Handbook of X-Ray Spectrometry Revised and Expanded

Handbook of X-Ray Spectrometry Second Edition, Revised and Expanded

edited by René E. Van Grieken University of Antwerp Antwerp, Belgium

Andrzej A. Markowicz Vienna, Austria

Marcel Dekker, Inc.

New York • Basel

TM

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

Copyright © 2002 Marcel Dekker, Inc.

ISBN: 0-8247-0600-5 First edition was published as Handbook of X-Ray Spectrometry: Methods and Techniques This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http:==www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales=Professional Marketing at the headquarters address above. Copyright # 2002 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

Copyright © 2002 Marcel Dekker, Inc.

Preface to the Second Edition

The positive response to the first edition of Handbook of X-Ray Spectrometry: Methods and Techniques and its commercial success have shown that in the early 1990s there was a clear demand for an exhaustive book covering most of the specialized approaches in this field. Therefore, some five years after the first edition appeared, the idea of publishing a second edition emerged. In the meantime, remarkable and steady progress has been made in both instrumental and methodological aspects of x-ray spectrometry. This progress includes considerable improvements in the design and production technology of detectors and in capillary optics applied for focusing the primary photon beam. The advances in instrumentation, spectrum evaluation, and quantification have resulted in improved analytical performance and in further extensions of the applicability range of xray spectrometry. Consequently, most of the authors who contributed to the first edition of this book enthusiastically accepted the invitation to update their chapters. The progress made during the last decade is reflected well in the chapters of the second edition, which were all considerably revised, updated, and expanded. A completely new chapter on microbeam x-ray fluorescence analysis has also been included. Chapter 1 reviews the basic physics behind x-ray emission techniques, and refers to extensive appendices for all the basic and generally applicable x-ray physics constants. New analytical expressions have been introduced for the calculation of fundamental parameters such as the fluorescence yield, incoherent scattering function, atomic form factor, and total mass attenuation coefficient. Chapter 2 outlines established and new instrumentation and discusses the performances of wavelength-dispersive x-ray fluorescence (XRF) analysis, which, with probably 15,000 units in operation worldwide today, is still the workhorse of x-ray analysis. Its applications include process control, materials analysis, metallurgy, mining, and almost every other major branch of science. The additional material in this edition covers new sources of excitation and comprehensive comparisons of the technical parameters of newly produced wavelength-dispersive spectrometers. Chapter 3 has been completely reconsidered, modified, and rewritten by a new author. The basic principles, background, and recent advances are described for the tubeexcited energy-dispersive mode, which is invoked so frequently in research on environmental and biological samples. This chapter is based on a fresh look and follows a completely different approach.

Copyright © 2002 Marcel Dekker, Inc.

Chapter 4 reviews in depth the available alternatives for spectrum evaluation and qualitative analysis. Techniques for deconvolution of spectra have enormously increased the utility of energy-dispersive x-ray analysis, but deconvolution is still its most critical step. The second edition includes discussions of partial least-squares regression and modified Gaussian shape profiles. Chapter 5 reviews quantification in XRF analysis of the classical and typical ‘‘infinitely thick’’ samples. In addition to being updated, the sections on calibration, quality control, and mathematical correction methods have been expanded. Chapter 6, on quantification for ‘‘intermediate-thickness’’ samples, now also includes the presentation of a modified version of the emission-transmission method and a discussion of both the accuracy and limitations of such methods. Chapter 7 is a completely original treatment by a new author of radioisotope-induced and portable XRF. It discusses semiconductor detectors, including the latest types, analyzes in detail the uncertainty sources, and reviews the recent and increasingly important applications. Since the appearance of the first edition, synchrotron-induced x-ray emission analysis has increased in importance. Chapter 8 was updated and modified by including a comprehensive review of the major synchrotron facilities. Although its principles have been known for some time, it is only since the advent of powerful commercial units and the combination with synchrotron sources that total reflection XRF has rapidly grown, mostly now for characterization of surfaces and of liquid samples. This is the subject of the substantially modified and expanded Chapter 9. The new authors have taken a radically different approach to the subject. Polarized-beam XRF and its new commercial instruments are treated in detail in a substantially revised and expanded Chapter 10. Capillary optics combined with conventional fine-focus x-ray tubes have enabled the development of tabletop micro-XRF instruments. The principles of the strongly growing microbeam XRF and its applications are now covered thoroughly in an additional chapter, Chapter 11. Particle-induced x-ray emission analysis has grown recently in its application types and particularly in its microversion. Chapter 12 discusses the physical backgrounds, instrumentation, performance, and applications of this technique. The sections dealing with the applications were substantially expanded. Although the practical approaches to electron-induced x-ray emission analysis— a standard technique with wide applications in all branches of science and technology— are often quite different from those in other x-ray analysis techniques, a treatment of its potential for quantitative and spatially resolved analysis is given in Chapter 13. The new and expanded sections deal with recent absorption correction procedures and with the quantitative analysis of samples with nonstandard geometries. Finally, the completely updated and revised Chapter 14 reviews the sample preparation techniques that are invoked most frequently in XRF analysis. The second edition of this book is again a multiauthored effort. We believe that having scientists who are actively engaged in a particular technique covering those areas in which they are particularly qualified outweighs any advantages of uniformity and homogeneity that characterize a single-authored book. The editors (and one coworker) again wrote three of the chapters in the new edition. For all the other chapters, we were fortunate to have the cooperation of truly eminent specialists, some of whom are new contributors (see Chapters 3, 7, 9, 10 and 11). We wish to thank all the contributors for their considerable and (in most cases) timely efforts. Copyright © 2002 Marcel Dekker, Inc.

We hope that novices in x-ray emission analysis will find this revised and expanded handbook useful and instructive, and that our more experienced colleagues will benefit from the large amount of readily accessible information available in this compact form, some of it for the first time. An effort has been made to emphasize the fields and developments that have come into prominence lately and have not been covered in other general books on x-ray spectrometry. We also hope this book will help analytical chemists and other users of x-ray spectrometry to fully exploit the capabilities of this powerful analytical tool and to further expand its applications in material and environmental sciences, medicine, toxicology, forensics, archeometry, and many other fields. Rene´ E. Van Grieken Andrzej A. Markowicz

Copyright © 2002 Marcel Dekker, Inc.

Preface to the First Edition

Scientists in recent years have been somewhat ambivalent regarding the role of x-ray emission spectrometry in analytical chemistry. Whereas no radically new and stunning developments have been seen, there has been remarkably steady progress, both instrumental and methodological, in the more conventional realms of x-ray fluorescence. For the more specialized approaches—for example, x-ray emission induced by synchrotron radiation, radioisotopes and polarized x-ray beams, and total-reflection x-ray fluorescence— and for advanced spectrum analysis methods, exponential growth and=or increasing acceptance has occurred. Contrary to previous books on x-ray emission analysis, these latter approaches make up a large portion of the present Handbook of X-Ray Spectrometry. The major milestone developments that shaped the field of x-ray spectrometry and now have widespread applications all took place more than twenty years ago. After wavelength-dispersive x-ray spectrometry had been demonstrated and a high-vacuum x-ray tube had been introduced by Coolidge in 1913, the prototype of the first modern commercial x-ray spectrometer with a sealed x-ray tube was built by Friedmann and Birks in 1948. The first electron microprobe was successfully developed in 1951 by Castaing, who also outlined the fundamental concepts of quantitative analysis with it. The semiconductor or Si(Li) detector, which heralded the advent of energy-dispersive x-ray fluorescence, was developed around 1965 at Lawrence Berkeley Laboratory. Acceleratorbased particle-induced x-ray emission analysis was developed just before 1970, mostly at the University of Lund. The various popular matrix correction methods by Lucas-Tooth, Traill and Lachance, Claisse and Quintin, Tertian, and several others, were all proposed in the 1960s. One may thus wonder whether the more conventional types of x-ray fluorescence analysis have reached a state of saturation and consolidation, typical for a mature and routinely applied analysis technique. Reviewing the state of the art and describing recent progress for wavelength- and energy-dispersive x-ray fluorescence, electron and heavy charged-particle-induced x-ray emission, quantification, and sample preparation methods is the purpose of the remaining part of this book. Chapter 1 reviews the basic physics behind the x-ray emission techniques, and refers to the appendixes for all the basic and generally applicable x-ray physics constants. Chapter 2 outlines established and new instrumentation and discusses the performances of wavelength-dispersive x-ray fluorescence analysis, which, with probably 14,000 units in operation worldwide today, is still the workhorse of x-ray analysis with applications in a wide range of disciplines including process control, materials analysis, metallurgy, Copyright © 2002 Marcel Dekker, Inc.

mining, and almost every other major branch of science. Chapter 3 discusses the basic principles, background, and recent advances in the tube-excited energy-dispersive mode, which, after hectic growth in the 1970s, has now apparently leveled off to make up approximately 20% of the x-ray fluorescence market; it is invoked frequently in research on environmental and biological samples. Chapter 4 reviews in depth the available alternatives for spectrum evaluation and qualitative analysis; techniques for deconvolution of spectra have enormously increased the utility of energy-dispersive x-ray analysis, but deconvolution is still its most critical step. Chapters 5 and 6 review the quantification problems in the analysis of samples that are infinitely thick and of intermediate thickness, respectively. Chapter 7 is a very practical treatment of radioisotope-induced x-ray analysis, which is now rapidly acquiring wide acceptance for dedicated instruments and field applications. Chapter 8 reviews synchrotron-induced x-ray emission analysis, the youngest branch, with limited accessibility but an exponentially growing literature due to its extreme sensitivity and microanalysis potential. Although its principles have been known for some time, it is only since the advent of powerful commercial units that total reflection x-ray fluorescence has been rapidly introduced, mostly for liquid samples and surface layer characterization; this is the subject of Chapter 9. Polarized beam x-ray fluorescence is outlined in Chapter 10. Particle-induced x-ray emission analysis is available at many accelerator centers worldwide; the number of annual articles on it is growing and it undergoes a revival in its microversion; Chapter 11 treats the physical backgrounds, instrumentation, performance, and applications of this technique. Although the practical approaches to electron-induced x-ray emission analysis, now a standard technique with wide applications in all branches of science and technology, are often quite different from those in other x-ray analysis techniques, a separate treatment of its potential for quantitative and spatially resolved analysis is given in Chapter 12. Finally, Chapter 13 briefly reviews the sample preparation techniques that are invoked most frequently in combination with x-ray fluorescence analysis. This book is a multi-authored effort. We believe that having scientists who are actively engaged in a particular technique covering those areas for which they are particularly qualified and presenting their own points of view and general approaches outweighs any advantages of uniformity and homogeneity that characterize a single-author book. Three chapters were written by the editors and a coworker. For all the other chapters, we were fortunate enough to have the cooperation of eminent specialists. The editors wish to thank all the contributors for their efforts. We hope that novices in x-ray emission analysis will find this book useful and instructive, and that our more experienced colleagues will benefit from the large amount of readily accessible information available in this compact form, some of it for the first time. This book is not intended to replace earlier works, some of which were truly excellent, but to supplement them. Some overlap is inevitable, but an effort has been made to emphasize the fields and developments that have come into prominence lately and have not been treated in a handbook before. Rene´ E. Van Grieken Andrzej A. Markowicz

Copyright © 2002 Marcel Dekker, Inc.

Contents

Preface to the Second Edition Preface to the First Edition Contributors 1 X-ray Physics Andrzej A. Markowicz I. Introduction II. History III. General Features IV. Emission of Continuous Radiation V. Emission of Characteristic X-rays VI. Interaction of Photons with Matter VII. Intensity of Characteristic X-rays VIII. IUPAC Notation for X-ray Spectroscopy Appendixes I. Critical Absorption Wavelengths and Critical Absorption Energies II. Characteristic X-ray Wavelengths (A˚) and Energies (keV) III. Radiative Transition Probabilities IV. Natural Widths of K and L Levels and Ka X-ray Lines (FWHM), in eV V. Wavelengths of K Satellite Lines (A˚) VI. Fluorescence Yields and Coster–Kronig Transition Probabilities VII. Coefficients for Calculating the Photoelectric Absorption Cross Sections t (Barns=Atom) Via ln–ln Representation VIII. Coefficients for Calculating the Incoherent Collision Cross Sections sc (Barns=Atom) Via the ln–ln Representation IX. Coefficients for Calculating the Coherent Scattering Cross Sections sR (Barns=Atom) Via the ln–ln Representation X. Parameters for Calculating the Total Mass Attenuation Coefficients in the Energy Range 0.1–1000 keV [Via Eq. (78)] XI. Total Mass Attenuation Coefficients for Low-Energy Ka Lines XII. Correspondence Between Old Siegbahn and New IUPAC Notation X-ray Diagram Lines References Copyright © 2002 Marcel Dekker, Inc.

56

2 Wavelength-Dispersive X-ray Fluorescence Jozef A. Helsen and Andrzej Kuczumow I. II. III. IV. V. VI. VII.

Introduction Fundamentals of Wavelength Dispersion Layout of a Spectrometer Qualitative and Quantitative Analysis Chemical Shift and Speciation Instrumentation Future Prospects References

3 Energy-Dispersive X-ray Fluorescence Analysis Using X-ray Tube Excitation Andrew T. Ellis I. II. III. IV. V.

Introduction X-ray Tube Excitation Systems Semiconductor Detectors Semiconductor Detector Electronics Summary References

4 Spectrum Evaluation Piet Van Espen I. II. III. IV. V. VI. VII. VIII. IX. X.

Introduction Fundamental Aspects Spectrum Processing Methods Continuum Estimation Methods Simple Net Peak Area Determination Least-Squares Fitting Using Reference Spectra Least-Squares Fitting Using Analytical Functions Methods Based on the Monte Carlo Technique The Least-Squares-Fitting Method Computer Implementation of Various Algorithms References

5 Quantification of Infinitely Thick Specimens by XRF Analysis Johan L. de Vries and Bruno A. R. Vrebos I. II. III. IV. V. VI.

Introduction Correlation Between Count Rate and Specimen Composition Factors Influencing the Accuracy of the Intensity Measurement Calibration and Standard Specimens Converting Intensities to Concentration Conclusion References

Copyright © 2002 Marcel Dekker, Inc.

6 Quantification in XRF Analysis of Intermediate-Thickness Samples Andrzej A. Markowicz and Rene´ E. Van Grieken I. II. III. IV.

Introduction Emission-Transmission Method Absorption Correction Methods Via Scattered Primary Radiation Quantitation for Intermediate-Thickness Granular Specimens References

7 Radioisotope-Excited X-ray Analysis Stanislaw Piorek I. II. III. IV. V. VI. VII. VIII.

Introduction Basic Equations Radioisotope X-ray Sources and Detectors X-ray and g-ray Techniques Factors Affecting the Overall Accuracy of XRF Analysis Applications Future of Radioisotope-Excited XRF Analysis Conclusions Appendix: List of Companies that Manufacture Radioisotope-Based X-ray Analyzers and Systems References

8 Synchrotron Radiation-Induced X-ray Emission Keith W. Jones I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Introduction Properties of Synchrotron Radiation Description of Synchrotron Facilities Apparatus for X-ray Microscopy Continuum and Monochromatic Excitation Quantitation Sensitivities and Minimum Detection Limits Beam-Induced Damage Applications of SRIXE Tomography EXAFS and XANES Future Directions References

9 Total Reflection X-ray Fluorescence Peter Kregsamer, Christina Streli, and Peter Wobrauschek I. II. III. IV. V. VI. VII.

Introduction Physical Principles Instrumentation Chemical Analysis Surface Analysis Thin Films and Depth Profiles Synchrotron Radiation Excitation

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VIII. IX.

Light Elements Related Techniques References

10 Polarized Beam X-ray Fluorescence Analysis Joachim Heckel and Richard W. Ryon I. II. III. IV. V. VI. VII.

Introduction Theory Barkla Systems Bragg Systems Barkla-Bragg Combination Systems Secondary Targets Conclusion References

11 Microbeam XRF Anders Rindby and Koen H. A. Janssens I. II. III. IV. V.

Introduction and Historical Perspective Theoretical Background Instrumentation for Microbeam XRF Collection and Processing of m-XRF Data Applications References

12 Particle-Induced X-ray Emission Analysis Willy Maenhaut and Klas G. Malmqvist I. II.

III. IV. V. VI. VII. VIII.

Introduction Interactions of Charged Particles with Matter, Characteristic X-ray Production, and Continuous Photon Background Production Instrumentation Quantitation, Detection Limits, Accuracy, and Precision Sample Collection and Sample and Specimen Preparation for PIXE Analysis Applications Complementary Ion-Beam-Analysis Techniques Conclusions References

13 Electron-Induced X-ray Emission John A. Small, Dale E. Newbury, and John T. Armstrong I. II. III. IV. V.

Introduction Quantitative Analysis Microanalysis at Low Electron Beam Energy Analysis of Samples with Nonstandard Geometries Spatially Resolved X-ray Analysis References

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14 Sample Preparation for X-ray Fluorescence Martina Schmeling and Rene´ E. Van Grieken I. II. III. IV. V. VI. VII.

Introduction Solid Samples Fused Specimen Liquid Specimen Biological Samples Atmospheric Particles Sample Support Materials References

Copyright © 2002 Marcel Dekker, Inc.

Contributors

John T. Armstrong, Ph.D. Gaithersburg, Maryland

National Institute of Standards and Technology,

Eindhoven, The Netherlands

Johan L. de Vries, Ph.D.*

Oxford Instruments Analytical Ltd., High Wycombe,

Andrew T. Ellis, Ph.D. Buckinghamshire, England Joachim Heckel, Ph.D. many

Spectro Analytical Instruments, GmbH & Co. KG, Kleve, Ger-

Jozef A. Helsen, Ph.D.

Catholic University of Leuven, Leuven, Belgium University of Antwerp, Antwerp, Belgium

Koen H. A. Janssens, Ph.D. Keith W. Jones, Ph.D.

Brookhaven National Laboratory, Upton, New York

Peter Kregsamer, Dr. techn., Dipl. Ing.

Lublin Catholic University, Lublin, Poland

Andrzej Kuczumow, Ph.D. Willy Maenhaut, Ph.D.

Ghent University, Ghent, Belgium

Klas G. Malmqvist, Ph.D. Sweden

Lund University and Lund Institute of Technology, Lund,

Andrzej A. Markowicz, Ph.D. Dale E. Newbury, Ph.D. Maryland

Atominstitut, Vienna, Austria

Vienna, Austria

National Institute of Standards and Technology, Gaithersburg,

Copyright © 2002 Marcel Dekker, Inc.

Stanislaw Piorek, Ph.D.{ Anders Rindby, Ph.D. Go¨tebo¨rg, Sweden

Niton Corporation, Billerica, Massachusetts

Chalmers University of Technology and University of Go¨tebo¨rg,

Lawrence Livermore National Laboratory, Livermore, Califor-

Richard W. Ryon, B.A. nia

Martina Schmeling, Ph.D.

Loyola University Chicago, Chicago, Illinois

John A. Small, Ph.D. Maryland

National Institute of Standards and Technology, Gaithersburg,

Christina Streli, Ph.D.

Atominstitut, Vienna, Austria

Piet Van Espen, Ph.D.

University of Antwerp, Antwerp, Belgium

Rene´ E. Van Grieken, Ph.D.

University of Antwerp, Antwerp, Belgium

Bruno A. R. Vrebos, Dr. Ir.

Philips Analytical, Almelo, The Netherlands

Peter Wobrauschek, Ph.D.

Copyright © 2002 Marcel Dekker, Inc.

Atominstitut, Vienna, Austria

1 X-ray Physics Andrzej A. Markowicz Vienna, Austria

I.

INTRODUCTION

In this introductory chapter, the basic concepts and processes of x-ray physics that relate to x-ray spectrometry are presented. Special emphasis is on the emission of the continuum and characteristic x-rays as well as on the interactions of photons with matter. In the latter, only major processes of the interactions are covered in detail, and the cross sections for different types of interactions and the fundamental parameters for other processes involved in the emission of the characteristic x-rays are given by the analytical expressions and=or in a tabulated form. Basic equations for the intensity of the characteristic x-rays for the different modes of x-ray spectrometry are also presented (without derivation). Detailed expressions relating the emitted intensity of the characteristic x-rays to the concentration of the element in the specimen are discussed in the subsequent chapters of this handbook dedicated to specific modes of x-ray spectrometry.

II.

HISTORY

X-rays were discovered in 1895 by Wilhelm Conrad Ro¨ntgen at the University of Wu¨rzburg, Bavaria. He noticed that some crystals of barium platinocyanide, near a discharge tube completely enclosed in black paper, became luminescent when the discharge occurred. By examining the shadows cast by the rays. Ro¨ntgen traced the origin of the rays to the walls of the discharge tube. In 1896, Campbell-Swinton introduced a definite target (platinum) for the cathode rays to hit; this target was called the anticathode. For his work x-rays, Ro¨ntgen received the first Nobel Prize in physics, in 1901. It was the first of six to be awarded in the field of x-rays by 1927. The obvious similarities with light led to the crucial tests of established wave optics: polarization, diffraction, reflection, and refraction. With limited experimental facilities, Ro¨ntgen and his contemporaries could find no evidence of any of these; hence, the designation ‘‘x’’ (unknown) of the rays, generated by the stoppage at anode targets of the cathode rays, identified by Thomson in 1897 as electrons. The nature of x-rays was the subject of much controversy. In 1906, Barkla found evidence in scattering experiments that x-rays could be polarized and must therefore by waves, but W. H. Bragg’s studies of the produced ionization indicated that they were Copyright © 2002 Marcel Dekker, Inc.

corpuscular. The essential wave nature of x-rays was established in 1912 by Laue, Friedrich, and Knipping, who showed that x-rays could be diffracted by a crystal (copper sulfate pentahydrate) that acted as a three-dimensional diffraction grating. W. H. Bragg and W. L. Bragg (father and son) found the law for the selective reflection of x-rays. In 1908, Barkla and Sadler deduced, by scattering experiments, that x-rays contained components characteristic of the material of the target; they called these components K and L radiations. That these radiations had sharply defined wavelengths was shown by the diffraction experiments of W. H. Bragg in 1913. These experiments demonstrated clearly the existence of a line spectrum superimposed upon a continuous (‘‘White’’) spectrum. In 1913, Moseley showed that the wavelengths of the lines were characteristic of the element of the which the target was made and, further, showed that they had the same sequence as the atomic numbers, thus enabling atomic numbers to be determined unambiguously for the first time. The characteristic K absorption was first observed by de Broglie and interpreted by W. L. Bragg and Siegbahn. The effect on x-ray absorption spectra of the chemical state of the absorber was observed by Bergengren in 1920. The influence of the chemical state of the emitter on x-ray emission spectra was observed by Lindh and Lundquist in 1924. The theory of x-ray spectra was worked out by Sommerfeld and others. In 1919, Stenstro¨m found the deviations from Bragg’s law and interpreted them as the effect of refraction. The anomalous dispersion of x-ray was discovered by Larsson in 1929, and the extended fine structure of x-ray absorption spectra was qualitatively interpreted by Kronig in 1932. Soon after the first primary spectra excited by electron beams in an x-ray tube were observed, it was found that secondary fluorescent x-rays were excited in any material irradiated with beams of primary x-rays and that the spectra of these fluorescent x-rays were identical in wavelengths and relative intensities with those excited when the specimen was bombarded with electrons. Beginning in 1932, Hevesy, Coster, and others investigated in detail the possibilities of fluorescent x-ray spectroscopy as a means of qualitative and quantitative elemental analysis. III.

GENERAL FEATURES

X-rays, or Ro¨ntgen rays, are electromagnetic radiations having wavelengths roughly within the range from 0.005 to 10 nm. At the short-wavelength end, they overlap with g-rays, and at the long-wavelength end, they approach ultraviolet radiation. The properties of x-rays, some of which are discussed in detail in this chapter, are summarized as follows: Invisible Propagated in straight lines with a velocity of 36108 m=s, as is light Unaffected by electrical and magnetic fields Differentially absorbed while passing through matter of varying composition, density, or thickness Reflected, diffracted, refracted, and polarized Capable of ionizing gases Capable of affecting electrical properties of liquids and solids Capable of blackening a photographic plate Able to liberate photoelectrons and recoil electrons Capable of producing biological reactions (e.g., to damage or kill living cells and to produce genetic mutations) Copyright © 2002 Marcel Dekker, Inc.

Emitted in a continuous spectrum whose short-wavelength limit is determined only by the voltage on the tube Emitted also with a line spectrum characteristic of the chemical elements Found to have absorption spectra characteristic of the chemical elements

IV.

EMISSION OF CONTINUOUS RADIATION

Continuous x-rays are produced when electrons, or other high-energy charged particles, such as protons or a-particles, lose energy in passing through the Coulomb field of a nucleus. In this interaction, the radiant energy (photons) lost by the electron is called bremsstrahlung (from the German bremsen, to brake, and Strahlung, radiation; this term sometimes designates the interaction itself). The emission of continuous x-rays finds a simple explanation in terms of classic electromagnetic theory, because, according to this theory, the acceleration of charged particles should be accompanied by the emission of radiation. In the case of high-energy electrons striking a target, they must be rapidly decelerated as they penetrate the material of the target, and such a high negative acceleration should produce a pulse of radiation. The continuous x-ray spectrum generated by electrons in an x-ray tube is characterized by a short-wavelength limit lmin , corresponding to the maximum energy of the exciting electrons: lmin ¼

hc eV0

ð1Þ

where h is Planck’s constant, c is the velocity of light, e is the electron charge, and V0 is the potential difference applied to the tube. This relation of the short-wavelength limit to the applied potential is called the Duane–Hunt law. The probability of radiative energy loss (bremsstrahlung) is roughly proportional to q2 Z2 T=M20 , where q is the particle charge in units of the electron charge e, Z is the atomic number of the target material, T is the particle kinetic energy, and M0 is the rest mass of the particle. Because protons and heavier particles have large masses compared to the electron mass, they radiate relatively little; for example, the intensity of continuous x-rays generated by protons is about four orders of magnitude lower than that generated by electrons. The ratio of energy lost by bremsstrahlung to that lost by ionization can be approximated by
2 m0 ZT ð2Þ M0 1600m0 c2 where m0 the rest mass of the electron.

A.

Spectral Distribution

The continuous x-ray spectrum generated by electrons in an x-ray tube (thick target) is characterized by the following features: 1. Short-wavelength limit, lmin [Eq. (1)]; below this wavelength, no radiation is observed. Copyright © 2002 Marcel Dekker, Inc.

2.

3.

Wavelength of maximum intensity lmax , approximately 1.5 times lmin ; however, the relationship between lmax and lmin depends to some extent on voltage, voltage waveform, and atomic number. Total intensity nearly proportional to the square of the voltage and the first power of the atomic number of the target material.

The most complete empirical work on the overall shape of the energy distribution curve for a thick target has been of Kulenkampff (1922, 1933), who found the following formula for the energy distribution;   ð3Þ IðvÞ dv ¼ i aZðv0
vÞ þ bZ2 dv where IðnÞ dn is the intensity of the continuous x-rays within a frequency range ðn; n þ dvÞ; i is the electron current striking the target, Z is the atomic number of the target material, n0 is the cutoff frequency ð¼ c=lmin Þ above which the intensity is zero, and a and b are constants independent of atomic number, voltage, and cutoff wavelength. The second term in Eq. (3) is usually small compared to the first and is often neglected. The total integrated intensity at all frequencies is I ¼ iða0 ZV20 þ b0 Z2 V0 Þ

ð4Þ

in which a0 ¼ aðe2 =h2 Þ=2 and b0 ¼ bðe=hÞ. An approximate value for b0 =a0 is 16.3 V; thus, I ¼ a0 iZV0 ðV0 þ 16:3ZÞ

ð5Þ

The efficiency Eff of conversion of electric power input to x-rays of all frequencies is given by Eff ¼

I ¼ a0 ZðV0 þ 16:3ZÞ V0 i

ð6Þ

where V0 is in volts. Experiments give a0 ¼ ð1:2  0:1Þ  10
9 (Condon, 1958). The most complete and successful efforts to apply quantum theory to explain all features of the continuous x-ray spectrum are those of Kramers (1923) and Wentzel (1924). By using the correspondence principle, Kramers found the following formulas for the energy distribution of the continuous x-rays generated in a thin target: pffiffi AZ e dv; IðvÞ dv ¼ 316p 3m0 V0 c3 IðvÞ dv ¼ 0; 2

2 5

v < v0 v > v0

ð7Þ

where A is the atomic mass of the target material. When the decrease in velocity of the electrons in a thick target was taken into account by applying the Thomson–Whiddington law (Dyson, 1973), Kramers found, for a thick target, 8pe2 h Zðv0
vÞ dv IðvÞ dv ¼ pffiffiffi 3 3lm0 c3

ð8Þ

where l is approximately 6. The efficiency of production of the x-rays calculated via Kramers’ law is given by Eff ¼ 9:2  10
10 ZV0

ð9Þ

which is in qualitative agreement with the experiments of Kulenkampff (Stephenson, 1957); for example, Copyright © 2002 Marcel Dekker, Inc.

Eff ¼ 15  10
10 ZV0

ð10Þ

It is worth mentioning that the real continuous x-ray distribution is described only approximately by Kramers’ equation. This is related, inter alia, to the fact that the derivation ignores the self-absorption of x-rays and electron backscattering effects. Wentzel (1924) used a different type of correspondence principle than Kramers, and he explained the spatial distribution asymmetry of the continuous x-rays from thin targets. An accurate description of continuous x-rays is crucial in all x-ray spectrometry (XRS). The spectral intensity distributions from x-ray tubes are of great importance for applying fundamental mathematical matrix correction procedures in quantitative x-ray fluorescence (XRF) analysis. A simple equation for the accurate description of the actual continuum distributions from x-ray tubes was proposed by Tertian and Broll (1984). It is based on a modified Kramers’ law and a refined x-ray absorption correction. Also, a strong need to model the spectral Bremsstrahlung background exists in electron-probe x-ray microanalysis (EPXMA). First, fitting a function through the background portion, on which the characteristic x-rays are superimposed in an EPXMA spectrum, is not easy; several experimental fitting routines and mathematical approaches, such as the Simplex method, have been proposed in this context. Second, for bulk multielement specimens, the theoretical prediction of the continuum Bremsstrahlung is not trivial; indeed, it has been known for several years that P the commonly used Kramers’ formula with Z directly substituted by the average Z
¼ i Wi Zi (Wi and Zi are the weight fraction and atomic number of the ith element, respectively) can lead to significant errors. In this context, some improvements are offered by several modified versions of Kramers’ formula developed for a multielement bulk specimen (Statham, 1976; Lifshin, 1976; Sherry and Vander Sande, 1977; Smith and Reed, 1981). Also, a new expression for the continuous x-rays emitted by thick composite specimens was proposed (Markowicz and Van Grieken, 1984; Markowicz et al., 1986); it was derived by introducing the compositional dependence of the continuum x-rays already in the elementary equations. The new expression has been combined with known equations for the self-absorption of x-rays (Ware and Reed, 1973) and electron backscattering (Statham, 1979) to obtain an accurate description of the detected continuum radiation. A third problem is connected with the description of the x-ray continuum generated by electrons in specimens of thickness smaller than the continuum x-ray generation range. This problem arises in the analysis of both thin films and particles by EPXMA. A theoretical model for the shape of the continuous x-rays generated in multielement specimens of finite thickness was developed (Markowicz et al., 1985); both composition and thickness dependence have been considered. Further refinements of the theoretical approach are hampered by the lack of knowledge concerning the shape of the electron interaction volume, the distribution of the electron within the interaction volume, and the anisotropy of continuous radiation for different x-ray energies and for different film thickness. B.

