.g. Nauer-Gerhardt and Bunge, 1986). Especially, anodical oxidation of aluminium and aluminium alloys ('Barker etch') causes the epitacticai growth of an optically anisotropic oxide layer on the sample surface. For pure aluminium and many aluminium alloys anodisation for about l minute in a reagent of 49% H 2 0 , 49% methanol and 2% H F with a voltage of 60-70V gives good results. Under polarised light the grains then appear with different brightness in accordance with their crystallographic orientation ( S ~ t r eet al., 1986; Kroger et al.. 1988). As an example, Figure 12.5 shows a micrograph of a partially recrystallised AI-Cu single crystal obtained by anodical oxidation. The sample was rotated such that the (single crystalline) matrix appears black, whereas the recrystallised grains that have nucleated at shear bands display different brightness according to their respective orientation (Engler et d . , 1993b). Anodical oxidation is used to assess the quality of aluminium foils for high voltage electrolytic capacitors, as the capacity of the final product is determined by the ratio between two recrystallisation texture orientations, the cube-orientation {001)<100> and the R-orientation {124)<21 l > , which can easily be distinguished under polarised light (e.g. Hasenclever and Scharf, 1996). Another optical method to determine individual orientations, which has been known for many decades, is based on the evaluation of etch pits (Kostron, 1950; Tucker and Murphy, 1952i53; Nauer-Gerhardt and unge, 1986). This technique has recently regained some interest via analysing the evolution of the Goss orientation during secondary recrystallisation in iron-silicon transformer steels (Bottcher et al., 1992; Baudin et al., 1994; Lee et al., 1995). Here: the etching attack is concentrated at points where dislocations intersect with the sample surface. Only those crystallographic planes whose normals correspond to the direction of minimum dissolution velocity are attacked. In the metallographic micrographs the etch pits appear as characteristic polygons, whose shape is determined by the crystallographic orientation (Figure 12.6). However, both the spatial resolution and the accuracy of orientation determination by this technique is limited, so that generally only qualitative or semi-quantitative conc!usions can be drawn. Case Study No. 11 provides more details on tile methodology. For copper and some copper alloys very satisfactory results can be obtained by a more recent method, the '(l l l)-etching' (Kohlhoff et al.; 1988). If copper is etched in hot concentrated H N 0 3 , only the { l l l)-planes are attacked. The < l lO>-directions along which the corresponding { l l l $-planes intersect form the valleys and rims of the deep etched surface. Studying such samples in an SEM permits analysis of
COMPARISONS BETWEEN METHODS
Figure 12.5 Microstructure of a partially recrystallised AI-1.8%Cu single crystal obtained by anodical oxidation. investigated with polarised light (80% cold rolled, annealed for 100s at 300°C). New recrystallised _grainsapparently nucleated at shear bands that proceed at an angle of approximately 35" to the rolling direction. RD.
structures down to a size of 1 pm. As an example, Figure 12.7 shows a micrograph obtained from a partially recrystallised copper sheet. It is obvious that the etch patterns differ in the various deformed grains which is a result of their different crystallographic orientations. Furthermore, some layers of the sheet reveal a totally different etching pattern. Here, recrystallisation has already started, leading to the formation of cube-oriented regions. A major advantage of this novel technique is the quasi-continuous imaging of orientations in the microstructure, which enables one readily to draw conclusions on the orientation topography. By comparing the etch patterns with an atlas that contains the patterns of all possible orientations the local orientations can be evaluated to an accuracy of 5"-10" (Wang et al., 1995). The main drawback of the { l 1 l)-etching technique - and in general of all chemical techniques for orientation determination - is their strong materials dependence. Suitable etchants to evaluate crystallographic features are only available for a few
INTRODUCTION TO TEXTURE ANALYSIS
Figure 12.6 Etch pits in recrqstallised A1-0.5%Mn (Courtesy of S. WeiB)
Figure 12.7 { l l l )-Etching of partially recrystall~sedcopper (0546 reduction)
materials. Methods based on the diffraction of radiation, in contrast. are not only much less material sensitive. but in general also much more precise, and therefore superior for orientation determination.
COMPARISONS BETWEEN METHODS
12.4 SUMMATION: COMPARISON AND ASSESSMENT OF THE EXPERIMENTAL METHODS FOR TEXTURE ANALYSIS
In this book, a variety of common techniques to derive the crystallographic texture of crystalline samples has been introduced. The vast majority of techniques for texture analysis is based on the difraction of radiation - neutrons, X-rays or electrons - at the crystal lattice, and the various techniques were summarised on Figure 3.12. A few other techniques make use of a crystal property being anisotropic in different crystallographic directions (Section 12.3). In the following, all the various techniques will be summarised and their main advantages and disadvantages will be compared. Techniques to obtain the texture of a polycrystalline sample by means of X-ray or neutron diffraction provide the integral texture of the polycrystalline array (Chapter 4). This nzacrotexture discloses information about the volume fraction associated with each orientation. which can be used to predict texture-related anisotropy of polycrystalli~ie behaviour during elastic and plastic straining, and also physical properties as magnetic permeability, thermal expansion. etc. (Chapter 1). Furthermore. analysis of the texture changes during the thermomechanical treatment of materials yields valuable, statistically relevant information about the mechanisms of the underlying metallurgical processes, particularly deformation, recrystallisation or phase transformations. In geology, macrotexture analysis can provide insight into the mechanisms that have led to rock formation millions of years ago. Because of the very limited access to neutron beam lines, macrotexture measurements will in most cases be performed by X-ray difJiaction. However, the much greater depth of penetration into the sample by neutrons offers some distinct advantages for texture analysis in large-grained or irregularly-shaped specimens, snlall volume fractions of a second-phase, samples with low crystal symmetry or multi-phase systems (Sections 4.4 and 4.5). On the other hand with a view to determine textures of small volumes X-rays are superior. Whereas in neutron difi-action the entire specimen volunle of typically several cubic centimetres contributes to the diffracted intensities, in standard X-ray techniques only volumes of the order of cubic millimetres are sampled, so that e.g. the textures of specific layers in a sheet can be discriminated. Synchrotrons yield a white spectrum of X-rays with intensities of 6 to 7 orders of magnitude higher than conventional X-ray tubes combined with minimum angular divergence. This combination of high intensity, small beam size and free choice of wavelength opens a wide range of possibilities for texture analysis (Section 12.2). Experiments that make use of sj~nclzrotrondiffraction represent a hybrid between the macrotexture techniques covering large bulk samples and the microtexture techniques (see below) that record the texture of volumes down to the individual grain level. As in the case of neutron diffraction, the main drawback of synchrotronbased techniques is the very limited availability of synchrotron sources so that experiments using this radiation will most likely be limited to special cases that cannot be accolnplished by other techniques but necessitate one of the characteristic advantages of synchrotron radiation. In contrast to the macrotexture methods. nzicrotexture or local texture analysis is performed by sampling the orientations of individz~ulsingle crystallites, e.g. grains, subgrains, etc. (The only exception to this is the selected area diffraction pole figure
290
INTRODUCTION TO TEXTURE ANALYSTS
technique (Section 8.3.2) which records the texture in a small contiguous polycrystalline volume). The main requirements for an experimental technique to determine local, usually individual, orientations are: e
The technique yields the crystallographic orientation unambiguously and with high accuracy
e
The spatial resolution of the applied technique is higher than the size of the microstructural regions of interest, i.e. in general better than 1 pm
e
It is possible to locate and image specific regions of interest in the microstructure of the specimen, from which orientation measurements can be obtained
9
Orientation measurement and evaluation is easy to perform and, to facilitate large-scale investigation and/or orientation microscopy and mapping, should be automated
The main characteristics which can be addressed by the methods for microtexture or local texture analysis are: a
The spatial arrangement of the crystallographic orientations in the microstructure, i.e. orientation topography
e
The intensity distribution of orientations in specific subregions of the microstructure
e
Orientation correlations, i.e. misorientations, between (usually) neighbouring grains in the microstructure
The most popular techniques for local orientation determination are summarised in Table 12.1. The Table, and other work on this subject (e.g. Dingley, 1981; Humphreys, 1984; Schwarzer, 1989; Gottstein and Engler, 1993), clearly show that individual orientations are determined most easily and with ~naximumaccuracy from electron diffraction patterns in an electron microscope (SEM, TEM). These methods are distinguished from others in Table 12.1 by high accuracy with errors of only a few degrees and a high spatial resolution in the sub-micrometre range. A further advantage of electron microscopy techniques is based on the fact that orientation and microstructure of the site of interest are analysed simultaneously with the same spatial resolution. The corresponding evaluation methods are nowadays highly standardised and automated (Chapter 6), so that local orientation and microtexture determination can be used for routine purposes. Use of other methods, such as selective etching tecliriiques (Section 12.3), Laue X-ray diffi-action (Section 3.4), and the micro-Kossel technique (Section 7.2), are restricted to some special applications. The main advantage of TEM-based techniques (Chapter 8) is their excellent spatial resolution, which is at least one order of magnitude better than for the SEM. With regard to the different techniques for determination of local orientations, evaluation of Kikuchi patterns obtained by means of microdiJ;f;actiol or convergeuzt beam electron d$ffi.uction (CBED) is advantageous compared to the selected area difiactiori (SAD) technique, as the former yield higher spatial resolution as well as much higher angular accuracy. However, only SAD allows direct measurement of
COMPARISONS BETWEEN METHODS Table 12.1 Overview of the most common techniques for determination of local orientations -
Method
Technique
Spatial Angular Application resolution* accuracy
TEM
CBED microdiffraction SAD
polycrystals (grains, subgrains. deformed microstructure, inhomogeneities. recrystallisation nuclei)
SEM
EBSD SAC mlcro-Kossel
polycrystals (grains, subgrams) polycrystals (grains) polycrystals (grains)
X-ray diffraction
Laue synchrotron conventional Laue
polycrystals (grains) s~nglecrystals, very coarse grams
Optlcal techniques selectlve ({l l l}) etchlng etch plts
polycrystals (grains) polqcrystals (coarse grains)
*Diameter of the minimum detectable sample area.
pole figures of small volumes in the TEM. With CBED, Kikuchi patterns from regions as small as a few nanometres can be obtained. This high spatial resolution also allows investigation of orientations in highly deformed samples. Finally, the TEM offers much more additional information on the microstructure, i.e. on precipitates, dislocations and dislocation arrangements, etc., than it is possible to obtain in the SEM. The major disadvantages of the TEM are the difficult and laborious sample preparation and the very small area of view, which is usually much less than 100 pm in diameter. This generally restricts the application of the TEM to few selected examples with only limited information on the statistical relevance of the effects observed. The limitations of TEM provided great incentive to develop SEM-based techniques to perform single grain orientation measurements in thick bulk samples, and the available techniques, miuo-Kossel difiaction, selected area chlnnneling (SAC) and electron back-scatter diffraction (EBSD), have been described in Chapter 7. A comparison of these three techniques clearly proves that the EBSD technique offers by far the most advanced possibilities due to its ease of performance combined with high accuracy and high spatial resolution. and is the principal technique used in the SEM nowadays. Sample preparation is not critical, electrolytical polishing or slight chemical etching is usually sufficient. The evaluation of the patterns is very much facilitated by the large steric angle obtained, which speeds up evaluation and reduces ambiguity of the determined orientation. A spatial resolution better than 0.5 pm even allows investigation of deformed samples, if the microstructure is comprised of undistorted (dislocation-free) volumes e.g. cells or subgrains - in excess of this size. Because of the lower signal-to-noise ratio of the EBSD patterns, the accuracy of orientation determination is somewhat inferior to TEM, but with an accurate calibration a relative error of 0.5"-1" can be obtained, which is sufficient for most texture applications. The development of fully automated EBSD has led to the implementation of orientation microscopy and orientation mupping (Chapter 10). The ability to produce automatically an orientation map, i.e. a depiction of the microstructure in terms of -
292
INTRODUCTION TO TEXTURE ANALYSIS
its orientation constituents, has greatly enhanced the popularity and potential of EBSD. Whereas the visualisation aspects of mapping contribute to its appeal, the opportunity for quantification of the orientation/microstructure relationship on a very fine scale is scientifically the much more powerful feature. Orientation microscopy and mapping in the TEM is not yet viable (Section 10.1). A further key feature of microtexture determination is that crystullogruphic unalj.sis of surjkces can be undertaken (Chapter 11). Such surfaces include grain or phase boundaries. cracks, slip traces and external surfaces such as facets or fracture surfaces. A limited amount of analysis of grainlphase boundaries can be carried out in the TEM, but most analysis, because of the three-dimensional nature of the surface/specimen geometry, is the province of SEM and EBSD. At present crystallographic analysis of surfaces by EBSD, especially in three-dimensions, is not exploited very widely but will almost certainly be a growth area in the future. As a resume of the comparison between microtexture techniques, it can be summarised that nowadays basically two techniques prevail for measurement of local orientations: TEM and EBSD in the SEM. Maximum spatial resolution is obtained by evaluation of Kikuchi patterns in a TEM. As soon as the microstructural regions exceed a size of about 1 ym, however, the EBSD technique in the SEM is most suitable due to its ease of use and full automatisation. Thus, EBSD is able to fill the gap between the TEM based techniques, which have highest spatial resolution but mostly lack good statistics: and the X-ray macrotextures yielding statistically reliable data but no information on the spatial arrangement of the orientations. Already much work is proceeding to expand further the capabilities of EBSD for microtexture analysis, and also to introduce EBSD as an alternative to macrotexture analysis to obtain overall texture data. These developments include improving the spatial resolution, accuracy and speed of operation via use of FEG SEM; and providing enhanced solve algorithms and faster processors respectively (Section 7.9). Hence, EBSD will find more application in areas which have traditionally been the province of either TEM or X-rays. It should be emphasised that the different techniques should not be considered as competitive but should be applied in a complementary manner. A combination of different techniques for orientation determination with increasing spatial resolution yields much more valuable information on the underlying mechanisms than is accessible by the sole use of the individual techniques. The Case Studies demonstrate a wide range of applications of macrotexture, microtexture and orientation mapping, and in several examples show the value of combining various techniques and analysis methodologies.
T CASE STUDIES
RI
E
In Section 2.2.3 the concept of ‘orientation variants’ was introduced. This Case Study presents extractions from two investigations where inhomogeneity in the orientation variants was observed, and illustrates how these can be recognised and quantified. The texture of as-drawn tungsten wire is important because it influences the high temperature properties of lamp filaments. Until the advent of EBSD, X-ray diffraction had been used to obtain texture information from large numbers of wires, 100-200 pm thick and having grain widths of less than 1 pm, mounted together (see Section 4.6). More recently EBSD has been used to obtain microtexture information from individual wires. Although the grain size in these specimens is small. it is clearly not possible to prepare TEM specimens from them, and so EBSD is the ideal choice of experimental tool. EBSD patterns were obtained from etched tips of 21 5 tungsten wires, and individual data points were smoothed to contours in pole figures (Troost et al., 1994). Analysis of the <110> pole figure obtained indicated. from the appearance of a pronounced maximum in the centre of the pole figure. the presence of a strong < 110> fibre texture. Furthermore (1 lk) and ( h h l ) tend to align parallelly to the wire circumference, producing texture components [l lO](l lk) and 1110](hhl). However, further inspection of the pole figure revealed that the intensity distribution on the inner ring was inhomogeneous. This means that, of the two possible variants of (1 lk) aiid (hhl), (11E) or (1 Ik) and (IzlzT) or (hlzl), only one of each was occurring. Turning now to a similar effect in a different material, the texture of superplastically deformed 8090 aluminium-lithium sheet changes considerably during superplastic deformation, aiid the texture changes are indicative of the processes which are occurring, i.e. grain rotation,’sliding or slip/dislocation creep. Measurement of microtextures, using EBSD, allows more insight into these processes. Grain sizes were generally less than 10 pm (Randle, 1995a). Figures l a and b show the microtexture 295
INTRODUCTION TO TEXTURE ANALYSIS
296
pole figures from contiguous grains in specimens deformed to true strains of 0.25 and 2.4 respectively. The orientations were analysed quantitatively by extracting peaks from the pole figures and calculating the proportion of each orientation,
Figure 1 (1 11) EBSD pole figures from contiguous grains in aluminium deformed to (a) 0.25 and (b) 2.4 true strain.
50
S1
S2
S3
S4
B1
B2
T1
T2
Texture category Figure 2 Proportions of textuie Lariants extracted from Figme 1
T3
T4
CASE STUDY NO. 1
297
allowing a tolerance of 15" (Section 9.9). The analysis revealed that the orientations present were S; (123)<412> or {123}<634>; brass, ( 1 10)<112> and (102)<221>, which is a twin of the rotated cube texture {120)<001>, and so will be designated T. The S and T orientation comprise four variants each. S1. S2, S3, S4 and T1. T2, T3, T4 respectively whereas the higher symmetry brass orientation comprises two variants, B1, B2. These ten variants are:
The pole figures in Figure 1 indicate considerable weakening of the texture as superplastic deformation proceeded. Moreover, the asymmetry in the pole figures indicates that not all the texture variants listed above are present in the microtexture. The proportions of each texture variant are shown on Figure 2. and it is clear that the specimen which has undergone only 0.25 true strain contains virtually only two different orientations in the region sampled. S4 and B2. whereas the specimen which has been highly superplastically deformed shows a greater spread of orientations. One implication of these differences is that the very high proportion of the same orientation variant will feed through to give a high proportion of low angle boundaries (62% for the low-strained specimen) which will in turn affect the ability of grains to rotate during superplastic deformation.
e
Suitability of E SD for wire specimens
e
Appraisal of microtexture pole figures and extraction of information
References Troost. K.Z., Slangen, .H.J. and Gerristsen. E., 1994, Mc~t.Sci. F O I W ~ , 1299. Randle. V., 1995a, Actu ,Wet. Mut..
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According to a recent literature survey, approximately only 8% of the reported investigations on the distribution of grain boundary populations in polycrystals analyse the boundary plane in addition to the niisorientation parameters (Randle, 1996). This statistic is partly a reflection of the technical difficulties involved in obtaining plane crystallography measurements with acceptable accuracy (Section 11.2). However, several workers have stressed the cruciality of deriving the boundary plane distribution to provide necessary insight into grain boundary behaviour. This Case Study outlines one such investigation in 99.99% pure copper with an average grain size of 240 pm (Randle et al., 1997). Twenty 1 kg Vickers diamond hardness indents were placed onto the sample surface, to be used as location markers, and grain orientations were obtained using EBS from approximately fifteen grains around each of the indents. Grain inisorientations were calculated for 255 boundaries which exhibited the classic straight appearance associated with an annealing twin (C3). In the sample population 207 boundaries were confirmed to be C3s. The SEM was directly linked to an image analysis system and each hardness indent and surrounding grain boundaries were recorded in bitniap format. The sample had two 10 kg indents placed into its surface which were then recorded using the largest possible magnification for best accuracy. Sectioning by metallographic grinding and polishing was conducted on the sample, the two indents were rephotographed and the difference in size of the indents before and after sectioning was then used to establish the depth of material removed. Each of the 1 kg indents were again recorded using image analysis software, with precautions taken to ensure that all microscope conditions were the same as for the ‘before grinding’ images. Figure 1 shows examples of the indents and traces of grain boundaries before and after grinding. The image analysis software allowed the images to be rotated such that the ‘before grinding’ and ‘after grinding’ indents could easily be superimposed onto each other. The resultant composite image was then used to measure the grain 299
INTRODUCTION TO TEXTURE AN-ALYSIS
300
Figure 1 Calibration indents and grain boundaries in copper (a) before and (b) after serial sectioning.
PROPORTION (%)
0
5
10
15
20
25
30
11Ill 11 23,17,17/775 332110,77 7741855 2211744 21 11552 5221441 31 11771 1101411 88111 1,22 551171 1 v)
w
5
a
772110,11 322111,44 3111755 11,77113,55 2111211 5441722 2211100 77511 1,11 111/511 Other ATB Twist Symmetrical Irrational
Figure 2 Distribution statistics for C3 grain boundarj planes in annealed copper and nickel
CASE STC'DY hTO.2
30 1
boundary trace displacement and hence calculate the boundary inclination. These data, plus the grain orientation. were used to obtain the grain boundary plane Miller indices in the coordinate system of both interfacing grains (Section 11.2). Errors in the measurement of boundary plane crystallography can arise from the three parameters which comprise the input data - the orientation measurement, the angle cl between the boundary trace and a reference axis. and the inclination angle 3 (Figure 11.3). With a carefully calibrated EBSD system a misorientation is accurate to 50.5". This feeds through to an estimated accuracy of &4" on the n~easurementof grain boundary plane crystallography. Three-quarters of the C3s in the sample population from copper were tilt boundaries on the {011$ zone. with 43% of these being coherent twins or nearcoherent twins on <2?1.17,17>~<775>~ planes. i.e. 8" from < l 11 >1<1 1l>,. The high population of near-coherent twins was thought to be due to modification of existing twins via oxygen take-up during the heat treatment. There is a general trend for lower energy C3s to occur more frequently although some plane coinbinations, such as <744>,<855>,. tend not to occur frequently even though they are associated with low energy. Figure 2 illustrates the distribution statistics along with data previously acquired for nickel for comparison. This case study denzonstrates .
..
e
The procedure for obtaining the Miller indices of grain boundary planes bp use of EBSD and a calibrated serial sectioning procedure
Q
Analysis of grain boundary plane data in terms of the interface-plane scheme
References
Randle. V.. 1996, T/w Role qf flze Coincidence Site Lattice iri Grain Bozi~zdcry Engii~eering,The Institute of Materials, London. Randle, V., Caul, M. and Fiedler. J.: 1997, Miwos. Mia-ouml., 3, 224.
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CASE STUDY NO. 3 USE OF ORIENTATION MAPPING TO INVES GRAIN BOUNDARY STRUCTURE EFFECT GRAIN GROWTH
Orientation mapping aids grain growth studies in two ways, which this Case Study will demonstrate: e
Concomitant grain size and microtexture components can be extracted
0
Grain boundary parameters, including CSL analysis (Section 11.2), can be related to grain size
Most such experiments are conducted in a 'post mortem' mode, that is, after grain growth has taken place. One example is the development of the Goss texture in Fe-Si transformer steels (Section 1.1) where CSL statistics have been compiled from a mapped area of specimen measuring 400 pm X 360 pm and an average grain size of 13 pm (Lin et al., 1996). Figure 1 shows the proportion of CSLs compiled from the potential orientation relationship between the experimentally determined primary matrix and the reference hypothetical nucleus orientation. The Goss orientation, { l 10)<001>, shows a higher potential for forming low-C boundaries during random growth than do the other reference orientations. Another experiment which took advantage of the capabilities of orientation mapping concerned grain growth in rolled A1-Mg sheet (Field et al., 1996). Orientation mapping was performed over a region of microstructure measuring 150pm X 150pm. The microtexture of the resulting grain size distribution was analysed by grain size. The data were 'texture by number' related rather than 'texture by volume' related (Section 9.2). The largest grain size category (> 50 pm2) had a random texture and notably no cube grains. Most of the grains were in the small category, i.e. < 20 pm2. The misorientations of neighbouring grains were analysed to determine if the mobile boundaries, i.e. those around large grains, exhibited a preferred misorientation dependence. The number of grain boundaries extracted from the map was over 2500, with 400 boundaries in small grain clusters and 250 boundaries defined as
INTRODUCTION TO TEXTURE ANALYSIS 14
Reference orientation Figure 1 Proportions of low C boundaries (C = 3-9) between several reference orientations and the 1996). experimentally determined primary matrix in a Fe-Si all01 (Adapted from Lin et d..
lI
5
-small grains
- r- large grains
10
15
20
25
30
35
40
45
50
55
60
65
Misorientationangle " Figure 2 Misorientation angle distributions for all grains, small grains and largest grains in a AI-Mg alloy (Adapted from Field et U / . , 1996).
CASE STUDY NO. 3
305
bordering larger grains. Boundaries were extracted from the map by an interactive 'point and click' procedure. Figure 2 shows misorientation angle distributions separately for the total distribution, the sinall grains and the large grains. Both the total distribution and the small grain distribution are similar to the Mackenzie plot for the uncorrelated. random case (Section 9.5 and Figure 9.13) whereas the large grain set shows a non-random distribution of misorientation angles. Conspicuously absent from the mobile boundary (large grain) distribution are low angle boundaries. These statistics were f~lrtherunderlined by plotting in Rodrigues space. which represents the whole inisorientation rather than only the misorientation angle (Section 9.6.4). The peak in the large grain distribution is 16 times random and occurs at a high misorientation angle near the < l 1l > misorientation axis. i.e. near C3. The results from this investigation suggest that in this alloy low-angle boundaries are less mobile than others and there inay be some tendency for boundaries near C3 to have high mobility. This case strrrly demonstr.ates. .