Spatial Distribution and Polarization

The spatial distribution of the continuous x-rays emitted by thin targets has been investigated by Kulenkampff (1928). The author made an extensive survey of the intensity at angles between 22 and 150 to the electron beam in terms of dependence on wavelength and voltage. The target was a 0.6-mm-thick Al foil. Figure 1 shows the continuous x-ray intensity observed at different angles for voltages of 37.8, 31.0, 24.0, and 16.4 kV filtered by 10, 8, 4, and 1.33 mm of Al, respectively (Stephenson, 1957). Curve (a) is repeated as a dotted line near each of the other curves. The angle of the maximum intensity varied from Copyright © 2002 Marcel Dekker, Inc.

Figure1 Intensity of continuous x-rays as a function of direction for different voltages. (Curve (a) is repeated as dotted line.) (From Stephenson, 1957.)

50 for 37.8 kV to 65 for 16.4 kV. Figure 2 illustrates the intensity of the continuous x-rays observed in the Al foil for different thicknesses as a function of the angle for a voltage of 30 kV (Stephenson, 1957). The theoretical curve is from the theory of Scherzer (1932). The continuous x-ray intensity drops to zero at 180 , and although it is not zero at 0 as the theory of Scherzer predicts, it can be seen from Figure 2 that for a thinner foil, a lower intensity at 0 is obtained. Summarizing, it appears that the intensity of the continuous x-rays emitted by thin foils has a maximum at about 55 relative to the incident electron beam and becomes zero at 180 . The continuous radiation from thick targets is characterized by a much smaller anisotropy than that from thin targets. This is because in thick targets the electrons are rarely stopped in one collision and usually their directions have considerable variation. The use of electromagnetic theory predicts a maximum energy at right angles to the incident electron beam at low voltages, with the maximum moving slightly away from perpendicularity toward the direction of the elctron beam as the voltage is increased. In general, an increase in the anisotropy of the continuous x-rays from thick targets is observed at the short-wavelength limit and for low-Z targets (Dyson, 1973). Copyright © 2002 Marcel Dekker, Inc.

Figure 2 Intensity of continuous x-rays as a function of direction for different thicknesses of the A1 target together with theoretical prediction. (From Stephenson, 1957.)

Continuous x-ray beams are partially polarized only from extremely thin targets; the angular region of polarization is sharply peaked about the photon emission angle y ¼ m0 c2 =E0 , where E0 is the energy of the primary electron beam. Electron scattering in the target broadens the peak and shifts the maximum to larger angles. Polarization is defined by (Kenney, 1966) Pðy; E0 ; En Þ ¼

ds?ðy; E0 ; En Þ
dskðy; E0 ; En Þ ds?ðy; E0 ; En Þ þ dskðy; E0 ; En Þ

ð11Þ

where an electron of energy E0 radiates a photon of energy En at angle y; ds?ðy; E0 ; En Þ and dskðy; E0 ; En Þ are the cross sections for generation of the continuous radiation with the electric vector perpendicular (?) and parallel (k) to the plane defined by the incident electron and the radiated photon, respectively. Polarization is difficult to observe, and only thin, low-yield radiators give evidence for this effect. When the electron is relativistic before and after the radiation, the electrical vector is most probably in the ? direction. Practical thick-target Bremsstrahlung shows no polarization effects whatever (Dyson, 1973; Stephenson, 1957; Kenney, 1966).

V.

EMISSION OF CHARACTERISTIC X-RAYS

The production of characteristic x-rays involves transitions of the orbital electrons of atoms in the target material between allowed orbits, or energy states, associated with ionization of the inner atomic shells. When an electron is ejected from the K shell by electron bombardment or by the absorption of a photon, the atom becomes ionized and the ion is left in a high-energy state. The excess energy the ion has over the normal state of the atom is equal to the energy (the binding energy) required to remove the K electron to a state of rest outside the atom. If this electron vacancy is filled by an electron coming from an L level, the transition is accompanied by the emission of an x-ray line known as the Ka line. This process leaves a vacancy in the L shell. On the other hand, if the atom contains sufficient electrons, the K shell vacancy might be filled by an electron coming from an M level that is accompanied by the emission of the Kb line. The L or M state ions that remain may also give rise to emission if the electron vacancies are filled by electrons falling from further orbits. Copyright © 2002 Marcel Dekker, Inc.

A.

Inner Atomic Shell Ionization

As already mentioned, the emission of characteristic x-ray is preceded by ionization of inner atomic shells, which can be accomplished either by charged particles (e.g., electrons, protons, and a-particles) or by photons of sufficient energy. The cross section for ionization of an inner atomic shell of element i by electrons is given by (Bethe, 1930; Green and Cosslett, 1961; Wernisch, 1985) Qi ¼ pe4 ns bs

ln U UE2c;i

ð12Þ

where U ¼ E=Ec;i is the overvoltage, defined as the ratio of the instantaneous energy of the electron at each point of the trajectory to that required to ionize an atom of element i, Ec;i is the critical excitation energy, and ns and bs are constants for a particular shell: s ¼ K: s ¼ L:

ns ¼ 2; bs ¼ 0:35 ns ¼ 8; bs ¼ 0:25

The cross section for ionization Qi is a strong function of the overvoltage, which shows a maximum at U ffi 3–4 (Heinrich, 1981; Goldstein et al., 1981). The probability (or cross section) of ionization of an inner atomic shell by a charged particle is given by (Merzbacher and Lewis, 1958) ss ¼

8pr20 q2 fs Z 4 Zs

ð13Þ

where r0 is the classic radius of the electron equal to 2.818610
15 m, q is the particle charge, Z is the atomic number of the target material, fs is a factor depending on the wave functions of the electrons for a particular shell, and Zs is a function of the energy of the incident particles. In the case of electromagnetic radiation (x or g), the ionization of an inner atomic shell is a result of the photoelectric effect. This effect involves the disappearance of a radiation photon and the photoelectric ejection of one electron from the absorbing atom, leaving the atom in an excited level. The kinetic energy of the ejected photoelectron is given by the difference between the photon energy hn and the atomic binding energy of the electron Ec (critical excitation energy). Critical absorption wavelengths (Clark, 1963) related to the critical absorption energies (Burr, 1974) via the equation l(nm) ¼ 1.24=E(ke V) are presented in Appendix I. The wavelenghts of K, L, M, and N absorption edges can also be calculated by using simple empirical equations (Norrish and Tao, 1993). For energies far from the absorption edge and in the nonrelativistic range, the cross section tK for the ejection of an electron from the K shell is given by (Heitler, 1954) pffiffiffi
7=2 32 2 2 Z5 m 0 c2 pr0 tK ¼ 3 ð137Þ4 hv

ð14Þ

Equation (14) is not fully adequate in the neighborhood of an absorption edge; in this case, Eq. (14) should be multiplied by a correction factor f(X ) (Stobbe, 1930):
1=2
4X arccot X D e fðXÞ ¼ 2p hv 1
e
2pX where Copyright © 2002 Marcel Dekker, Inc.

ð15Þ


X¼

D hv
D

1=2 ð15aÞ

with 1 m 0 c2 D ffi ðZ
0:3Þ2 2 ð137Þ2

ð15bÞ

When the energy of the incident photon is of the order m0 c2 or greater, relativistic cross sections for the photoelectric effect must be used (Sauter, 1931).

B.

Spectral Series in X-rays

The energy of an emission line can be calculated as the difference between two terms, each term corresponding to a definite state of the atom. If E1 and E2 are the term values representing the energies of the corresponding levels, the frequency of an x-ray line is given by the relation v¼

E1
E2 h

ð16Þ

Using the common notations, one can represent the energies of the levels E by means of the atomic number and the quantum numbers n, l, s, and j (Sandstro¨m, 1957): ! 2 E ðZ
Sn;l Þ2 1 3 2 ðZ
dn;l; j Þ ¼ þa
Rh n2 n3 l þ 12 4n
a2

ðZ
dn;l; j Þ4 jð j þ 1Þ
lðl þ 1Þ
sðs þ 1Þ n3 2lðl þ 12Þðl þ 1Þ

ð17Þ

where Sn;l and dn;l; j are screening constants that must be introduced to correct for the effect of the electrons on the field in the atom, R is the universal Rydberg constant valid for all elements with Z > 5 or throughout nearly the whole x-ray region, and a is the finestructure constant given by a¼

2pe2 hc

ð17aÞ

The theory of x-ray spectra reveals the existence of a limited number of allowed transitions; the rest are ‘‘forbidden.’’ The most intense lines create the electric dipole radiation. The transitions are governed by the selection rules for the change of quantum numbers: Dl ¼ 1;

Dj ¼ 0 or  1

ð18Þ

The j transition 0 ? 0 is forbidden. According to Dirac’s theory of radiation (Dirac, 1947), transitions that are forbidden as dipole radiation can appear as multipole radiation (e.g., as electric quadrupole and magnetic dipole transitions). The selection rules for the former are Dl ¼ 0 or  2;

Dj ¼ 0; 1; or  2

The j transitions 0 ? 0, and 0 $ 1 are forbidden. The selection rules for magnetic dipole transitions are 1 1 2? 2,

Copyright © 2002 Marcel Dekker, Inc.

ð19Þ

Dl ¼ 0;

Dj ¼ 0 or  1

ð20Þ

The j transition 0 ? 0 is forbidden. The commonly used terminology of energy levels and x-ray lines is shown in Figure 3. A general expression relating the wavelength of an x-ray characteristic line with the atomic number of the corresponding element is given by Moseley’s law (Moseley, 1914): 1 ¼ kðZ
sÞ2 l

ð21Þ

where k is a constant for a particular spectral series and s is a screening constant for the repulsion correction due to other electrons in the atom. Moseley’s law plays an important role in the systematizing of x-ray spectra. Appendix II tabulates the energies and wavelengths of the principal x-ray emission lines for the K, L, and M series with their approximate relative intensities, which can be defined either by means of spectral line peak intensities or by area below their intensity distribution curve. In practice, the relative

Figure 3

Commonly used terminology of energy levels and x-ray lines. (From Sandstro¨m, 1957.)

Copyright © 2002 Marcel Dekker, Inc.

intensities of spectral lines are not constant because they depend not only on the electron transition probability but also on the specimen composition. Considering the K series, the Ka fraction of the total K spectrum is defined by the transition probability pKa, which is given by (Schreiber and Wims, 1982) pKa ¼

IðKa1 þ Ka2 Þ IðKa1 þ Ka2 Þ þ IðKb1 þ Kb2 Þ

ð22Þ

Wernisch (1985) proposed a useful expression for the calculation of the transition probability pKa for different elements: 8 11  Zi  19 < 1:052
4:39  10
4 Z2i ; ð23Þ pKa;i ¼ 0:896
6:575  10
4 Zi ; 20  Zi  29 : 1:0366
6:82  10
3 Zi þ 4:815  10
5 Z2i ; 30  Z2i  60 For the L level, split into three subshells, several electron transitions exist. The transition probability pLa, defined as the fraction of the transitions resulting in La1 and La2 radiation from the total of possible transitions into the L3 subshell, can be calculated by the expression (Wernisch, 1985) 8 39  Zi  44 < 0:944; ð24Þ pLa;i ¼
4:461  10
1 þ 5:493  10
2 Zi :
7:717  10
4 Z2i þ 3:525  10
6 Z3i ; 45  Zi  82 Radiative transition probabilities for various K and L x-ray lines (West, 1982–83) are presented in detail in Appendix III. The experimental results, together with the estimated 95% confidence limits, for the relative intensity ratios for K and L x-ray lines for selected elements with atomic number from Z ¼ 14 to 92 have been reported by Stoev and Dlouhy (1994). The values are in a good agreement with other published experimental data. Because the electron vacancy created in an atom by charged particles or electromagnetic radiation has a certain lifetime, the atomic levels E1 and E2 [Eq. (16)] and the characteristic x-ray lines are characterized by the so-called natural widths (Krause and Oliver, 1979). The natural x-ray linewidths are the sums of the widths of the atomic levels involved in the transitions. Semiempirical values of the natural widths of K, L1 , L2 and L3 levels, Ka1 and Ka2 x-ray lines for the elements 10  Z  110 are presented in Appendix IV. Uncertainties in the level width values are from 3% to 10% for the K shell and from 8% to 30% for the L subshell. Uncertainties in the natural x-ray linewidth values are from 3% to 10% for Ka1;2 . In both cases, the largest uncertainties are for low-Z elements (Krause and Oliver, 1979).

C.

X-ray Satellites

A large number of x-ray lines have been reported that do not fit into the normal energylevel diagram (Clark, 1955; Kawai and Gohshi, 1986). Most of the x-ray lines, called satellites or nondiagram lines, are very weak and are of rather little consequence in analytical x-ray spectrometry. By analogy to the satellites in optical spectra, it was supposed that the origin of the nondiagram x-ray lines is double or manyfold ionization of an atom through electron impact. Following the ionization, a multiple electron transition results in emission of a single photon of energy higher than that of the characteristic x-rays. The majority of nondiagram lines originate from the dipole-allowed deexcitation of multiply Copyright © 2002 Marcel Dekker, Inc.

ionized or excited states and are called multiple-ionization satellites. A line where the initial state has two vacancies in the same shell, notably the K shell, is called a hypersatellite. In practice, the most important nondiagram x-ray lines occur in the Ka series; they are denoted as the Ka3 ;a4 doublet, and their origin is a double electron transition. The probability of a multiple-electron transition resulting in the emission of satellite x-ray lines is considerably higher for low-Z elements than for heavy and medium elements. For instance, the intensity of the AlKa3 satellite line is roughly 10% of that of the AlKa1 ; a2 characteristic x-rays. Appendix V tabulates wavelengths of the K satellite lines. A new class of satellites that are inside the natural width of the parent lines was observed by Kawai and Gohshi (1986). The origin of these satellites, called parasites or hidden satellites, is multiple ionization in nonadjacent shells. D.

Soft X-ray Emission-Band Spectra

In the soft x-ray region, the characteristic emission spectra of solid elements include continuous bands of width varying from 1 to 10 electron volts (eV); the same element in vapor form produces only the usual sharp spectral lines. The bands occur only when an electron falls from the outermost or valency shell of the atom, the levels of which are broadened into a wide band when the atoms are packed in a crystal lattice. Investigation of the emission-band spectra is of great significance in understanding the electronic structure of solid metals, alloys, and complex coordination compounds. E.

Auger Effect

It has already been stated that the excess of energy an atom possesses after removing one electron from an inner shell by whatever means may be emitted as characteristic radiation. Alternatively, however, an excited atom may return to a state of lower energy by ejecting one of its own electrons from a less tightly bound state. The radiationless transition is called the Auger effect, and the ejected electrons are called Auger electrons (Auger, 1925; Burhop, 1952). Generally, the probability of the Auger effect increases with a decrease in the difference of the corresponding energy states, and it is the highest for the low-Z elements. Because an excited atom already has one electron missing (e.g., in the K shell) and another electron is ejected in an Auger process (e.g., from the L shell), the atom is left in a doubly-ionized state in which two electrons are missing. This atom may return to its normal state by single- or double-electron jumps with the emission of diagram or satellite lines, respectively. Alternatively, another Auger process may occur in which a third electron is ejected from the M shell. The Auger effect also occurs after capture of a negative meson by an atom. As the meson changes energy levels in approaching the nucleus, the energy released may be either emitted as a photon or transferred directly to an electron that is emitted as a high-energy Auger electron (in the keV range for hydrogen and the MeV range for heavy elements). Measurements of the energy and intensity of the Auger electrons are applied extensively in surface physics studies (Auger electron spectroscopy). F.

FluorescenceYield

An important consequence of the Auger effect is that the actual number of x-ray photons produced from an atom is less than expected, because a vacancy in a given shell might be Copyright © 2002 Marcel Dekker, Inc.

filled through a nonradiative transition. The probability that a vacancy in an atomic shell or subshell is filled through a radiative transition is called the fluorescence yield. The application of this definition to the K shell of an atom is straightforward, and the fluorescence yield of the K shell is oK ¼

IK nK

ð25Þ

where IK is the total number of characteristic K x-ray photons emitted from a sample and nK is the number of primary K shell vacancies. The definition of the fluorescence yield of higher atomic shells is more complicated, for the following two reasons: 1. Shells above the K shell consist of more than one subshell; the average fluorescence yield depends on how the shells are ionized. 2. Coster–Kronig transitions occur, which are nonradiative transitions between the subshells of an atomic shell having the same principal quantum number (Fink, 1974; Bambynek et al., 1972). In case Coster–Kronig transitions are absent, the fluorescence yield of the ith subshell of a shell, whose principal quantum number is indicated by XðX ¼ L; M; . . .Þ, is given as oX i ¼

IX i nX i

ð26Þ


X for the shell X is defined as An average or mean fluorescence yield o
X ¼ o

k X

ð27Þ

X NX i oi

i¼1

where NX i is the relative number of primary vacancies in the subshell i of shell X: nX i NX i ¼ Pk

X i¼1 ni

;

k X

NX i ¼1

ð28Þ

i¼1

The summations in Eqs. (27) and (28) extend over all k subshells of shell X. For the definition of the average fluorescence yield, the primary vacancy distribution must be fixed;
X generally is not that is, Coster–Kronig transitions must be absent. It is noteworthy that o a fundamental property of the atom, but depends both on the atomic subshell fluorescence X yields oX i and on the relative number of primary vacancies Ni characteristic of the method used to ionize the atoms. In the presence of Coster–Kronig transitions, which modify the primary vacancy distribution by the transfer of ionization from one subshell with a given energy to a subshell with less energy, the average fluorescence yields can be calculated by using two
X is regarded as a alternative approaches. In the first, the average fluorescence yield o with a vacancy distribution linear combination of the subshell fluorescence yields oX i modified by Coster–Kronig transitions:
X ¼ o

k X i¼1

X VX i oi ;

k X i¼1

Copyright © 2002 Marcel Dekker, Inc.

VX i >1

ð29Þ

where VX i is the relative number of vacancies in the subshell i of shell X, including vacancies shifted to each subshell by Coster–Kronig transitions. The VX i values can be expressed in terms of the relative numbers NX i of primary vacancies and the Coster–Kronig transition probability for shifting a vacancy from a subshell Xi to a higher subshell Xj , denoted as f X ij (Bambynek et al., 1972): X VX 1 ¼ N1

ð30Þ

X X X VX 2 ¼ N2 þ f 12 N1

VX 3

¼

NX 3

þ

X fX 23 N2

þ

ðfX 13

þ

fX 12

X fX 23 ÞN1


X is a linear combination of In an alternative approach, the mean fluorescence yield o : the relative numbers of primary vacancies NX i
X ¼ o

k X

X NX i ni

ð31Þ

i¼1

where nX i represents the total number of characteristic x-rays that result per primary vacancy in the Xi subshell. The transformation relations between the coefficients nX i and the subshell fluorescence yields oX i follow from Eqs. (29) through (31) and are given in Fink (1974) and Bambynek et al. (1972). X Among the fluorescence yield oX i , the Auger yield ai , and the Coster–Kronig X transition probabilities fij , the following relationship must hold (Krause, 1979): X oX i þ ai þ

k X

fX ij ¼ 1

ð32Þ

i¼1

The mean Auger yield a
X is given by a
x ¼

k X

X VX i ai

ð33Þ

i¼1

The values of the K, L, and M shell fluorescence yields, the Coster–Kronig transition probabilities, as well as the Auger yields are given in Appendix VI. Although, in principle, the K shell fluorescence yield ok can be calculated theoretically, experimental data are applied in practice. The following semiempirical equation, due to Burhop (1952), gives values correct to a few percent between Z ¼ 23 and Z ¼ 57 and less accurate values outside these limits:
1=4 oK ¼ 6:4  10
2 þ 3:40  10
2 Z
1:03  10
6 Z3 ð34Þ 1
oK The fluorescence yield for the K series can also be calculated from a different equation:
1=4 oK ¼
0:217 þ 0:03318Z
1:14  10
6 Z3 ð35Þ 1
oK which gives quite good agreement with the experimental values for almost all elements. Based on a critical review of the available experimental and theoretical data, a recom
L ; and o
M values together with the following analytical expressions mended set of oK ; o were given by Hubbell (1989) and Hubbell et al. (1994): Copyright © 2002 Marcel Dekker, Inc.

" oK ð1  Z  100Þ ¼

3 X

Ci Zi

i¼0

#4 8 < :

" 1þ

3 X i¼0

Ci Zi

#4 9
1 = ;

ð36Þ

with C0 C1 C2 C3

¼ 0:0370  0:0052 ¼ 0:03112  0:00044 ¼ ð5:44  0:11Þ  10
5 ¼
ð1:25  0:07Þ  10
6


L ð3  Z  36Þ ¼ 1:939  10
8  Z3:8874 o " #4 8 " #4 9
1 < = 3 3 X X
L ð37  Z  100Þ ¼ Ci Zi 1þ Ci Zi o : ; i¼0 i¼0

ð37Þ

with C0 C1 C2 C3

¼ 0:17765 ¼ 0:00298937 ¼ 8:91297  10
5 ¼
2:67184  10
7


M ð13  Z  100Þ ¼ 1:29  10
9 ðZ
13Þ4 o

ð38Þ

Other useful expressions for the calculation of the fluorescence yields oK ð12  Z  42Þ and oL3 ð38  Z  79Þ have been proposed by Hanke et al. (1985), based on literature and experimental data: oK ¼ 3:3704  10
1
6:0047  10
2 Z þ 3:3133  10
3 Z2
3:9215  10
5 Z3 oL3 ¼ 4:41  10
2
4:7559  10
3 Z þ 1:1494  10
4 Z2
1:8594  10
7 Z3

ð39Þ ð40Þ


N data is a theoretical work of McGuire (1974) For the N shell, the best source of o which provides oN1 ; oN2 , and oN3 values for 25 elements over the range 38  Z  103 and oN4 ; oN5 , and oN6;7 values for 20 elements over the range 50  Z  103.
N can be calculated from (Hubbell, 1989) The average fluorescence yield o
N ¼ o

7 X 1 NN oN 32 i i i¼1

ð41Þ

where NNi are the numbers of electrons in each Ni subshell. A comparison of the total x-ray yields for bulk samples (including both the probability of ionization and the fluorescence yield) in terms of photons per steradian per incident quantum for electrons, protons, and x-ray photons is shown in Figure 4. Copyright © 2002 Marcel Dekker, Inc.

Figure 4 Total x-ray yields for excitation by electrons, protons, and primary x-ray photons as a function of energy of the exciting quantum. (From Birks, 1971a.)

G.

Fine Features of X-ray Emission Spectra (Valence or Chemical Effects)

Because characteristic x-ray emission is a process in which the innermost electrons in the atom are concerned, it is reasonable to suppose that the external, or valence, electrons have little or no effect on the x-ray emission lines. However, this is not fully true for K lines of low-Z elements and L or M lines of higher-Z elements, where the physical state and chemical combination of the elements affect the characteristic x-rays (Clark, 1955). The changes in fine features of x-ray emission spectra with chemical combination can be classified into three groups: (1) shifting in wavelength (Kallithrakas-Kontos, 1996), (2) distortion of line shape, and (3) intensity changes (Kawai et al., 1993; Rebohle et al., 1996). Wavelength shifts to both longer and shorter wavelengths result from energy-level changes due to electrical shielding or screening of the electrons when the valence electrons are drawn into a bond. Generally, the so-called last or highest-energy member of a given series is most affected by chemical combination; maximum energy shifts are of the order of a few electron volts. Distortion of an x-ray emission line shape gives some indication of the energy distribution of the electrons occupying positions in or near the valence shell. The changes in the characteristic x-ray intensity are a result of alterations in excitation Copyright © 2002 Marcel Dekker, Inc.

probabilities of the electrons undergoing transitions. Certain x-ray lines or bands appear or disappear with chemical combinations. In the case of the K series, the most noticeable chemical effects on x-ray emission are seen in spectra from low-Z elements (4  Z  17). The L series shows as large or even larger changes with chemical combination of the elements than K series. The valence effects in L spectra have been observed for elements of the first transition series and others nearby in the periodic table. Because the fine features of x-ray emission spectra may be applied to determine how each element is chemically combined in the sample (speciation), the valence effects found numerous applications in such fields as physics of solids and surface or near-surface characterization. VI.

INTERACTION OF PHOTONS WITH MATTER

Interactions of photons with matter, by which individual photons are removed or deflected from a primary beam of x or g radiation, may be classified according to the following: The kind of target, such as electrons, atoms or nuclei, with which the photon interacts The type of event, such as absorption, scattering, or pair production, that takes place These interactions are thought to be independent of the origin of the photon (nuclear transition for g-rays versus electronic transition for x-rays); hence, we use the term ‘‘photon’’ to refer to both g- and x-rays here. Possible interactions are summarized in Table 1 (Hubbell, 1969), where t is the total photoelectric absorption cross section per atom (t ¼ tK þ tL þ   ) and sR and sC are Rayleigh and Compton collision cross sections, respectively. The probability of each of these many competing independent processes can be expressed as a collision cross section per atom, per electron, or per nucleus in the absorber. The sum of all these cross sections, normalized to a per atom basis, is then the probability stot that the incident photon will have an interaction of some kind while passing through a very thin absorber that contains one atom per square centimeter of area normal to the path of the incident photon: stot ¼ t þ sR þ sC þ   

ð42Þ

The total collision cross section per atom stot , when multiplied by the number of atoms per cubic centimeter of absorber, is then the linear attenuation coefficient m per centimeter of travel in the absorber: 

 

 1 cm2 g N0 atoms  m r ¼ stot ð43Þ cm cm3 g atom A where r is the density of the medium and N0 is Avogadro’s number (6.0225261023 atoms=g atom). The mass attenuation coefficient m (cm2=g) is the ratio of the linear attenuation coefficient and the density of the material. It is worth mentioning that the absorption coefficient is a much more restricted concept than the attenuation coefficient. Attenuation includes the purely elastic process in which the photon is merely deflected and does not give up any of its initial energy to the absorber; in this process, only a scattering coefficient is involved. In a photoelectric interaction, the entire energy of the incident photon is absorbed by an atom of the medium. Copyright © 2002 Marcel Dekker, Inc.

Table 1

Classification of Photon Interactions Scattering

Type of interaction Interaction with atomic electrons

Interaction with nucleus or bound nucleons

Interaction with electrical field surrounding charged particles

Interactions with mesons

Absorption

Elastic (coherent)

Inelastic (incoherent)

Multiphoton effects

Photoelectric effecta Z4 low energy t Z5 high energy Nuclear photoelectric effect: reactions (g, n) (g, p), photofission Z (E 10 MeV) 1. Electron–positron pair production in field of nucleous, Z2 (E 1.02 MeV) 2. Electron–positron pair production in electron field Z (E 2.04 MeV) 3. Nucleon–antinucleon pair production (E 3 GeV) Photomeson production (E 150 MeV)

Rayleigh scatteringa sR Z2

Compton scatteringa sC Z

Two-photon Compton scattering Z

Nuclear coherent scattering (g, g) Z2

Nuclear Compton scattering (g, g0 ) Z

Delbru¨ck scattering Z4

Coherent resonant scattering (g, g)

a Major effects of photon attenuation in matter, which are of great importance in practical x-ray spectrometry. Source: From Hubbell, 1969.

Copyright © 2002 Marcel Dekker, Inc.

In the Compton effect, some energy is absorbed and appears in the medium as the kinetic energy of a Compton recoil electron; the balance of the incident energy is not absorbed and is present as a Compton-scattered photon. Absorption, then, involves the conversion of incident photon energy into the kinetic energy of a charged particle (usually an electron), and scattering involves the deflection of incident photon energy. For narrow, parallel, and monochromatic beams, the attenuation of photons in homogeneous matter is described by the exponential law: I ¼ I0 e
m



t

ð44Þ

where I is the transmitted intensity, I0 is the incident intensity, and t is the absorber thickness in centimeters. If the absorber is a chemical compound or a mixture, its mass attenuation coefficient can be approximately evaluated from the coefficients mi for the constituent elements according to the weighted average: m¼

n X

Wi mi

ð45Þ

i¼1

where Wi is the weight fraction of the ith element and n is the total number of the elements in the absorber. The ‘‘mixture rule’’ [Eq. (45)] ignores changes in the atomic wave function resulting from changes in the molecular, chemical, or crystalline environment of an atom. Above 10 keV, errors from this approximation are expected to be less than a few percent (except in the regions just above absorption edges), but at very low energies (10–100 eV), errors of a factor of 2 can occur (Deslattes, 1969). For situations more complicated than the narrow-beam geometry, the attenuation is still basically exponential, but it is modified by two additional factors. The first of these, sometimes called a geometry factor, depends on the source absorber geometry. The other factor, often called the buildup factor, takes into account secondary photons produced in the absorber, mainly as the result of one or more Compton scatters, which finally reach the detector. The determination of the buildup factor, defined as the ratio of the observed effect to the effect produced only by the primary radiation, constitutes a large part of g-ray transport theory (Evans, 1963). In subsequent sections, only major effects of photon attenuation are discussed in detail.

A.

Photoelectric Absorption

In the photoelectric absorption described partially in Sec. V.A, a photon disappears and an electron is ejected from an atom. The K shell electrons, which are the most tightly bound, are the most important for this effect in the energy region considered in XRS. If the photon energy drops below the binding energy of a given shell, however, an electron from that shell cannot be ejected. Hence, a plot of t versus photon energy exhibits the characteristic ‘‘absorption edges.’’ The mass photoelectric absorption coefficient tN0 =A at the incident energy E (keV) can approximately be calculated based on Walter’s equations (Compton and Allison, 1935): Copyright © 2002 Marcel Dekker, Inc.