.
a
Examples of how user-specified data is extracted from an orientation map, often by interactive means
e
'Texture by number' representation of microtextures with respect to grain slze classes Analysis of grain boundary inisorientations in terms of angles, CSLs and Rodrigues space
References
Lin. P,, Pcilumbo, G., Harase, J. and Aust. K.T.. 1996. Actn Mntrr.., 44, 4677. Field, D.P., Nelson, T.W. and Dingley. D J . . 1996, Mat. Sci. For-urn,204-2
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CASE STUDY NO. 4 CRACK PROPAGATION IN A TITANIUM ALLOY
The fatigue crack path in a-titanium alloys has often been found to relate to the size and crystallographic orientation of the a colonies. These in turn are dependent on the orientation of the bcc 0-titanium grain which existed prior to the 3 to a phase transformation which takes place on cooling the alloy. The crystallographic relationship between the two phases is described by the Burgers relation (Burgers, 1934) which states that: {llO)~//{OOOl)a and < I l l > 3//<1120>a If the Burgers relation is obeyed, the prediction is that neighbouring a colonies will have an angle of O0, 60" or 90" between their basal planes, occurring with a frequency of 1:4:1. This implies that {110)8 planes have been chosen at random. There has been little evidence to confirm or otherwise adherence to the Burgers relation in titanium alloys. This Case Study shows how EBSD has been used to measure the orientation of individual a colonies in the titanium alloy IMI685, and hence gather more information about the validity of the Burgers relation and the effect of fatigue fracture (Wilson et al., 1997). In this investigation orientations were measured from near to each fracture surface and perpendicular to the stress axis. Figure 1 shows a typical example of a fracture surface. Similar 'control' data were also obtained from identical unfatigued material. Having obtained orientation matrices using EBSD. a computer program was written to manipulate the data in order to obtain, amongst other parameters, the angle between (0001) plane normals in neighbouring colonies. The processing steps of the algorithm consisted of: 1. The user-defined direction was converted into Miller indices (V,,,) and then into the orthonormal coordinate system (Vorth),by means of the matrix operator Q (Section 2.2.1):
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 T)pical fracture surface and microstructure in the titanium alloy (Courtesy of R.J. Wilson).
2. The orthonormal direction was calculated in terms of the colony's coordinate system (V,,). This is defined by the matrix G (Gt is the transposed matrix):
3. A pair of vectors were identified and the angle (Q) between them calculated:
This algorithm was activated twice on each pair. firstly to calculate the angle between the basal planes and secondly to calculate the angle between the < l 120> direction and the basal plane. The resulting list of angles was then transferred to a spreadsheet package for analysis. Figure 2a shows a frequency plot of angles between basal planes in neighbouring colonies for the control specimen. The ratio between angles differences of O C , 60" and 90" is close to the 1:4:1 ratio, as predicted by the Burgers relation. On the other hand Figure 2b shows that for EBSD data collected from near the crack path the ratio has changed to approximately 5:8:3. This change represents a marked increase in common basal planes for fractured specimens, suggesting that the crack chooses
CASE STUDY NO. 4
309
Control data
I
UCrack data
1
57.5-62 5
Angle range
Figure 2 Plot of' angles between basal planes: (a) in the control specimen: and (b) near the crack path.
a 'meak path' for propagation. The basal plane is an easy slip and common cleavage plane for these alloys. This case study demonstrates.. S .
Application of EBSD to a structure'property relationship in a hexagonal material
a
Application of customised software to augment standard EBSD routines to obtain specific crystallographic parameters, i.e. those connected with the Burgers relation
References Burgers, W.G., 1934, Pl~ysicu,1. 561. Wilson, R.J., Randle, V. and Evans, W.J., 1997; Plzil. Mug., 76A, 471.
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CASE STUDY NO. 5 ORIENTATION-RELATED EFFECTS IN T MINERALS
Although an understanding of orientation distributions (and related spatial effects such as interface geometry) is as important in studies of the physical properties of minerals as it is for metallic materials, the former have been studied relatively infrequently. This is because of practical difficulties in specimen preparation and microscopy (e.g. specimen charging), and because minerals often have lower symmetries than metals which can render unambiguous diffraction pattern indexing a challenging task. This Case Study illustrates two separate EBSD investigations on the minerals olivine (Faul and Fitz Gerald, 1999) and omphacite (Mauler et al., 1998), which have orthorhombic and monoclinic symmetry respectively. It has been noted from TEM studies that in partially molten aggregates of polycrystalline olivine the melt phase was distributed inhomogeneously at interfaces. In order to investigate a possible correlation between boundary misorientation and wetting it was more convenient to use EBSD than TEM, because of the usual specimen preparation difficulties and time penalties associated with the latter. EBSD patterns were recorded at 20 kV accelerating voltage and 10 nA beam current, from specimens which had been diamond/colloidal silica polished, with approximately 10 nm of carbon coating on the specimen surface to avoid charging. Fully automated pattern indexing did not prove to be possible; instead, the correct indexing was chosen interactively by comparing detected bands with simulated patterns as shown in Figure 1 for a correctly indexed pattern. Data were obtained for grain pairs at both wetted and non-wetted boundaries. Figure 2 shows the distribution of misorientation angles and axes. For orthorhombic crystals the maximum disorientation angle is approximately 117" and the stereographic unit triangle is li8 of the reference sphere. A random distribution of disorientation axes is located towards the centre of the triangle (Faul and Fitz Gerald, 1999). It was concluded from the misorientation distributions that the melt-free grain boundaries show a greater tendency for a non-random distribution.
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 EBSD pattern from olivine with correct pattern simulation overlaid
(4
(b) Melt-Free Grain Boundaries
4 12 20 28 36 44 52 60 68 76 84 9210QL08116
Misorientation Angle Figure 2
Wetted Grain Boundaries
4 1 2 2 0 2 8 3 6 4 1 52606876849210QO8Ll6
Misorientation Angle
Disorientation distributions for (a) 402 unwetted boundaries and (b) 260 wetted boundaries.
A second example of the study of orientations in geological materials is a comparison of universal-stage (U-stage) optical microscopy (Section 12.3.2) and
CASE STUDY NO. 5
3 13
U-S
yA
Figure 3 Pole figures (equal area projections) for 100 omphacite grains obtained by (a) U-stage optical microscopy and (b) EBSD.
EBSD to obtain orientation distributions in omphacite. Optical measurements are time-consuming, whereas EBSD measurenlents are complicated not only by the fact that omphacite has a very low symmetry but also that it has a natural variance in composition which leads to variations in lattice constants and scattering amplitudes, which feed through to changes in the diffraction pattern. Specimens were prepared by polishing thin sections in colloidal silica for ten hours. Carbon coating was not necessary for EBSD and diffraction patterns were acquired at an accelerating voltage of l 5 kV. In order to index the diffraction patterns the structure data were compiled and electron intensities were calculated for thirty reflection families (Miller indices) in two possible space groups. The changes in composition and space group did not have a strong impact on the reflection table. which somewhat simplified the indexing process and allowed EBSD patterns to be indexed with an acceptable reliability. Figure 3a shows U-stage nleasurements for 100 grains displayed on a pole figure showing both contoured and discrete data; compared to data obtained by EBSD from the same population of grains (Figure 3b). Although there is quite good agreement between the two pole figures. there are some discrepancies, possibly caused by ambiguities in locating the same grains or by the presence of subgrains.
..
This case study denzonstrates . e
Successful solving of EBSD patterns from low-symmetry n~aterials,although pattern indexation is a fairly complex procedure
e
The form of grain disorientation angle,axis distributions for orthorhombic symmetry
References Faul. U.H. and Fitz Gerald. J.D., 1999, Phj.s. & Chetw. ofA2.il~~er.als. 26. 187. Mauler, A., Kunze, K., Burg, J.-P. and Philippot, P., 1998, Mat. Sci. Forzmz, 273275. 705.
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CASE STUDY NO. 6 MICROTEXTURE MEASURE MULTI-PHASE MATERIALS
To date microtexture analysis been applied principally to single phase materials or to one phase in a multi-phase material. It can be of interest also to obtain orientations of each phase in a microstructure, for instance so that orientation relationships between the phases can be calculated. This Case Study shows examples of an early application of EBSD to the phases in both high chromium white irons (Powell and Randle, 1997; Randle and Laird, 1993) and ferrite with retained austenite (Field et al., 1997). The microstructure of high chromium white irons consists predominantly of an austenite matrix and large (several microns in size) (Cr, Fe),C3 carbides which have a hexagonal crystal structure. The 'in situ composite' nature of the material renders it inappropriate for TEM work, so EBSD was employed to infer connectivity of the carbide rods from measurement of their orientations to assess concurrently the microtexture of both phases and to link potentially these factors to variations in fracture toughness in two specimens. The results showed that there were clear microtextural differences between white iron specimens having undergone different cooling regimes, which were linked to the morphology of the carbides. The carbides in a Fe-CS-C alloy show a distinct texture close to [0lT0] in the specimen normal direction whereas those in a 1.3% Si coinrnercial white iron had a diffuse texture, with regions in the inverse pole figure near to major crystal directions, i.e. [OOOl], [T~To],[OlTO], unpopulated. Both inverse pole figures and orientation micrographs showed that carbides were inore connected in the Fe-Cr-C alloy than in 1.3% Si commercial white iron. Figure 1 illustrates part of these results, showing a stronger microtexture in the Fe-Cr-C alloy than in the commercial alloy. Less connectivity in the carbides in the 1.3% Si coininercial white iron is the probable cause of the higher fracture toughness in the as-cast condition when 1.2-1.6% Si is added. White irons inay contain a third phase, orthorhombic (Fe, Cr),C. Such irons have also been analysed using EBSD and it has been shown that the orientation distribution of all three phases can be obtained. Figure 2 shows that the inicrostructure of these irons consists of duplex carbides which have a (Fe, Cr),C3
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 In\erse pole figures (specimen normal directloll) from carbides in (a) Fc-Cr-C alloy and (b) l .jo%Si commercial m hitr Iron.
core surrounded by a (Fe,Cr);C shell. EBSD indicated that there was no coupled growth of the two phases (Randle and Laird, 1993). Whereas the example described above relied on interactive EBSD. a steel specimen of retained austenite (fcc iron) in a ferrite matrix (bcc iron) has been analysed using automated EBSD and mapping. Some of the technical aspects of automated multiphase microtexture analysis are described in Section 10.3.3. The specimen contained about 10% retained austenite. as determined by optical metallography. The mapping analysis. with a scan of 20,000 measurements over an area of 40 X 40 pm', gave a retained austenite area fraction of 12.6%. This value was obtained directly from the number of sampling points which was recognised as ferrite (and shaded dark in the map) and those recognised as austenite (light shading). The data were also analysed for various statistical measures of the distributio~~. For example, from the
CASE STUDY NO. 6
317
Figure 2 Microstructure of a white iron a-here the carbide morphology consists of (Cr, Fe),C; cores surrounded b! (Fe. C S ) ~ Cshells. The matrix is predominantly austenite.
~nisorientationdistribution of the austenite and ferrite it hvas evident that a strong orientation relationship of 45" existed (Section 9.8). This case study demoiistr.utes.. . Application of E SD to multi-phase materials Progress in full automation, mapping and extraction of data from multiphase materials References Field, D.P.: Wright. S.I. and Dingley, D J . , 1997: Proc. 11th Iizternational Corzference on Textzcres qf Materiuls: I I C O T O M l 1 ) (Ed. Z. Liang et al.) International Academic Publishers. China: 94. Powell, G. and Randle. V., 1997. J. Mut. Sci., 32, 561. Randle, V. and Laird, G., 11.. 1993 J. Mat. Sci., 28, 4245.
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CASE STUDY NO. 7 IN SZTU INVESTIGATION OF NUCLEATION OF RECRYSTALLISATION IN COLD ROLLED BORON DOPED NICKEL ALUMINIDE
This Case Study pertains to the nucleation of recrystallisation in cold rolled polycrystalline boron doped Ni3A1 (ICl5). In order to elucidate the incipient stages of nucleation, in situ annealing experiments using a hot stage in a TEM were performed and the orientations of newly forming grains and the surrounding matrix were determined by microdiffraction (Section 8.4). During in situ experiments, the possibility of a fast on-line orientation determination helps to track orientation changes during the annealing process, as the development of new orientations can easily be recorded and the further evolution of the respective grains can be traced (Section 6.3.4). The intermetallic compound Ni3A1 is very well suited for this purpose, because neither cell formation nor recovery were observed prior to the onset of recrystallisation and the high stored energy of the material results in a small critical nucleus size. Hence, the nucleation process can be observed within the transparent areas of the TEM samples (Escher and Gottstein, 1998). In the first example (Figure 1) nucleation of recrystallisation occurred at a former high-angle grain boundary. After annealing for 150 min at 600°C, and some minutes at 700°C, the grain boundary started to bulge and tilting of the specimen exposed contrast changes at the tip of the bulges (Figure la). This example gives also evidence of an additional benefit of the TEM investigations: the possibility of specimen tilt. The tilting procedure permits the detection of grain contrast changes and thus makes the discovery of new orientations easier. Orientation measurements exhibited twin relationships between the tips of the new grains (1 and 1') and the parent grain A (in Figure lb, the common < 111> axis of the twin relation is encircled). At higher rolling deformations Ni3Al develops an inhomogeneous microstructure which can give rise to nucleation of recrystallisation at deformation heterogeneities as transition bands, microbands, shear bands in addition to nucleation at grain boundaries (Ball and Gottstein, 1993). An example of new recrystallised grains emerging at microbands is shown in Figure 2a. In this instance, the temperature was
INTRODUCTION TO TEXTURE ANALYSIS
320
Figure 1 (a) Grain boundary bulging and generation of new orientalions in a 10% cold rolled sample; (b) j l l l } pole figure of the determined orlentations (Courtesy of C. Escher).