8 30:3Z3:94 > < AE3

tN0 4:30 ¼ 0:978Z AE3 > A : 0:78Z3:94 AE3

for E > EK for EL1 < E < EK for EM1 < E
ð46Þ

Based on available experimental and theoretical information for approximately 10,000 combinations of Z and E covering 87 elements and the energy range 1 keV to 1 MeV, the following ln–ln polynomials for the photoeffect cross section tj have been fitted in incident photon energy between each absorption-edge region (Hubbell et al., 1974): ln tj ¼

1;2;or P3

Aij fln½E ðkeVÞgi

ð47Þ

i¼0

In this polynomial, the total photoeffect cross section tj represents one of the following sums: t1 ¼ tM þ tN þ tO þ   

or

tN þ tO þ    ;

EM1 < E < EL3 or E < EM5 t2 ¼ tL þ tM þ tN þ tO þ    ; EL1 < E < EK t3 ¼ tK þ tL þ tM þ tN þ tO þ    ;

ð48Þ

E > EK

The values of the fitted coefficients Aij for the ln–ln representation are given in Appendix VII (McMaster et al., 1969). In multiple-edge regions (e.g., between L1 and L3 edge energies), the photoelectric absorption cross sections are also obtained via Eq. (47) by using the following constant ‘‘jump ratios’’ j (t just above an absorption edge divided by t just below that absorption edge): j ¼ 1:16 j ¼ 1:64 ¼ 1:16  1:41 j ¼ 1:1 j ¼ 1:21 ¼ 1:1  1:1 j ¼ 1:45 ¼ 1:1  1:1  1:2 j ¼ 2:18 ¼ 1:1  1:1  1:2  1:5

for for for for for for

EL2 < E < EL1 EL3 < E < EL2 EM2 < E < EM1 EM3 < E < EM2 EM4 < E < EM3 EM5 < E < EM4

ð49Þ

Simple expressions for calculating the values of the energies of all photoabsorption edges are given in Sec. VI.D. The experimental ratio of the total photoelectric absorption cross section t to the K shell component tK can be fitted with an accuracy of 2–3% by the equation (Hubbell, 1969) t ffi 1 þ 0:01481 ln2 Z
0:000788 ln3 Z tK

ð50Þ

Based on the tables of McMaster et al. (1969), Poehn et al. (1985) found a useful approximation for the calculation of the jump ratios (called also jump factors) for the K shell ( jK ) and L3 subshell ( jL3 ): jK ¼ 1:754  10
6:608  10
1 Z þ 1:427  10
2 Z2
1:1  10
4 Z3

for 11  Z  50

jL3 ¼ 2:003  10
7:732  10
1 Z þ 1:159  10
2 Z2
5:835  10
5 Z3 Copyright © 2002 Marcel Dekker, Inc.

for 30  Z  83

ð51Þ

As already mentioned [Eq. (49)], the values of the jump factors at the L2 and L1 absorption edges are constant for all elements and equal to 1.41 and 1.16, respectively. Tabulated values for the photoelectric absorption cross sections for the elements 1  Z  100 in the energy range of 1 keV to 100 MeV are also available in the work of Storm and Israel (1970), which provides the photon cross sections for all major interaction processes as well as the atomic energy levels, K and L x-ray line energies, weighted average energies for the K and L x-ray series, and relative intensities for K and L x-ray lines. When the apparently sharp x-ray absorption discontinuities are examined at high resolution, they are found to contain a fine structure that extends in some cases to about a few hundred electron volts above the absorption edge. The fine structure very close to an absorption edge (less than or equal to 50 eV above the edge) is generally referred to as the Kossel structure and is designated as XANES (x-ray absorption near-edge structure). Peaks and trenches in this region, which can differ by a factor of 2 or more from the smoothly extrapolated data, can be described in terms of transitions of the (very low energy) ejected electrons to unfilled discrete energy states of the atom (or molecule), rather than to the continuum of states beyond a characteristic energy (Sandstro¨m, 1957; Koningsberger and Prins, 1988; Behrens, 1992b). Superimposed on the Kossel structure is the so-called Kronig structure [extended x-ray absorption fine structure (EXAFS)], which usually extends to about 300 eV above the absorption edge (occasionally to nearly 1 keV above an edge). The Kronig structure can be described in terms of interference effects on the de Broglie waves of the ejected electrons by the molecular or crystalline spatial ordering of neighboring atoms (Hasnain, 1991; Behrens, l992a). The oscillations of the absorption coefficient are of the order of 50% in the energy region 50–60 eV above an absorption edge and of the order of 15% in the region beyond 200 eV above the edge. Modulations of the absorption coefficient in the energy region above an absorption edge can be described theoretically in terms of the electronic parameters (Lee and Pendry, 1975). Through a Fourier transform relationship, the modulations are closely related to the radial distribution function around the element of interest (Sayers et al., 1970). Because both the Kossel and the Kronig fine structures can vary in magnitude and in energy displacement of the features, depending on the molecular, crystalline, or thermal environment of the atom, they can be applied for local structural analysis of various materials, including powders, disordered solids, and liquid and amorphous substances (Lagarde, 1983, Behrens, 1992a, 1992b; Koningsberger and Prins, 1988).

B.

Compton Scattering

Compton scattering (Compton, 1923a, 1923b) is the interaction of a photon with a free electron that is considered to be at rest. The weak binding of electrons to atoms may be neglected, provided the momentum transferred to the electron greatly exceeds the momentum of the electron in the bound state. Considering the conservation of momentum and energy leads to the following equations: hn ¼

hn0 1 þ gð1
cos yÞ

gð1
cos yÞ T ¼ hn0
hn ¼ hn0 1 þ gð1
cos yÞ
 1 y tan j ¼ cot 1þg 2 Copyright © 2002 Marcel Dekker, Inc.

ð52Þ ð53Þ ð54Þ

with g¼

hn0 m 0 c2

where hn0 and hn are the energies of the incident and scattered photon, respectively, y is the angle between the photon directions of travel before and following a scattering interaction, and T and f are the kinetic energy and scattering angle of the Compton recoil electron, respectively. For f ¼ 180 , Eqs. (52) and (53) reduce to ðhnÞmin ¼ hn0

1 1 þ 2g

ð55Þ

and Tmax ¼ hn0

2g 1 þ 2g

ð56Þ

The differential Klein–Nishina collision cross section dsKN =dO (defined as the ratio of the number of photons scattered in a particular direction to the number of incident photons) for unpolarized photons striking unbound, randomly oriented electrons is given by (Klein and Nishina, 1929) 

dsKN r20 hn 2 hn0 hn ¼ þ
sin2 y dO 2 hn0 hn hn0

cm2 electron  sr

ð57Þ

where sr is an abbreviation for steradian. Substitution of Eq. (52) for Eq. (57) gives the differential cross section as a function of the scattering angle y: dsKN r20 1 þ cos2 y ¼ dO 2 ½1 þ gð1
cos yÞ2 ( ) g2 ð1
cos yÞ2  1þ ð1 þ cos2 yÞ½1 þ gð1
cos yÞ

cm2 electron  sr

ð58Þ

For very small energies hn0 m0 c2 , the expression reduces to the classic Thompson scattering cross section for electromagnetic radiation on an electron: dsTh r20 ¼ ð1 þ cos2 yÞ dO 2

cm2 electron  sr

ð59Þ

For low energies of incident photons (approximately less than a few tens of a kiloelectron volt, the angular distribution of Compton-scattered photons is symmetrical about y ¼ 90 ; at higher incident photons energies, the Compton scattering becomes predominantly forward. The differential Klein–Nishina scattering cross section dsSKN =dO for unpolarized radiation, defined as the ratio of the amount of energy scattered in a particular direction to the energy of incident photons, is given by dssKN hn dsKN ¼ hn0 dO dO

cm2 electron  sr

Copyright © 2002 Marcel Dekker, Inc.

ð60Þ

The average (or total) collision cross section sKN gives the probability of any Compton interaction by one photon while passing normally through a material containing one electron per square centimeter: Zp

dsKN 2p sin y dy dO 0 ( )   lnð1 þ 2gÞ lnð1 þ 2gÞ 1 þ 3g 2 1 þ g 2ðg þ 1Þ

¼ 2pr0 þ g2 1 þ 2g g 2g ð1 þ 2gÞ2

sKN ¼

cm2 electron ð61Þ

Again, at the low-energy limit, this cross section reduces to the classic Thomson cross section: 8 sTh ¼ pr20 ¼ 0:6652  10
24 3

cm2 electron

ð62Þ

At extremely high energies hn0 m0 c2 , Eq. (61) reduces to
 1 1 cm2 ln 2g þ sKN ¼ pr20 electron g 2

ð63Þ

The average (or total) scattering cross section, defined as the total scattered energy in photons of various energies hn, is given by Zp

ssKN

dssKN 2p sin y dy dO 0 " # 2ð1 þ gÞð2g2
2g
1Þ 8g2 2 lnð1 þ 2gÞ ¼ pr0 þ þ g3 3ð1 þ 2gÞ3 g2 ð1 þ 2gÞ2

¼

cm2 electron

ð64Þ

The usual Klein–Nishina theory that assumes that the target electron is free and at rest cannot be directly applicable in some cases. Departures from it occur at low energies because of electron-binding effects and, at high energies, because of the possibility of emission of an additional photon (double Compton effect) and radiative corrections associated with emission and reabsorption of virtual photons; these corrections are discussed in the work of Hubbell (1969). The total incoherent (Compton) collision cross section per atom sC, involving the binding corrections by applying the so-called incoherent scattering function S(x, Z ), can be calculated according to Z1 ( 1 2 ½1 þ gð1
cos yÞ
2 sC ¼ r0 2
1 " # ) g2 ð1
cos yÞ2 cm2 2  1 þ cos y þ ð65Þ ZSðx; ZÞ 2pdðcos yÞ 1 þ gð1
cos yÞ atom where x ¼ sinðy=2Þ=l is the momentum transfer parameter and l is the photon wavelength (in angstroms). Copyright © 2002 Marcel Dekker, Inc.

The values of the incoherent scattering function S(x, Z ) and the incoherent collision cross section sC are given by Hubbell et al. (1975). A useful combination of analytical functions for calculating S(x, Z ) has recently been proposed by Szalo´ki (1996): 8 3 X > < s1 ðx; ZÞ ¼ di ½expðxÞ
1i ; 0  x  xi Sðx; ZÞ ¼ i¼1 > : s2 ðx; ZÞ ¼ ½Z
s1 ðx; ZÞ
t2 g1 ðxÞ þ t2 g2 ðxÞ þ s1 ðx1 ; ZÞ x1 < x ð66Þ where g1 ðxÞ ¼ 1
exp½t1 ðx1
xÞ and g2 ðxÞ ¼ 1
exp½t3 ðx1
xÞ The parameters for the calculation of the S(x, Z ), including the critical value of x1, are given by Szalo´ki (1996) for all elements (Z ¼ 1–100) and any values of x. The average deviation between the calculated [Eq. (66)] and tabulated data (Hubbell et al., 1975) is slightly above 1%. The incoherent collision cross sections sC can also be calculated by using ln–ln polynomials already defined by Eq. (47) (by simply substituting tj with sC and taking i ¼ 3). The values of the fitted coefficients for the ln–ln representation for sC valid in the photon energy range 1 keV to 1 MeV are given in Appendix VIII. To complete this subsection, it is worth mentioning the Compton effect for polarized radiation. The differential collision cross section ðdsKN =dOÞpp for the plane-polarized radiation scattered by unoriented electrons has also been derived by Klein and Nishina. It represents the probability that a photon, passing through a target containing one electron per square centimeter, will be scattered at an angle y into a solid angle dO in a plane making an angle b with respect to the plane containing the electrical vector of the incident wave: 


 dsKN r20 hn 2 hn0 hn 2 2 þ ¼
2 sin y cos b dO pp 2 hn0 hn hn0

cm2 electron  sr

ð67Þ

The cross section has its maximum value for b ¼ 90 , indicating that the photon and electron tend to be scattered at right angles to the electrical vector of the incident radiation. The scattering of circularly polarized (cp) photons by electrons with spins aligned in the direction of the incident photon is described by

2 dsKN 2 hn ¼ r0 dO cp hn0 

  hn0 hn hn0 hn cm2 þ
ð68Þ 
sin2 y  cos y hn hn0 hn hn0 electron  sr The first term is the usual Klein–Nishina formula for unpolarized radiation. The þ sign for the additional term applies to right circularly polarized photons. Copyright © 2002 Marcel Dekker, Inc.

C.

Rayleigh Scattering

Rayleigh scattering is a process by which photons are scattered by bound atomic electrons and in which the atom is neither ionized nor excited. The incident photons are scattered with unchanged frequency and with a definite phase relation between the incoming and scattered waves. The intensity of the radiation scattered by an atom is determined by summing the amplitudes of the radiation coherently scattered by each of the electrons bound in the atom. It should be emphasized that, in Rayleigh scattering, the coherence extends only over the Z electrons of individual atoms. The interference is always constructive, provided the phase change over the diameter of the atom is less than one-half a wavelength; that is, whenever
 4p y ra sin <1 l 2

ð69Þ

where ra is the effective radius of the atom. Rayleigh scattering occurs mostly at the low energies and for high-Z materials, in the same region where electron binding effects influence the Compton scattering cross section. The differential Rayleigh scattering cross section for unpolarized photons is given by (Pirenne, 1946) dsR 1 2 ¼ r0 ð1 þ cos2 yÞ j Fðx; ZÞj2 2 dO

cm2 atom  sr

ð70Þ

where Fðx; ZÞ is the ‘‘atomic form factor,’’ Z1 Fðx; ZÞ ¼

rðrÞ4pr

sin½ð2p=lÞrs dr ð2p=lÞrs

ð71Þ

0

where rðrÞ is the total density, r is the distance from the nucleus, and s ¼ 2 sinðy=2Þ. The atomic form factor has been calculated for Z < 26 using the Hartree electronic distribution (Pirenne, 1946) and for Z > 26 using the Fermi–Thomas distribution (Compton and Allison, 1935). At high photon energies, Rayleigh scattering is confined to small angles; at low energies, particularly for high-Z materials, the angular distribution of the Rayleigh-scattered radiation is much broader. A useful simple criterion for judging the angular spread of Rayleigh scattering is given by (Evans, 1958).


0:0133Z1=3 yR ¼ 2 arcsin E ðMevÞ

 ð72Þ

where yR is the opening half-angle of a cone containing at least 75% of the Rayleighscattered photons. In the forward direction, jFðx; ZÞj2 ¼ Z2 , so that Rayleigh scattering becomes appreciable in magnitude and must be accounted for in any g- or x-ray scattering experiments. The total coherent (Rayleigh) scattering cross section per atom sR can be calculated from Copyright © 2002 Marcel Dekker, Inc.

1 sR ¼ r20 2

Z1 ð1 þ cos2 yÞjFðx; ZÞj2 2p dðcos yÞ
1

3 ¼ sTh 8

ð73Þ

Z1 ð1 þ cos2 yÞjFðx; ZÞj2 dðcos yÞ
1

2

cm atom

The values of the atomic form factor F(x, Z ) and the coherent scattering cross section sR are given in the work of (Hubbell et al., 1975). Recently, Szalo´ki (1996) proposed a useful combination of analytical functions to calculate F(x, Z): 8 0  x  x1 f11 ðx; ZÞ ¼ a expð
b1 xÞ þ ðZ
aÞ expð
cxÞ; > > > f ðx; ZÞ ¼ f ðx ; ZÞ expð
b ðx
xÞ; < x 1  x  x2 12 11 1 2 1 ð74Þ F1 ðx; ZÞ ¼ f13 ðx; ZÞ ¼ f12 ðx2 ; ZÞ exp½
b3 ðx2
xÞ; x 2  x  x3 > h ib4 > > : f14 ðx; ZÞ ¼ f13 ðx3 ; ZÞ x ; x 3  x  x4 x3 where 1  Z  7 8 < f21 ðx; ZÞ ¼ a expð
b1 xÞ þ ðZ
aÞ expð
cxÞ; 0  x  x1 x 1  x  x2 F2 ðx; ZÞ ¼ f22 ðx; ZÞ ¼ f21 ðx1 ; ZÞ exp½
b2 ðx1
xÞ; : f23 ðx; ZÞ ¼ f22 ðx2 ; ZÞ exp½
b3 ðx2
xÞ; x 2  x  x3

ð75Þ

where 8  Z  100 The parameters for the calculation of F(x, Z ), including the critical values of x1, x2, and x3, are given by Szalo´ki (1996) for all elements (Z ¼ 1–100) and the momentum transfer x from 0 to 15 A˚71. The average deviation between the calculated [Eqs. (74) and (75)] and tabulated data (Hubbell et al., 1975) is less than 2%. The simplest method for calculating the coherent scattering cross section sR consists in applying the ln–ln representation [see Eq. (47) with sR instead of tj and i ¼ 3]. The values of the fitted coefficients for ln–ln polynomials for calculating sR in the photon energy range 1 keV to 1 MeV are given in Appendix IX.

D. Total Mass Attenuation Coefficient An extensive review of current tabulations of x-ray attenuation coefficients has been given by Hubbell (1984). Differences between various compilations of total mass attenuation coefficients result from uncertainties in our knowledge of partial cross sections for the interaction of photons with matter as a function of elemental atomic number Z and photon energy E. Present discrepancies are disturbing, to say the least, frequently amounting to 5–10% in the photon energy region below 10 keV and rising to as much as 30% near an absorption edge. Hubbell (1982) has tabulated mass attenuation coefficients and mass energy absorption coefficients for photon energies from 1 keV to 20 MeV for 40 elements ranging from hydrogen (Z ¼ 1) to uranium (Z ¼ 92) and for 45 mixtures and compounds of dosimetric interest. The uncertainty ranges for the total mass attenuation coefficient values in the tabulation of McMaster et al. (1969) have been estimated by Hubbell et al. (1974). These ranges of uncertainties fall into four categories. Category I (uncertainty below 2%) applies over the energy region 6–40 keV (except near absorption edges) for the following elements: C, Copyright © 2002 Marcel Dekker, Inc.

Mg, Al, Ti, Fe, Ni, Cu, Zn, Zr, Mo, Pd, Ag, Cd, Sn, La, Gd, Ta, W, Pt, Au, Pb, Th, and U. In this category, the photon energy region above 100 keV is also included for all elements in which incoherent scattering comprises more than 90% of the total cross section. Category II (uncertainty of 2–5%) applies to the energy region 2–6 keV for all elements, 6–40 keV for all elements not specified in category I, and above 40 keV except for the scattering-dominated region specified in category I. In category III (uncertainty of 5–15%), the authors (Hubbell et al., 1974) included (1) the elements hydrogen, helium, and lithium, (2) the energy region 1–2 keV for elements, and (3) the regions containing K, L, M, and N absorption edges, and the fine-structure regions extending from 200 eV to 1 keV above each of these regions. The experimental uncertainties in cases 1 and 2 greatly exceed 15%. Category IV (uncertainty above 15%) applies to the photon energy region about 200 eV above an absorption edge (Kossel and Kronig fine-structure regions) for all elements. Based on the tables published by McMaster et al. (1969), Wernisch et al. (1984) developed an algorithm for the calculation of the total mass attenuation coefficient valid for the photon energy range from 1 to 50 keV and for 73 elements (11  Z  83). The authors have applied the simple expression m

cm2 ¼ Hedþk g

ð76Þ

ln E

Values of H, d, and k have been obtained from least-squares fits applied to the data published by McMaster et al. (1969); they are given in Figure 5. The values of the edge energies EK, EL1, EL2, EL3, EM1, EM2, EM3, EM4, and EM5 can simply be calculated from (Wernisch et al., 1984) Ei ¼ ri þ si Z þ ti Z2 þ ni Z3

keV

ð77Þ

the parameters ri, si, ti, and ni for various absorption edges i are given in Table 2. Another flexible semiempirical scheme to calculate the total mass attenuation coefficient m for a very wide photon energy range (0.1–1000 keV) has been proposed by Orlic et al. (1993) m ¼ exp½ p1 þ p2 ðln lÞ þ p3 ðln lÞ2 þ p4 ðln lÞ3  þ sKN ZN0 A
1

ð78Þ

where sKN is the average incoherent collision cross section gives by Eq. (61). The values of the fitting parameters p1, p2, p3, and p4 are constant for each element and within the energy regions defined by two adjacent absorption edges or for energies beyond the K absorption edge; the parameters are given in Appendix X. The experimental and theoretical values of total interaction cross sections [stot, Eq. (42)] and the mass attenuation coefficients m for the elements (1  Z  98) in the energy range 4.9–24.9 keV have been collected and compared by Arndt et al. (1992). Most of the available tabulations of x-ray attenuation coefficients do not include the photon energy region below 1 keV that corresponds to the energies of characteristic K x-rays of light elements (Z < 11). Experimental data in this energy region are incomplete and it should not be assumed that the accuracy of the available tabulated values is better than 15% (Veigele, 1974; Appendix XI). E.

Diffraction, Refraction, and Dispersion

When a beam of monochromatic x-rays falls onto a crystal lattice, a regular periodic arrangement of atoms, a diffracted beam only results in definite directions. The phenomenon of x-ray diffraction at an ordered array of atoms (or molecules) can also be Copyright © 2002 Marcel Dekker, Inc.

Figure 5 Definition of the energy ranges (a) and values of the parameters H, d, and k (b) applied for the calculation of the total mass attenuation coefficients according to Eq. (76). (From Wernisch et al., 1984.) Copyright © 2002 Marcel Dekker, Inc.

Table 2 Values of the Parameters ri,si,ti, and ni Applied for Calculating the Energies of Absorption Edges via Eq. (77) Applicability range i K L1 L2 L3 M1 M2 M3 M4 M5

ri

si
1


1.304610
4.506610
1
6.018610
1 3.390610
1
8.645
7.499
6.280
4.778
2.421

ti
3


2.633610 1.566610
2 1.964610
2
4.931610
2 3.977610
1 3.459610
1 2.831610
1 2.184610
1 1.172610
1

ni
3

9.718610 7.599610
4 5.935610
4 2.336610
3
5.963610
3
5.250610
3
4.117610
3
3.303610
3
1.845610
3


5

4.144610 1.792610
5 1.843610
5 1.836610
6 3.624610
5 3.263610
5 2.505610
5 2.115610
5 1.397610
5

%a

Zmin

Zmax

3.5 2.2 2.3 1.9 0.4 0.4 0.4 0.4 0.4

11 28 30 30 52 55 55 60 61

63 83 83 83 83 83 83 83 83

a Standard deviation of calculated energies [Eq. (77)] relative to the energies from the tables of McMaster et al. (1969). Source: From Wernisch et al., 1984.

interpreted as a reflection of an incident x-ray beam by the interior planes of a crystal (Bragg reflection). By elementary calculation of the difference in path between two coherent rays, W. L. Bragg found the reinforcement condition for reflection (known as Bragg’s equation or law):
 yn nl ¼ 2d sin ð79Þ 2 where n is the order of reflection, d is the interplanar spacing, and yn=2 is the angle of reflection (or Bragg angle) defined as the angle between the reflecting plane of the crystal and the incident or reflected beam. The first-order reflection (n ¼ 1) is normally strongest, and the reflected intensity decreases as n increases. Bragg’s law as given in Eq. (79) is only a first approximation, as the refraction in the crystal interferes with the angle of reflection. Because the refractive index of x-rays is slightly less than unity, the deviations from Bragg’s law, Eq. (79), were not observed in the early years until methods were found for precise measurements of x-ray wavelengths. The refraction is accounted for by ascribing a slightly different value dn of the lattice constant to each order of reflection; the simple Bragg’s law [Eq. (79)] can thus be written as (Sandstro¨m, 1957)

 4d2 d yn nl ¼ 2d 1
2 2 sin ð80Þ n l 2 where d ¼ 1
n0 for small photon absorption (l < 1 A˚) and n0 is the refractive index for x-rays. d is a small positive number of the order 10
5 for heavy elements and 10
6 for light elements at l ¼ 1 A˚ and is proportional to l2. Because the value of d is positive, total reflection occurs back into air when an x-ray beam meets a surface at a large enough angle of incidence. Provided no absorption occurs, the critical glancing angle ytr (tr, total reflection) is defined by (Sandstro¨m, 1957) pffiffiffiffiffi sin ytr ¼ 2d ð81Þ Copyright © 2002 Marcel Dekker, Inc.

For l ¼ 0.1 nm, the value of ytr is of the order 10
3 for light elements and 5610
3 for heavy elements, increasing in proportion to l The refractive index n0 for a medium containing one type of atom can be calculated from (Hirsch, 1962) n0 ¼ 1


Nl2 e2 Fð0Þ 2p m0 c2

ð82Þ

where N is the number of atoms per unit volume and Fð0Þ is the atomic scattering factor at zero scattering angle. Equation (82) shows that n0 depends on the wavelength; this phenomenon is called dispersion. The anomalous dispersion causes the quantity d=l2 , Eq. (80), to vary slightly with the wavelength. The variations become important only in the neighborhood of the absorption edges of the constituents of the crystal. Dividing the quantity d=l2 into one normal part ½ðd=l2 Þn  and one anomalous part ½ðd=l2 Þa , the theory of anomalous dispersion leads to an expression of Bragg’s law that can be written as (Sandstro¨m, 1957)
 

 4d2 d 4d2 d yn nl ¼ 2d 1
2
sin ð83Þ 2 2 2 n n 2 l n l a Combined with Bragg’s law in its uncorrected form, Eq. (79), this expression becomes (Sandstro¨m, 1957)
 
 
 4d2 d yn =2 4d2 d
l ¼ 2d 1
2 l ð84Þ sin n n n2 l 2 a l2 n The values of ðd=l2 Þa giving a correction for anomalous dispersion can be determined experimentally. The theory of anomalous dispersion has been applied by Sparks (1974, 1975) to explain the inelastic angular-independent scattering from elements having an absorptionedge energy just above the energy of the incident x-rays. The observed intensity of the inelastically scattered radiation was found to be dependent on the nearness of the energy of an absorption edge to the energy of the incident x-rays. The energy of the inelastic peaks is shifted from the incident energy by the binding energy of the most tightly bound shell from which electrons could be photoejected by the incident radiation. F.

X-ray Raman Scattering

Immediately after the discovery of Raman scattering in the visible-wavelength region, a similar effect concerning x-ray radiation was experimentally examined (Davis and Mitchell, 1928; Krishnan, 1928). X-ray Raman scattering appears as a band spectrum having a short-wavelength edge corresponding to a definite energy loss equal to the K electron-binding energy EK of the element. This inelastic effect was observed, for example, when CrKa and CuKa radiation was scattered by solids of light elements, such as lithium, beryllium , boron, and by graphite (Suzuki, 1966; Suzuki et al., 1970; Suzuki and Nagasawa, 1975). The shape of the Raman band is similar to that of the soft x-ray K absorption spectrum of the solids. Mizuno and Ohmura (1967) have found the following two conditions for x-ray Raman scattering: 4p
a <1 l0 Copyright © 2002 Marcel Dekker, Inc.

ð85Þ

and hv0 EK

ð86Þ

where a
is the mean radius of charge distribution of the K electrons and l0 is the wavelength of the incident x-rays. The intensity of the x-ray Raman scattering I(y, l) is given by (Suzuki and Nagasawa, 1975) ( 
2 
2 ) 4p
a y 4p
a y 2 ð87Þ sin þT2 ðlÞ sin Iðy; lÞ ffi ð1 þ cos yÞ T1 ðlÞ l0 2 l0 2 where T1(l) and T2(l) factors are related to the dipole and multipole transitions, respectively. Although according to this equation a slight displacement of the peak position of the Raman band is expected with the scattering angle, the peak position does not shift in most experiments. In general, x-ray Raman scattering gives information about the unoccupied states above the Fermi level of the relevant solids. Moreover, this kind of inelastic scattering by electrons in solids should sometimes be taken into account in x-ray spectrum evaluation.

VII.

INTENSITY OF CHARACTERISTIC X-RAYS

This section provides some of the necessary background information for subsequent chapters dealing with quantitative x-ray analysis. Derivation of any relationship between excitation source intensity and measured characteristic x-rays is sometimes complex and is presented in detail in many relevant books, such as those by Jenkins et al. (1981) and Tertian and Claisse (1982) on XRF analysis and by Goldstein et al. (1981) and Heinrich (1981) on EPXMA. A.

Photon Excitation

When continuous (polychromatic) radiation is used to excite the characteristic x-rays of element i in a completely homogeneous sample of thickness T (cm) and when enhancement effects are neglected, the intensity of the fluorescent radiation Ii (Ei) is described by Ii ðEi Þ dO1 dO2 ¼

dO1 dO2 eðEi Þ 4p sin C1 E Zmax 1
exp½
rTðmðE0 Þ csc C1 þ mðEi Þ csc C2 Þ  ai ðE0 Þ I0 ðE0 Þ dE0 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð88Þ

Ec;i

with ai ðE0 Þ ¼

Wi t0i ðE0 Þoi pi


 1 1
ji

ð89Þ

where dO1 and dO2 are the differential solid angles for the incident (primary) and emerging (characteristic) radiation, respectively; eðEi Þ is the intrinsic detector efficiency for recording a photon of energy Ei ; Ec;i and Emax are the critical absorption energy of element i and the maximum energy in the excitation spectrum; r is the density of the specimen (in g=cm3); C1 and C2 are the effective incidence and takeoff angles, respectively; mðE0 Þ and mðEi Þ are the total mass attenuation coefficients (in cm2=g) for the whole specimen [Eq. (45)] at Copyright © 2002 Marcel Dekker, Inc.

energies E0 and Ei , respectively; I0 ðE0 Þ dE0 is the number of incident photons per second per steradian in the energy interval E0 to E0 þ dE0 ; Wi is the weight fraction of the ith element; and t0i ðE0 Þ is the total photoelectric mass absorption coefficient for the ith element at the energy E0 (in cm2=g). Because large solid angles O1 and O2 are used for the excitation and characteristic x-rays in practical spectrometers, Eq. (88) should also be integrated over these finite solid angles. Such calculations can often be omitted; however, and for a given measurement geometry, an experimentally determined geometry factor G can be applied. As seen from Eq. (88), the intensity of characteristic x-rays is modified by the effects of primary [mðE0 Þ] and secondary [mðEi Þ] absorption in the specimen; this is a major source of the so-called matrix effects in XRF analysis. If the excitation source is monochromatic (emits only one energy), Eq. (88) simplifies to eðEi Þai ðE0 ÞI0 ðE0 Þ sin C1 1
exp½
rTðmðE0 Þ csc C1 þ mðEi Þ csc C2 Þ  mðE0 Þ csc C1 þ mðEi Þ csc C2

Ii ðEi Þ ¼ G

ð90Þ

The enhancement effect, consisting of an extra excitation of the element of interest by the characteristic radiation of some matrix elements, modifies the equations for the intensity Ii ðEi Þ. In the case of monochromatic photon excitation, a factor, 1 þ Hi , should be included in Eq. (90), where Hi is the enhancement term defined as Hi ¼


 m X 1 1 Wk o k 1
m ðEk Þmk ðE0 Þ 2mi ðE0 Þ k¼1 jk i   lnð1 þ mðE0 Þ=½mðEk Þ sin C1 Þ lnð1 þ mðEi Þ=½mðEk Þ sin C2 Þ  þ mðE0 Þ= sin C1 mðEi Þ= sin C2

ð91Þ

where mi ðE0 Þ and mi ðEk Þ are the total mass attenuation coefficients for the ith element at the energy of incident radiation ðE0 Þ and characteristic radiation of the element k ðEk Þ, respectively, mk ðE0 Þ is the total mass attenuation coefficient for the element k at the energy E0, and mðEk Þ is the total mass attenuation coefficient for the whole specimen at the energy Ek . 1. Thin-SampleTechnique If the total mass per unit area of a given sample m ¼ rT is small, Eq. (90) simplifies to Iithin ðEi Þ ¼

G eðEi Þai ðE0 ÞI0 ðE0 Þm sin C1

ð92Þ

The relative error resulting from applying Eq. (92) instead of Eq. (90) does not exceed 5% when the total mass per unit area satisfies the condition m thin 

0:1 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð93Þ

A major feature of the thin-sample technique is that the intensity of the characteristic x-rays, Iithin , depends linearly on the concentration of the ith element (or on its mass per unit area); it is equivalent to the fact that, in the thin-sample technique, matrix effects can safely be neglected. Copyright © 2002 Marcel Dekker, Inc.