Figure 2 (a) Xucleation of new grains at a microband in a 40% cold rolled sample; (h) [ 11 1 ) pole figure of the determined orientations (Courtes] of C. Escller).
slowly increased to 400°C and then raised in steps of 50 "C up to 600°C. with a 30 min hold at every temperature level. The orientation measurements again revealed several twin relationships between the matrix and the new grains. Moreover. grain 2 and grain 4 were related by a C27b misorientation. i.e. a third order twin relation. Thus in both examples, irrespective of the nucleation site - grain boundaries or microbands twin relationships were observed already at the very beginning of grain generation. This leads to the conclusion that twinning occurs easily and that the formation of twin boundaries plays a vital role in the nucleation process of the intermetallic conlpound Ni3A1. -
CASE STUDY NO. 7
This case study denzonstrates
32 1
...
e
The use of microdiffraction in the TEM to obtain orientation data from very small grains
e
Exploitation of the TEM specimen tilt facility to locate new grains
References
Ball, J. and Gottstein, G., 1993, Intermetallics. 1, 171, 191. Escher, C. and Gottstein, G., 1998. Acta Mnter., 46, 525.
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CASE STUDY NO. 8 DEFORMATION AND LOCAL ORIENTATION EVOLUTION
During deformation, particularly at large strains, very small crystallites (typically 40 nm-300 nm) develop in metals such as nickel or aluminium. Characterisation of both the evolution of these local orientation patterns and the associated dislocation substructures is crucial to understanding the deformation process. TEM is the optimal technique for microtexture evaluation of such materials where a probe size in the tens of nanometers range is required. This Case Study highlights not only the application in this manner of microdiffraction in the TEM, but also the use of orientation mapping and Rodrigues space as tools to facilitate analysis of the data. The patterns of local orientation and dislocation structure have been quantitatively characterised in pure nickel following cold rolling to 98% reduction (Hughes and Kumar, 1997). Long lamellar dislocation boundaries and equiaxed subgrains were formed by the process. The rotation axes and disorientations across the dislocation boundaries in adjacent crystallites, measured linearly in the normal to the rolling direction, are shown in the orientation map in Figure 1. The different ideal texture components which occurred (within 15') are denoted by combinations of various grey scales and patterns. All the components are: 0
Cube, labelled D, {100}<001>
S I , Sz, S3,
Sq,
{123}<634>
0
Copper C I and C2 {112}<111>
0
Brass BI and BZ {110}<112>
0
Goss {110}<001>
Both the diameter of each crystallite and the disorientation angle are superimposed directly on Figure 1 as the x- and y-axis respectively. Note also that the disorientation angles are expressed with a handedness (Section 2.6). Hence a large concentration of relevant information is compacted into a single representation. 323
INTRODUCTION TO TEXTURE ANALYSIS
324
60 n 0
40
W
E; 0
20
.r(
.U
5
s
0
.d
5
6
-20
-40
-60
0
3 4 5 60 Distance along ND (pm) 1
2
3 2
Figure 1 Orientation map showing disorientation across dislocation boundaries in highly deformed nickel as a function of distance in the normal direction. (Courtesy of D.A. Hughes).
Figure 1 shows that a large number of high angle disorientations accompany a very fine scale pattern of local texture components and, remarkably, nearly all the ideal texture components and their variants were found over a distance of only a few microns. Figure 2 shows the statistical aspects of the microtexture displayed in Rodrigues space. The exact location of the ideal components is labelled in the diagram, and a grey scale is used to denote intensity. These ideal components lie on or near the surface of the fundamental zone of Rodrigues space. The experimental data were interpreted in terms of lattice spins and active deformation modes (Hughes and Kumar, 1997). In another investigation real space coordinates and Rodrigues space coordinates have been superimposed in one dimension (Weiland, 1992). Microdiffraction in TEM was used to analyse the misorientations between crystallites in deformed single crystals of high purity aluminium. The substructure consisted of elongated subgrains varying in size from 0.15 pm to 4.5 pm depending on the deformation level. Figure 3 shows a perspective representation of misorientations between adjacent subgrains along one specimen direction: this direction and one axis of Rodrigues space, X 1, are made to be parallel. The misorientations were calculated with respect to a reference grain positioned at the origin of the spatial axis. This means, of course, that the misorientation distribution depends on the choice of the reference grain. The data in Figure 3 show that both the rotation axis (direction of the vector) and the rotation angle (length of the vector) vary in a systematic manner along the extension direction of the specimen. Clearly, there are long range misorientations present in the material
CASE STUDY NO. 8
Figure 2 Distribution in Rodrigues space of deformed crystallites in nickel. Major texture variants are labelled (Courtesy of D.A. Hughes).
Extension direction
--)
1
50pm
lOOpm
Figure 3 Representation of misorientation in deformed aluminium as Rodrigues vectors along the specimen extension direction. The axes of Rodrigues space are labelled XI, X2, X3 (Adapted from Weiland, 1992).
INTRODUCTION TO TEXTURE ANALYSIS
326
which are highlighted by this representation method. An accumulation of misorientation exists on a local scale up to the fifth or seventh neighbour.
This case study demonstmtes . Validity of microdiffraction in the TEM to analyse deformed microstructures e
Application of customised orientation mapping to demonstrate succinctly spatial and orientation components
e
Use of Rodrigues space to display orientation variants and, moreover, to link with spatial coordinates
References
Hughes. D.A. and Kumar, A., 1997, Proc. 11th International Conference on Textwes of Materials IICO TOM1l ) (Ed. Z . Liang et al.) Int. Academic Publishers. Beijing, 1376. Weiland, H., 1992. Acta Met. Mat., 40, 1083.
CASE STUDY NO. 9 FORMATION OF THE CUBE RECRYSTALLIS TEXTURE
N
This Case Study pertains to the evolution of the cube orientation, the main recrystallisation texture component in most aluminium sheet products. A cold rolled sheet prepared from high purity aluminium with 1.3wt% manganese was isothermally annealed for various times in a salt bath (Engler et al., 1996~).After complete recrystallisation, the X-ray macrotexture comprised a quite strong cube orientatioii {001}<100> with some scatter about RD (Figure la). Since most of the characteristic features of recrystallisation textures in AI-alloys can be found in the p2 = 0"-section of the QDF. in the following this representation of texture data is preferred to showing the entire QDF (see Section 5.4.2). To study the nucleation mechanisms leading to this recrystallisation texture, samples in the as-deformed and partially recrystallised states were subjected to TEM local texture analysis (Section 8.4). After a short anneal of 30 s at 350 "C less than 1% of the entire volume was recrystallised, so that nucleation sites of first recrystallised grains could easily be distinguished from each other.
Levels: 2 - 4 Figure 1 9 2 at 350°C).
= 0"-section
of the X-ray inacrotexture of the fully recrystallised sample (annealed for 1000s
327
INTRODUCTION TO TEXTURE ANALYSIS (P1
(P2=O0
Levels: 2 - 4 - 7 - 10
(b) new grains
(a) subgrains
Figure 2 ODFs of TEM single grain orientation measurements at the cube bands (annealed for 30 s at 350 "C; q2 = P-sections); (a) subgrains in the cube bands; (b) new grains originating from the cube bands.
It turned out that most recrystallisation nuclei formed within band-like structures existing already in the as-deformed microstructure, which are commonly referred to as cube bands (Ridha and Hutchinson, 1982; Dons and Nes, 1986). TEM orientation measurements showed that the bands mainly exhibited cube orientations which were significantly shifted around R D or TD, whereas the subgrains in the deformed grains on either side of the bands comprised rolling texture orientations. During subsequent annealing, some of these cube oriented subgrains started growing into the neighbouring microstructure. In order to study the influence of nucleation in the cube bands on the final recrystallisation textures in more detail, the orientations of about 50 subgrains within the cube bands and the new grains which had emerged from these bands were determined in the TEM. Two separate ODFs were computed, and the p 2 = 0'-sections are shown in Figure 2. Both microtextures revealed strong cube orientations which were significantly rotated either around R D or around TD, whereas grains with a near-exact cube orientation were only scarcely detected. The resemblance of these two ODFs points at the nucleation of cube and particularly of rotated cube orientations in the cube bands. However, after completed recrystallisation of aluminium alloys the texture maximum is typically found at the exact cube position. i.e. at O", O", 0" in Euler space (Section 2.5). Thus, with progressing recrystallisation an orientation shift from the rotated cube orientations observed in Figure 2 to the exact position prevalent in the X-ray macrotexture (Figure 1) takes place. In order to examine this orientation shift, the individual orientations of a large number of new grains stemming from cube bands were determined at various stages of recrystallisation; now, the grains were sufficiently large to permit analysis by EBSD rather than TEM. With progressing recrystallisation as indicated by the increase in recrystallised volume fraction X(t) - the position of the cube peak shifted remarkably to its exact position, and only some RD- and TD-scatter is retained (Figure 3); such that the O D F of the (nearly) completely recrystallised sample strongly resembles the corresponding X-ray macrotexture (Figure 1). This orientation shift strongly suggests a growth selection -
CASE STUDY NO. 9
(a) X(t)
bevels: 2 - 4 - 7 (b) X(f)=90°/o Figure 3 ODFs of EBSD single grain orientation measurements of the cube grains with progressing recrystallisation (3:= 0'-sections).
of grains having an exact cube orientation out of the scattered spectrum of nucleus orientations (Engler et d.. 1996~). This case studj' dernonstrates . . . e
The benefits of combining. in the same investigation. different techniques for texture measurement - X-rays. TEM and EBSD - to cover different ranges of spatial resolution
e
The use of ODFs for microtexture analysis
References
Dons. A.L. and Nes. E., 1986. Mat. Sci. Tech.. 2. 8. Engler. 0..Yang, P. and Kong, X.W.. 1996c. Acta Mat., 44, 3349. Ridha, A.A. and Hutchinson, W.B.. 1982, Acta Met.. 30, 1929.
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CASE STUDY NO. 10 TEXTURE AND OXIDATION OF METALS
It is known that the oxidation resistance of crystallographic planes in metals is highly anisotropic (Cathcart et al., 1969). For example, at 700°C the oxide on (111) is approxin~atelyone order of magnitude thinner than that on (100) in pure nickel. Hence, it is of interest to investigate the relationship between oxidation and texture in metals, as demonstrated in this Case Study. Recent work has concentrated on the influence of nickel substrate crystallography, and the addition of reactive elements such as cerium, on oxide growth (Czerwinski and Szpunar, 1997). The experiments involved preparing specimens of high purity nickel either by chemical polishing or by mechanical polishing which introduced surface deformation, followed by oxidation. Cerium was added to some specimen surfaces by techniques such as ion implantation. The texture of both metallic substrates and oxidised surfaces was measured using standard X-ray diffractometry (Section 4.3) and the microtexture of small regions of cerium-implanted oxide was investigated by TEM. This technique was also used to study concurrently the fine microstructural changes brought about in the oxide by cerium implantation. Figure 1 shows the surface morphology after oxidation in both the mechanically polished specimen (Figure l a) and the chemically polished specimen (Figure I b). In the latter case epitaxial oxide growth has been promoted on some grains, and the oxide thickness and morphology strongly depends on the orientation of the underlying grain. By contrast deformation of the nickel surface destroys the crystallographic influence. The macrotextures of the nickel substrate and the oxidation product grown for 15 hours at 800 "C after both chemical polishing and mechanical polishing are shown as (1 11) pole figures in Figures 2a, b and c respectively. The texture of the nickel substrate is mainly {100}<023>, i.e. rotated or 'off'-cube (Figure 2a). The texture in the oxidation product on the chemically polished specimen is also off-cube, (100)<023>, which supports the influence of epitaxy between the oxide and substrate (Figure 2b). Conversely, the pole figure of nickel oxide formed on mechanically polished nickel does not show the influence of the substrate, since only a weak (1 10) fibre texture is present (Figure 2c).
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 SEM imcrographs showing the surface morphology of nickel oxide on a nickel substrate grown after (a) mechanical polishing and (b) chemical polishing.
CASE STUDY NO. 10
ldent.: Ni Contour: 1;1.2;1.4;1.6;1.8;2;2.2
(4
Ident.: NiO
Contour: 1;1.2;1.4;1.6;1.8;2;2.2
ideni.: NiO Contour: 1; 1.2;1.4;1.6;1.8;2.2.2
Figure 2 X-ray (111) pole figures for (a) nickel substrate and nickel oxide grown after 15h at 800°C; (b) after chemical polishing: (c) after mechanical polishing.
Traces of oxygen-active elements, such as cerium, are known to reduce greatly oxidation. However, it was found that there was a strong influence of substrate crystallography, and the oxidation resistance was only effective when surface modification was carried out using sol-gel coatings or sputtered coatings rather than ion implantation which affects the crystallography of the substrate surface. The effects were investigated by TEM methods. These investigations show, therefore, that controlling the surface texture may be a very effective way to extend the lifetime of reactive element coatings at high temperatures in corrosive environments.