2. Thick-SampleTechnique By the term ‘‘thick sample,’’ we mean a sample whose mass per unit area (thickness) is greater than the so-called saturation mass (thickness). The saturation thickness is defined as a limiting value above which practically no further increase in the intensity of the characteristic radiation is observed as the sample thickness is increased. If the total mass per unit area of a given sample is sufficiently large, Eq. (90) simplifies to Iithick ðEi Þ ¼

GeðEi Þai ðE0 ÞI0 ðE0 Þ mðE0 Þ þ sin C1 =ðsin C2 ÞmðEi Þ

ð94Þ

The relative error resulting from applying Eq. (94) instead of Eq. (90) does not exceed 1% when the total mass per unit area satisfies the condition m thick

B.

4:61 mðE0 Þ csc C1 þ mðEi Þ csc C2

ð95Þ

Electron Excitation

For a thin foil (defined as a foil of thickness such that the beam electron undergoes only one scattering act), the intensity of the characteristic x-rays Iitf (in photons per second) is given by Iitf ¼ Ki I0 Qi oi N0

1 rTWi Ai

ð96Þ

where Ki is a factor depending on the measurement geometry and detection efficiency and the other symbols have the following dimensions: I0 ¼ electrons=s, Qi ¼ ionizations per electron for 1 atom=cm2, oi ¼ photons=ionization. N0 ¼ atoms=mol, 1=Ai ¼ mol=g, r ¼ g=cm3, and T ¼ cm. For a bulk target, Eq. (96) must be integrated over the electron path, taking into account the loss of energy by the electron beam:

Iibulk

Wi ¼ I0 oi Ki N0 Ai

ZEc;i

Qi ðEÞ dE dE=dðrxÞ

ð97Þ

E0

where dE=dðrxÞ is the electron energy loss per unit of distance traveled in a material, given by the Bethe equation (Heinrich, 1981; Bethe and Ashkin, 1953):
 n dE 1 X Wj Z j E ¼
78; 500 ln 1:166 dðrxÞ E j¼1 Aj Jj

keV=g cm
2

ð98Þ

where Jj is the mean ionization potential of element jðJj ¼ 9:76Zj þ 58:5Zj
0:19 eVÞ (Goldstein et al., 1981). Strictly speaking, the Bethe equation is only valid for the electron energy E > 6:3Jj . Below this limit, the energy loss of the electrons should be described either by the modification of Rao-Sahib and Wittry (1972) [Eq. (99)] or by the modification of Love et al. (l978) [Eq. (l00)]: Copyright © 2002 Marcel Dekker, Inc.

n dE 1 X Wj Z j pffiffiffiffi keV=g cm
2 ¼
0:6236  105 pffiffiffiffi dðrxÞ A E j¼1 j Jj n X dE Wj Z j 1 ¼
1=2
5 A J dðrxÞ j j 1:18  10 ðE=Jj Þ þ 1:47  10
6 E=Jj j¼1

ð99Þ keV=g cm
2

ð100Þ

However, to describe the intensity of characteristic x-rays emitted, three effects should be considered additionally (Heinrich, 1981; Goldstein et al., 1981; Love and Scott, 1981): 1. 2. 3.

C.

Absorption of characteristic x-rays within the specimen Electron backscattering Secondary fluorescence by characteristic x-rays and=or bremsstrahlung continuum produced by an electron beam

Particle Excitation

For a thin, uniform, homogeneous target, the intensity of characteristic x-rays Itti is given by the simple formula I tti ¼ Ki I0 N0 ss;i ðE0 Þoi

1 mi Ai

ð101Þ

where mi is the areal density of the element with atomic number Zi and atomic mass Ai . For a thick, homogeneous target, the intensity of characteristic x-rays, Iithick , from the element i of concentration Wi can be calculated by (Campbell and Cookson, 1984) Iithick ¼

0 X Ki oi N0 ss;i ðEÞTi ðEÞ dE I 0 Wi Ai SðEÞ E

ð102Þ

0

where S(E) is the stopping power and Ti ðEÞ is the photon attenuation factor. The latter is given by 0 1 ZE sin C1 dE C B Ti ðEÞ ¼ exp@
mðEi Þ ð103Þ A SðEÞ sin C2 E0

More details on various x-ray analytical techniques are provided in subsequent chapters.

VIII.

IUPAC NOTATION FOR X-RAY SPECTROSCOPY

The nomenclature commonly used in XRS to describe x-ray emission spectra was introduced by M. Sieghahn in the l920s and is based on the relative intensity of lines from different series. A new and more systematic notation for x-ray emission lines and absorption edges, based on the energy-level designation, was developed by the International Union of Pure and Applied Chemistry (Jenkins et al. 1991). Because the new notation, called the IUPAC notation, replaces the Siegbahn’s notation, some characteristic features of the new nomenclature must be mentioned. The IUPAC notation prescribes Arabic numerals for subscripts; the original notation uses Roman numerals (e.g., L2 and L3 , Copyright © 2002 Marcel Dekker, Inc.

instead of LII and LIII ). In the IUPAC notation, states with double or multiple vacancies should be denoted by, for example, K2 ; KL1 , and Ln2;3 , which correspond to the electron configurations 1s
2 ; 1s
1 2s
1 , and 2p
n , respectively. X-ray transitions and x-ray emission diagram lines are denoted by the initial (placed first) and final x-ray levels separated by a hyphen. To conform with the IUPAC notation of x-ray spectra, the hyphen separating the initial and final state levels should also be introduced into the notation for Auger electron emission process. The IUPAC notation is compared with the Sieghahn notation in Appendix XII.

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX I: CRITICAL ABSORPTION WAVELENGTHS AND CRITICAL ABSORPTION ENERGIES K edge Atomic number Element 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu

L1 edge

L2 edge

L3 edge

M4 edge

A˚

keV

A˚

keV

A˚

keV

A˚

keV

918 504 226.953 106.9 64.6 43.767 31.052 23.367 18.05 14.19 11.48 9.512 7.951 6.745 5.787 5.018 4.397 3.871 3.437 3.070 2.757 2.497 2.269 2.070 1.896 1.743 1.608 1.488 1.380

0.014 0.025 0.055 0.116 0.192 0.283 0.399 0.531 0.687 0.874 1.08 1.303 1.559 1.837 2.142 2.470 2.819 3.202 3.606 4.037 4.495 4.963 5.462 5.987 6.535 7.109 7.707 8.329 8.978

258 225 197 143 105 81.0 64.2 52.1 43.2 36.4 30.7 26.8 23.4 20.5 18.3 16.3 14.6 13.3 12.22 11.27

0.048 0.055 0.063 0.087 0.118 0.153 0.193 0.238 0.287 0.341 0.399 0.462 0.530 0.604 0.679 0.762 0.849 0.929 1.015 1.100

564 365 248 170 125 96.1 75.6 61.1 50.2 41.8 35.2 30.2 27.0 23.9 21.3 19.1 17.2 15.6 14.2 13.0

0.022 0.034 0.050 0.073 0.099 0.129 0.164 0.203 0.247 0.297 0.352 0.411 0.460 0.519 0.583 0.650 0.721 0.794 0.871 0.953

564 365 253 172 127 96.9 76.1 61.4 50.6 42.2 35.5 30.8 27.3 24.2 21.6 19.4 17.5 15.9 14.5 13.3

0.022 0.034 0.049 0.072 0.098 0.128 0.163 0.202 0.245 0.294 0.349 0.402 0.454 0.512 0.574 0.639 0.708 0.779 0.853 0.933

Copyright © 2002 Marcel Dekker, Inc.

A˚

keV

M5 edge A˚

keV

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu

1.283 1.196 1.117 1.045 0.980 0.920 0.866 0.816 0.770 0.728 0.689 0.653 0.620 0.589 0.561 0.534 0.509 0.486 0.464 0.444 0.425 0.407 0.390 0.374 0.359 0.345 0.331 0.318 0.307 0.295 0.285 0.274 0.265 0.256

Copyright © 2002 Marcel Dekker, Inc.

9.657 10.365 11.100 11.860 12.649 13.471 14.319 15.197 16.101 17.032 17.993 18.981 19.996 21.045 22.112 23.217 24.341 25.509 26.704 27.920 29.182 30.477 31.800 33.155 34.570 35.949 37.399 38.920 40.438 41.986 43.559 45.207 46.833 48.501

10.33 9.54 8.73 8.107 7.506 6.97 6.46 5.998 5.583 5.232 4.867 4.581 4.298 4.060 3.83 3.626 3.428 3.254 3.085 2.926 2.777 2.639 2.511 2.389 2.274 2.167 2.068 1.973 1.889 1.811 1.735 1.665 1.599 1.536

1.200 1.30 1.42 1.529 1.651 1.78 1.92 2.066 2.220 2.369 2.546 2.705 2.883 3.054 3.24 3.418 3.616 3.809 4.018 4.236 4.463 4.695 4.937 5.188 5.451 5.719 5.994 6.282 6.559 6.844 7.142 7.448 7.752 8.066

11.87 10.93 9.94 9.124 8.416 7.80 7.21 6.643 6.172 5.755 5.378 5.026 4.718 4.436 4.180 3.942 3.724 3.514 3.326 3.147 2.982 2.830 2.687 2.553 2.429 2.314 2.204 2.103 2.011 1.924 1.843 1.767 1.703 1.626

1.045 1.134 1.248 1.358 1.473 1.59 1.72 1.865 2.008 2.153 2.304 2.467 2.627 2.795 2.965 3.144 3.328 3.527 3.726 3.938 4.156 4.380 4.611 4.855 5.102 5.356 5.622 5.893 6.163 6.441 6.725 7.018 7.279 7.621

12.13 11.10 10.19 9.39 8.67 8.00 7.43 6.89 6.387 5.962 5.583 5.223 4.913 4.632 4.369 4.130 3.908 3.698 3.504 3.324 3.156 3.000 2.855 2.719 2.592 2.474 2.363 2.258 2.164 2.077 1.995 1.918 1.845 1.775

1.022 1.117 1.217 1.32 1.43 1.55 1.67 1.80 1.940 2.079 2.220 2.373 2.523 2.677 2.837 3.001 3.171 3.351 3.537 3.728 3.927 4.131 4.340 4.557 4.780 5.010 5.245 5.488 5.727 5.967 6.213 6.466 6.719 6.981

15.56

0.7967

15.89

0.7801

13.122 12.459

0.9448 0.9951

13.394 23.737

0.9257 0.9734

11.288 10.711

1.0983 1.1575

11.552 11.013

1.0732 1.1258

Appendix I

Continued L1 edge

K edge Atomic number 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

L2 edge

L3 edge

Element

A˚

keV

A˚

keV

A˚

keV

A˚

keV

Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

0.247 0.238 0.231 0.223 0.216 0.209 0.202 0.196 0.190 0.184 0.178 0.173 0.168 0.163 0.158 0.153 0.149 0.415 0.141 0.137 0.133 0.130 0.126 0.123

50.215 51.984 53.773 55.599 57.465 59.319 61.282 63.281 65.292 67.379 69.479 71.590 73.856 76.096 78.352 80.768 83.046 85.646 88.037 90.420 93.112 95.740 98.418 101.147

0.116 0.113 0.110 0.108

106.759 109.741 112.581 115.610

1.477 1.421 1.365 1.317 1.268 1.222 1.182 1.140 1.100 1.061 1.025 0.990 0.956 0.923 0.893 0.863 0.835 0.808 0.782 0.757 0.732 0.709 0.687 0.665 0.645 0.625 0.606 0.588 0.56

8.391 8.722 9.081 9.408 9.773 10.141 10.487 10.870 11.271 11.681 12.097 12.524 12.968 13.427 13.875 14.354 14.837 15.338 15.858 16.376 16.935 17.490 18.058 18.638 19.229 19.842 20.458 21.102 21.764

1.561 1.501 1.438 1.390 1.338 1.288 1.243 1.199 1.155 1.114 1.075 1.037 1.001 0.967 0.934 0.903 0.872 0.843 0.815 0.789 0.763 0.739 0.715 0.693 0.671 0.650 0.630 0.611 0.592

7.938 8.256 8.619 8.918 9.260 9.626 9.972 10.341 10.732 11.128 11.533 11.953 12.380 12.817 13.266 13.731 14.210 14.695 15.205 15.713 16.244 16.784 17.337 17.904 18.478 19.078 19.677 20.311 20.938

1.710 1.649 1.579 1.535 1.482 1.433 1.386 1.341 1.297 1.255 1.216 1.177 1.140 1.106 1.072 1.040 1.008 0.979 0.950 0.923 0.897 0.872 0.848 0.825 0.803 0.782 0.761 0.741 0.722

7.250 7.517 7.848 8.072 8.361 8.650 8.941 9.239 9.554 9.874 10.196 10.529 10.867 11.209 11.556 11.917 12.3 12.655 13.041 13.422 13.817 14.215 14.618 15.028 15.439 15.865 16.293 16.731 17.160

Copyright © 2002 Marcel Dekker, Inc.

M4 edge

M5 edge

A˚

keV

A˚

keV

8.601

1.4415

8.847 8.487

1.4013 1.4609

6.87 6.59 6.33 6.073 5.83 5.59 5.374 5.157 4.952 4.757 4.572

1.804 1.880 1.958 2.042 2.126 2.217 2.307 2.404 2.504 2.606 2.711

7.11 6.83 6.560 6.30 6.05 5.81 5.584 5.36 5.153 4.955 4.764

1.743 1.814 1.890 1.967 2.048 2.133 2.220 2.313 2.406 2.502 2.603

3.557 3.436 3.333

3.485 3.608 3.720

3.729 3.618 3.497

3.325 3.436 3.545

93 94 95 96 97 98 99 100

Np Pu Am Cm Bk Cf Es Fm

0.105 0.102 0.099 0.097 0.094 0.092 0.090 0.088

118.619 121.720 124.816 128.088 131.357 134.683 138.067 141.510

Source: From Clark, 1963 and Burr, 1974.

Copyright © 2002 Marcel Dekker, Inc.

0.553 0.537 0.521 0.506 0.491 0.477 0.464 0.451

22.417 23.097 23.793 24.503 25.230 25.971 26.729 27.503

0.574 0.557 0.540 0.525 0.509 0.494 0.480 0.466

21.596 22.262 22.944 23.640 24.352 25.080 25.824 26.584

0.704 0.686 0.669 0.653 0.637 0.622 0.607 0.593

17.614 18.066 18.525 18.990 19.461 19.938 20.422 20.912

APPENDIX II: CHARACTERISTIC X-RAY WAVELENGTHS (A
) AND ENERGIES (keV) Table 1

K Series Diagram Lines (A˚)a

Line

a1,2

a1

a2

Approximate intensity

150

100

50

Li B B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

230 113 67 44 31.603 23.707 18.307 14.615 11.909 9.889 8.339 7.126 6.155 5.373 4.729 4.192 3.744 3.360 3.032 2.750 2.505 2.291 2.103 1.937 1.791 1.659 1.542 1.437 1.341 1.256 1.177 1.106 1.041 0.981 0.927 0.877 0.831 0.788 0.748 0.710 0.676 0.644 0.614 0.587

Copyright © 2002 Marcel Dekker, Inc.

8.338 7.125 6.154 5.372 4.728 4.191 3.742 3.359 3.031 2.749 2.503 2.290 2.102 1.936 1.789 1.658 1.540 1.435 1.340 1.255 1.175 1.105 1.040 0.980 0.926 0.875 0.829 0.786 0.747 0.709 0.675 0.643 0.613 0.585

8.341 7.127 6.157 5.375 4.731 4.194 3.745 3.362 3.034 2.753 2.507 2.294 2.105 1.940 1.793 1.661 1.544 1.439 1.344 1.258 1.179 1.109 1.044 0.984 0.930 0.880 0.833 0.791 0.751 0.713 0.679 0.647 0.617 0.590

b1

b3 15

14.460 11.574 9.559 7.960

b2

b4

5

<1

11.726 9.667 8.059 6.778

5.804 5.032 4.403 3.886 3.454 3.089 2.780 2.514 2.285 2.085 1.910 1.757 1.621 1.500 1.392 1.296 1.207 1.129 1.057 0.992 0.933 0.879 0.829 0.783 0.740 0.701 0.665 0.632

1.393 1.208 1.129 1.058 0.993 0.933 0.879 0.830 0.784 0.741 0.702 0.666 0.633 0.601

0.572 0.546 0.521

0.573 0.546 0.521

1.489 1.381 1.284 1.196 1.117 1.045 0.980 0.921 0.866 0.817 0.771 0.728 0.690 0.654 0.621 0.590 0.562 0.535 0.510

0.866 0.815 0.770 0.727 0.689 0.653 0.620 0.561 0.534

Table 1

Continued

Line

a1,2

a1

a2

Approximate intensity

150

100

50

0.559 0.535 0.512 0.491 0.470 0.451 0.433 0.416 0.401 0.385 0.371 0.357 0.344 0.332 0.321 0.309 0.299 0.289 0.279 0.270 0.261 0.253 0.244 0.236 0.229 0.222 0.215 0.209 0.202 0.196 0.191 0.158 0.180 0.175 0.170 0.165 0.161 0.156 0.152 0.148 0.144 0.144 0.140 0.133 0.131 0.126

0.564 0.539 0.517 0.495 0.475 0.456 0.438 0.421 0.405 0.390 0.376 0.362 0.349 0.337 0.325 0.314 0.304 0.294 0.284 0.275 0.266 0.258 0.250 0.241 0.234 0.227 0.220 0.213 0.207 0.201 0.196 0.190 0.185 0.180 0.175 0.170 0.165 0.161 0.157 0.153 0.149 0.149 0.145 0.138 0.136 0.131

Ag Cs In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

0.561 0.536 0.514 0.492 0.472 0.453 0.435 0.418 0.402 0.387 0.373 0.359 0.346 0.334 0.322 0.311 0.301 0.291 0.281 0.272 0.263 0.255 0.246 0.238 0.231 0.224 0.217 0.211 0.204 0.198 0.193 0.187 0.182 0.177 0.172 0.167 0.162 0.185

0.135 0.128

a Conversion equation: E (keV) ¼ 12.4=l (A˚). Source: Copyright ©From 2002 Clark, Marcel1963. Dekker, Inc.

b1

b3

b2

b4

5

<1

0.498 0.476 0.455 0.436 0.418 0.401 0.385 0.360 0.355 0.342 0.329 0.317 0.305 0.294

0.487 0.465 0.445 0.426 0.408 0.391 0.376

0.486

0.274 0.265 0.256 0.246 0.238 0.231 0.223 0.216 0.208 0.203 0.196 0.191 0.185 0.179 0.174 0.169 0.164 0.160 0.155 0.151 0.147 0.143

0.267 0.258 0.249 0.239 0.231

15 0.497 0.475 0.455 0.435 0.417 0.400 0.384 0.369 0.355 0.341 0.328 0.316 0.305 0.294

0.346 0.333 0.320 0.309 0.297 0.287

0.444 0.425 0.407

0.319 0.307

0.283 0.274 0.264 0.255 0.246 0.237 0.230 0.222 0.215 0.208 0.202 0.195 0.190 0.184 0.179 0.173 0.168 0.163 0.159 0.154 0.150 0.146 0.142 0.138 0.134 0.131 0.127 0.127 0.124 0.117 0.115 0.111

0.135 0.132 0.128 0.128 0.125 0.118 0.116 0.112

0.217 0.203 0.197 0.190 0.185 0.179 0.174 0.169 0.164 0.159 0.155 0.150 0.146 0.147 0.138 0.133

0.184 0.179 0.174 0.168 0.163 0.159 0.154 0.150 0.146 0.141 0.138

0.114

0.114

0.108

0.108

Table 2

L Series Diagram Lines (A˚)

Line Approximate intensity Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

a

a1

a2

b1

b2

b3

b4

b5

b6

b7

b9

110

100

10

50

20

6

4

1

<1

<1

<1

36.393 31.393 27.445 24.309 21.713 19.489 17.602 16.000 14.595 13.357 12.282 11.313 10.456 9.671 8.990 8.375

36.022 31.072 27.074 23.898 21.323 19.158 17.290 15.698 14.308 13.079 12.009 11.045 10.194 9.414 8.735 8.126

7.318 6.863 6.449 6.070 5.725 5.406

7.325 6.870 6.456 6.077 5.732 5.414

7.075 6.623 6.211 5.836 5.492 5.176

4.846 4.597 4.368 4.154 3.956 3.752 3.600 3.439 3.290 3.148

4.854 4.605 4.376 4.162 3.965 3.781 3.609 3.448 3.299 3.157

2.892

2.902

Copyright © 2002 Marcel Dekker, Inc.

19.429 17.757 15.742 14.269 13.167 12.115 11.225

8.930

5.586 5.238 4.923

6.788 6.367 5.983 5.632 5.310 5.013

6.821 6.403 6.018 5.668 5.346 5.048

6.984 6.519 6.094 5.710 5.361 5.048

4.620 4.374 4.146 3.935 3.739 3.555 3.385 3.226 3.077 2.937

4.372 4.130 3.909 3.703 3.514 3.339 3.175 3.023 2.882 2.751

4.487 4.253 4.034 3.834 3.644 3.470 3.306 3.152 3.009 2.874

4.532 4.289 4.071 3.870 3.681 3.507 3.344 3.190 3.046 2.912

4.487 4.242 4.016 3.808 3.614 3.436 3.270 3.115 2.971 2.837

3.155 3.005 2.863 2.730

3.792 3.605 3.430 3.268 3.115 2.973 2.839 2.713

2.683

2.511

2.628

2.666

2.593

2.485

2.478

b10

b15

b17

g1

g2

g3

g4

g5

g6

g8

‘

Z

s

t

<1

<1

<1

10

1

2

<1

<1

<1

<1

3

1

<1

<1

410 260 180

5.384 5.036 4.726

3.799 3.611 3.437 3.274 3.121 2.979 2.847 2.720

4.182 3.944 3.725 3.523 3.336 3.162 3.001 2.852 2.712 2.582

2.492

2.348

67.84 56.212 47.835 41.042 36.671 31.423 27.826 24.840 22.315 20.201 18.358 16.693 15.297 14.081 12.976 11.944 11.069 10.293 9.583

67.25 56.813 47.325 40.542 35.200 30.942 27.375 24.339 21.864 19.73 17.86 16.304 14.940 13.719 12.620 11.608 10.732 9.959 9.253

6.754 6.297 5.875 5.497 5.151 4.387

8.363 7.836 7.356 6.918 6.517 6.150

8.042 7.517 7.040 6.606 6.210 5.847

2.026 2.778 2.639 2.511 2.391

4.288 4.045 3.822 3.616 3.426 3.249 3.085 2.932 2.790 2.657

5.503 5.217 4.952 4.707 4.480 4.269 4.071 3.888 3.716 3.557

5.204 4.922 4.660 4.418 4.193 3.983 3.789 3.607 3.438 3.280

2.174

2.417

3.267

2.994

6.045 5.644 5.283 4.953 4.654 4.380 3.897 3.685 3.489 3.307 3.137 2.980 2.835 2.695 2.567 2.447 2.237

Copyright © 2002 Marcel Dekker, Inc.

2.233

Table 2

Continued

Line Approximate intensity Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

a

a1

a2

b1

b2

b3

b4

b5

b6

b7

b9

110

100

10

50

20

6

4

1

<1

<1

<1

2.776 2.665 2.561 2.463 2.370 2.283 2.199 2.120 2.046 1.976 1.909 1.845 1.785 1.726 1.672 1.619 1.569 1.522 1.476 1.433 1.391 1.352 1.313 1.277 1.242 1.207 1.175 1.144 1.114

2.785 2.674 2.570 2.473 2.382

2.404 2.303 2.208 2.119 2.035

2.516 2.410 2.311 2.216 2.126

2.555 2.449 2.349 2.255 2.166

2.482 2.379 2.282 2.190 2.103

2.382 2.275 2.180 2.091 2.009

2.376 2.282 2.188 2.100 2.016

2.210 2.131 2.057 1.986 1.920 1.856 1.796 1.738 1.682 1.630 1.580 1.533 1.487 1.444 1.402 1.363 1.325 1.288 1.253 1.218 1.186 1.155 1.126

2.567 2.458 2.356 2.259 2.166 2.081 1.998 1.920 1.847 1.777 1.710 1.647 1.587 1.530 1.476 1.424 1.374 1.327 1.282 1.238 1.197 1.158 1.120 1.083 1.049 1.015 0.982 0.952 0.921

1.882 1.812 1.746 1.682 1.623 1.567 1.514 1.463 1.416 1.370 1.327 1.285 1.245 1.206 1.169 1.135 1.102 1.070 1.040 1.010 0.983 0.955 0.929

1.962 1.887 1.815 1.747 1.681 1.619 1.561 1.505 1.452 1.402 1.353 1.307 1.263 1.220 1.179 1.141 1.104 1.068 1.034 1.001 0.969 0.939 0.908

2.000 1.926 1.853 1.785 1.720 1.658 1.601 1.544 1.491 1.441 1.392 1.346 1.302 1.260 1.218 1.179 1.142 1.106 1.072 1.039 1.007 0.977 0.948

1.779

1.856 1.788 1.723 1.659 1.599

1.862 1.792

1.494

1.485

1.387 1.342 1.298 1.256 1.215 1.177 1.140 1.106 1.072 1.040 1.010 0.981 0.953 0.926 0.900

1.946 1.875 1.807 1.742 1.681 1.622 1.567 1.515 1.466 1.419 1.374 1.331 1.290 1.252 1.213 1.179 1.143 1.111 1.080 1.050 1.021 0.993 0.967

1.395 1.350 1.306 1.264 1.224 1.186 1.149 1.115 1.082 1.050 1.019 0.990 0.962 0.935

1.384 1.336 1.291 1.247 1.204 1.165 1.126 1.090 1.054 1.021 0.986 0.957 0.927 0.898

1.030 1.005

1.017

0.840 0.814

0.858 0.836

0.803

0.841

0.807

0.871

0.817

0.769

0.956 0.933 0.911 0.890 0.868 0.849

0.968 0.945 0.923 0.901 0.880 0.860

0.766 0.742 0.720 0.698 0.678 0.658

0.794 0.774 0.755 0.735 0.719 0.701

0.755 0.732 0.710

0.793 0.770 0.748

0.765 0.746 0.726

0.828 0.803 0.789

0.775 0.755 0.736

0.723 0.701 0.681

0.669

0.707

0.691

Source: From Clark, 1963.

Copyright © 2002 Marcel Dekker, Inc.

1.577

b10

b15

b17

g1

g2

g3

g4

g5

g6

g8

‘

Z

s

t

<1

<1

<1

10

1

2

<1

<1

<1

<1

3

1

<1

<1

2.387 2.290 2.195 2.107 2.023

2.442 2.141 2.048 1.961 1.878

2.138 2.046 1.960 1.879 1.801

2.134 2.041 1.955 1.874 1.797

2.075 1.983 1.899 1.819 1.745

1.870 1.800 1.731 1.667

1.726 1.657 1.592 1.530 1.473 1.417 1.364 1.316 1.268 1.222 1.179 1.138 1.098 1.061 1.025 0.991 0.958 0.927 0.897 0.868 0.840 0.814 0.786

1.659 1.597 1.534 1.477 1.423 1.371 1.321 1.274 1.228 1.185 1.144 1.105 1.068 1.032 0.998 0.966 0.934 0.905 0.876 0.848 0.822 0.796 0.771

1.655 1.591 1.529 1.471 1.417 1.364 1.315 1.268 1.222 1.179 1.138 1.099 1.062 1.026 0.992 0.959 0.928 0.898 0.869 0.842 0.815 0.790 0.764

1.606 1.544 1.485 1.427 1.374 1.323 1.276

1.494 1.392 1.343 1.299 1.254 1.212 1.172 1.133 1.097 1.062 1.028 0.996 0.964 0.934 0.905

1.372 1.328 1.437 1.287 1.247 1.339 1.208 1.293 1.171 1.137 1.166 1.072 1.128 1.041 1.090 1.012 1.056 0.984 1.022 0.957 0.989 0.931

1.185 1.143 1.103 1.065 1.028 0.993 0.959 0.928 0.897 0.867 0.839 0.812 0.761

2.309 2.222 3.135 2.862 2.205 3.006 2.740 2.110 2.023 2.892 2.620 2.020 1.936 2.784 2.512 1.935 1.855 2.675 2.409

1.708

1.518 1.462 1.406 1.355 1.307 1.260 1.215 1.173 1.132 1.094 1.057 1.022 0.988 0.956 0.925 0.895 0.867 0.840

1.243 1.198 1.155 1.114 1.074 1.037 1.001 0.967 0.934 0.903 0.873 0.845 0.817 0.791 0.765

2.482 1.632 2.395 2.312 2.234 2.158 2.086 2.019 1.955 1.250 1.894 1.204 1.836 1.161 1.782 1.120 1.728 1.081 1.678 1.044 1.630 1.008 1.585 0.974 1.541 0.941 1.499 0.010 1.460 0.880 1.422 0.852 1.385 0.824 1.350 0.799 1.317 1.283

2.218 2.049 1.898 1.826 1.757 1.695 1.635 1.478 1.523 1.471 1.421 1.374 1.328 1.285 1.243 1.202 1.164 1.127 1.092 1.058

1.831 1.776 1.663 1.723 1.612 1.672

1.352 1.414 1.279 1.342 1.244 1.308 1.210

0.716 0.776 0.838 0.844 0.694 0.682 0.675 0.649 0.717 0.673 0.680 1.167 0.908 0.730 0.708 0.687

0.653 0.642 0.635 0.611 0.675 0.632 0.640 1.115 0.855 1.011 1.080 0.634 0.624 0.617 0.594 0.655 0.613 1.091 0.830 0.615 0.605 0.598 0.577 0.635 0.595 0.601 1.067 0.806 0.904 1.035 0.597 0.579 0.562

Copyright © 2002 Marcel Dekker, Inc.