INTRODUCTION TO TEXTURE ANALYSIS
334
Furthermore. in the case of nickel and its alloys, a (1 11) surface texture is beneficial for oxidation resistance.
This case study demonstrates.
..
o
A combined approach to a texture investigation of use of X-rays and TEM
0
A direct link between texture and a specific property (oxidation)
References Cathcart, J.V., Petersen, G.F. and Sparks, C.J., 1969, J. Electroclzem. Soc., 116. 664. Czerwinski, F. and Szpunar, J.A.. 1997. Pmc. 11th Interrzationul Conference on Textures of Materials IICOTOMIIJ (Ed. Z . Liang et al.) Academic Publishers, Beijing, 723.
CASE STUDY NO. l 1 ICROTEXTURE DET NATION I IRON-SILICON BY PITTING
As described in Section 12.3.2, etching techniques can be used to provide orientation information, since there are different responses of certain crystallographic planes to etchants. This Case Study illustrates how etch pitting has been used to reveal the microtexture in Fe-Si transformer steel, based on the discernment of essentially three types of etch figures: e
Square etch figures, denoting (h001 Rectangular etch figures, denoting (hkO)
e
Triangular etch figures, denoting {hkl)
Fe-Si alloys respond particularly well to crystallographic etching to form etch pits. However, their formation in terms of size and number is very sensitive to precise conditions. A multi-stage etching procedure is used such as immersion in 5% H F + H 2 0 2 solution for 20s for surface conditioning, followed by 2 s in 15% HF + H 2 0 2 to make the etch pits, followed by 1% H F t 4% H 2 0 2 + 20% H 3 P 0 4 + 75% H 2 0 for 10 s to sharpen the etch pits and reveal grain boundaries. Images of the etch pits can be viewed using optical microscopy (Lee and Szpunar, 1995) or SEM (Baudin et al., 1994). Figure 1 shows an SEM image of etch pits in three grains around a triple junction (Cruz et d., 1995). The practical use of etch pits for orientation determination involves capturing the etch pit image, then measuring the ratio of the pro-jected sides of the etch pit and the angles between the sides. If an 'absolute' orientation is required, it is also necessary to know the angle between one direction in the etch pit and a reference axis in the specimen such as the rolling direction. Usually, these data are obtained from a digitised image via a computer controlled procedure. Figure 2 shows a (100) pole figure comprising data from 92 grains obtained using both the etch pit method and EBSD. Although the agreement appears to be very good, the accuracy for orientation determination from etch pits overall is estimated
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 SEM image (inverse imaging) of etch pits in three grains in Fe-3%Si (Courtesy of T. Baudin).
Figure 2 {loo} pole figures of 92 individual orientations in Fe-3%Si from (a) EBSD and (b) etch pitting (Courtesy of T. Baudin).
to be 5-8" compared to 2" for EBSD (Baudin et al., 1994). A major part of this error in both cases relates to the accurate specification of the specimen axes. If only relative orientation are considered, there is a greater accuracy available. One experiment has used etch pits to obtain grain boundary parameters and classify them
CASE STUDY NO. 11
337
according to the CSL model (Section 11.2.1). The results indicated that in the initial stages of secondary recrystallisation Goss grains (i.e. grains having {110}<001> orientation) having more C5 boundaries grew faster than other grains (Lee and Szpunar, 1995). To sum up, the etch pit method of orientation determination is a useful (semi-) quantitative method for orientation determination which allows a rapid evaluation of microtexture using optical microscopy or SEM without any requirement for further hardware such as EBSD. Qualitative estimates of orientation can also be made 'by eye' from etch pits. This case study demonstvates
...
Examples of the practical use of etch pitting to obtain orientation and misorientation data using simple optical or SEM techniques 0
Direct comparison of data obtained by etch pitting and EBSD
References Baudin, T., Paillard, P., Cruz, F. and Penelle, R., 1994, J. Appl. Cryst., 27, 924. Cruz, F., Caleyo, F., Baudin, T., Estevez, E. and Penelle, R., 1995, Mats. Char., 34, 189. Lee, K.T. and Szpunar, J.A., 1995, Can. Met. Quart., 34, 257.
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CASE STUDY N RE AND EARING IN ALUMINIU
One of the most prominent applications of quantitative texture analysis is the prediction and, eventually, the control of mechanical properties of sheet materials. In a polycrystalline assembly properties that vary in different crystallographic directioiis are controlled by the distribution of all orientations in the assembly, that is by its crystallographic texture. In the present Case Study we illustrate this relationship in terms of the earing behaviour of Al-sheets. When a cylindrical cup is drawn from a circular blank of a textured sheet, the riin of this cup will not be level but will depict a number of distinct high points - usually called ears - and an equal number of low points. called troughs: as shown in Figure 1. This earing is very undesirable, as it causes inhomogeneous wall thickness reduction and it may interfere with automated deep drawing operations. Therefore, inany attempts have been performed to suppress or reduce earing by proper texture control (e.g. Inagaki, 1991). In most recrystallised Al-sheets the texture is dominated by the cube-orientation (001)<100>. which results in the formation of ears at 0" and 90" to the foriner rolling direction. Rolled sheets display ears at 645" to the rolling direction, in contrast. which can be utilised in practical applications to control the earing behaviour by balancing the opposite 0"/90" ears caused by the cube recrystallisation texture component (Oscarsson et al., 1991). The demands for weight reduction in automotive constructions led to an increasing interest in sheets made from the age-hardening AA6xxx AI-Mg-Si alloys in automotive body applications. In order to achieve good formability, the material will be used in a solution treated condition so that a recrystallised structure is present. Thus, understanding and control of the recrystallisation is of great importance in order to optimise the sheet formability. The recrystallisation textures are strongly influenced by a variety of factors like alloy composition, constitution and, in particular, thermomechanical treatment. Thus, there are a number of process parameters that can be modified in order to control the final recrystallisation textures and the resulting anisotropy. In the following, this is demonstrated by control of the precipitation state in the Al-Mg-Si alloy AA6016. 339
INTRODUCTION TO TEXTURE ANALYSIS
Figure 1 Cup formed by deep drawing of a rolled sheet of Al, showing earing as a consequence of the sheet texture (Courtesy of J. Hirsch).
Two specimens of this alloy were subjected to different thermomechanical treatments so as to achieve different state of precipitation/supersaturation of the material prior to cold rolling by 75% thickness reduction (Engler et al., 1596). The final treatment of the sheets consisted of a continuous solution anneal for 30s at 540°C and subsequent water quench (condition T4). Microstructural investigations yielded a completely recrystallised structure with slightly elongated grains; the recrystallisation textures of the two differently treated sheets are shown in Figure 2. One alloy was treated such that most alloying elements were precipitated in the form of rather coarse particles. Upon the final recrystallisation anneal those precipitates gave rise to particle stimulated nucleation (PSN), resulting in a very weak texture with some characteristic intensities of a rotated cube-orientation and the Porientation (011)<122> (Figure 2a). The other specimen was subjected to an additional hot band anneal which gave rise to heavy precipitation reactions of finely dispersed Mg,Si precipitates. Such dispersoids are known to interfere strongly with the progress of recrystallisation and in particular they have been shown to suppress efficiently PSN in favour of a pronounced cube recrystallisation texture (Figure 2b) (Vatne et al., 1997). Figure 3 shows the earing profiles of the two differently pre-treated sheets in terms of the cup height lz, plotted as a function of the angle a to the rolling direction, symmetrised to the range 0" < a < 90" due to the orthonormal rolling symmetry. Whereas the sheet with the strong cube-texture develops
CASE STUDY NO. 12
h(p,
(p2=const.
(a) without dispersoids
compl. ODF
(p,
341 cP,=const.
(b) with dispersoids
Figure 2 Recrystallisation textures of the two different specimens.
compl. ODF
INTRODUCTION TO TEXTURE ANALYSIS
342
a protruding earing profile with the typical 0"/90° ears, the other sheet displays a much more levelled profile with weak ears at 45". This case study demonstrates. .. e
Relation between crystallographic texture and plastic anisotropy (earing)
e
Influence of dispersoids on the recrystallisation textures
e
Possibility to improve material properties by proper texture control
References Engler, 0. and Hirsch, J., 1996. Mat. Sci. Forwn, 217-222. 479. Inagaki, H., 1991, Z. Metallk., 82, 361. Oscarsson, A., Hutchinson, W.B. and Ekstrorn, H.E., 1991, Mat. Sci. Tech., 7, 554. Vatne. H.E., Engler, 0. and Nes, E., 1997, Mat. Sci. Tech., 13, 93.
APPENDIX 1 MILLER AND
Within the crystal coordinate system, both crystallographic directions and lattice planes are commonly described in terms of integer indices, called the Miller indices, which are derived as follows. A crystallographic direction is given by the vector r in the crystal:
r
= tla
+ vb +
lee,
( A l.1)
where a, b, c are the base vectors of the unit cell (Figure 2.2) and thus the direction is unambiguously described by the values of U, v and IV. It is common practice to characterise a direction by the triplet of smallest integers having the same ratios which is denoted by the symbol [ZLVIV]. A lattice plane is described according to: .Y
1'
-
h-+/i-+l-= a b c
1.
(A1.2)
where x: y and z are the coordinates of any point on that plane. Crystal planes are usually denoted by the reciprocal multiples of the axis intercepts 12, k and I, again reduced to the smallest integers with the same ratios; these are the Miller indices (hkl). Figure A l . l a shows the (1 12) plane in an orthorhombic crystal. The normal to this plane called its pole is commonly indexed (1 12), too. Note that this normal (1 12) is not parallel to the direction [112]. Only in cubic crystals are plane normals and directions with same indices parallel, whereas for other crystals this only holds for special cases. Because of the symmetry elements of most crystal structures, a lattice direction or a lattice plane is symmetrically equivalent to some others. For instance, planes like (112), (121); (121) all have the same atomic arrangement, lattice spacing etc. and thus they are summarised by the family (1 12) - denoted by curly brackets rather than parentheses. Likewise, a generic set of symmetrically equivalent directions is denoted by angular brackets >. Though directions and planes can be described by the Miller indices in all crystal structures, for hexagonal and trigonal crystals it is often more convenient to use a coordinate system with four axes. Commonly, the z-axis is chosen along the 6-fold or 3-fold c-axis such that the x- and y-axes come to lie in the basal plane. In order to ~
~
343
INTRODUCTION TO TEXTURE ANALYSIS
Figure A l . l Example of the crystallographic indices for directions and planes. (a) (1 12) plane and [l 121 direction in an orthorhombic crystal. Note that the (1 12) plane normal is not parallel to the [l 121 direction: (b) planes and directions in the (0001) basal plane of a hexagonal lattice expressed as both Miller-Bravais and Miller indices.