Table 3

M Series Diagram Lines (A˚)

Line K Cu Ru Rh Pd Ag Cd In Sn Sb Te Ba La Ce Pr Nd Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Th Pa U

a1 19 29 44 45 46 47 48 49 50 51 52 56 57 58 59 60 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 90 91 92

a2

b

g

l 680 170

26.85 25.00 21.80 20.46

14.88 14.06

14.51 13.78

12.675 Band Band Band Band Band Band Band 8.460

Band Band 10.744 10.253 9.792 9.364 8.965 8.593 8.246 7.909 7.600 7.304 7.022 6.756 6.504 6.267 6.037 5.828 5.623 5.452 5.250 5.075 4.909 3.942 3.827 3.715

8.139

8.155 7.840

7.539 7.251 6.983 6.528

7.546 7.258 6.990 6.490

6.261 6.046 5.840

6.275 6.057 5.854 5.666

5.461 5.285 5.118 4.138 4.022 3.910

Source: From Clark, 1963.

Copyright © 2002 Marcel Dekker, Inc.

5.472 5.299 5.129 4.151 4.035 3.924

17.94 16.92 15.93 12.700 12.064 11.534 10.997 10.504 9.599 9.211 8.844 8.485 8.144 7.865 7.545

18.38

14.22 13.57 12.98 12.43 11.86 11.37

7.023 6.761 6.543 6.312 6.088 5.887 5.681 5.501 5.320 5.145

10.48 10.07 9.69 9.32 8.96 8.63

4.825 4.674 4.531 3.679 3.577 3.480

6.97 6.74 6.52 5.24 5.08 4.95

8.02 7.74 7.47

Table 4 Atomic number 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Energies of Principal K and L X-ray Emission Lines (keV) Element Li Be B C N C F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn

Kb2

8.328 8.976 9.657 10.365 11.100 11.863 12.651 13.465 14.313 15.184 16.083 17.011 17.969 18.951 19.964 21.012 22.072 23.169 24.297 25.454 26.641 27.859 29.106

Copyright © 2002 Marcel Dekker, Inc.

Kb1

1.067 1.297 1.553 1.832 2.136 2.464 2.815 3.192 3.589 4.012 4.460 4.931 5.427 5.946 6.490 7.057 7.647 8.264 8.904 9.571 10.263 10.981 11.725 12.495 13.290 14.112 14.960 15.834 16.736 17.666 18.621 19.607 20.585 21.655 22.721 23.816 24.942 26.093 27.274 28.483

Ka1

Ka2

0.052 0.110 0.185 0.282 0.392 0.523 0.677 0.851 1.041 1.254 1.487 1.486 1.40 1.739 2.015 2.014 2.309 2.306 2.622 2.621 2.957 2.955 3.313 3.310 3.691 3.688 4.090 4.085 4.510 4.504 4.952 4.944 5.414 5.405 5.898 5.887 6.403 6.390 6.930 6.915 7.477 7.460 8.047 8.027 8.638 8.615 9.251 9.234 9.885 9.854 10.543 10.507 11.221 11.181 11.923 11.877 12.648 12.597 13.394 13.335 14.164 14.097 14.957 14.882 15.774 15.690 16.614 16.520 17.478 17.373 18.410 18.328 19.278 19.149 20.214 20.072 21.175 21.018 22.162 21.988 23.172 22.982 24.207 24.000 25.270 25.042

Lg1

2.302 2.462 2.623 2.792 2.964 3.144 3.328 3.519 3.716 3.920 4.131

Lb2

Lb1

2.219 2.367 2.518 2.674 2.836 3.001 3.172 3.348 3.528 3.713 3.904

0.344 0.399 0.458 0.519 0.581 0.647 0.717 0.790 0.866 0.943 1.032 1.122 1.216 1.517 1.419 1.526 1.638 1.752 1.872 1.996 2.124 2.257 2.395 2.538 2.683 2.834 2.990 3.151 3.316 3.487 3.662

La1

La2

0.341 0.395 0.492 0.510 0.571 0.636 0.704 0.775 0.849 0.928 1.009 1.096 1.166 1.282 1.379 1.480 1.587 1.694 1.692 1.806 1.805 1.922 1.920 2.042 2.040 2.166 2.163 2.293 2.290 2.424 2.420 2.558 2.554 2.696 2.692 2.838 2.833 2.994 2.978 3.133 3.127 3.287 3.279 3.444 3.436

Table 4

Continued

Atomic number

Element

Kb2

51 Sb 30.387 52 To 31.698 53 I 33.016 54 Xe 34.446 55 Cs 35.819 56 Ba 37.255 57 La 38.728 58 Ce 40.231 59 Pr 41.772 60 Nd 43.298 61 Pm 44.955 62 Sm 46.553 63 Eu 48.241 64 Gd 49.961 65 Tb 51.737 66 Dy 53.491 67 Ho 55.292 68 Er 57.088 69 Tm 58.969 70 Yb 60.959 71 Lu 62.946 72 Hf 64.936 73 Ta 66.999 74 W 69.090 75 Re 71.220 76 Os 73.393 77 Ir 75.605 78 Pt 77.866 79 Au 80.165 80 Hg 82.526 81 Tl 84.904 82 Pb 87.343 83 Bi 89.833 84 Po 92.386 85 At 94.976 86 Rn 97.616 87 Fr 100.305 88 Ra 103.048 89 Ac 105.838 90 Th 108.671 91 Pa 111.575 92 U 114.549 93 Np 117.533 94 Pu 120.592 95 Am 123.706 96 Cm 126.875 97 Bk 130.101 98 Cf 133.383 99 Es 136.724 100 Fm 140.122 Copyright © 2002 Marcel Dekker, Inc.

Kb1

Ka1

Ka2

Lg1

Lb2

Lb1

La1

La2

29.723 30.993 32.292 33.644 34.984 35.376 37.799 39.255 40.746 42.269 43.945 45.400 47.027 48.718 50.391 52.178 53.934 55.690 57.576 59.352 61.282 63.209 65.210 67.233 69.298 71.404 73.549 75.736 77.968 80.258 82.558 94.922 87.335 89.809 92.319 94.877 97.483 100.136 102.846 105.592 108.408 111.289 114.181 117.146 120.163 123.235 126.362 129.544 132.781 136.075

26.357 27.471 28.610 29.802 30.970 32.191 33.440 34.717 36.023 37.359 38.649 40.124 41.529 42.983 44.470 45.985 47.528 49.099 50.730 52.360 54.063 55.757 57.524 59.310 61.131 62.991 64.886 66.820 68.794 70.821 72.860 74.957 77.097 79.296 81.525 83.800 86.119 88.485 90.894 93.334 95.851 98.428 101.005 103.653 106.351 109.098 111.896 114.745 118.646 120.598

26.109 27.200 28.315 29.485 30.623 31.815 33.033 34.276 35.548 36.845 38.160 39.523 40.877 42.280 43.737 45.193 46.686 48.205 49.762 51.326 52.959 54.579 56.270 57.973 59.707 61.477 63.278 65.111 66.980 68.894 70.320 72.794 74.805 76.868 78.956 81.080 83.243 85.446 87.681 89.942 92.271 94.648 97.023 99.457 101.932 104.448 107.023 109.603 112.244 114.926

4.347 4.570 4.800 5.036 5.280 5.531 5.789 6.052 6.322 6.602 6.891 7.180 7.478 7.788 8.104 8.418 8.748 9.089 9.424 9.779 10.142 10.514 10.892 11.283 11.684 12.094 12.509 12.939 13.379 13.828 14.288 14.762 15.244 15.740 16.248 16.768 17.301 17.845 l8.405 18.977 19.559 20.163 20.774 21.401 22.042 22.699 23.370 24.056 24.758 25.475

4.100 4.301 4.507 4.720 4.936 5.156 5.384 5.613 5.850 6.090 6.336 6.587 6.842 7.102 7.368 7.638 7.912 8.188 8.472 8.758 9.048 9.346 9.649 9.959 10.273 10.596 10.918 11.249 11.582 11.923 12.268 12.620 12.977 13.338 13.705 14.077 14.459 14.839 l5.227 15.620 16.022 16.425 16.837 17.254 17.667 18.106 18.540 18.980 19.426 19.879

3.543 4.029 4.220 4.422 4.620 4.828 5.043 5.262 5.489 5.722 5.956 6.206 6.456 6.714 6.979 7.249 7.528 7.810 8.103 8.401 8.709 9.021 9.341 9.670 10.008 10.354 10.706 11.069 11.439 11.823 12.210 12.611 13.021 13.441 13.873 14.316 14.770 15.233 15.712 16.200 16.700 17.218 17.740 18.278 18.829 19.393 19.971 20.562 21.l66 21.785

3.605 3.769 3.937 4.111 4.286 4.467 4.651 4.840 5.034 5.230 5.431 5.636 5.846 6.039 6.275 6.495 6.720 6.948 7.181 7.414 7.654 7.898 8.145 8.396 8.651 8.910 9.173 9.441 9.711 9.987 10.266 10.549 10.836 11.128 11.424 11.724 12.029 12.338 12.650 12.966 13.291 13.613 13.945 14.279 14.618 14.961 15.309 15.661 16.018 16.379

3.595 3.758 3.926 4.098 4.272 4.451 4.635 4.823 5.014 5.208 5.408 5.609 5.816 6.027 6.241 6.457 6.680 6.904 7.135 7.367 7.604 7.843 8.087 8.333 8.584 8.840 9.098 9.360 9.625 9.896 10.170 10.448 10.729 11.014 11.304 11.597 11.894 12.194 12.499 12.808 13.120 13.438 12.758 14.082 14.411 14.743 15.079 15.420 15.764 16.113

APPENDIX III: RADIATIVE TRANSITION PROBABILITIES Radiative Transition Probabilities for K X-ray Lines

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100

Ca Ti Cr Fe Ni Zn Ge Sc Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu Cm Cf Em

Ka2 =Ka1 0.505 0.505 0.506 0.506 0.507 0.509 0.511 0.513 0.515 0.518 0.520 0.523 0.526 0.528 0.531 0.533 0.536 0.537 0.542 0.545 0.549 0.551 0.556 0.560 0.565 0.568 0.572 0.576 0.580 0.585 0.590 0.595 0.600 0.605 0.612 0.619 0.624 0.631 0.638 0.646 0.652

e e

Kb1 ¼ KM2 þ KM3 þ KM4;5 : Kb2 ¼ KN2;3 þ KO2;3 : c Kb ¼ Kb1 þ Kb2 : d Ka ¼ Ka1 þ Ka2 : Source: From West, 1982–83. e

b

e

a

Copyright © 2002 Marcel Dekker, Inc.

e

Atomic number Element

Kb3 =Kb1

ðKb1 þ Kb3 Þ=Ka1

Kb1 a =Ka1

0.516 0.517 0.518 0.519 0.521 0.522 0.523 0.525 0.526 0.527 0.529 0.531 0.532 0.534 0.535 0.537 0.539 0.541 0.542 0.544 0.546 0.548 0.550 0.552 0.554 0.556

0.116 0.137 0.155 0.172 0.189 0.202 0.215 0.225 0.235 0.244 0.251 0.258 0.264 0.270 0.275 0.280 0.285 0.290 0.294 0.298 0.303 0.307 0.310 0.314 0.317 0.320 0.324 0.326 0.330 0.333 0.336 0.339 0.342 0.345 0.348 0.351 0.354 0.356 0.359 0.362 0.364

0.116 0.137 0.156 0.171 0.187 0.202 0.215 0.225 0.235 0.244 0.252 0.259 0.265 0.271 0.277 0.282 0.287 0.292 0.297 0.301 0.306 0.311 0.314 0.318 0.322 0.325 0.329 0.332 0.336 0.339 0.343 0.346 0.350 0.353 0.356 0.360 0.363 0.366 0.370 0.374 0.377

e

Table 1

Kb2 b =Ka1

Kbc =Kad

0.006 0.013 0.022 0.034 0.043 0.048 0.051 0.054 0.056 0.060 0.064 0.670 0.076 0.082 0.085 0.088 0.089 0.090 0.090 0.091 0.092 0.094 0.097 0.100 0.103 0.106 0.110 0.113 0.118 0.123 0.125 0.130 0.134 0.138

0.069 0.095 0.114 0.128 0.133 0.137 0.142 0.153 0.164 0.175 0.185 0.193 0.201 0.209 0.216 0.222 0.226 0.232 0.240 0.244 0.247 0.250 0.253 0.256 0.259 0.261 0.263 0.267 0.270 0.274 0.277 0.282 0.285 0.288 0.291 0.295 0.299 0.301 0.305 0.309 0.312

Table 2 Atomic number 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

Radiative Transition Probabilities for L1 X-ray Lines Normalized to Lb3 ¼ 100 Element

Lb3

Lb4

Lg3

Lg2

Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

— — — 71.0 64.9 61.9 61.0 61.4 62.6 64.1 65.5 66.7 67.5 69.7 68.0 67.8 67.6 67.5 67.6 68.3 69.6 71.7 74.8 78.8 83.9 89.8 95.0 102.0 110.0 120.0

10.8 14.7 18.1 21.0 23.4 25.5 27.2 28.5 29.5 30.3 30.8 31.2 31.4 31.5 31.5 31.4 31.4 31.4 31.4 31.6 31.9 32.3 33.0 34.0 35.2 36.8 38.7 41.0 43.8 47.5

— — — — — — — — — — — — — — — — — 18.5 20.0 21.5 23.2 25.0 26.9 28.8 30.9 33.1 35.3 37.7 40.2 44.0

Copyright © 2002 Marcel Dekker, Inc.

Table 3 Atomic number 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

L2 X-ray Lines Normalized to Lb1 ¼ 100 Element

Lb1

LZ

Lg1

Lg6

Zn Ge Se Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sn Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

12.3 10.4 8.75 7.40 6.25 5.25 4.35 3.63 3.0 2.6 2.35 2.25 2.2 2.16 2.12 2.10 2.10 2.10 2.10 2.10 2.12 2.13 2.16 2.20 2.23 2.28 2.33 2.40 2.45 2.50 2.60 2.80 2.30

— — — — — 0.91 6.71 10.6 13.1 14.5 15.3 15.4 15.6 15.6 15.7 15.8 15.9 16.1 16.5 17.0 17.6 18.4 19.2 20.1 20.9 21.7 22.3 22.8 23.2 23.4 23.7 24.0 24.2

— — — — — — — — — — — — — — — — — — — — — — 0.375 1.73 2.42 2.98 3.45 3.88 4.29 4.74 5.25 5.88 6.65

Copyright © 2002 Marcel Dekker, Inc.

Table 4

L3 X-ray Lines Normalized to La1 ¼ 100

Atomic number 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94

Element Ti Cr Fe Ni Zn Ge Se Kr Sr Zr Mo Ru Pd Cd Sn Te Xe Ba Ce Nd Sm Gd Dy Er Yb Hf W Os Pt Hg Pb Po Rn Ra Th U Pu

Source: From West, 1982–83.

Copyright © 2002 Marcel Dekker, Inc.

La1

Lb2,15

La2

Lb5

Lb6

Ll

100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

— — — — — — — — — 2.43 6.40 9.55 12.1 13.9 15.4 16.4 17.2 17.8 18.2 18.5 18.8 19.2 19.6 20.0 20.5 21.2 21.9 22.7 23.5 24.4 25.3 26.2 26.8 27.3 27.5 27.5 27.0

— — — — — — — — — — 12.5 12.2 12.1 11.9 11.7 11.5 11.3 11.2 11.1 11.1 11.0 11.1 11.1 11.2 11.2 11.3 11.3 11.4 11.4 11.5 11.5 11.4 11.4 11.3 11.1 11.0 10.5

— — — — — — — — — — — — — — — — — — — — — — — — — — 0.242 0.873 1.74 2.62 3.24 3.85 4.28 4.69 4.94 5.20 5.40

— — — — — — — — — — — — — — — — — — — — — — — — — 1.15 1.28 1.38 1.46 1.55 1.59 1.65 1.70 1.75 1.80 1.85 1.89

40.37 26.13 15.35 10.29 7.56 5.96 4.98 4.36 3.98 3.75 3.65 3.58 3.55 3.56 5.59 3.62 3.67 3.73 3.79 3.86 3.92 3.99 4.07 4.15 4.23 4.32 4.42 4.53 4.65 4.78 4.93 5.09 5.27 5.46 5.69 5.93 6.18

APPENDIX IV: NATURAL WIDTHS OF K AND L LEVELS AND Ka X-RAY LINES (FWHM), IN eV Level Element

X-ray line

K

L1

L2

L3

Ka1

Ka2

(0.000) (0.000) 0.00l 0.004 0.015 0.032 0.054 0.083 0.126 0.152

(0.000) (0.000) 0.001 0.004 0.014 0.033 0.054 0.087 0.128 0.156

0.24 0.30 0.36 0.43 0.49 0.57 0.65 0.72 0.81 0.89

0.24 0.30 0.36 0.43 0.49 0.56 0.64 0.72 0.80 0.89

10 11 12 13 14 15 16 17 18 19

Ne Na Mg Al Si P S Cl Ar K

0.24 0.30 0.36 0.42 0.48 0.53 0.59 0.64 0.68 0.74

( < 0.1) (0.2) 0.41 0.73 1.03 1.26 1.49 1.58 1.63 1.92

20 21 22 23 24 25 26 27 28 29

Ca Sc Ti V Cr Mn Fe Co Ni Cu

0.81 0.86 0.94 1.01 1.08 1.16 1.25 1.33 1.44 1.55

2.07 2.21 2.34 2.41 2.54 2.62 2.76 2.79 2.89 3.06

0.17 0.19 0.24 0.26 0.29 0.34 0.37 0.43 0.52 0.62

0.17 0.19 0.22 0.24 0.27 0.32 0.36 0.43 0.48 0.56

0.98 1.05 1.16 1.26 1.35 1.48 1.61 1.76 1.94 2.11

0.98 1.06 1.18 1.28 1.37 1.50 1.62 1.76 1.96 2.17

30 31 32 33 34 35 36 37 38 39

Zn Ga Ge As Se Br Kr Rb Sr Y

1.67 1.82 1.96 2.14 2.33 2.52 2.75 2.99 3.25 3.52

3.28 3.38 3.53 3.79 3.94 4.11 4.28 4.44 4.67 4.71

0.72 0.83 0.95 1.03 1.13 1.21 1.31 1.43 1.54 1.65

0.65 0.76 0.82 0.94 1.00 1.08 1.17 1.27 1.39 1.50

2.32 2.59 2.78 3.08 3.33 3.60 3.92 4.26 4.63 5.02

2.39 2.66 2.92 3.17 3.46 3.73 4.06 4.92 4.79 5.18

40 41 42 43 44 45 46 47 48 49

Zr Nb Mo Tc Ru Rh Pd Ag Cd In

3.84 4.14 4.52 4.91 5.33 5.77 6.24 6.75 7.28 7.91

4.78 3.94 4.25 4.36 4.58 4.73 4.93 4.88 4.87 5.00

1.78 1.87 1.97 2.08 2.23 2.35 2.43 2.57 2.62 2.72

1.57 1.66 1.78 1.91 2.00 2.13 2.25 2.40 2.50 2.65

5.40 5.80 6.31 6.82 7.33 7.90 8.49 9.16 9.79 10.56

5.62 6.01 6.49 6.99 7.56 8.12 8.67 9.32 9.91 10.63

50 51 52 53

Sn Sb Te I

8.49 9.16 9.89 10.6

2.97 3.13 3.32 3.46

2.84 3.00 3.12 3.25

2.75 2.87 2.95 3.08

11.2 12.0 12.8 13.7

11.3 12.2 13.0 13.8

Copyright © 2002 Marcel Dekker, Inc.

Appendix IV

Continued Level

Element

X-ray line

K

L1

L2

L3

Ka1

Ka2

54 55 56 57 58 59

Xe Cs Ba La Ce Pr

11.4 12.3 13.2 14.1 15.1 16.2

3.64 3.78 3.92 4.06 4.21 4.34

3.40 3.51 3.57 3.68 3.80 3.89

3.13 3.25 3.32 3.41 3.48 3.60

14.6 15.6 16.5 17.6 I8.6 19.8

14.8 15.8 16.8 17.8 18.9 20.1

60 61 62 63 64 65 66 67 68 69

Nd Pm Sm Eu Gd Tb Dy Ho Er Tm

17.3 18.5 19.7 21.0 22.3 23.8 25.2 26.8 28.4 30.1

4.52 4.67 4.80 4.91 5.05 5.19 5.25 5.33 5.43 5.47

3.97 4.06 4.15 4.23 4.32 4.43 4.55 4.66 4.73 4.79

3.65 3.75 3.86 3.91 4.01 4.12 4.17 4.26 4.35 4.48

20.9 22.2 23.6 24.9 26.4 27.9 29.4 31.1 32.7 34.6

21.3 22.5 23.8 25.2 26.7 28.2 29.8 31.5 33.1 34.9

70 71 72 73 74 75 76 77 78 79

Yb Lu Hf Ta W Re Os Ir Pt Au

31.9 33.7 35.7 37.7 39.9 42.1 44.4 46.8 49.3 52.0

5.53 5.54 5.63 5.58 5.61 6.18 7.25 8.30 9.39 10.5

4.82 4.92 5.02 5.15 5.33 5.48 5.59 5.69 5.86 6.00

4.60 4.68 4.80 4.88 4.98 5.04 5.16 5.25 5.31 5.41

36.5 38.4 40.5 42.6 44.9 47.2 49.6 52.1 54.6 57.4

36.7 38.7 40.7 42.9 45.2 47.6 50.0 52.5 55.2 58.0

80 81 82 83 84 85 86 87 88 89

Hg Tl Pb Bi Po At Rn Fr Ra Ac

54.6 57.4 60.4 63.4 66.6 69.8 73.3 76.8 80.4 84.1

11.3 12.0 12.2 12.4 12.6 12.8 13.1 13.3 13.4 13.6

6.17 6.32 6.48 6.67 6.83 7.01 7.20 7.47 7.68 7.95

5.50 5.65 5.81 5.98 6.13 6.29 6.41 6.65 6.82 6.98

60.1 63.1 66.2 69.4 72.7 76.1 79.7 83.4 87.2 91.1

60.8 63.8 66.8 70.1 73.4 76.8 80.5 84.2 88.1 92.0

90 91 92 93 94 95 96 97 98 99

Th Pa U Np Pu Am Cm Bk Cf Es

88.0 91.9 96.1 100 105 109 114 119 124 129

13.7 14.3 14.0 14.0 13.5 13.3 13.6 13.8 14.0 14.3

8.18 8.75 9.32 9.91 10.5 10.9 11.4 11.8 12.2 12.7

7.13 7.33 7.43 7.59 7.82 8.04 8.26 8.55 8.75 9.04

95.2 99.3 103.5 108 113 117 122 127 132 138

96.2 100.7 105.4 110 115 120 125 130 136 141

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX IV

Continued Level

Element 100 101 102 103 104 105 106 107 108 109 110

Fm Md No Lr

X-ray line

K

L1

L2

L3

Ka1

Ka2

135 140 145 150 156 162 168 174 181 187

14.4 14.8 15.1 15.9 16.2 16.5 16.8 17.0 17.2 17.6

13.1 13.6 14.0 14.4 14.9 15.5 15.9 16.4 17.0 17.5

9.33 9.61 9.90 10.1 10.5 10.8 11.2 11.6 12.0 12.3

144 150 155 161 167 173 179 185 193 200

148 153 159 165 171 177 184 190 198 205

193

18.1

18.1

12.7

206

211

Source: From Krause and Oliver, 1979.

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX V: WAVELENGTHS OF K SATELLITE LINES (A
) Atomic number 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Element Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag

aII

aI

a3

11.860 11.830 9.845 9.820 8.302 8.283 7.100 7.095 7.082 6.140 6.131 6.117 5.370 5.354 5.341 4.725 3.739 3.356 3.029 2.747 2.502

3.728 3.347 3.021 2.740 2.496 2.282 2.095 1.930 1.784 1.653 1.536 1.431 1.138 1.250 1.173

Source: From Clark, 1963.

Copyright © 2002 Marcel Dekker, Inc.

a3I

a3II

a4

11.810

5.339

9.804 8.267 7.073 6.109 5.344 5.334

a4I

a5

a6

a7

b0

11.742 11.711 9.749 9.724 8.226 8.206 7.069 7.035 7.017 7.026 6.075 6.063

bI

6.816 6.838 4.415

3.721 3.340 3.015 2.733 2.491 2.279 2.093 1.928 1.782 1.651 1.535 1.430 1.335 1.250 1.172 1.101 1.036

3.718 3.338 3.013 2.731 2.489 2.277 2.091 1.926 1.780 1.650 1.533 1.428 1.334 1.248 1.170 1.100 1.035

3.724 3.343 3.018 2.737 2.493 2.280 2.094 1.929 1.783 1.652

3.717 3.337 3.013 2.731 2.490 2.278 2.091 1.927 1.781 1.650 1.534 1.429 1.335 1.249 1.171 1.101 1.036

0.8727

0.8646

0.8721

0.7836

0.7828 0.7432 0.7065

0.7832 0.7436 0.7070

3.496 3.133 2.819 2.551 2.320 2.118 1.783 1.645 1.522

1.061 0.9958 0.9367 0.8832 0.8835 0.7881 0.7457 0.7056 0.6706 0.6369 0.5766 0.5493

0.5833

0.5837 0.5007

3.101 2.789 2.522 2.291 2.090 1.914 1.760 1.623 1.502 1.394 1.296 1.209

b1x

bx

5.800 5.028

6.753 5.792 5.023

bV

bII

b IIIII

5.712 4.400 3.882 3.449 3.087

bIII

bIV

b5

b6

b7

b8

b9

b10

1.116 1.044 0.9786 0.9194

1.064 0.9996 0.9403

1.050 0.9854 0.9629

1.054 0.9889 0.9297

1.047 0.9228

0.8155 0.7699 0.7218 0.6890 0.6531

0.8365 0.7911 0.7492 0.7105 0.6741

0.8234 0.7778 0.7362 0.6973 0.6619 0.6288

0.8259 0.7802 0.7382 0.6993 0.6636 0.6303

0.8193 0.7736 0.7315 0.6924 0.6568 0.6235

0.6256

0.6519

0.5693 0.5427

0.5708 0.5439 0.5189 0.4954

0.5639 0.5374

0.5399

0.5607 0.5452

0.4902

0.4920

5.691

4.395 3.441 3.082 2.772 2.506 2.277 2.078

3.412 3.054 2.749 2.489 2.262 2.066 1.895 1.742 1.607 1.487 1.380 1.282

1.749 1.614 1.494 1.391 1.295 1.207 1.128 1.042 0.9770 0.9177 1.8745 0.8135 0.7681 0.7261 0.6870 0.6510

0.5589

3.404 3.048 2.744

0.5808

0.4943

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX VI: FLUORESCENCE YIELDS AND COSTER^KRONIG TRANSITION PROBABILITIES Table 1

K Shell Fluorescence Yield oK

Atomic number 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Element C N O Ne Na Mg Al Si P S Cl Ar K Sc Ca Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru

oK

Atomic number

Element

oK

0.0009 0.0015 0.0022 0.0100 0.020 0.030 0.040 0.055 0.070 0.090 0.105 0.125 0.140 0.165 0.190 0.220 0.240 0.26 0.285 0.32 0.345 0.375 0.41 0.435 0.47 0.50 0.53 0.565 0.60 0.635 0.665 0.685 0.71 0.72 0.755 0.77 0.785 0.80

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 92

Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb In Hf Ta W Re Os Ir Pd Au Hg Pb U

0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.875 0.88 0.89 0.895 0.90 0.905 0.91 0.915 0.92 0.925 0.93 0.93 0.935 0.94 0.94 0.945 0.945 0.95 0.95 0.95 0.955 0.955 0.96 0.96 0.96 0.96 0.965 0.965 0.965 0.97 0.97

Source: From Birks, 1971b and Fink, 1974.

Copyright © 2002 Marcel Dekker, Inc.

Table 2

Experimental L Subshell Fluorescence Yields oi

Atomic number

Element

o1

o2

54 56 65 67

Xe Ba Tb Ho

68

Er

70

Yb

71

Lu

72

Hf

73

Ta

74

W

75 76 77

Re Os Ir

78

Pt

79

Au

80

Hg

81

Tl

0.07  0.02

0.319  0.010 0.373  0.025

82

Pb

0.07  0.02 0.09  0.02

0.363  0.015

83

Bi

0.12  0.01 0.095  0.005

0.32  0.04 0.38  0.02

0.06 0.18

0.165  0.018 0.170  0.055 0.185  0.060 0.188  0.011

Copyright © 2002 Marcel Dekker, Inc.

0.25  0.02 0.257  0.013

0.331  0.021

0.39  0.03 0.319  0.010

o3 0.10  0.01 0.05  0.01 0.188  0.016 0.22  0.03 0.169  0.030 0.21  0.03 0.172  0.032 0.20  0.02 0.183  0.011 0.22  0.03 0.251  0.035 0.22  0.03 0.228  0.025 0.27  0.01 0.25  0.03 0.191 0.228  0.013 0.254  0.025 0.207 0.272  0.037 0.284  0.043 0.290  0.030 0.244 0.262  0.036 0.262 0.31  0.04 0.317  0.029 0.291  0.018 0.276 0.31  0.04 0.317  0.025 0.40  0.02 0.32  0.05 0.367  0.050 0.300  0.010 0.37  0.07 0.386  0.053 0.306  0.010 0.330  0.021 0.337 0.315  0.013 0.32 0.35  0.05 0.354  0.028 0.367 0.36 0.37  0.05 0.362  0.029 0.40  0.05

Table 2

Continued

Atomic number

Element

90

Th

91 92

Pa U

96

Cm

o1

0.28  0.06

o2

o3

0.552  0.032 0.55  0.02

0.340  0.018 0.42 0.517  0.042 0.46  0.05 0.44 0.500  0.040 0.515  0.034 0.63  0.02

f13

f23

Source: From Bambynek et al., 1972.

Table 3

Measured L Shell Coster–Kronig Yields

Atomic number

Element

f12 0.66  0.07 0.41  0.36

56 65 67 68 70 73

Ba Tb Ho Er Yb Ta

74 75 77 78 79

W Re Ir Pt Au

80

Hg

81

Tl

0.17  0.05 0.14  0.03

82

Pb

83

Bi

0.15  0.04 0.17  0.05 0.19  0.05

0.76  0.10 0.57  0.10 0.56  0.07 0.56  0.05 0.57  0.03 0.61  0.08 0.58  0.05

0.18  0.02

0.58  0.02

92 93

U Np

0.10  0.04

0.55  0.09

94

Pu

96

Cm

0.038  0.022

0.68  0.04

Source: From Bambynek et al., 1972.

Copyright © 2002 Marcel Dekker, Inc.