account for the symmetry of the basal plane an additional, redundant axis is introduced in such a way that the three axes a , , a2 and a3 all lie at 120" to each other in the basal plane (Figure A l . lb), leading to the description of directions and planes by a four index notation, the Miller-Bravais indices. Lattice planes can still be described by their reciprocal intercepts on all four axes; they are denoted by the Miller-Bravais indices (hkiT) with h k + i = 0. When directions are described by the Miller-Bravais indices [uvt~'](with u + v t = 0), the third index has to be considered as an additional vector component (Figure A1 .lb). Since the third index can readily be derived, it is often replaced by a dot, e.g. (27.0), or sometimes even completely omitted. However, the resulting three-index set is not to be confused with the Miller indices. The four index Miller-Bravais notation has the advantage that symmetrically equivalent directions and planes have similar indices. As an example, the directions a , = [2n0], a2 = [T~To]and a3 = L11201 would be indexed in the Miller notation as [IOO], [OlO] and [ n o ] so that symmetry is no longer apparent (Figure Al.lb). However, some calculations that are important for indexing Kikuchi-patterns, including determination of a common zone from two planes, a plane from two directions and interzonal angles (e.g. Section 6.3.1), can be performed more easily by using the three-index Miller indices. The transformations linking the two conventions are (Partridge, 1969):
+
+
APPENDIX 1
For planes, i.e. (hkil) + (HKL):
For directions, i.e. [uvtw]+ [ U V W ] :
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c
LL
ELAT
As discussed in Section 2.2.3. there are a number of c~~.stallogl.nphicall~-related solutioizs for an orientation 01- misorientation, depending on the crystal system. These crystallographically-related solutions A4' are generated by premultiplying the orientation or misorientation matrix, e.g. M , by a symmetry operator T,: M ' = T,M. where the value of i depends on the symmetry of the crystal system. There are 24 matrices T, for cubic symmetry, which are: 100 010 001
010 001 100
ooi oio
ooi
-
100
010 100
ooi
oio
100 010
-
-
001 100
-
100 010
001 010
ooi ioo 010 ooi o o i 100 100 oio
-
100 loo oio Too oio 001 100 oio ooi oio 001 001 oio 010 Too 001 ooi 100 001 oio 100 ooi 010 100
100
ooi 010
001 100
-
oio
010 100 001
001 loo 010
oio
-
-
100
-
ooi oio
100
ooi
The 12 symmetry operation matrices for hexagonal symmetry are:
1 0
0 1
0 0
0
0
1
-0.5 -0.87 0 0.87-0.5 0 1 0 0 -0.5 0.87 0 0.87 0.5 0 0 -1 0
-0.5 0.87 0 -0.87 -0.5 0 0 0 1
0.5 0.87 0 -0.87 0.5 0 1 0 0
0.5 -0.87 0 0.87 0.5 0
1 0 0
0
0
1
-1 0 0
0 1 0
0 0
0 -1 0
0 0 -1
-0.5 -0.87 0 -0.87 0.5 0 0 0 -1
-1 347
-1 0 0
0 -1
0
0 0 1
0.5 0.87 0 0.87-0.5 0 0
0
-1
0.5 -0.87 0 -0.87 -0.5 0 0
0
-1
348
INTRODUCTION TO TEXTURE ANALYSIS
The 4 symmetry operation matrices for orthorhombic symmetry are (Faul and Fitz Gerald, 1999): loo 010 001
oTo
100
Too Too 010 oTo
007
007
001
APPENDIX 3 THE 24 CRYSTALLOGRAPHICAELY-RELATED SOLUTIONS FOR THE FOUR S-TEXTURE VARIANTS
s1
s2
s3
54
(123)[634]* (321) [a361 (123)[634] (321)[436] (132)[6a] (lZ)[rn] (132)[643]
(123)[634] (327)[436] (123)[6 3 1 (321)[436] (1 32)[643] ( 123)@34] ( 132)[643] (273)[364] (n3)[634]* (213)[364] (23 1)[346] (3 12)[463] (231)[346] (712)[463] (312)[463] (231)[%6] (231)[3461 (3n)[463] (2 13)[364] (T32)[643] (32 1)[436] (213)[3641 (m)[643] (321)[4361
(213)[364] (312)[463] (213)[364] (312)[463] (231)[346] (213)[%z] (231)[346] (123)[6%]* (x3)[%%] (i23)[634] (1 32)[643] (321)[436] (i32)[m] (%1)[436] (321)[436] (1%)[643] (n2)[643]
(21 3)[364] (3 12)[463] (213)[3641
(2T3)[ 3641 (n3)[634] (213)[364] (231)[346] (3 12)[463] (23i)[%] (q2)[463] (31)[46?] (231)[3461 (x1)[346] (3n) [463] (2 13)[3641 (732)[643] (321)[436] (m)[364] (132)E6431 (321)[436]
(321)[436] (123)[634] (23 1)[%6] (312)[463] (123)[ a 4 1 (m)[346] (m)[4a]
*is the lowest angle solution.
349
(312)[463] (237)[346] (213) [36?] (231)[rn] (123)[634] (n3)[364] (723)[6341* (1 32)[643] (321)[436] (732)[643] (=1)[436]
(321)[a6] (1%)[643] (n2)[ 6431 (321)[436] (123)[634] (23 1)[346] (3%2)[463] (123)[634] (m)[346] (m)[463]
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APPENDIX 4 SPHERICAL PROJECTION AND THE STEREOGRAPHIC, EQUAL-AREA AND GNOMONIC PROJECTIONS
Projection is used in many branches of crystallography to represent the key features of crystals, namely the relationships between lattice planes and directions. The subject of this book, texture, is an aspect of crystallography and in order to understand such attributes as Kikuchi patterns (Chapter 6) and pole figures (Sections 2.4, 5.2 and 9.2), it is crucial to gain an elementary knowledge of projection geometry. A crystal can be represented by a set of normals, one from each crystallographic plane. Angular relationships between planes are therefore the same as those between their respective normals. If a sphere having the same origin as the crystal itself is circun~scribedon the crystal, the intersections of the plane normals, known as poles, with the sphere surface form a spherical projection. otherwise known as a reference sphere (Figure A4.1). The symmetry of the crystal itself is carried through to the representation on the reference sphere. This means that the reference sphere is divided, by the crystal symmetry planes, into spherical triangles. For example, in the cubic, hexagonal and orthorhombic systems there are 48, 24 and 8 crystallographically equivalent stereographic unit triangles respectively. Various projections are then introduced to project the three-dimensional reference sphere onto a more manageable two-dimensional flat form.
A4.1
THE STEREOGRAPHIC PROJECTION
The stereographic projection is a means of representing three-dimensional angular relationships in two dimensions. In the present context, the angular relationships of interest are those between crystallographic plane normals or between crystallographic directions. The main application of the stereographic projection in texture analysis is that pole figures/inverse pole figures, and anglelaxis space (Section 9.5) are usually stereographic projections (although equal area projections, Section A4.2, are also sometimes used).
INTRODUCTION TO TEXTURE ANALYSIS L ,
Reference sphere
Figure A4.1 A spherical projection (reference sphere) with origin, 'north pole' and 'south pole' labelled 0, N , S respectively. A crystal is imagined to lie at 0. The normal of a plane in the crystal intersects the reference sphere at P. P is projected from the south pole onto the equatorial plane at p; this is the stereographic projection of P.
The intermediate stage in the construction of a stereographic projection is spherical projection of poles onto the reference sphere as described above. Using terminology relating to the Earth, the point of projection is the south pole and the plane of projection is the equatorial plane as shown on Figure A4.1. It can be seen that poles in the northern hemisphere fall inside the equator, or 'primitive circle', poles on the primitive circle project onto themselves and poles in the southern hemisphere fall outside the primitive circle. It is more convenient to project poles in the southern hemisphere from the north pole so that they also fall inside the primitive circle. They are denoted in the stereographic projection by open circles to distinguish them from poles arising from the northern hemisphere which are denoted by smaller, filled circles. In Figure A4.1 the projection p of the pole P onto the equatorial plane is given by r-tan LOSP, where r is the radius of the sphere and i 0 S P = INOP. The main attraction for crystallographers of the stereographic projection is that angular relationships in the crystal are preserved in the projection. Consideration of the projection geometry shows that the angle between the centre of the projection, 0, and the primitive circle is 90". For crystals with orthogonal symmetry, it is conventional to denote the 'axes' of the standard stereogram (i.e. the three crystal axes 100, 010 and 001) in the positions shown on Figure A4.2. Figure A4.2 shows a standard stereographic projection, i.e. the equatorial plane, illustrating some of its basic features. p is a pole of a plane, and the trace of the plane itself is a 'great circle'. A great circle is the projection of any plane which passes through the origin of the reference sphere (whereas a plane which does not pass through the origin projects as a 'small circle').
APPENDIX 4
Figure 44.2 Stereographic projection showing the projection of the great circle (zone) of which p is the pole. A and B lie on the zone of p and the angle between them is measured along the great circle which connects them.
A plane is usually represented by its normal (pole), although sometimes it is more convenient to represent it by the trace of the plane, i.e. the great circle or zone (e.g. Figures 11.5 and A4.2). The angle between two poles/directions is measured along a great circle which passes through both of them. For example on Figure A4.2 A and B are directions which lie in the same plane, and the angle between them is measured along the great circle connecting them. Sincep is the plane normal, the angle between p and any points on the plane (e.g. A and B) is 90". Angles on the stereogram are measured with the aid of grid known as a Wulff net which is marked in 2" intervals as shown in Figure A4.3a. The projection of the stereographic unit triangles for the cubic system is shown on Figure A4.4. There are 24 each of right-handed and left-handed triangles. A single stereographic unit triangle is usually used for an inverse pole figure (Sections 2.4.2, 5.2.2 and 9.2) and misorientation axis distributions (Section 9.5) or interface plane distributions (Section 11.2). In general, if the specimen axis which is to be plotted in the inverse pole figure is a mirror (symmetry) plane of the specimen, a single unit triangle suffices. Otherwise, a right- and left-handed triangle should be used. A pole figure is usually plotted in the whole of the stereographic projection, although sometimes 'reduced' pole figures in one-quarter of the stereogram are seen. This very brief description of the construction and some of the features of the stereographic projection which are pertinent to texture analysis is amplified and discussed rigorously elsewhere (Johari and Thomas, 1969; McKie and McKie, 1974).
INTRODUCTION TO TEXTURE ANALYSIS
Figure A4.3 (a) Wulff net for measuring angles in stereographic projection; (b) Sclimidt net for measuring angles in equal-area projection
Figure A4.4 Unit triangles in stereographic projection for cubic crystal symmetry. The apices of one triangle are labelled.
A4.2 THE EQUAL-AREA PROJECTION In the equal-area projection equal areas on the reference sphere have equal areas in projection. Such a projection is particularly appropriate for the measurement of population densities. For example, a random distribution of points (orientations)
APPENDIX 4
355
Figure A 4 5 Projection of a random distribution of 500 poles on the surface of a sphere. (a) Stereographic projection with superimposed Wulff net; (b) equal-area projection with superimposed Schmidt net.
appears as a uniform density of points on the equal area projection, whereas this distribution is distorted on the stereographic projection. With reference to Figure A4.1: in an equal-area projection the projection of P onto the equatorial plane is given by rfisin LOSP. Figure A4.3b shows the Schnzidt net which is used to determine angular relationships in equal-area projection. and comparison between them and the Wulff net in Figure A4.3a illustrates the equal-area characteristics of the former. Here, the size of the net units represents correctly the fraction of orientation space covered by each. Figure A4.5 shows the projection of 500 randomly oriented poles in both stereographic and equal area projection. Whereas in the stereographic projection the poles are clearly clustered in the central regions of the pole figure (Figure A4.5a), they appear homogeneously distributed in the equal area projection (Figure A4.5b). Disadvantages of the equal area projection include that circles on the sphere do not appear as circles in the projection and that angles between great circles are not preserved. Despite the advantages of equal-area projection for assessing population densities, stereographic projection remains popular in metallurgy and equal-area projection is most widely used in geology (Wenk and Kocks, 1987).
A4.3 THE GNOMONIC PROJECTION In the gnomonicprojecrion poles are projected from the centre of the reference sphere to a tangent plane at the north pole as shown in Figure A4.6 (McKie and McKie, 1974). Since every great circle passes through the projection point O all zones project as straight lines. The projection of P is now given by tan LNOP. This has the disadvantage that as LNOP approaches 90" the distance NP in the projection becomes unmanageably large, until when NOP = 90" P projects at infinity. The link
356
INTRODUCTION TO TEXTURE ANALYSIS
Projection plane
-___-----
Figure A4.6 The relationship between the reference sphere and the projection plane for gnomonic projection. The pole P is projected from 0 to a tangent plane at the north pole.
between the gnomonic projection and texture analysis is that a Kikuchi diffraction pattern is a gnomonic projection. The distortion at large projection angles is particularly apparent in EBSD patterns, since they have a capture angle of approximately 60" (Section 6.2 and Figure 6.4).
APPENDIX 5 X-RAY COUNTERS AND PULSE HEIGHT ANALYSIS
In this Appendix the different types of counters that may be used for recording the Geiger, scintillation, diffracted intensities in X-ray diffraction experiments proportional and semiconductor counters - will be described, as well as the use of pulse height analysers to improve the peak-to-background ratio. (For a more thorough treatment see e.g. Cullity, 1978; Barrett and Massalski, 1980). All electrical counters suitable for X-ray diffraction experiments depend on the power of the X-rays to ionise atoms, either of a gas (Geiger and proportional counters) or of a solid (scintillation and semiconductor counters). With regard to quantitative texture analysis, mainly three aspects of the different counters are of importance: -
An incoming X-ray quantum results in an electrical pulse of the counter. With increasing number of X-ray quanta, i.e. with increasing counting rates, the time interval between the pulses decreases and may become so small that successive pulses cannot any longer be resolved by the counter, which leads to counting losses at high counting rates. The counter eficiency describes the fraction of X-ray quanta that is indeed able to initiate an electrical pulse in the counter. Both absorption of X-rays in the counter window as well as passing of X-rays through the entire counter without being absorbed by the recording medium decrease the efficiency. In most counters the output voltage is proportional to the energy of the X-ray photon (see below), and the energy resolutiorz is a measure for this proportionality between the size of the voltage pulse generated by the counter and the energy of the absorbed X-ray quantum.
The Geiger-counter (also called Geiger-Miiller counter) was the first electrical counter used to detect radiation. It consists of a wire electrode in a gas filled chamber with a voltage at the wire which is sufficiently high (- 1500 V) to attract an electron in the gas so strongly that it would ionise many atoms of the gas along its path (Figure A5. la). Thus, an ionisation effect by a single incident X-ray photon results in a chain
INTRODUCTION TO TEXTURE ANALYSIS
insulator
crystal glass photocathode
photomultiplier tube
Figure A5.1 Schemat~crepresentation of X-ra) counters (adapted from Culllt), 1978)
of ~onisingevents that culminate in a strong discharge through the gas. The number of such pulses in a given time interval is a measure of the radiation intensity. The main disadvantage of the Geiger-counters for X-ray texture analysis is the long deadtime of 10P4s between initiation of the ionisation chain and resetting the counter. which may lead to losses in the order of 10% at counting rates of only 500sP'. Another drawback is that the output of Geiger-counters does not depend on the incident wavelength, i.e. it is not proportional, so that Geiger-counters are today virtually obsolete in diffractometry.
-
APPENDIX 5
359
Scirzti&.ztion counters
Much more often scintillation counters are in use, which consist of a fluorescent crystal mounted at the end of a photomultiplier tube (Figure A5.lb). An X-ray photon absorbed in this crystal - typically a sodium iodide crystal containing l % thalliurn ejects photoelectrons which energise fluorescence of the thallium-ions. The fluorescent light ejects electrons in the photomultiplier tube which are multiplied and finally transferred to the measuring electronics. Scintillation detectors stand out by a constant. very high efficiency throughout the range of wavelengths used in X-ray diffractometry and by a short dead-time of only 1 0 - ~s caused by the decay of the fluorescence from a photon absorption. Thus, X-ray data with rates of 10000 S-' can be counted with losses below 1%. The output voltage is proportional to the energy of the X-ray photon, so that the reflected peak-intensity can be distinguished from background and fluorescent radiation with different wavelengths by setting an appropriate energy window of a subsequent pulse-height analyser (see below).
-
-
Proportional counters
In principle. a proportional counter resembles a Geiger-counter. but it works at a lower voltage of lO00V to ensure proportionality between the energy of the absorbed photon and the amplitude of the output signal. The dead-time of proportional counters is comparable to that of scintillation counters, permitting similar counting rates. Their efficiency is lower, however, and depends more strongly on the wavelength of the X-rays. Its amplification is smaller. requiring stronger amplification by the subsequent electronics. The main advantage of proportional counters is given by the fact that for monochro~naticradiation the distribution of amplitudes of the pulses is narrower than with scintillation counters, thus allowing a better separation of diffracted peak and background intensity.
-
Semiconductor detectors
Nowadays. increasingly solid-state senzicorzductor detectors are used, since they produce pulses proportional to the absorbed X-ray energy with better energy resolution than any other counter. Furthermore, they stand out by a high efficiency and by low counting losses. A semiconductor detector basically consists of a silicon single crystal containing small amounts of lithium, and therefore, these counters are designated Si(Li). Germanium counters are in use for detection of radiation with shorter wavelengths. The crystal is produced so as to comprise very thin surface layers of n-type on one side, meaning it contains free electrons from donor impurities. and p-type on the opposite side, i.e. containing free holes from acceptor impurities (Figure A5. lc). The central region is neither 71- nor p-type. i.e. it is intrinsic (i), which means that it has a very high resistivity because only few electrons are thermally excited from the valence band across the energy gap into the conduction band. However, an incident X-ray quantum can cause excitation and create a multitude (about 1000) of pairs of free electrons (-) in the conduction band and free holes (+) in the valence band. If a high voltage is applied to the opposite faces of the
360
INTRODUCTION TO TEXTURE ANALYSIS
crystal, the electrons and holes will be swept to these faces, creating a small electrical pulse. Subsequently, these very small pulses are amplified by a field-effect transistor (FET). Disadvantages are that the semiconductor counters must be cooled with liquid nitrogen or with a Peltier-cooler and that they require more complex electronics.
In all counters that are commonly used today, scintillation, proportional and semiconductor, the output voltage depends on the wavelength of the X-ray photon in so far as the size of the resulting pulses is proportional to the energy of the triggering X-ray quanta. This means that if an X-ray quantum of CuKa-radiation (E = hu = 9 keV) generates a pulse of 1 mV, then a MoKa-quantum (lzv = 20 keV) would produce a pulse of (2019) = 2.2mV. Therefore, electrical units that distinguish between pulses of different size, so-called pulse-height analyse^.^, can distinguish between X-rays of different energy and, therewith, background and fluorescent radiation with different wavelengths can be reduced. Let us assume that X-rays with three different wavelengths a > b > c are detected by a counter, which in turn produces pulses of different sizes A < B < C (Figure A5.2). If an electronic circuit that transmits only pulses with size larger than a certain value (given by the voltage V 1 )is inserted between counter and rate meter, then pulses with larger wavelengths, e.g. pulse A, are not counted. This can be used to reduce background intensities resulting from the continuous spectrum as well as fluorescence with long
Figure A5.2 Pulse-height analysis (adapted from Cullity, 1978).
APPENDIX 5
361
wavelengths. If the device is additionally able to reject pulses with size larger than a voltage V*, e.g. pulse C, then diffracted intensities from low wavelengths will be ignored as well (Figure A5.2). Thus, with a proper setting of the baseline and the window of the pulse-height analyser the peak/background ratio can markedly be improved.
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APPENDIX 6 KIKUCHI MAPS OF FCC, BCC AND HEXAGONAL CRYSTAL STRUCTURES
This Appendix comprises Kikuchi maps for fcc, bcc and hexagonal crystal structures, which can be used to aid manual indexing or to check automatic indexing of Kikuchi patterns (Section 6.3). The maps are constructed from the most prominent reflectors usually visible in EBSD patterns of reasonable quality. Note however that SAC and TEM Kikuchi patterns usually contain higher-index reflectors as well (Sections 6.2 and 6.3). Figure A6.la shows the Kikuchi map for fcc crystal structures, constructed from the four most prominent reflectors (111), (2001, (220) and (311). The map covers one stereographic unit triangle ([001]-[011]-[Ill]), which approximately corresponds to the capture angle in standard EBSD applications. The corresponding Kikuchi map for bcc crystal structures -constructed from the reflectors { l 10). 1200) (21 1) and (3 10) - is shown in Figure A6. lb. The Kikuchi map for hexagonal crystal structures (Figure A6.2) covers two stereographic triangles so as to facilitate indexing. Here, the seven most prominent reflectors (0001), {1i00), (2201), {1T01), (1%03), {1120), and (1122) (in MillerBravais notation) which can usually be observed in good quality EBSD patterns are considered. Note that interplanar and interzonal angles in hexagonal structures depend on the c l a ratio; Figure A6.2 is derived for clu = 1.59 (titanium). EBSD patterns of other hexagonal metals with different c l a ratios therefore look slightly different, though the main geometrical features remain unaffected.
I N T R O D U C T I O h T O TEXTURE ANALYSIS
Figure ,461 K~kiichimaps for ( a ! k c and (h) hcc crhstal stsucrii~-es(one iteseo~raphict r ~ n i i ~ l e )
APPENDIX 6
Figure A6.2 Kikuchi map for hexagonal crystal structures (Ti, c / n
-
1.59, two stereographic triangles).
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GENERAL BIBLIOGRAP
CONFERENCES Proc. 7th Inte~xationalConference on Textures of Materials J I C O T O M 7 ) (Ed. C.M. Brakman et al.), Netherlands, Soc. for Mat. Sci., Zwijndrecht (1984). Proc. 8th International Corference on Textures of Materials ( I C O T O M S ) (Ed. J.S. Kallend and G . Gottstein), TMS, Warrendale Pa (1988). Proc. 9th International Conference on Textures of Materials ( I C O T O M 9 ) (Ed. H.-J. Bunge), Tex. and Micros.; 14/18 (1991). Proc. 10th International Conference on Textures of Materials IICOTOMIO) (Ed. H-J Bunge), Mat. Sci. Forum, 157-162 (1994). Proc. 11th International Conference on Textures of Materials ( I C O T O M I I ) (Ed. Z . Liang et al.), Academic Publishers, Beijing (1997). Proc. Microscale Textures ofMaterials (Ed. B.L. Adains et al.), Tex. and Micros., 2 (1993). Proc. Intel;facial Engineering jor Optinzised Properties, Proc. No. 458, Materials Research Sociefy Fall Meeting (Ed. C.L. Briant et al.), Pittsburgh (1996). PI^. Textures on a Microscale, Mat. Sci. Tech., Institute of Materials, London (1997). Proc. Int. C o ~ f on . Texture and Anisotropj, qf Polycrystals~(Ed. R.A. Schwarzer), Trans. Tech. Pub. Ltd., Switzerland. 183-190.
BOOKS Bacon, G.E.. 1975, Neutron DiJfiaction Clarendon Press, Oxford. Barrett, C.S. and Massalski, T.B. 1980, Structure of Metals: Crystallograplzic Methods, Principles and Data, McGraw-Hill, New York. Bunge. H.-J. and Esling C. (Ed.), 1986, Quantitative Texture Analj>sis, D G M Informationsgesellschaft, Oberursel, Germany. Bunge, H.-J. (Ed.), 1986, Experimental Techniques of Texture Analysis, D G M Informationsgesellschaft, Oberursel; Germany.
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Bunge, H.-J. (Ed.), 1987, Theoretical Methods of Texture Analysis, D G M Informationsgesellschaft, Oberursel, Germany. Bunge, H.-J., 1982, Texture Anal.vsis in Material Science Butterworths, London. Cullity, B.D. 1978, Elenzents qf X-ray diffraction (2nd edition), Addison-Wesley Inc., USA. Dingley, D.J. Baba-Kishi, K. and Randle: V. 1994, An Atlas of Backscatter Kikuclzi Difiaction Puttenis, Institute of Physics Publishing, Bristol. Edington, J.W.: 1975, Electron Diffraction in the Electron Microscope, MacMillan, London. Edington, J.W., 1976, Prucfical Electron Microscopy in Materials Science, Van Nostrand Reinhold Co., New York. Goodhew P.J. and Humphreys F.J., 1988, Electron Microscopj~and Analj~sis(2nd edition), Taylor and Francis, London. Hatherly, M. and Hutchinson, W.B., 1979, An Introduction to Textures in Metals (Monograph no. 5), The Institute of Metals, London. Hirsch, P.B., Howie, A., Nicholson, R.B., Pashley, D.W. and Whelan, M.J., 1977, Electron Microscopy of Thbz Crystals, Krieger Pub.: Malabar Fla. Johari, 0. and Thomas, G., 1969, The Stereographic Projection and its Applicatiom, Vol. IIA of Techniques of Metal Research (Ed. R.F. Bunshah), Interscience, New York. Kocks, U.F. et al. (Eds.), 1998, Texture and Anisotropy, Preferred Orientatiorzs, and their EfJct on Materials Properties, Cambridge Univ. Press. Loretto, M.H., 1994, Electron Beurn Analysis of Materials, Chapman and Hall, LondonlNew York. McKie, C. and McKie, D.. 1974, C~ystallineSolids, Nelson, London. Phillips, F.C.. 1971, A72 Introduction to C~ystallogrcihy (4th edition). Longman Group Ltd., London. Randle, V., 1992, Microtexture Deterr?zination and its Applications, Institute of Materials, London. Randle, V., 1993, The Measurement of Grain Bourzdaq~ Geometry, Institute of Physics Publishing, Bristol. Randle, V., 1996, The Role oj" the Coincidence Site Lattice in Grain Bozmdary Engineering, Institute of Materials, London. Reimer, L., 1997, Transmission Electron Microscopj: Physics of' Iniage Formation and Microannlysis, Springer-Verlag, BerlinINew York. Reimer, L., 1998, Scanning Electron Microscopy: Physics qf Imnge Fornzation and Microannlysis, Springer-Verlag, BerlinINew York. Smallman, R.E. and Bishop, R.J., 1995, Metals and Materials, ButterworthHeinemann Ltd., Oxford. Wenk, H.-R. (Ed.), 1985, Preferred Orientation in Defornzed Metals and Rocks: Academic Press Inc., UK.