< 0.14

0.25  0.13

0.43  0.28

0.19 < 0.36 0.27  0.03 0.30  0.04 0.46  0.06 0.50  0.05 0.51  0.13 0.61  0.07

0.74  0.04

0.066  0.014 0.205  0.034 0.255  0.025 0.142  0.009 0.148  0.010 0.20  0.04

0.22 0.22  0.04 0.08  0.02 0.188  0.010 0.25  0.13 0.169  0.010 0.159  0.013 0.164  0.016 0.156  0.010 þ0.14 0.06 7 0.06 0.164 0.23  0.12 þ0.05 0.02 70.02 0.22  0.08 0.24  0.08 0.188  0.019

Table 4

Theoretical L Subshell Fluorescence Yields oi and Coster–Kronig Yields fij a

Atomic number

Element

o1

13 14 15 16 17 18 19 20 22 24 26

Al Si P S Cl Ar K Ca Ti Cr Fe

3.05-6 9.77-6 2.12-5 3.63-5 5.60-5 8.58-5 1.15-4 1.56-4 2.80-4 2.97-4 3.84-4

28 29 30 32 33 34 35 36

Ni Cu Zn Ge As Se Br Kr

4.63-4

37 38 40

Rb Sr Zr

42

Mo

44 47

Ru Ag

50

Sn

51 54 56 60

Sb Xe Ba Nd

65 67

Tb Ho

70 74

Yb W

79 80 83 85 90 93

Au Hg Bi At Th Np

a

5.23-4 7.70-4 1.40-3 1.30-3 1.85-3 2.19-3 1.32-2 3.00-3 3.97-3 3.96-3 5.75-3 6.34-3 7.74-3 1.02-2 1.01-2 1.30-2 1.30-2 3.11-2 5.84-2 4.46-2 7.46-2 6.00-2 0.112 0.094 0.112 0.115 0.138 0.105 0.098 0.120 0.129 0.197

o2

1.43-3 2.69-3 3.57-3 7.72-3 8.85-3 9.94-3 1.09-2 2.20-2 1.19-2

o3 2.40-3 1.08-3 4.1-4 2.9-4 2.3-4 1.9-4 2.1-4 2.1-4 1.18-3 3.29-3 5.59-3 1.49-3 8.02-3 3.83-3 1.08-2 1.44-2 9.74-3 1.78-2

f12

f13

0.629 0.636 0.652

0.325

0.622

0.322 0.266 0.282 0.302

0.624 0.671 0.547 0.616

2.36-2 1.23-2

0.230 0.225

0.686 0.585

2.43-2 2.95-2 2.01-2 3.73-2 2.59-2 4.50-2 6.02-2 4.49-2 7.37-2

0.249 0.236 0.271 0.166 0.048 0.057 0.052 0.064 0.052 0.072 0.164 0.179 0.168 0.207 0.165

0.646 0.648 0.522 0.689 0.692 0.779 0.786 0.695 0.784 0.693 0.316 0.274 0.336 0.303 0.332

0.202 0.178 0.180 0.195 0.160 0.083 0.101 0.069 0.082 0.069

0.309 0.317 0.316 0.332 0.324 0.644 0.618 0.656 0.612 0.575

6.33-2 9.70-2 8.99-2 0.135 0.120 0.160 0.201

0.287 0.271 0.357 0.352 0.417 0.422 0.529 0.460

0.268 0.253 0.327 0.321 0.389 0.380 0.461 0.472

f23

0.982 0.975 0.971 0.968 0.964 0.965 0.962 0.955 0.313 0.317 0.302

2.24-2 2.94-2 1.89-2 3.50-2 2.45-2 4.18-2 5.47-2 4.30-2 6.56-2 5.67-2 6.16-2 9.12-2 9.07-2 0.133 0.120 0.166 0.203

f12 þ f13

Figures following a sign indicate powers of 10. For example, 3.05-6 means 3.0561076. Copyright 2002 Bambynek Marcel Dekker, Source:©From et al., Inc. 1972.

7.24-2 9.97-2 0.109 2.49-2 4.13-2 5.95-2 7.64-2 8.97-2 9.22-2 0.107 0.115 0.118 0.123 0.124 0.126 0.136 0.152 0.130 0.162 0.136 0.138 0.173 0.151 0.141 0.142 0.131 0.138

0.123 0.117 0.132 0.108 0.101 0.100 0.102 0.209

Table 5 Atomic number 76 79 79 82 82 83 83 83 92 93

96

Measured M Shell Fluorescence Yields and Coster–Kronig Probabilities Element Os Au Au Pb Pb Bi Bi Bi U Np


M o

oLMa

oLMb

0.013  0.0024

0.016  0.003

0.024  0.005

0.030  0.006

0.026  0.005

0.032  0.006

0.030  0.006

0.037  0.005

o1 þ f12o2

ni

oi

0.023  0.001 0.029  0.002 0.037  0.007 0.035  0.002 0.06

Cm

a

Corrected for a 20% contribution from double M shell vacancies. Uncorrected values. Source: From Bambynek et al., 1972. b

Copyright © 2002 Marcel Dekker, Inc.

þ 0.003 0.002 70.002

þ 0.0089 0.0075 70.0075

n1 ¼ 0.065  0.014 n2 ¼ 0.0080  0.029 n3 ¼ 0.062  0.005 n4 ¼ 0.065  0.012 n4 ¼ 0.065  0.012 n4,5 ¼ 0.081  0.016

o5 ¼ 0.06  0.012

þ 0.0051 n1 ¼ 0.068  0.023 o2 ¼ 0.0046 n2 ¼ 0.062  0.019 70.0046 n3 ¼ 0.080  0.006 n4,5 ¼ 0.075  0.012 o5 ¼ 0.075  0.012

Table 6

Adopted Values of Fluorescence and Coster-Kronig Yieldsa ok

Z 5 6 7 8 9

o1

o2

o3

f1

f12

f13

f23

B C N O F

1.7E-03 2.8E-03 5.2E-03 8.3E-03 0.013

10 11 12 13 14

Ne Na Mg Al Si

0.018 0.023 0.030 0.039 0.050

2.9E-05 2.6E-05 3.0E-05

1.2E-03 7.5E-04 3.7E-04

1.2E-03 7.5E-04 3.8E-04

0.962 0.965 0.959

0.32 0.32 0.32

0.64 0.64 0.64

15 16 17 18 19

P S Cl Ar K

0.063 0.078 0.097 0.118 0.140

3.9E-05 7.4E-05 1.2E-04 1.8E-04 2.4E-04

3.1E-04 2.6E-04 2.4E-04 2.2E-04 2.7E-04

3.1E-04 2.6E-04 2.4E-04 2.2E-04 2.7E-04

0.951 0.944 0.939 0.934 0.929

0.32 0.32 0.32 0.31 0.31

0.63 0.62 0.62 0.62 0.62

20 21 22 23 24

Ca Sc Ti V Cr

0.163 0.188 0.214 0.243 0.275

3.1E-04 3.9E-04 4.7E-04 5.8E-04 7.1E-04

3.3E-04 8.4E-04 1.5E-03 2.6E-03 3.7E-03

3.3E-04 8.4E-04 1.5E-03 2.6E-03 3.7E-03

0.920 0.912 0.902 0.894 0.885

0.31 0.31 0.31 0.31 0.31

0.61 0.60 0.59 0.58 0.57

25 26 27 28 29

Mn Fe Co Ni Cu

0.308 0.340 0.373 0.406 0.440

8.4E-04 1.0E-03 1.2E-03 1.4E-03 1.6E-03

5.0E-03 6.3E-03 7.7E-03 8.6E-03 0.010

5.0E-03 6.3E-03 7.7E-03 9.3E-03 0.011

0.877 0.868 0.856 0.847 0.839

0.30 0.30 0.30 0.30 0.30

0.58 0.57 0.56 0.55 0.54

0.028 0.028

30 31 32 33 34

Zn Ga Ge As Se

0.474 0.507 0.535 0.562 0.589

1.8E-03 2.1E-03 2.4E-03 2.8E-03 3.2E-03

0.011 0.012 0.013 0.014 0.016

0.012 0.013 0.015 0.016 0.018

0.831 0.822 0.815 0.809 0.804

0.29 0.29 0.28 0.28 0.28

0.54 0.53 0.53 0.53 0.52

0.026 0.032 0.050 0.063 0.076

35 36 37 38 39

Br Kr Rb Sb Y

0.618 0.643 0.667 0.690 0.710

3.6E-03 4.1E-03 4.6E-03 5.1E-03 5.9E-03

0.018 0.020 0.022 0.024 0.026

0.020 0.022 0.024 0.026 0.028

0.800 0.797 0.794 0.790 0.785

0.28 0.27 0.27 0.27 0.26

0.52 0.52 0.52 0.52 0.52

0.088 0.100 0.109 0.117 0.126

40 41 42 43 44

Zr Nb Mo Tc Ru

0.730 0.747 0.765 0.780 0.794

6.8E-03 9.4E-03 0.010 0.011 0.012

0.028 0.031 0.034 0.037 0.040

0.031 0.034 0.037 0.040 0.043

0.779 0.713 0.712 0.711 0.709

0.26 0.10 0.10 0.10 0.10

0.52 0.61 0.61 0.61 0.61

0.132 0.137 0.141 0.144 0.148

45 46 47 48 49

Rh Pd Ag Cd In

0.808 0.820 0.831 0.843 0.853

0.013 0.014 0.016 0.018 0.020

0.043 0.047 0.051 0.056 0.061

0.046 0.049 0.052 0.056 0.060

0.705 0.700 0.694 0.688 0.681

0.10 0.10 0.10 0.10 0.10

0.60 0.60 0.59 0.59 0.59

0.150 0.151 0.153 0.155 0.157

Copyright © 2002 Marcel Dekker, Inc.

Table 6

Continued

Z

ok

o1

o2

o3

f1

f12

f13

f23

50 51 52 53 54

Sn Sb Te I Xe

0.862 0.870 0.877 0.884 0.891

0.037 0.039 0.041 0.044 0.046

0.065 0.069 0.074 0.079 0.083

0.064 0.069 0.074 0.079 0.085

0.439 0.448 0.495 0.461 0.466

0.17 0.17 0.18 0.18 0.19

0.27 0.28 0.28 0.28 0.28

0.157 0.156 0.155 0.154 0.154

55 56 57 58 59

Cs Ba La Ce Pr

0.897 0.902 0.907 0.912 0.917

0.049 0.052 0.055 0.058 0.061

0.090 0.096 0.103 0.110 0.117

0.091 0.097 0.104 0.111 0.118

0.470 0.474 0.478 0.482 0.485

0.19 0.19 0.19 0.19 0.19

0.28 0.28 0.29 0.29 0.29

0.154 0.153 0.153 0.153 0.153

60 61 62 63 64

Nd Pm Sm Eu Gd

0.921 0.925 0.929 0.932 0.935

0.064 0.066 0.071 0.075 0.079

0.124 0.132 0.140 0.149 0.158

0.125 0.132 0.139 0.147 0.155

0.488 0.490 0.492 0.493 0.493

0.19 0.19 0.19 0.19 0.19

0.30 0.30 0.30 0.30 0.30

0.152 0.151 0.150 0.149 0.147

65 66 67 68 69

Tb Dy Ho Er Tm

0.938 0.941 0.944 0.947 0.949

0.083 0.089 0.044 0.100 0.106

0.167 0.178 0.189 0.200 0.211

0.164 0.174 0.182 0.192 0.201

0.493 0.492 0.490 0.487 0.483

0.19 0.19 0.19 0.19 0.19

0.30 0.30 0.30 0.30 0.29

0.145 0.143 0.142 0.140 0.139

70 71 72 73 74

Yb Lu Hf Ta W

0.951 0.953 0.955 0.957 0.958

0.112 0.120 0.128 0.137 0.147

0.222 0.234 0.246 0.258 0.270

0.210 0.220 0.231 0.243 0.255

0.478 0.472 0.465 0.457 0.447

0.19 0.19 0.18 0.18 0.17

0.29 0.28 0.28 0.28 0.28

0.138 0.136 0.135 0.134 0.133

75 76 77 78 79

Re Os Ir Pt Au

0.959 0.961 0.962 0.963 0.964

0.144 0.130 0.120 0.114 0.107

0.283 0.295 0.308 0.321 0.334

0.268 0.281 0.294 0.306 0.320

0.485 0.552 0.603 0.640 0.672

0.16 0.16 0.15 0.14 0.14

0.33 0.39 0.45 0.50 0.53

0.130 0.128 0.126 0.124 0.122

80 81 82 83 84

Hg Tl Pb Bi Po

0.965 0.966 0.967 0.968 0.968

0.107 0.107 0.112 0.117 0.122

0.347 0.360 0.373 0.387 0.401

0.333 0.347 0.360 0.373 0.386

0.690 0.696 0.696 0.694 0.689

0.13 0.13 0.12 0.11 0.11

0.56 0.57 0.58 0.58 0.58

0.120 0.118 0.116 0.113 0.111

85 86 87 88 89

At Rn Fr Ra Ac

0.969 0.969 0.970 0.970 0.971

0.128 0.134 0.139 0.146 0.153

0.415 0.429 0.443 0.456 0.468

0.399 0.411 0.424 0.437 0.450

0.685 0.682 0.677 0.672 0.664

0.10 0.10 0.10 0.09 0.09

0.59 0.58 0.58 0.58 0.58

0.111 0.110 0.109 0.108 0.108

90 91 92 93 94

Th Pa U Np Pu

0.971 0.972 0.972 0.973 0.973

0.161 0.162 0.176 0.187 0.205

0.479 0.472 0.467 0.466 0.464

0.463 0.476 0.489 0.502 0.514

0.660 0.664 0.652 0.642 0.605

0.09 0.08 0.08 0.07 0.05

0.57 0.58 0.57 0.57 0.56

0.108 0.139 0.167 0.192 0.198

Copyright © 2002 Marcel Dekker, Inc.

Table 6

Continued

Z

ok

o1

o2

o3

f1

f12

f13

f23

95 96 97 98 99

Am Cm Bk Cf Es

0.974 0.974 0.975 0.975 0.975

0.218 0.228 0.236 0.244 0.253

0.471 0.479 0.485 0.490 0.497

0.526 0.539 0.550 0.560 0.570

0.595 0.587 0.580 0.573 0.565

0.05 0.04 0.04 0.03 0.03

0.55 0.55 0.54 0.54 0.54

0.203 0.200 0.198 0.197 0.196

100 101 102 103 104

Fm Md No Lr

0.976 0.976 0.976 0.977 0.977

0.263 0.272 0.280 0.282 0.291

0.506 0.515 0.524 0.533 0.544

0.579 0.588 0.596 0.604 0.611

0.556 0.548 0.540 0.538 0.531

0.03 0.02 0.02 0.01 0.01

0.53 0.53 0.52 0.53 0.52

0.194 0.191 0.189 0.185 0.181

105 106 107 108 109

0.977 0.978 0.978 0.978 0.978

0.300 0.310 0.320 0.331 0.343

0.553 0.562 0.573 0.584 0.590

0.618 0.624 0.630 0.635 0.640

0.522 0.513 0.505 0.497 0.488

0.01

0.51 0.51 0.50 0.50 0.49

0.178 0.174 0.171 0.165 0.163

110

0.979

0.354

0.598

0.644

0.477

0.48

0.158

a Note: 1.7E-03 ¼ 1.761073; designation for L shell is omitted, read, for example, o1 ¼ oL1 ; f1 ¼ f12 þ f13 . Source: From Krause, 1979.

Copyright © 2002 Marcel Dekker, Inc.

Table 7 Z

Auger Yields for K, L1, L2 and L3 Levelsa aK

a1

a2

a3

Z

aK

a1

a2

a3

B C N O F

0.998 0.997 0.995 0.992 0.987

50 51 52 53 54

Sn Sb Te I Xe

0.138 0.130 0.123 0.116 0.109

0.524 0.513 0.504 0.495 0.488

0.778 0.775 0.771 0.767 0.763

0.936 0.931 0.926 0.921 0.915

10 11 12 13 14

Ne Na Mg Al Si

0.982 0.977 0.970 0.961 0.950

0.038 0.035 0.041

0.999 0.999 1.000

0.999 0.999 1.000

55 56 57 58 59

Cs Ba La Ce Pr

0.103 0.098 0.093 0.088 0.083

0.481 0.474 0.467 0.460 0.454

0.756 0.751 0.744 0.737 0.730

0.909 0.903 0.896 0.889 0.882

15 16 17 18 19

P S Cl Ag K

0.937 0.922 0.903 0.882 0.860

0.049 0.056 0.061 0.066 0.071

1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000

60 61 62 63 64

Nd Pm Sm Eu Gd

0.079 0.075 0.071 0.068 0.065

0.448 0.444 0.437 0.432 0.428

0.724 0.717 0.710 0.702 0.695

0.875 0.868 0.861 0.853 0.845

20 21 22 23 24

Ca Sc Ti V Cr

0.837 0.812 0.786 0.757 0.725

0.080 0.088 0.098 0.105 0.114

1.000 0.999 0.999 0.997 0.996

1.000 0.999 0.999 0.997 0.996

65 66 67 68 69

Tb Dy Ho Er Tm

0.062 0.059 0.056 0.053 0.051

0.424 0.419 0.416 0.413 0.411

0.688 0.679 0.669 0.660 0.650

0.836 0.826 0.818 0.808 0.799

25 26 27 28 29

Mn Fe Co Ni Cu

0.692 0.660 0.627 0.394 0.560

0.122 0.131 0.143 0.152 0.159

0.995 0.994 0.992 0.963 0.962

0.995 0.994 0.992 0.991 0.989

70 71 72 73 74

Yb Lu Hf Ta W

0.049 0.047 0.045 0.043 0.042

0.410 0.408 0.407 0.406 0.406

0.640 0.630 0.619 0.608 0.597

0.790 0.780 0.769 0.757 0.745

30 31 32 33 34

Zn Ga Ge As Se

0.526 0.493 0.465 0.438 0.411

0.167 0.176 0.183 0.188 0.193

0.963 0.956 0.937 0.923 0.908

0.988 0.987 0.985 0.984 0.982

75 76 77 78 79

Re Os Ir Pt Au

0.041 0.039 0.038 0.037 0.036

0.371 0.318 0.277 0.246 0.221

0.587 0.577 0.566 0.555 0.344

0.732 0.719 0.706 0.694 0.680

35 36 37 38 39

Br Kr Rb Sr Y

0.382 0.357 0.333 0.310 0.290

0.196 0.199 0.201 0.205 0.209

0.894 0.880 0.869 0.859 0.848

0.980 0.978 0.976 0.974 0.972

80 81 82 83 84

Hg Tl Pb Bi Po

0.035 0.034 0.033 0.032 0.032

0.203 0.197 0.192 0.189 0.189

0.533 0.522 0.511 0.500 0.488

0.667 0.653 0.640 0.627 0.614

40 41 42 43 44

Zr Nb Mo Tc Ru

0.270 0.253 0.235 0.220 0.206

0.214 0.278 0.278 0.278 0.279

0.840 0.832 0.825 0.819 0.812

0.969 0.966 0.963 0.960 0.957

85 86 87 88 89

At Rn Fr Ra Ac

0.031 0.031 0.030 0.030 0.029

0.187 0.184 0.184 0.182 0.183

0.474 0.461 0.448 0.436 0.424

0.601 0.589 0.576 0.563 0.590

45 46 47 48 49

Rh Pd Ag Cd In

0.192 0.180 0.169 0.157 0.147

0.282 0.286 0.290 0.294 0.299

0.807 0.802 0.796 0.789 0.782

0.954 0.951 0.948 0.944 0.940

90 91 92 93 94

Th Pa U Np Pu

0.029 0.028 0.028 0.027 0.027

0.179 0.174 0.172 0.171 0.190

0.413 0.389 0.366 0.342 0.338

0.537 0.524 0.511 0.498 0.486

5 6 7 8 9

Copyright © 2002 Marcel Dekker, Inc.

Table 7 Z

Continued aK

a1

a2

a3

Z

aK

a1

a2

a3

95 96 97 98 99

Am Cm Bk Cf Es

0.026 0.026 0.025 0.025 0.025

0.167 0.185 0.184 0.183 0.182

0.326 0.321 0.317 0.313 0.307

0.474 0.461 0.450 0.440 0.430

105 106 107 108 109

0.023 0.022 0.022 0.022 0.022

0.178 0.177 0.175 0.172 0.169

0.269 0.264 0.256 0.251 0.247

0.382 0.376 0.370 0.365 0.360

100 101 102 103 104

Fm Md No Lr

0.024 0.024 0.024 0.023 0.023

0.181 0.180 0.180 0.180 0.178

0.300 0.294 0.287 0.282 0.275

0.421 0.412 0.404 0.396 0.389

110

0.021

0.169

0.244

0.356

a Designation for L shell is omitted; read, for example, a1  aL1. Source: From Krause, 1979.

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX VII: COEFFICIENTS FOR CALCULATING THE PHOTOELECTRIC ABSORPTION CROSS SECTIONS
(BARNS=ATOM) VIA ln^ln REPRESENTATION Table 1

E > EK a

Atomic number

Element

Atomic weight

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh

1.008 4.003 6.940 9.012 10.811 12.010 14.008 16.000 19.000 20.183 22.997 24.320 26.970 28.086 30.975 32.066 35.457 39.944 39.102 40.080 44.960 47.900 50.942 51.996 54.940 55.850 58.933 58.690 63.540 65.380 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Copyright © 2002 Marcel Dekker, Inc.

A0 2.44964 6.06488 7.75370 9.04511 9.95057 1.06879 þ 1 1.12765 þ 1 1.17130 þ 1 1.20963 þ 1 1.24485 þ 1 1.26777 þ 1 1.28793 þ 1 1.31738 þ 1 1.32682 þ 1 1.33735 þ 1 1.37394 þ 1 1.36188 þ 1 1.39491 þ 1 1.37976 þ 1 1.42950 þ 1 1.39664 þ 1 1.43506 þ 1 1.47601 þ 1 1.48019 þ 1 1.48965 þ 1 1.43456 þ 1 1.47047 þ 1 1.42388 þ 1 1.45808 þ 1 1.44118 þ 1 1.36182 þ 1 1.39288 þ 1 1.34722 þ 1 1.30756 þ 1 1.32273 þ 1 1.35927 þ 1 1.30204 þ 1 1.35888 þ 1 1.34674 þ 1 1.27538 þ 1 1.33843 þ 1 1.39853 þ 1 1.28214 þ 1 1.26658 þ 1 1.21760 þ 1

A1

A2

A3

73.34953 73.29055 72.81801 72.83487 72.74173 72.71400 72.65400 72.57229 72.44148 72.45819 72.24521 72.12574 72.18203 71.98174 71.86342 72.04786 71.71937 71.82276 71.54015 71.88644 71.40872 71.66322 71.88867 71.82430 71.79872 71.23491 71.38933 79.67736-1 71.18375 79.33083-1 73.18459-1 74.79613-1 77.73513-2 1.83235-1 1.37130-1 73.05214-2 3.82736-1 2.20194-3 1.91023-1 6.97409-1 2.81028-1 71.17426-1 7.51993-1 8.85020-1 1.19682

74.71370-2 71.07256-1 72.41738-1 72.10021-1 72.15138-1 72.00530-1 72.00445-1 72.05893-1 72.34461-1 72.12591-1 72.74873-1 72.99392-1 72.58960-1 73.16950-1 73.39440-1 72.73259-1 73.54154-1 73.28827-1 73.94528-1 72.83647-1 74.14365-1 73.31539-1 72.71861-1 72.79116-1 72.83664-1 74.18785-1 73.86631-1 74.78070-1 74.13850-1 74.77357-1 76.11348-1 75.72897-1 76.60456-1 76.94264-1 76.83203-1 76.51340-1 77.32427-1 76.38940-1 76.86616-1 77.89307-1 76.86607-1 75.91094-1 77.87006-1 78.11144-1 78.66697-1

7.09962-3 1.44465-2 2.62542-2 2.29526-2 2.27845-2 2.07248-2 2.00765-2 1.99244-2 2.19537-2 1.96489-2 2.50270-2 2.67643-2 2.22840-2 2.73928-2 2.88858-2 2.29976-2 2.90841-2 2.74382-2 3.23561-2 2.26263-2 3.34355-2 2.62065-2 2.15792-2 2.17324-2 2.22095-2 3.21662-2 3.03286-2 3.66138-2 3.12088-2 3.62829-2 4.58138-2 4.31277-2 4.92177-2 5.02280-2 4.95424-2 4.77616-2 5.29874-2 4.60070-2 4.97356-2 5.64531-2 4.86607-2 4.17843-2 5.58668-2 5.73759-2 6.06931-2

Table 1

Continued

Atomic number

Element

Atomic weight

Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130 140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

A0 1.39389 þ 1 1.33926 þ 1 1.15254 þ 1 1.18198 þ 1 1.30323 þ 1 9.06999 1.16656 þ 1 1.21075 þ 1 1.10857 þ 1 1.13757 þ 1 1.02250 þ 1 1.09780 þ 1 1.02725 þ 1 1.10156 þ 1 1.17632 þ 1 1.13864 þ 1 1.19223 þ 1 1.16168 þ 1 9.91968 1.13818 þ 1 1.14845 þ 1 8.75203 1.20195 þ 1 1.25613 þ 1 7.42791 1.26387 þ 1 7.58160 8.65271 7.57541 1.36944 1.37534 þ 1 1.25506 þ 1 1.27882 þ 1 4.96352 1.97594 þ 1 1.52879 þ 1 8.63374 9.44293 1.51782 þ 1 1.34336 þ 1 1.37951 þ 1 1.82787 þ 1

Notation abbreviated as in Appendix VI, Table 4.

Copyright © 2002 Marcel Dekker, Inc.

A1

A2

A3

1.64528-1 4.41380-1 1.07714 1.45768 7.90788-1 3.28791 1.71052 1.43635 2.08356 1.94161 2.67835 2.23814 2.74562 2.22056 1.79481 2.05593 1.79546 1.97533 3.03111 2.14447 2.10451 3.71822 1.84815 1.57523 4.28955 1.55476 4.47037 3.73117 4.28874 7.79444 1.02122 1.63090 1.63605 5.79212 71.97990 2.73664-1 3.69400 3.44965 3.49021-1 1.34805 1.23983 71.17371

76.62170-1 76.93711-1 78.31424-1 78.88529-1 77.62349-1 71.26203 79.48281-1 78.82038-1 71.01209 79.83232-1 71.12648 71.03549 71.14174 71.02216 79.36661-1 79.88180-1 79.42902-1 79.70901-1 71.17520 79.99222-1 79.89870-1 71.29273 79.39582-1 78.90467-1 71.35167 78.81094-1 71.42808 71.26359 71.34998 71.99822 77.77126-1 78.75676-1 78.98523-1 71.61842 72.76981-1 76.38890-1 71.21312 71.19886 76.37638-1 78.13282-1 78.01545-1 73.68344-1

4.76289-2 4.82085-2 5.79120-2 6.05982-2 5.27872-2 8.53470-2 6.53213-2 6.03575-2 6.90310-2 6.7l986-2 7.62669-2 7.02339-2 7.74162-2 6.90465-2 6.35332-2 6.69106-2 6.44202-2 6.58459-2 7.86751-2 6.75569-2 6.69382-2 8.55026-2 6.38106-2 6.09779-2 8.66136-2 6.02036-2 9.39044-2 8.23539-2 8.65200-2 1.26225-1 5.38811-2 5.92011-2 6.18550-2 1.02911-1 2.68856-2 4.57495-2 7.74601-2 7.83484-2 4.51377-2 5.55664-2 5.53596-2 2.98738-2

Table 2

EL1 < E < EKa

Atomic number

Element

Atomic weight

Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce

22.997 24.310 26.970 28.086 30.975 32.066 35.457 39.944 39.102 40.080 44.960 47.900 50.942 51.996 54.040 55.850 58.933 58.690 63.540 65.380 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910 106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

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A0 1.02355 þ 1 1.05973 þ 1 1.08711 þ 1 1.12237 þ 1 1.15508 þ 1 1.18181 þ 1 1.20031 þ 1 1.22960 þ 1 1.24878 þ 1 1.27044 þ 1 1.28949 þ 1 1.31075 þ 1 1.32514 þ 1 1.34236 þ 1 1.35761 þ 1 1.36696 þ 1 1.38699 þ 1 1.39848 þ 1 1.42439 þ 1 1.43221 þ 1 1.44795 þ 1 1.46813 þ 1 1.46431 þ 1 1.47048 þ 1 1.48136 þ 1 1.49190 þ 1 1.49985 þ 1 1.50114 þ 1 1.51822 þ 1 1.52906 þ 1 1.52088 þ 1 1.53494 þ 1 1.55086 þ 1 1.54734 þ 1 1.55757 þ 1 1.55649 þ 1 1.56869 þ 1 1.59668 þ 1 1.62101 þ 1 1.58638 þ 1 1.57557 þ 1 1.61087 þ 1 1.64086 þ 1 1.63098 þ 1 1.65418 þ 1 1.66217 þ 1 1.63134 þ 1 1.65862 þ 1

A1

A2

72.55905 72.89818 72.77860 72.73694 72.92200 72.64618 72.41694 72.63279 72.53656 72.55011 72.40609 72.53576 72.49765 72.51532 72.49761 72.39195 72.50669 72.48080 72.58677 72.62384 72.54469 72.69285 72.48397 72.38853 72.42347 72.42418 72.39108 72.28169 72.38946 72.38703 72.20278 72.26646 72.33733 72.23080 72.24976 72.17229 72.22636 72.38363 72.51838 72.19010 72.04460 72.27876 72.48214 72.31679 72.46363 72.48972 72.20156 72.36288

71.19524-1 2.34506-1 1.75853-1 1.27557-1 2.54262-1 79.68049-2 72.40897-1 77.36600-2 71.04892-1 79.43195-2 71.77791-1 79.57177-2 7 1.06383-1 71.01999-1 71.05943-1 71.37648-1 78.69945-2 78.88115-2 76.67398-2 72.64926-2 77.57204-2 72.08355-2 77.96180-2 71.05877-1 79.14590-2 78.76447-2 79.59473-2 71.26485-1 78.81174-2 79.12292-2 71.36759-1 71.16881-1 79.87857-1 71.19454-1 71.13377-1 71.27652-1 71.12223-1 78.01104-2 75.49951-2 71.13539-1 71.40745-1 79.29405-2 75.07179-2 78.54498-2 75.42849-2 74.49623-2 79.80569-2 76.54708-2

Table 2 Atomic number 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94 a

Continued Element

Atomic weight

Pr Nd Pm Sm Eu Cd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

Notation abbreviated as in Appendix VI, Table 4.

Copyright © 2002 Marcel Dekker, Inc.