INDEX
Absorption 41, 51-5. 65-70, 72-4, 85. 89, 95, 100, 157. 197 Absorption edge 52-4, 68, 280 Angle axls p a x 20, 31-9. 225-33 263-4 Anglelax~s of misoiientatlon 34, 36, 186, 266 of rotation 13. 31-6. 40 Atomic scattering amphtude 42-3, 47 Automated crystal orientat~onm'lpping. ACOM ree oiientatlon mappmg and orientation microscopy Axes crystal 16, 31, 34-8, 86. 174. 271 specimen 16, 24. 33-40. 134, 176, 335 symmetry 39, 234 lefeience 34-8. 103 130 137, 179, 185. 210, 269
Crystal coordinate system see coordinate system and axes. crystal Crystal orientation distribution, C O D 114-17 Crystal orientation mapping, COM see orientation mapping and orientation inicroscopy Crystallographically-related solution 17-21, 28-30, 33. 37-8, 113-4. 1 16-7, 220-2, 228. 241, 267, 347-9 Cylindrical angleiaxis space 13, 36-7, 225, 230, 2 3 3 4 . 240-1
Back-scatter detector, forward-mounted 160, 167-8 Backscatter Kikuchi diffraction, BKD see electron backscatter diffraction Bragg's Law 42-6, 56-7. 61-2, 69, 80, 129, 154 -7, 173, 1 9 2 4 . 199 Brandon criterion 265 Burns algorithm 142
Debjre-Schelrer method 48-51, 61, 195-7, 279-84 Diffraction Bragg-Bientano condition 79. 90 electron 49, 55-8. 127-8, 154, 189. 290 neutlon 54-5, 61-3, 88-92. 95-6. 100-1, 206, 244. 280. 289 X-ray 5. 40. 52-8. 61, 65-100, 214, 279, 281, 289, 291, 295. 357 Diffractometi y, angulai dispels~ve 80 energy d~speisive 80-2, 90 Disorientation 225-9. 3 11, 323
Camera constant 194 Characteristic radiation see X-lays Clustering 6, 10, 254-5 274 Coincidence plot 264 Coincidence site lattice. CSL 186. 224-36, 254 -5,257-9.264-8.299-301,303-5.337 consti ained 266 in Euler space 231-2, 265 in Rodrgues space 233-5, 265 C-value 264-5, 275 Connectivity 7, 207, 243, 254-8, 265. 274-8. 315 Convergent beam electron diffract~on, CBED 128. 189, 198-203, 290-1 Cooidinate system 14-37. 105. 112-14, 121. 136, 140-1. 208, 210, 266, 272-3, 301, 307. 343
Electron backscatter diffract~on, EBSD 10-1 1. 157-88, 291-2. 295-317, 328-9, 335-7 accelerating voltage 171-6 accuracy 160, 162. 167, 171. 176, 180, l87 camera 158-60, 165-85 components 162 deslgn ph~losophy 168 pattern enhancement 160, 166, 176-8 evolution 157-61 hardware 159-60. 165-8. 172, 176, 180 piobecuirent 166. 174-6, 185 resolution 162. 171-6, 180, 187 softwaie 139-45, 159. 160, 178 specimen preparation 162-5 working distance 172, 175. 179
386
INDEX
Electron backscatter diffraction calibration 178-85 iterative pattern fitting 184-5 known crystal 180-3 pattern magnification 183 shadow casting 180 Electron backscatter patterns, EBSP see electron backscatter diffraction Electron channeling diffractiou see selected area channeling Electrons. Auger 56 backscattered 11, 56, 157, 167, 172 secondary 56 Equal area projection w e projection, equal area Euler angles 13. 18-20, 25-36, 40, 112-19, 138, 186. 213, 230-6. 255, 263 Euler space 13, 25-31, 38. 101, 106, 112-23, 186. 21 3-20, 230-43 cylindrical 115-1 8 Fibre axis 119, 223 plot 119-21 texture 40, 97, 103, 109. 121, 208-10, 221-3. 254, 283. 295, 331 Friedel's Law 65. 280 g-space see orientation space Gnomonic projection see projection. gnomonic Grain boundary 4. 34. 186, 191, 208, 224-38, 249, 254, 257, 260-1: 2 6 4 7 4 , 299, 303, 319 Asymmetrical tilt. ATGB 266 general see random random 266 special 232, 257 symmetrical tilt, STGB 266 twist, TWGB 266 Grain boundary design see grain boundary engineering Grain boundary engineering 7, 238 Grain boundary plane 266-74, 299 Grain junction 229, 250, 254, 260. 275 H ~ g hresolut~onelectron microscopy, HREM 189-91 Higher order Laue zone, HOLZ 200 Hough transform 142-7, 28 1-2 Ideal orientation 20-33. 40, 186 Image quality 134. 148-51, 186. 258, 26 1
Intercrystalline structure distribution function, ISDF 277 Interface damage function, IDF 277 Interface-plane scheme 266-8. 301 Intra-grain 186, 224, 229, 251, 257 Inverse pole figure 25. 82, 86-8, 102-3, 106. 116, 121, 186,208-10,213, 225.283, 315 Kikuchi band 130. 134-8; 145, 147, 152, 155. 173, 182 Kikuchi diffraction pattern see Kikuchi pattern Kikuchi line 128-32, 134, 146-8, 155, 198-200 Kikuchi map 139, 363 Kikuchi pattern 127-52. 198-203, 281, 290-2, 351 automatic indexing 132, 1 3 9 4 8 beam normal 136-9 evaluation 1 3 3 4 8 formation 128-3 1 indexing 127, 130, 134-6 projection 132-3 Kossel technique 57, 127, 153-5. 290-1 Laue method 48-51, 274, 279-82, 290-1 Light 41.58 polarised 285-7 Mackenzie distribution 225. 305 cell 234, 236 triangle see Mackenzie distribution Mass absorption coefficient 51-3, 69 Matrix, crystal 16, 136. 141 identity 20. 33. 275 misorientation 34, 347 orientation 13; 1 6 4 0 , 134-42, 186, 210: 27 1, 347 rotation see orientation matrix symmetry 17-21, 347-8 Mesotexture 7. 224 Microdiffraction 128, 189, 198-203, 290-2, 319, 323 Micro-Kossel technique see Kossel technique Microscopy, optical 5, 162-3. 312-13 Miller indices 13, 16-22, 40; 46. 138, 220, 266-8, 301, 307, 313, 343-5 Miller-Bravais indices see Miller indices Misorientation 7, 30-1, 3 4 4 0 , 137, 148, 169, 186, 190, 205-7, 220, 224-44, 253, 257-8, 263-78, 280, 299, 303, 310, 323, 335
INDEX Misor~entat~on d i s t i ~ b u t ~ ofunction. n MODF, M D F 101.224,2314,23641.277 phys~cal 237 stat~stlcal 101, 236 texture-derived Aee stat~st~cnl MODF hlonochiomator 69, 82, 88. 90, 280, 282 Multl-phase 8, 5 4 5 , 77, 92-6, 100, 171, 252-3. 256-8, 289, 315-17 Neutron diffraction see diffraction; neutron Normal direction, N D 14, 21; 24-6. 31, 87, 115-19, 136-~7;169, 210, 220. 327 Normal to rolling plane see normal direction Optical nlethods 279. 285-8, 291 (111) etching 286-7, 291 anodical oxidation 286 etch pits 286. 291, 335-7 U-stage 286, 312 Orientation, absolute 34, 137, 180 Orientation connectivity see connectivity Orientation correlation function, OCF 237-9 Orientation difference distribution function, O D D F 236-9 Orientation distribution, continuous 38. 214-17, 220 Orientation distribution function, O D F 5, 69, 46, 82. 86-7. 9 2 4 , 101, 104-23, 197, 205, 213-19, 236-8, 285, 327 C-coefficient 82, 105-12, 122.215-16.240 component method 112 direct methods 109-12, 122, 213-17, 231, 236 fundamental equation 105-10 ghost error 65. 106-9. 215 harmonic method 105. 109; 112. 122 maximum entropy method 109 positivity method 108 quadratic method 107-8 series expansion method 75. 105-9, 112. 122, 205, 213-17, 231, 236, 240, 243 spherical harmonic function 105 truncation error 106, 112, 122 vector method 111 WIMV method 94, 110-1 1 zero-range method 107 Orientation imaging microscopy, OIM see orientation microscopy and orientation mapping Orientation, local 153, 155, 198, 202. 207, 230. 243, 279. 287, 290-2, 323 Orientation map 11, 186, 207, 219-20, 245-62
387
Orientation mapping 1 1-12, 186. 212, 245-62, 291, 303; 323, drawbacks 262 misorientations and interfaces 253, 257-8 pattern quality 253, 261 strengths 261 texture components 253. 255-7 Orientation microscopy 11 245-62, 142-5, 202, 291 beam control 248-9 stage control 248 Orientation perturbation see intra-grain Orientation relationship see misorientation and phase relationship Orientation space 28, 101, 109-23, 21 3-1 6, 231-4, 2 3 6 9 Orientation topography 5; 245-7. 287, 290 Orientations. single, statistical relevance 206, 215. 217-20, 291 ;
Pattern centre. PC 132-6, 140, 158-9. 178-84 Pattern clar~ty see m a g e quality Peak separation 77-8. 88, 90, 92-5 Penetrat~ondepth 52-6. 69-73. 95-9, 206, 280-1 Phase analysis 78. 155 boundary 34, 134. 191, 240, 257, 263, 268. 292 ident~fication 153, 160 reldtionsh~p 6. 8-9, 34. 224, 240, 315 transformation 4, 90, 112, 289, 307 Photoglainmetry 2 7 2 4 Pole see zone axis Pole figure 5. 22-5, 51. 61-127, 186, 189, 195-7. 206-13. 222, 225, 240. 243, 255. 2 8 3 4 , 295, 311. 315, 327. 331, 335, 353 angles 24-5. 63, 71. 73, 89. 101. 104-5, 210. 212 background 82-3, 92, 197 complete 74; 86, 89, 197 contouring 102-3 counting statistics 82, 85-6; 89: 95 defocusing 69, 71-2, 82-5 incomplete 86, 104, 106; 112. 122. 197 inverse see inverse pole figure macrotexture 93 magnetic 55 microtexture 240, 295 normalisation 82, 86, 102-3, 112, 212-13 scanning 7 4 6 selected area diffraction 195-7 window 75
388
INDEX
Pole figure inversion 101. 105, 122 Position sensitive detector, PSD neutron radiation 90-2, 94, 100 X-radiation see X-ray detector Powder sample 50-1, 71, 85, 282 Preferred orientation 3, 6 51. 61 Projection. equal-area 23, 352-5 gnomoiiic 13241. 155, 178, 183. 281, 355-6 spherical 351 stereographic 22; 34, 36, 63, 210. 225, 234. 351-6 R-vector .see Rodrigues vector Radon traiisform 142-4 Reciprocal lattice 194, 198 Reference sphere 22-3, 132, 178, 234, 311; 351-6 Reflector 46, 81, 132. 363 R F space .see Rodrigues space Rodrigues space 3 7 4 0 . 102, 206. 220-5, 230, 234-8, 240, 243, 265. 305, 323-6 fundamental zone 38-40. 220-2, 234-6 Rodrigues vector 13; 18-20, 34, 3 6 4 0 , 138, 186, 230, 235. 255 Rodrigues-Frank space see Rodrigues space Rolling direction, R D 14, 22--6. 31; 87, 114, 118-19, 137-8, 169. 210; 220, 327-8 Sample coordinate system see coordinate system and axes. specimen Sample orientation distribution, SOD 116 Sample symmetry see crystallographicallyrelated solution Sample preparation see specimen preparation Scattering 42-6, diffuse 128, 198-9 elastic 42; 55. 56; 128. 198-9 inelastic 56, 199 magnetic 55 nuclear 55 Schmidt net 354-5 Selected area channeling. SAC 57, 128. 132, 136, 141, 148; 153. 155-8; 162, 179. 202, 291, 363 Selected area diffraction, SAD 56, 112. 145-7, 189, 191-203 Single crystal 49, 62, 64. 68-9, 104, 113, 134, 164, 175, 177, 180-2, 192. 198. 210, 217, 281, 286, 289, 324
Skeleton line 119 Slip trace 263. 268, 271. 292 Specimen coordinate system Tee coordinati system and axes. specimen Specimen preparation 96-9. 162-5, 191, 202. 282, 291 Stereogram Tee projection, stereographic Structure factor 46-8 Synchrotron radiation 79, 279-84, 289 Texture by number 212, 216, 243, 252, 303 by volume 211. 216, 243, 252, 303 local see orientation, local raiidom 61, 84-7, 102, 195, 275, 282, 303 Time-of-flight, TOF 88. 90-2. 95, 100 Topology 2 7 4 5 Triple line see grain junction Twin 168-9, 171, 226, 138-9. 231, 254-5, 265. 297-301. 319-20 Ultrasonic measurements Wulff net
58, 279, 284-5
353-5
X-rays 5, 41-58. 61-100, 153-5, 187, 206, 279-84. 289-92, 295. 327, 331, 357 characteristic radiation 67-8, 88-9, 154 continuous radiation 66. 89, 92 filter 54, 67-9, 82 fluoiescence 53. 69-70, 80-2, 93, 359 monochioniatic 44. 50. 52, 54. 62, 68-9 80, 82. 280, 282. 359 peak broadening 71. 77, 79, 83. 92 peneti'itioii 41. 51-2, 69. 71, 96-9 polychromatic. 'white' 49. 81, 90, 280-1 spectrum 54, 61, 66, 68 tube 53. 65-70. 82. 91 X-ray detector energy dispersive 81-2, 87-8. 94. 100 Geiger counter 61, 76-7. 357-8 position sensitive, PSD 77-9, 80, 82, 87, 94. 100 proportional counter 77, 359 pulse height andlysis 77. 80, 357-61 scintillation counter 76-7, 359 semiconductor counter 77. 81. 359 60 Zone axis 16. 130, 132, 136. 142. 148-9, 180-1, 192, 194