A0 1.67179 þ 1 1.65964 þ 1 1.68368 þ 1 1.68725 þ 1 1.70692 þ 1 1.71159 þ 1 1.71499 þ 1 1.73446 þ 1 1.76583 þ 1 1.77988 þ 1 1.74250 þ 1 1.69795 þ 1 1.72638 þ 1 1.64329 þ 1 1.72410 þ 1 1.72533 þ 1 1.78750 þ 1 1.73525 þ 1 1.65270 þ 1 1.73636 þ 1 1.74240 þ 1 1.71857 þ 1 1.77379 þ 1 1.77963 þ 1 1.75348 þ 1 1.75028 þ 1 1.85481 þ 1 1.75258 þ 1 1.75519 þ 1

A1

A2

72.40326 72.26073 72.38881 72.39051 72.48046 72.47838 72.45507 72.54821 72.72523 72.74671 72.51103 72.22577 72.37189 71.82851 72.30313 72.23874 72.61051 72.28550 71.76315 72.21112 72.23911 72.08470 72.37745 72.37691 72.23353 72.13876 72.61281 72.07237 72.02162

76.12619-2 78.72426-2 76.45041-2 76.01080-2 74.47055-2 74.37107-2 74.71370-2 73.17606-2 78.19409-4 72.87580-3 73.29454-2 77.32557-2 74.95994-2 71.32268-1 75.91006-2 77.27338-2 71.36093-2 75.88047-2 71.35232-1 77.30934-2 76.63720-2 78.53294-2 74.33223-2 74.55883-2 75.96161-2 77.24638-2 77.90574-3 77.23932-2 78.22940-2

Table 3

EM1 < E < EL3 a

Atomic number 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Element Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au

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Atomic weight 69.720 72.590 74.920 78.960 79.920 83.800 85.480 87.620 88.905 91.220 92.906 95.950 99.000 101.070 102.910 106.400 107.880 112.410 114.820 118.690 121.760 127.600 126.910 131.300 132.910 137.360 138.920 140.130 140.920 144.270 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200

A0

A1

1.22646 þ 1 1.24133 þ 1 1.25392 þ 1 1.26773 þ 1 1.27612 þ 1 1.28898 þ 1 1.30286 þ 1 1.31565 þ 1 1.32775 þ 1 1.34508 þ 1 1.35434 þ 1 1.36568 þ 1 1.37498 þ 1 1.38782 þ 1 1.40312 þ 1 1.41392 þ 1 1.41673 þ 1 1.43497 þ 1 1.44115 þ 1 1.45572 þ 1 1.46268 þ 1 1.47125 þ 1 1.47496 þ 1 1.47603 þ 1 1.49713 þ 1 1.50844 þ 1 1.51863 þ 1 1.52693 þ 1 1.53379 þ 1 1.54353 þ 1 1.55131 þ 1 1.56006 þ 1 1.57063 þ 1 1.57159 þ 1 1.58415 þ 1 1.59225 þ 1 1.60140 þ 1 1.60672 þ 1 1.61269 þ 1 1.61794 þ 1 1.62289 þ 1 1.62758 þ 1 1.63068 þ 1 1.62613 þ 1 1.63564 þ 1 1.64233 þ 1 1.65144 þ 1 1.67024 þ 1 1.64734 þ 1

72.68965 72.53085 72.41380 72.39750 72.37730 72.26021 72.38693 72.36655 72.43174 72.50201 72.50135 72.48982 72.44737 72.48066 72.61303 72.57206 72.48078 72.52756 72.49401 72.56792 72.55562 72.54324 72.48179 72.45068 72.53145 72.56341 72.58287 72.58174 72.57086 72.59006 72.59623 72.61328 72.63481 72.60843 72.64040 72.65289 72.67903 72.67587 72.67886 72.67715 72.67128 72.66622 72.66148 72.60672 72.62453 72.63163 72.64832 72.71631 72.57834

Table 3

Continued

Atomic number 80 81 82 83 86 90 92 94 a

Element

Atomic weight

Hg Tl Pb Bi Rn Th U Pu

200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

A0

A1

1.65903 þ 1 1.66564 þ 1 1.67131 þ 1 1.67078 þ 1 1.69000 þ 1 1.70483 þ 1 1.70353 þ 1 1.72953 þ 1

72.60670 72.61593 72.61538 72.58648 72.60945 72.58569 72.56903 72.62164

Notation abbreviated in Appendix VI, Table 4.

Table 4

E < EM5 a

Atomic number 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94

Element Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

a

Atomic weight 147.000 150.350 152.000 157.260 158.930 162.510 164.940 167.270 168.940 173.040 174.990 178.500 180.950 183.920 186.200 190.200 192.200 195.090 197.200 200.610 204.390 207.210 209.000 222.000 232.000 238.070 239.100

Notation abbreviated as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

Copyright © 2002 Marcel Dekker, Inc.

A0 1.55131 þ 1 1.56006 þ 1 1.57063 þ 1 1.57159 þ 1 1.58415 þ 1 1.59225 þ 1 1.60140 þ 1 1.60672 þ 1 1.61269 þ 1 1.39111 þ 1 1.39813 þ 1 1.40548 þ 1 1.41313 þ 1 1.42536 þ 1 1.42392 þ 1 1.42795 þ 1 1.43422 þ 1 1.43785 þ 1 1.44398 þ 1 1.45195 þ 1 1.45473 þ 1 1.45771 þ 1 1.46832 þ 1 1.47243 þ 1 1.47730 þ 1 1.49036 þ 1 1.48535 þ 1

A1 72.59623 72.61328 72.63481 72.60843 72.64040 72.65289 72.67903 72.67587 72.67886 72.40380 72.40841 72.42829 72.47214 72.32582 72.35326 72.21971 72.40183 72.34834 72.32838 72.33016 72.26773 72.25279 72.30940 72.12905 71.91192 72.12148 71.87733

APPENDIX VIII: COEFFICIENTS FOR CALCULATING THE INCOHERENT COLLISION CROSS SECTIONS C (BARNS=ATOM) VIA THE ln^ln REPRESENTATION Atomic Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Element H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd

Copyright © 2002 Marcel Dekker, Inc.

A0

A1

A2

A3

72.l5772 72.56357 71.08740 76.90079-1 77.91177-1 79.87878-1 71.23693 71.73679 71.87570 71.75510 79.67717-1 75.71611-1 74.39322-1 74.14971-1 74.76903-1 76.56419-1 77.18627-1 76.82105-1 73.44007-1 79.82420-2 71.59831-1 72.30573-1 73.08103-1 73.87641-1 72.47059-1 73.42379-1 74.28804-1 75.04360-1 75.70210-1 74.20535-1 73.58218-1 73.34383-1 73.39189-1 74.32927-1 74.48001-1 73.91810-1 71.28039-1 7.99161-2 6.29057-2 3.66697-2 2.02289-4 75.62860-2 7.57616-2 74.24981-2 71.60399-1 72.67564-1

1.32685 2.02536 1.03368 9.46448-1 1.21611 1.46693 1.74510 2.17686 2.32016 2.24226 1.61794 1.35498 1.30867 1.34868 1.46032 1.65408 1.74294 1.74279 1.49236 1.32829 1.39055 1.45848 1.52879 1.59727 1.49722 1.57245 1.64129 1.70040 1.75042 1.63400 1.60050 1.60237 1.62535 1.72833 1.76082 1.73010 1.53044 1.38397 1.41577 1.45207 1.49347 1.55778 1.44950 1.54639 1.64861 1.73740

73.05620-1a 74.48710-1 71.90377-1 71.71142-1 72.39087-1 72.93743-1 73.54660-1 74.49050-1 74.75412-1 74.47640-1 72.87191-1 72.22491-1 72.11648-1 72.22315-1 72.51331-1 72.98623-1 73.19429-1 73.17646-1 72.54135-1 72.13747-1 72.25849-1 72.39160-1 72.52768-1 72.66240-1 72.38781-1 72.53198-1 72.66013-1 72.76443-1 72.84555-1 72.53646-1 72.44908-1 72.45555-1 72.50783-1 72.77138-1 72.85099-1 72.76824-1 2.27403-1 71.92225-1 71.99713-1 72.08122-1 72.17419-1 72.33341-1 72.04890-1 72.26470-1 72.50238-1 72.69883-1

1.85025-2 2.79691-2 7.79955-3 6.51413-3 1.17686-2 1.56005-2 1.98705-2 2.64733-2 2.80680-2 2.55801-2 1.31526-2 8.30141-3 7.54210-3 8.41959-3 1.07202-2 1.42979-2 1.58429-2 1.56467-2 1.07684-2 7.73065-3 8.51954-3 9.38528-3 1.02571-2 1.11523-2 8.93208-3 9.85822-3 1.06512-2 1.12628-2 1.16930-2 9.27233-3 8.61898-3 8.71239-3 9.09103-3 1.11735-2 1.17865-2 1.11280-2 7.39033-3 4.78611-3 5.33312-3 5.95139-3 6.62245-3 7.85506-3 5.64745-3 7.18375-3 8.93818-3 1.03248-2

Appendix VIII Atomic Number 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94

Continued Element Ag Ce In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

a Notation as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

Copyright © 2002 Marcel Dekker, Inc.

A0

A1

A2

A3

71.66475-1 75.16701-2 78.17283-2 1.42151-2 1.56362-2 74.07579-2 74.04420-2 72.82407-3 1.84861-1 3.44376-1 4.09104-1 4.39881-1 4.49124-1 4.37283-1 4.05823-1 3.55383-1 2.80316-1 2.73133-1 2.57539-1 2.42685-1 2.28493-1 2.15233-1 2.02656-1 2.02248-1 1.97176-1 1.99469-1 1.96871-1 1.91015-1 1.89644-1 1.16448-1 7.19908-2 4.20186-2 1.56916-2 1.14587-1 1.47052-1 1.82167-1 1.89860-1 1.96619-1 1.70890-1 1.08277-1 3.88791-2

1.65794 1.57426 1.55865 1.55754 1.57175 1.64267 1.65596 1.64039 1.50030 1.38742 1.33075 1.30925 1.30351 1.31370 1.33837 1.37733 1.44016 1.43842 1.45064 1.46266 1.47438 1.48545 1.49625 1.48804 1.50264 1.50233 1.50623 1.51240 1.50867 1.57615 1.61204 1.63611 1.65406 1.58076 1.56695 1.54661 1.56125 1.60080 1.65561 1.74158 1.82229

72.48740-1 72.27646-1 72.24492-1 72.24736-1 72.28753-1 72.47897-1 72.51067-1 72.47642-1 72.13333-1 71.86356-1 71.70883-1 71.64548-1 71.61841-1 71.62866-1 71.67229-1 71.74941-1 71.88641-1 71.86137-1 71.87591-1 71.89102-1 71.90559-1 71.91908-1 71.93234-1 71.89143-1 71.92474-1 71.91385-1 71.91396-1 71.91922-1 71.89570-1 72.05532-1 72.13186-1 72.17964-1 72.20982-1 72.02968-1 72.00347-1 71.95793-1 72.00932-1 72.13800-1 72.29702-1 72.54104-1 72.76009-1

8.66218-3 7.05650-3 6.85776-3 6.91395-3 7.26386-3 8.80567-3 9.04874-3 8.82144-3 6.24264-3 4.24917-3 3.04111-3 2.52641-3 2.27394-3 2.29377-3 2.55570-3 3.06213-3 4.01226-3 3.75240-3 3.79932-3 3.85628-3 3.90903-3 3.95645-3 4.00233-3 3.62264-3 3.85751-3 3.74011-1 3.70889-3 3.71450-3 3.49584-3 4.66731-3 5.20497-3 5.52670-3 5.70751-3 4.35692-3 4.20901-3 3.90772-3 4.36768-3 5.51717-3 6.92516-3 8.95056-3 1.07392-2

APPENDIX IX: COEFFICIENTS FOR CALCULATING THE COHERENT SCATTERING CROSS SECTIONS R (BARNS/ATOM) VIATHE ln^ln REPRESENTATION Atomic number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Element H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag

Copyright © 2002 Marcel Dekker, Inc.

A0 71.19075-1 1.04768 1.34366 2.00860 2.61862 3.10861 3.47760 3.77239 4.00716 4.20151 4.26374 4.39404 4.51995 4.64678 4.76525 4.92707 5.07222 5.21079 5.25587 5.32375 5.43942 5.55039 5.65514 5.77399 5.84604 5.93292 6.01478 6.09204 6.17739 6.23402 6.28298 6.33896 6.39750 6.45637 6.51444 6.57129 6.59750 6.62203 6.67096 6.72275 6.79013 6.84600 6.87599 6.93136 6.97547 7.03216 7.06446

a

A1

A2

A3

79.37086-1 78.51805-2 1.81557-1 74.61920-2 72.07916-1 72.60580-1 72.15762-1 71.48539-1 75.60908-2 4.16247-2 1.34662-1 1.37858-1 1.40549-1 1.62780-1 1.68708-1 1.65746-1 1.49127-1 1.35618-1 1.88040-1 2.06685-1 2.00174-1 1.97697-1 1.99533-1 2.03858-1 2.13814-1 2.25048-1 2.37959-1 2.52277-1 2.73123-1 2.84312-1 2.91334-1 2.91512-1 2.88866-1 2.86737-1 2.86324-1 2.87711-1 3.02389-1 3.24559-1 3.25075-1 3.23964-1 3.11282-1 3.02797-1 3.26165-1 3.34794-1 3.46394-1 3.49838-1 3.63456-1

72.00538-1 74.03527-1 74.23981-1 73.37018-1 72.86283-1 72.71974-1 72.88874-1 73.07124-1 73.32017-1 73.56754-1 73.70080-1 73.59540-1 73.52441-1 73.58563-1 73.60383-1 73.59424-1 73.52858-1 73.47214-1 73.59623-1 73.61664-1 73.59064-1 73.57694-1 73.57487-1 73.59699-1 73.59718-1 73.61748-1 73.64056-1 73.66568-1 73.72360-1 73.72143-1 73.69391-1 73.65643-1 73.61747-1 73.58794-1 73.57027-1 73.56311-1 73.56755-1 73.61651-1 73.60613-1 73.59463-1 73.55233-1 73.51131-1 73.58969-1 73.63497-1 73.67794-1 73.70099-1 73.73597-1

1.06587-2 2.69398-2 2.66190-2 1.86939-2 1.44966-2 1.35181-2 1.51312-2 1.67303-2 1.87934-2 2.07585-2 2.14467-2 2.02380-2 1.93692-2 1.96926-2 1.97155-2 1.95505-2 1.89439-2 1.84333-2 1.93085-2 1.93328-2 1.91027-2 1.89866-2 1.89691-2 1.92225-2 1.91459-2 1.93024-2 1.94754-2 1.96586-2 2.01638-2 2.00525-2 1.97029-2 1.92895-2 1.88788-2 1.85618-2 1.83557-2 1.82470-2 1.81706-2 1.84800-2 1.83325-2 1.81890-2 1.78231-2 1.74403-2 1.80482-2 1.84429-2 1.87885-2 1.89983-2 1.92478-2

Appendix IX Atomic number 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 86 90 92 94

Continued Element

A0

A1

A2

A3

Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Rn Th U Pu

7.09856 7.12708 7.16085 7.19665 7.23464 7.27415 7.31469 7.33490 7.35812 7.39532 7.44255 7.48347 7.52334 7.56222 7.60020 7.63711 7.66938 7.70798 7.74188 7.77470 7.80643 7.83711 7.86662 7.89137 7.91803 7.94534 7.97266 7.99940 8.02574 8.05150 8.08084 8.10524 8.12542 8.14399 8.15996 8.17489 8.22553 8.27843 8.33010 8.38174

3.72199-1 3.82082-1 3.85512-1 3.85543-1 3.82493-1 3.77223-1 3.70315-1 3.76825-1 3.79361-1 3.69895-1 3.71328-1 3.68431-1 3.66462-1 3.65055-1 3.64134-1 3.63957-1 3.59752-1 3.65345-1 3.67107-1 3.69722-1 3.73226-1 3.77547-1 3.82933-1 3.86034-1 3.87021-1 3.87299-1 3.87704-1 3.88739-1 3.90458-1 3.93143-1 3.95790-1 4.00576-1 4.05858-1 4.08692-1 4.18031-1 4.27916-1 4.51478-1 4.79056-1 4.78314-1 4.77085-1

73.75345-1 73.76855-1 73.76481-1 73.75054-1 73.72715-1 73.69728-1 73.66280-1 73.65713-1 73.64099-1 73.59376-1 73.59642-1 73.57689-1 73.56048-1 73.54511-1 73.53086-1 73.51909-1 73.48899-1 73.50031-1 73.49433-1 73.49132-1 73.49147-1 73.49441-1 73.50126-1 73.49756-1 73.48881-1 73.47926-1 73.47155-1 73.46726-1 73.46658-1 73.47052-1 73.48032-1 73.49340-1 73.50329-1 73.49802-1 73.52330-1 73.55068-1 73.62056-1 73.67657-1 73.67250-1 73.66556-1

1.93481-2 1.94151-2 1.93305-2 1.91608-2 1.89194-2 1.86280-2 1.83025-2 1.81843-2 1.79817-2 1.75406-2 1.75852-2 1.74099-2 1.72620-2 1.71214-2 1.69894-2 1.68783-2 1.65890-2 1.66927-2 1.66273-2 1.65862-2 1.65710-2 1.65780-2 1.66173-2 1.65480-2 1.64406-2 1.63299-2 1.62372-2 1.61751-2 1.61455-2 1.61573-2 1.62345-2 1.63264-2 1.63772-2 1.62888-2 1.64660-2 1.66601-1 1.71556-2 1.74621-2 1.74129-2 1.73422-2

a

Notation as in Appendix VI, Table 4. Source: From McMaster et al., 1969.

Copyright © 2002 Marcel Dekker, Inc.

APPENDIX X: PARAMETERS FOR CALCULATING THE TOTAL MASS ATTENUATION COEFFICIENTS IN THE ENERGY RANGE 0.1^1000 keV [VIA EQ. (78)] Table1 Parameters for Calculating the Total Mass Attenuation Coefficients for Photon Energies Above the K Absorption Edge (k1, k2, k3, k4) and for Energies Between L1 and EK Absorption Edge (l1, l2, l3, l4); Z ¼ Atomic Number, Sy ¼ Symbol of Corresponding Element EL1 < E < EK Z

Sy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn

EL1

0.2292 0.2702 0.3200 0.3771 0.4378 0.5004 0.5637 0.6282 0.6946 0.7690

l1

77.00266 72.38938 72.50341 72.03564 72.91417 0.15632 2.10634 3.34768 3.81364 3.05305 2.76125 3.00797 2.62786 0.68453 2.44853 2.03061 2.33976 2.38673 2.53782 2.65431

Copyright © 2002 Marcel Dekker, Inc.

l2

5.19086 3.24818 3.43696 3.43789 4.13834 2.00149 0.28244 70.83777 71.04042 70.24003 0.29160 0.19834 0.68939 3.86200 1.46227 2.27680 1.88017 1.99445 1.85115 1.83552

E > EK l3

70.34413 70.11121 70.12806 70.13520 70.22931 0.32080 0.92539 1.34549 1.42378 1.25060 1.09998 1.11670 1.01331 70.43637 0.69759 0.31534 0.55084 0.50415 0.62868 0.65972

l4

70.04820 70.11707 70.16880 70.17980 70.17486 70.16451 70.16546 70.16625 0.04226 70.12878 70.07434 70.11829 70.11481 70.14231 70.15525

EK

k1

k2

k3

k4

0.2838 0.4016 0.5320 0.6854 0.8669 1.0721 1.3050 1.5596 1.8389 2.1455 2.4720 2.8224 3.2029 3.6074 4.0381 4.4928 4.9664 5.4651 5.9892 6.5390

76.48071 73.51566 72.36523 71.38914 70.58244 0.11476 0.63927 1.08730 1.42132 1.81194 2.07992 2.39196 2.62043 2.88077 3.06672 3.28801 3.42667 3.53219 3.75863 3.93329 4.00179 4.11127 4.21463 4.35405 4.44148

3.27538 2.60589 2.88866 3.05349 3.07947 3.09245 3.08731 3.07389 3.06182 3.04034 3.04059 3.00737 2.99387 2.96887 2.95078 2.92821 2.90289 2.88277 2.85955 2.83775 2.81618 2.79692 2.77190 2.75878 2.72589

0.04385 0.26361 0.15389 0.07919 0.05871 0.03213 0.01587 0.00035 70.00946 70.02320 70.02978 70.04492 70.05681 70.06174 70.07943 70.08514 70.09061 70.10191 70.09992 70.11226 70.11678 70.11885 70.13140 70.13335 70.14641

70.00866 70.03787 70.02935 70.02484 70.02647 70.02494 70.02538 70.02438 70.02708 70.02568 70.03575 70.02734 70.03063 70.02724 70.03227 70.03094 70.02881 70.03142 70.02689 70.02972 70.02898 70.02784 70.03049 70.03142 70.03205

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Te Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd

0.8461 0.9256 1.0081 1.0961 1.1936 1.2977 1.4143 1.5265 1.6539 1.7820 1.9210 2.0651 2.2163 2.3725 2.5316 2.6977 2.8655 3.0425 3.2240 3.4119 3.6043 3.8058 3.0180 4.2375 4.4647 4.6983 4.9392 5.1881 5.4528 5.7143 5.9888 6.2663 6.5488 6.8348 7.1260

2.41689 2.60591 2.69756 2.75840 2.87225 2.93846 3.02673 3.12353 3.19106 3.30117 3.36926 3.46256 3.54696 3.63965 3.71677 3.79880 3.86387 3.93865 3.99914 4.07189 4.12607 4.19974 4.24309 4.30447 4.35252 4.40598 4.43690 4.51763 4.53737 4.61780 4.65770 4.71490 4.77277 4.83442 4.87359

Copyright © 2002 Marcel Dekker, Inc.

2.80062 2.55718 2.72893 2.72647 2.72029 2.73406 2.74409 2.74414 2.76121 2.73599 2.74007 2.74320 2.73914 2.73646 2.73020 2.72885 2.72657 2.72262 2.70967 2.71531 2.71194 2.70350 2.70139 2.69926 2.69160 2.68835 2.68323 2.67963 2.66911 2.66299 2.65946 2.65505 2.64710 2.64088 2.63355

70.00696 0.18669 0.04244 0.04391 0.04441 0.03264 0.01747 0.01165 70.01683 0.01398 0.00224 70.10161 70.00741 70.01290 70.01273 70.01869 70.02278 70.02487 70.02706 70.02854 70.03442 70.03661 70.04152 70.04319 70.04781 70.04963 70.05424 70.05663 70.05780 70.06277 70.07172 70.07121 70.07113 70.07717 70.07273

70.01904 70.06696 70.03220 70.03423 70.03527 70.03438 70.03117 70.03096 70.02249 70.03428 70.03125 70.02723 70.03104 70.03036 70.03038 70.02896 70.02881 70.03004 70.03l74 70.03173 70.02941 70.02694 70.02776 70.03320 70.02689 70.03001 70.03161 70.03296 70.02658 70.02884 70.03577 70.03442 70.03077 70.03325 70.02815

7.1120 7.7089 8.3328 8.9789 9.6586 10.3671 11.1031 11.8667 12.6578 13.4737 14.3256 15.1997 16.1046 17.0384 17.9976 18.9856 19.9995 21.0440 22.1172 23.2199 24.3503 25.5140 26.7112 27.9399 29.2001 30.4912 31.8138 33.1694 34.5614 35.9846 37.4406 38.9246 40.4430 41.9906 43.5689

4.56600 4.64390 4.77308 4.81074 4.89850 4.93990 5.00743 5.07635 5.11759 5.19474 5.23284 5.29354 5.34695 5.41166 5.45290 5.50180 6.47503 5.57355 5.60095 5.63478 5.65532 5.68392 5.68137 5.72030 5.76598 5.74957 3.74700 5.79401 5.80690 5.82191 5.84551 5.85321 5.87476 5.90293 5.90905

2.70395 2.67587 2.65503 2.62267 2.60534 2.57045 2.55814 2.53376 2.50569 2.48145 2.44844 2.41378 2.38309 2.37364 2.33638 2.30251 3.03067 2.24202 2.21314 2.17349 2.14514 2.05134 2.09840 2.04363 2.05890 1.98639 1.96282 1.93222 1.91476 1.87715 1.87466 1.82364 1.79700 1.77311 1.74652

70.15515 70.16255 70.16936 70.18383 70.18393 71.19961 70.19972 70.20666 70.21581 70.22156 70.23505 70.24899 70.25992 70.25521 70.26879 70.28165 70.04965 70.29909 70.30745 70.32262 70.32915 70.34706 70.36341 70.36014 70.34949 70.37446 70.37979 70.38924 70.39242 70.040213 70.39966 70.41682 70.42193 70.42727 70.43341

70.03352 70.03403 70.03476 70.03729 70.03594 70.03856 70.03827 70.03872 70.03969 70.04024 70.04200 70.04399 70.04534 70.04393 70.04558 70.04737 70.03050 70.04910 70.04983 70.05195 70.05230 70.05473 70.05665 70.05573 70.05406 70.05693 70.05738 70.05845 70.05865 70.05948 70.05904 70.06098 70.06126 70.06168 70.06218

Table 1

Continued EL1 < E < EK

E > EK

Z

Sy

EL1

l1

l2

l3

l4

EK

k1

k2

k3

k4

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra

7.4279 7.7368 8.0520 8.3756 8.7080 9.0458 9.3942 9.7513 10.1157 10.4864 10.8704 11.2707 11.6815 12.0998 12.5267 12.9680 13.4185 13.8799 14.3528 14.8393 15.3467 15.8608 16.3875 16.9393 17.4930 18.0490 18.6390 19.2367

4.93318 4.95712 5.00931 5.03443 5.08055 5.11495 5.15537 5.19648 5.23938 5.26855 5.31163 5.34305 5.37806 5.41320 5.44913 5.47715 5.50865 5.54158 5.57564 5.60041 5.62415 5.65349 5.69400 5.73466 5.76863 5.74682 5.78319 5.80588

2.62474 2.61647 2.61226 2.60481 2.60115 2.59238 2.58071 2.57148 2.56466 2.55541 2.55490 2.54268 2.53595 2.53128 2.52042 2.52666 2.50428 2.50551 2.48980 2.47759 2.46173 2.45575 2.47086 2.46840 2.45600 2.43686 2.43220 2.41932

70.08002 70.07625 70.08680 70.08912 70.07597 70.08230 70.08885 70.09459 70.09047 70.10174 70.09208 70.10077 70.09548 70.10036 70.10577 70.09007 70.10492 70.09847 70.10793 70.11241 70.12251 70.12044 70.10349 70.09608 70.10153 70.11031 70.10993 70.11330

70.02998 70.02487 70.03205 70.03162 70.02286 70.02541 70.02644 70.02802 70.02454 70.03012 70.02484 70.02734 70.02359 70.02631 70.02724 70.02154 70.02495 70.02303 70.02561 70.02570 70.02850 70.02698 70.02277 70.01973 70.02099 70.02272 70.02259 70.02255

45.1840 46.8342 48.5190 50.2391 51.9957 53.7885 55.6177 57.4855 59.3896 61.3323 63.3138 65.3508 67.4164 69.5250 71.6764 73.8708 76.1110 78.3948 80.7249 83.1023 85.5304 88.0045 90.5259 93.1050 95.7299 98.4040 101.1370 103.9219

5.92214 5.92934 5.94975 5.92827 5.95498 5.95991 5.94804 5.96408 5.88779 6.04546 6.07395 6.00771 6.05159 6.00250 6.03030 6.11400 6.09889 6.15378 6.15303 6.19862 6.17641 6.17648 6.30427 6.21512 6.18945 6.24492 6.31471 6.26760

1.70715 1.69904 1.67827 1.64215 1.63471 1.61100 1.55826 1.54243 1.43854 1.57974 1.57891 1.48367 1.50856 1.43225 1.43738 1.50446 1.45990 1.53187 1.47494 1.50676 1.46912 1.45115 1.55736 1.43087 1.38290 1.47699 1.51865 1.45136

70.44407 70.44407 70.44795 70.45629 70.45467 70.46029 70.47717 70.47949 70.50808 70.45878 70.45631 70.48598 70.47273 70.49552 70.48952 70.46496 70.47717 70.44176 70.46441 70.45079 70.46009 70.46277 70.42520 70.46426 70.47598 70.43939 70.42408 70.44373

70.06326 70.06312 70.06335 70.06402 70.06359 70.06408 70.06595 70.06606 70.06870 70.06334 70.06301 70.06608 70.06439 70.06672 70.06580 70.06309 70.06429 70.05974 70.06253 70.06097 70.06183 70.06196 70.05792 70.06195 70.06300 70.05879 70.05721 70.05917

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89 90 91 92

Ac Th Pa U

19.8400 20.4721 21.1046 21.7574

5.83228 5.85156 5.87741 5.88224

2.39643 2.40038 2.35466 2.34307

70.12322 70.11549 70.14418 70.15054

70.02407 70.02204 70.02817 70.03027

106.7553 109.6509 112.6014 115.6061

6.32123 6.39823 6.28773 6.30701

1.48583 1.55418 1.41526 1.44776

70.42883 70.40447 70.44597 70.43028

70.05744 70.05479 70.05907 70.05716

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

Copyright © 2002 Marcel Dekker, Inc.

Table 2 Parameters for Calculating the Total Mass Attentuation Coefficients for Photon Energy Between L1, L2, and L3 Absorption Edges and for Energies Between L3 and Lower Energy Limit E0 or M Absorption Edge EL3 < E < EL2

E0<E<EL3 Z

Sy E0 =Eedge

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1085 0.1487 0.2122 0.2118 0.2370 0.2631

m1

m2

m3

m4

3.98381 72.71337 2.07772 8.53825 5.10006 7.28382 6.39592 2.83998 77.38556 2.13747 2.21508 4.76476 4.28916 2.32551 4.23694 2.60598 4.27624 5.14580 6.74588 7.26941 11.95287 15.89070 11.09104 5.18978 6.15438 5.83085 6.35098 5.57003

71.96207 2.78307 70.84611 74.62277 71.85097 73.26298 72.45282 0.44991 8.17416 1.11467 1.10224 71.05687 70.66344 1.22642 70.64439 0.84383 70.65284 71.51272 73.03410 73.69837 78.69443 712.8788 78.15432 71.99566 72.90381 72.56434 73.21302 72.03397

1.29140 0.23822 1.21097 1.97295 1.29907 1.60507 1.38399 0.61932 71.29292 0.46032 0.47160 1.09423 1.02071 0.45413 1.08840 0.68376 1.15806 1.47391 1.98313 2.28437 4.04104 5.51561 4.08776 2.09278 2.42622 2.36756 2.67445 2.13536

70.13371 70.05767 70.14523 70.19719 70.14560 70.16673 70.14698 70.08033 0.07794 70.06687 70.06697 70.12565 70.12207 70.06650 70.13570 70.10262 70.15244 70.19017 70.24674 70.28824 70.48832 70.65735 70.52451 70.32184 70.36813 70.37176 70.41914 70.33831

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EL3

1.0197 1.1154 1.2167 1.3231 1.4358 1.5499 1.6749 1.8044 1.9396 2.0800 2.2223 2.3705 2.5202 2.6769

l31

77.206 47.827 20.461 6.9330 0.0065 74.6062 75.6370 3.2959 2.0366 1.9840 2.2290 2.724 2.813 3.044

l32

727.639 716.433 75.2040 0.6640 3.8833 6.2403 6.9490 2.4609 3.1926 3.2996 3.2290 2.999 2.999 2.907

EL2 < E < EL1 EL2

1.0428 1.1423 1.2478 1.3586 1.4762 1.5960 1.7272 1.8639 2.0068 2.1555 2.3067 2.4647 2.6251 2.7932

l21

9.4520 6.2330 5.4560 5.1580 3.9491 3.8764 3.4100 4.0979 3.9355 3.9621 3.9270 3.981 3.996 4.051

l22

EL1

70.2320 1.1180 1.4580 1.6080 2.1968 2.2734 2.5440 2.2097 2.3328 2.3596 2.4170 2.425 2.447 2.452

0.2292 0.2702 0.3200 0.3771 0.4378 0.5004 0.5637 0.6282 0.6946 0.7690 0.8461 0.9256 1.0081 1.0961 1.1936 1.2977 1.4143 1.5265 1.6539 1.7820 1.9210 2.0651 2.2163 2.3725 2.5316 2.6977 2.8655 3.0425

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt

0.2901 0.3180 0.3425 0.3766 0.4129 0.4530 0.4946 0.5379 0.5827 0.6293 0.6775 1.2171 1.2928 1.3613 1.4346 1.5110 1.5753 1.6540 1.7228 1.8000 1.8808 1.9675 2.0468 2.1283 2.2065 2.3068 2.3981 2.4912 2.6009 2.7080 2.8196 2.9317 3.0485 3.1737 3.2960

9.99349 6.55219 6.32821 5.65378 3.12801 2.78289 2.57173 2.52338 4.98562 3.00387 3.00203 3.16785 3.19080 3.23265 3.29411 3.35891 3.41765 3.45084 3.49847 3.54762 3.58135 3.61719 3.67219 3.72181 3.77845 3.82800 3.86458 3.91793 3.94527 3.99165 4.02988 4.07043 4.10244 4.14636 4.18356

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78.27969 73.56318 73.29827 72.26740 1.54731 2.29146 2.70219 2.80299 71.29951 2.36374 2.56894 2.42799 2.48392 2.53710 2.53524 2.54117 2.51947 2.59000 2.55440 2.56798 2.57591 2.61430 2.57359 2.55757 2.52004 2.53261 2.50726 2.48402 2.52598 2.50855 2.51689 2.52499 2.53312 2.52893 2.53390

5.01647 2.98931 2.95809 2.49736 0.70511 0.30064 0.11208 0.07373 2.27938 0.26067 0.10775 0.12755 0.08776 0.05145 0.05465 0.04839 0.05777 0.00563 0.02671 0.01585 0.00022 70.03278 70.00002 0.010760 0.05588 0.03019 0.06374 0.08777 0.03404 0.05734 0.04574 0.03041 0.01448 0.01458 0.00520

70.76636 70.48554 70.49645 70.43066 70.16236 70.09544 70.07092 70.06562 70.44589 70.09483 70.06179 70.05218 70.04429 70.03744 70.03926 70.03842 70.04029 0.02894 70.03367 70.03181 70.02718 70.01918 70.02988 70.03537 70.04817 70.03886 70.05192 70.06146 70.04239 70.05289 70.05071 70.04512 70.03948 70.04009 70.03783

2.8379 3.0038 3.1733 3.3511 3.5375 3.7301 3.9288 4.1322 4.3414 4.5571 4.7822 5.0119 5.2470 5.4827 5.7234 5.9642 6.2079 6.4593 6.7162 6.9769 7.2428 7.5140 7.7901 8.0711 8.3579 8.6480 8.9436 9.2441 9.5607 9.8811 10.2068 10.5353 10.8709 11.2152 11.5637

3.320 3.653 4.146 4.111 3.971 4.293 4.834 4.637 4.618 4.577 4.572 4.215 4.363 4.401 4.449 4.469 4.463 4.500 4.536 4.553 4.583 4.633 4.655 4.695 4.727 4.767 4.802 4.847 4.866 4.911 4.940 4.979 5.006 5.050 5.084

2.770 2.586 2.250 2.321 2.454 2.217 1.753 1.940 1.955 2.048 2.073 2.528 2.384 2.397 2.402 2.438 2.500 2.528 2.502 2.556 2.543 2.537 2.572 2.558 2.581 2.592 2.569 2.560 2.629 2.584 2.624 2.629 2.661 2.673 2.666

2.9669 3.1469 3.3303 3.5237 3.7270 3.9380 4.1561 4.3804 4.6120 4.8521 5.1037 5.3594 5.6236 5.8906 6.1642 6.4404 6.7215 7.0128 7.3118 7.6171 7.9303 5.2516 8.5806 8.9178 9.2643 9.6169 9.9782 10.3386 10.7394 11.1361 11.5440 11.9687 12.3850 12.8241 13.2726

4.303 4.091 4.344 4.217 4.328 4.620 4.592 4.584 4.554 4.625 4.599 4.563 4.601 4.698 4.702 4.757 4.778 4.854 4.861 4.924 4.927 4.985 5.014 5.047 5.086 5.128 5.082 5.181 5.214 5.237 5.276 5.302 5.323 5.355 5.375

2.302 2.502 2.337 2.487 2.420 2.196 2.238 2.279 2.323 2.318 2.375 2.482 2.470 2.404 2.470 2.467 2.490 2.449 2.469 2.428 2.467 2.423 2.420 2.426 2.414 2.398 2.815 2.431 2.377 2.468 2.318 2.373 2.314 2.361 2.278

3.2240 3.4119 3.6043 3.8058 3.0180 4.2375 4.4647 4.6983 4.9392 5.1881 5.4528 5.7143 5.9888 6.2663 6.5488 6.8348 7.1260 7.4279 7.7368 8.0520 8.3756 8.7080 9.0458 9.3942 9.7513 10.1157 10.4864 10.8704 11.2707 11.6815 12.0998 12.5267 12.9680 13.4185 13.8799

Table 2

Continued EL3 < E < EL2

E0<E<EL3

EL2 < E < EL1

Z

Sy

E0 =Eedge

m1

m2

m3

m4

EL3

l31

l32

EL2

l21

l22

EL1

79 80 81 82 83 84 85 86 87 88 89 90 91 92

Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

3.4249 3.5616 3.7041 3.8507 3.9991 4.1494 4.3170 4.4820 4.6520 4.8220 5.0020 5.1823 5.3669 5.5480

4.22770 4.26093 4.29302 4.32934 4.36991 4.41866 4.46126 4.45433 4.49562 4.52666 4.56932 4.59153 4.63772 4.65216

2.52409 2.51954 2.51974 2.51778 2.52184 2.52158 2.51394 2.51411 2.51108 2.50733 2.50607 2.50190 2.49776 2.49170

0.01426 0.02092 0.01369 0.00460 70.01213 70.00971 0.00640 70.00666 70.01358 0.00692 70.01644 70.00815 70.00936 70.02703

70.04262 70.04665 70.04376 70.03614 70.02654 70.03298 70.04593 70.03871 70.02823 70.05016 70.03124 70.03948 70.03490 70.01144

11.9187 12.2839 12.6575 13.0352 13.4186 13.8138 14.2135 14.6194 15.0312 15.4444 15.8710 16.3003 16.7331 17.1663

5.125 5.156 5.184 5.220 5.257 5.301 5.347 5.334 5.376 5.410 5.454 5.490 5.525 5.541

2.690 2.643 2.630 2.644 2.642 2.625 2.642 2.637 2.647 2.653 2.662 2.696 2.666 2.672

13.7336 14.2087 14.6979 15.2000 15.7111 16.2443 16.7847 17.3371 17.9065 18.4843 19.0832 19.6932 20.3137 20.9476

5.406 5.430 5.451 5.512 5.536 5.555 5.574 5.535 5.650 5.644 5.761 5.883 5.798 5.829

2.303 2.337 2.323 2.502 2.453 2.387 2.335 2.281 2.494 2.411 2.594 2.800 2.534 2.571

14.3528 14.8393 15.3467 15.8608 16.3875 16.9393 17.4930 18.0490 18.6390 19.2367 19.8400 20.4721 21.1046 21.7574

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

Copyright © 2002 Marcel Dekker, Inc.

Table 3 Parameters for Calculating the Total Mass Attenuation Coefficients for Photon Energies Between M1, M2, . . ., M5 Absorption Edges and for Energies Between M5 and Lower Energy Limit E0 E0 < E < Eedge Z

E0

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

0.1085 0.1085 0.1085 0.1085 0.1085 0.1328 0.1328 0.1487 0.1717 0.1717 0.1511 0.1511 0.1511 0.1833 0.2122 0.2770 0.1926 0.2122 0.3117 0.3117 0.2770 0.2770 0.2770 0.2770 0.1717 0.2122 0.2770 0.2122

n1

n2

n3

EM5 < E < EM4 n4

EM5

m42

m41

EM4 < E < EM3 EM4

m32

m31

EM3 < E < EM2 EM3

m22

115.013 769.4208 14.9364 71.0543 106.662 767.3126 15.2602 71.1408 3.5107 2.3889 70.4178 0.0324 89.5750 757.9740 13.8424 71.1025 80.5898 757.4238 15.0813 71.3052 76.3873 1.0408 1.9749 70.3214 29.0389 718.5282 5.5664 70.5470 11.0289 76.3556 2.6658 70.3046 174.946 7138.963 38.3880 73.5077 42.9485 731.1679 9.1280 70.8669 39.8331 730.6663 9.5340 70.9502 37.0196 728.3109 8.8955 70.8939 14.2756 79.7352 3.9179 70.4554 23.1086 718.4405 6.7995 70.7732 32.5527 727.5567 9.7373 71.0874 0.7330 2.364 31.6390 728.1233 10.4351 71.2218 0.7905 0.7961 3.768 2.088 1.0622 3.897 28.6047 724.1290 8.8027 71.0079 0.8473 0.8485 3.554 2.190 1.1234 3.870 3.9181 70.9835 1.6398 70.2735 0.8861 0.9013 3.194 2.376 1.1854 3.852 4.6993 0.0130 0.6969 70.0975 0.9349 0.9511 3.317 2.355 1.2422 5.314 20.5246 716.5997 6.5042 70.7721 0.9843 0.9999 5.484 1.419 1.2974 3.879 16.9684 713.1825 5.4255 70.6587 1.0269 96.373 735.542 1.0515 6.609 0.921 1.3569 3.845 11.8798 77.7898 3.5371 70.4381 1.0802 89.021 733.267 1.1060 6.352 1.023 1.4198 3.825 5.6437 71.4828 1.4474 70.2096 1.1309 83.564 731.660 1.1606 6.316 1.039 1.4806 3.825 6.6183 72.4260 1.7586 70.2443 1.1852 76.154 729.235 1.2172 8.597
0.048 1.5440 3.792 75.6112 10.4330 72.6765 0.2602 1.2412 73.753 728.683 1.2750 6.289 1.062 1.6113 3.863 1.4535 3.1211 70.1907 70.0138 1.2949 69.682 727.456 1.3325 6.096 1.157 1.6756 3.809 4.077 0.1743 0.9232 70.1528 1.3514 65.046 725.898 1.3915 5.446 1.471 1.7412 3.811 1.6315 2.7196 0.0767 70.0607 1.4093 60.848 724.477 1.4533 5.353 1.523 1.8118 3.847

Copyright © 2002 Marcel Dekker, Inc.

EM2 < E < EM1

m21

EM2

m12

m11

2.621 2.098 2.132 2.162 1.551 2.193 2.234 2.255 2.278 2.308 2.296 2.342 2.363 2.368

1.0650 1.1367 1.2044 1.2728 1.3774 1.4028 1.4714 1.5407 1.6139 1.6883 1.7677 1.8418 1.9228 2.0058

3.983 3.957 3.970 4.007 2.489 3.993 4.039 3.986 4.005 4.032 4.027 4.060 4.092 4.087

2.072 2.097 2.114 2.120 2.863 2.167 2.170 2.206 2.220 2.217 2.242 2.243 2.245 2.269

EM1

1.2171 1.2928 1.3613 1.4346 1.5110 1.5753 1.6540 1.7228 1.8000 1.8808 1.9675 2.0468 2.1283 2.2065

Table 3

Continued E0 < E < Eedge

EM5 < E < EM4

EM4 < E < EM3

EM3 < E < EM2

EM2 < E < EM1

Z

E0

n1

n2

n3

n4

EM5

m42

m41

EM4

m32

m31

EM3

m22

m21

EM2

m12

m11

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

0.1926 0.1085 0.1085 0.1085 0.1085 1.1085 0.1085 0.1085 0.1085 0.1328 0.1511 0.2122 0.1926 0.1717 0.1926 0.2122 0.3117 0.3924 0.3924 0.3924 0.3924 0.2770 0.2770 0.2770

75.0355 71.8380 70.3276 70.1868 71.9405 71.1407 70.3842 71.8806 72.9368 1.5113 3.5698 3.6427 3.6296 4.6393 5.7697 5.6131 5.1757 4.2011 4.5198 5.5170 6.3317 4.1761 4.3146 3.9954

9.4382 6.2849 4.8522 4.8402 6.5977 5.9151 5.1929 6.8129 7.9106 3.4177 1.1260 1.0962 1.1014 70.1055 71.3933 71.3279 70.6452 0.6074 0.1763 71.1906 72.2638 0.8164 0.8409 1.2801

72.1146 71.0736 70.6175 70.6304 71.1795 70.9671 70.7225 71.2684 71.6078 70.1106 0.7499 0.7581 0.7912 1.2868 1.7810 1.8314 1.5552 1.0490 1.2693 1.9034 2.3875 1.0031 0.9488 0.7662

0.1726 0.0593 0.0113 0.0127 0.0672 0.0439 0.0155 0.0733 0.1054 70.0581 70.1652 70.1655 70.1770 70.2451 70.3078 70.3245 70.2925 70.2281 70.2654 70.3623 70.4354 70.2383 70.2273 70.2038

1.4677 1.5278 1.5885 1.6517 1.7351 1.8092 1.8829 1.9601 2.0404 2.1216 2.2057 2.2949 2.3893 2.4840 2.5796 2.6839 2.7867 2.8924 2.9999 3.1049 3.2190 3.3320 3.4418 3.5517

63.572 54.494 57.769 55.928 53.589 50.662 30.913 54.605 48.393 45.233 43.502 40.458 31.751 20.254 9.171 70.200 74.893 77.059 72.615 0.627 2.581 3.793 3.639 3.662

726.240 722.411 724.531 724.196 723.567 722.588 712.446 725.802 722.958 721.695 721.199 719.895 715.013 78.092 71.075 5.161 8.501 10.172 7.198 4.924 3.529 2.621 2.790 2.788

1.5146 1.5763 1.6394 1.7164 1.7932 1.8716 1.9489 2.0308 2.1161 2.2019 2.2911 2.3849 2.4851 2.5856 2.6876 2.7980 2.9087 3.0215 3.1362 3.2484 3.3702 3.4908 3.6112 3.7276

5.841 5.900 6.893 7.170 6.974 6.894 6.854 6.900 7.249 7.209 7.102 6.847 6.055 5.339 4.475 3.586 3.141 3.222 3.649 3.982 4.255 4.355 4.369 4.409

1.257 1.222 0.681 0.527 0.627 0.671 0.666 0.581 0.343 0.344 0.393 0.540 1.064 1.574 2.212 2.895 3.276 3.251 2.961 2.716 2.525 2.455 2.484 2.460

1.8845 1.9498 2.0236 2.1076 2.1940 2.2810 2.3673 2.4572 2.5507 2.6454 2.7430 2.8471 2.9566 3.0664 3.1769 3.3019 3.4260 3.5280 3.6638 3.7918 3.9098 4.0461 4.1738 4.3034

3.889 3.916 3.876 3.916 3.957 4.019 4.063 4.032 4.126 4.166 4.199 4.187 4.191 4.259 4.277 4.327 4.372 4.382 4.405 4.391 4.460 4.495 4.526 4.527

2.370 2.373 2.422 2.420 2.420 2.408 2.405 2.445 2.410 2.408 2.414 2.445 2.467 2.443 2.459 2.459 2.457 2.442 2.458 2.501 2.475 2.462 2.477 2.492

2.0998 2.1730 2.2635 2.3654 2.4687 2.5749 2.6816 2.7922 2.9087 3.0265 3.1478 3.2785 3.4157 3.5542 3.6963 3.8541 4.0080 4.1590 4.3270 4.4895 4.6560 4.8304 5.0009 5.1822

4.114 4.125 4.129 4.158 4.146 4.178 4.214 4.228 4.307 4.323 4.317 4.382 4.314 4.453 4.383 4.460 4.494 4.500 4.523 4.511 4.599 4.576 4.603 4.614

2.278 2.287 2.309 2.311 2.343 2.344 2.346 2.356 2.327 2.338 2.370 2.344 2.419 2.332 2.420 2.395 2.402 2.386 2.398 2.442 2.387 2.437 2.457 2.458

Source: From Orlic et al., 1993. Reprinted with kind permission from Elsevier Science – NL, Sara Burgehartstraat 25, 1055 KV Amsterdam, The Netherlands.

Copyright © 2002 Marcel Dekker, Inc.

EM1 2.3068 2.3981 2.4912 2.6009 2.7080 2.8196 2.9317 3.0485 3.1737 3.2960 3.4249 3.5616 3.7041 3.8507 3.9991 4.1494 4.3170 4.4820 4.6520 4.8220 5.0020 5.1823 5.3669 5.5480

APPENDIX XI: TOTAL MASS ATTENUATION COEFFICIENTS FOR LOW-ENERGY K LINES Emitter wavelength (A˚) energy (keV)

Absorber

Atomic number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Element

B 6.67 þ 1a 1.83 7 1

C 4.48 þ 1 2.77 7 1

N 3.16 þ 1 3.92 7 1

O 2.36 þ 1 5.25 7 1

F 1.83 þ 1 6.77 7 1

Ne 1.46 þ 1 8.49 7 1

Na 1.19 þ 1 1.04 þ 0

Mg 9.89 þ 0 1.25 þ 0

Al 8.34 þ 0 1.49 þ 0

H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co

1.81 þ 3 1.20 þ 4 2.98 þ 4 5.82 þ 4 3.93 þ 3 7.32 þ 3 1.19 þ 4 1.74 þ 4 2.45 þ 4 3.72 þ 4 4.25 þ 4 5.62 þ 4 6.44 þ 4 7.48 þ 4 7.49 þ 4 8.18 þ 4 7.44 þ 3 9.51 þ 3 1.14 þ 4 1.31 þ 4 1.43 þ 4 1.54 þ 4 1.68 þ 4 2.09 þ 4 2.23 þ 4 2.61 þ 4 2.86 þ 4

4.89 þ 2 4.06 þ 3 1.08 þ 4 2.29 þ 4 3.75 þ 4 2.75 þ 3 4.71 þ 3 6.90 þ 3 9.73 þ 3 1.36 þ 4 1.80 þ 4 2.44 þ 4 2.92 þ 4 3.47 þ 4 4.19 þ 4 4.95 þ 4 5.17 þ 4 5.24 þ 4 5.92 þ 3 7.15 þ 3 7.80 þ 3 8.45 þ 3 9.22 þ 3 1.11 þ 4 1.20 þ 4 1.40 þ 4 1.54 þ 4

1.51 þ 2 1.25 þ 3 3.67 þ 3 8.40 þ 3 1.53 þ 4 2.44 þ 4 1.83 þ 3 2.68 þ 3 3.73 þ 3 5.70 þ 3 7.26 þ 3 1.00 þ 4 1.23 þ 4 1.58 þ 4 1.93 þ 4 2.38 þ 4 2.67 þ 4 3.00 þ 4 3.51 þ 4 3.49 þ 4 3.97 þ 3 4.35 þ 3 4.77 þ 3 5.61 þ 3 6.13 þ 3 7.10 þ 3 7.85 þ 3

5.85 þ 1 5.07 þ 2 1.60 þ 3 3.78 þ 3 7.22 þ 3 1.22 þ 4 1.73 þ 4 1.27 þ 3 1.79 þ 3 2.71 þ 3 3.52 þ 3 4.89 þ 3 6.10 þ 3 7.93 þ 3 9.74 þ 3 1.21 þ 4 1.39 þ 4 1.59 þ 4 1.94 þ 4 2.21 þ 4 2.34 þ 4 2.20 þ 4 1.66 þ 4 3.13 þ 3 3.46 þ 3 4.00 þ 3 4.41 þ 3

2.62 þ 1 2.23 þ 2 7.57 þ 2 1.86 þ 3 3.63 þ 3 6.29 þ 3 9.23 þ 3 1.21 þ 4 9.12 þ 2 1.39 þ 3 1.83 þ 3 2.55 þ 3 3.22 þ 3 4.22 þ 3 5.19 þ 3 6.52 þ 3 7.50 þ 3 8.59 þ 3 1.07 þ 4 1.24 þ 4 1.34 þ 4 1.45 þ 4 1.57 þ 4 1.58 þ 4 1.68 þ 4 2.31 þ 3 2.58 þ 3

1.22 þ 1 1.06 þ 2 3.84 þ 2 9.59 þ 2 1.93 þ 3 3.43 þ 3 5.14 þ 3 6.94 þ 3 8.54 þ 3 7.54 þ 2 1.01 þ 3 1.42 þ 3 1.80 þ 3 2.37 þ 3 2.92 þ 3 3.69 þ 3 4.27 þ 3 4.89 þ 3 6.13 þ 3 7.19 þ 3 7.79 þ 3 8.59 þ 3 9.40 þ 3 1.09 þ 4 1.16 þ 4 1.31 þ 4 1.22 þ 4

6.32 þ 0 5.58 þ 1 2.09 þ 2 5.44 þ 2 1.10 þ 3 1.98 þ 3 3.02 þ 3 4.17 þ 3 5.23 þ 3 7.11 þ 3 5.93 þ 2 8.37 þ 2 1.07 þ 3 1.41 þ 3 1.74 þ 3 2.21 þ 3 2.57 þ 3 2.93 þ 3 3.71 þ 3 4.38 þ 3 4.74 þ 3 5.27 þ 3 5.81 þ 3 6.75 þ 3 7.27 þ 3 8.22 þ 3 8.84 þ 3

3.76 þ 0 3.15 þ 1 1.20 þ 2 3.26 þ 2 6.73 þ 2 1.21 þ 3 1.90 þ 3 2.67 þ 3 3.44 þ 3 4.55 þ 3 5.65 þ 3 5.06 þ 2 6.63 þ 2 8.93 þ 2 1.09 þ 3 1.39 þ 3 1.63 þ 3 1.85 þ 3 2.36 þ 3 2.75 þ 3 2.99 þ 3 3.34 þ 3 3.70 þ 3 4.30 þ 3 4.67 þ 3 5.31 þ 3 5.73 þ 3

2.23 þ 0 1.77 þ 1 6.82 þ 1 1.89 þ 2 3.94 þ 2 7.22 þ 2 1.14 þ 3 1.63 þ 3 2.11 þ 3 2.84 þ 3 3.53 þ 3 4.39 þ 3 4.07 þ 2 5.52 þ 2 6.78 þ 2 8.64 þ 2 1.01 þ 3 1.15 þ 3 1.47 þ 3 1.74 þ 3 1.90 þ 3 2.13 þ 3 2.36 þ 3 2.74 þ 3 3.00 þ 3 3.42 þ 3 3.71 þ 3

Copyright © 2002 Marcel Dekker, Inc.

Appendix XI

Continued Emitter wavelength (A˚) energy (keV)

Absorber

Atomic number 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Element Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs

B 6.67 þ 1a 1.83 7 1

C 4.48 þ 1 2.77 7 1

N 3.16 þ 1 3.92 7 1

O 2.36 þ 1 5.25 7 1

F 1.83 þ 1 6.77 7 1

Ne 1.46 þ 1 8.49 7 1

Na 1.19 þ 1 1.04 þ 0

Mg 9.89 þ 0 1.25 þ 0

Al 8.34 þ 0 1.49 þ 0

3.34 þ 4 3.65 þ 4 4.03 þ 4 4.18 þ 4 4.41 þ 4 4.41 þ 4 4.49 þ 4 4.19 þ 4 3.94 þ 4 3.52 þ 4 2.83 þ 4 2.04 þ 4 6.41 þ 3 4.28 þ 3 4.66 þ 3 4.99 þ 3 5.49 þ 3 5.87 þ 3 6.09 þ 3 6.34 þ 3 6.45 þ 3 6.61 þ 3 6.64 þ 3 6.71 þ 3 6.65 þ 3 5.77 þ 3 5.79 þ 3 5.91 þ 3

1.81 þ 4 1.96 þ 4 2.26 þ 4 2.42 þ 4 2.64 þ 4 2.86 þ 4 3.03 þ 4 3.22 þ 4 3.28 þ 4 3.56 þ 4 3.47 þ 4 3.10 þ 4 3.06 þ 4 2.96 þ 4 2.23 þ 4 1.41 þ 4 4.18 þ 3 4.64 þ 3 5.22 þ 3 5.48 þ 3 5.75 þ 3 6.06 þ 3 6.27 þ 3 6.50 þ 3 6.58 þ 3 6.99 þ 3 7.27 þ 3 7.52 þ 3

9.19 þ 3 9.96 þ 3 1.17 þ 4 1.26 þ 4 1.40 þ 4 1.55 þ 4 1.69 þ 4 1.86 þ 4 2.05 þ 4 2.17 þ 4 2.29 þ 4 2.34 þ 4 2.46 þ 4 2.67 þ 4 2.29 þ 4 2.29 þ 4 2.47 þ 4 2.48 þ 4 2.21 þ 4 1.10 þ 4 4.10 þ 3 4.42 þ 3 4.70 þ 3 4.94 þ 3 5.09 þ 3 5.52 þ 3 5.88 þ 3 6.13 þ 3

5.25 þ 3 5.94 þ 3 6.56 þ 3 7.10 þ 3 7.89 þ 3 8.80 þ 3 9.64 þ 3 1.07 þ 4 1.19 þ 4 1.28 þ 4 1.37 þ 4 1.51 þ 4 1.61 þ 4 1.78 þ 4 1.85 þ 4 1.78 þ 4 1.95 þ 4 1.94 þ 4 1.80 þ 4 1.91 þ 4 2.01 þ 4 2.23 þ 4 9.12 þ 3 3.36 þ 3 3.53 þ 3 3.88 þ 3 4.17 þ 3 4.40 þ 3

3.01 þ 3 3.20 þ 3 3.73 þ 3 4.08 þ 3 4.56 þ 3 5.10 þ 3 5.61 þ 3 6.28 þ 3 6.98 þ 3 7.57 þ 3 8.21 þ 3 9.06 þ 3 9.77 þ 3 1.08 þ 4 1.14 þ 4 1.19 þ 4 1.30 þ 4 1.38 þ 4 1.46 þ 4 1.47 þ 4 1.47 þ 4 1.36 þ 4 1.38 þ 4 1.48 þ 4 1.75 þ 4 1.18 þ 4 2.84 þ 3 3.02 þ 3

1.80 þ 3 2.01 þ 3 2.25 þ 3 2.47 þ 3 2.75 þ 3 3.08 þ 3 3.38 þ 3 3.97 þ 3 4.23 þ 3 4.63 þ 3 5.04 þ 3 5.58 þ 3 6.03 þ 3 6.69 þ 3 7.09 þ 3 7.45 þ 3 8.16 þ 3 8.79 þ 3 9.42 þ 3 1.00 þ 4 1.04 þ 4 1.10 þ 4 1.08 þ 4 1.13 þ 4 1.13 þ 4 1.02 þ 4 1.08 þ 4 1.15 þ 4

1.02 þ 4 9.47 þ 3 5.26 þ 3 1.52 þ 3 1.70 þ 3 1.95 þ 3 2.17 þ 3 2.44 þ 3 2.66 þ 3 2.92 þ 3 3.20 þ 3 3.55 þ 3 3.86 þ 3 4.27 þ 3 4.55 þ 3 4.90 þ 3 5.37 þ 3 5.80 þ 3 6.23 þ 3 6.66 þ 3 7.09 þ 3 7.51 þ 3 7.88 þ 3 8.26 þ 3 8.35 þ 3 8.24 þ 3 8.72 þ 3 9.79 þ 3

6.58 þ 3 7.04 þ 3 7.51 þ 3 6.92 þ 3 4.72 þ 3 1.30 þ 3 1.53 þ 3 1.68 þ 3 1.72 þ 3 1.89 þ 3 2.08 þ 3 2.32 þ 3 2.53 þ 3 2.80 þ 3 2.99 þ 3 3.57 þ 3 3.92 þ 3 4.23 þ 3 4.54 þ 3 4.86 þ 3 5.84 þ 3 6.20 þ 3 6.50 þ 3 6.82 þ 3 6.90 þ 3 6.93 þ 3 6.75 þ 3 6.67 þ 3

4.26 þ 3 4.52 þ 3 4.96 þ 3 5.23 þ 3 5.63 þ 3 5.32 þ 3 4.86 þ 3 1.01 þ 3 1.14 þ 3 1.25 þ 3 1.38 þ 3 1.52 þ 3 1.66 þ 3 1.84 þ 3 1.97 þ 3 2.12 þ 3 2.32 þ 3 2.51 þ 3 2.69 þ 3 2.90 þ 3 4.48 þ 3 4.75 þ 3 4.99 þ 3 5.25 þ 3 5.32 þ 3 5.39 þ 3 5.30 þ 3 5.29 þ 3

Copyright © 2002 Marcel Dekker, Inc.

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th

4.09 þ 3 3.76 þ 3 6.79 þ 3 8.74 þ 3 1.05 þ 4 1.26 þ 4 1.48 þ 4 1.58 þ 4 1.49 þ 4 1.87 þ 4 2.04 þ 4 2.16 þ 4 2.32 þ 4 2.43 þ 4 2.28 þ 4 2.15 þ 4 2.13 þ 4 2.04 þ 4 1.93 þ 4 1.76 þ 4 1.58 þ 4 1.38 þ 4 1.13 þ 4 9.09 þ 3 6.62 þ 3 5.42 þ 3 4.19 þ 3 3.17 þ 3 2.54 þ 3 2.28 þ 3 2.52 þ 3 2.69 þ 3 2.66 þ 3 2.49 þ 3 1.99 þ 3

Copyright © 2002 Marcel Dekker, Inc.

7.64 þ 3 7.74 þ 3 7.98 þ 3 8.77 þ 3 9.43 þ 3 1.04 þ 4 1.13 þ 4 1.20 þ 4 1.12 þ 4 1.24 þ 4 1.37 þ 4 1.47 þ 4 1.60 þ 4 1.72 þ 4 1.79 þ 4 1.75 þ 4 1.80 þ 4 1.82 þ 4 1.85 þ 4 1.84 þ 4 1.67 þ 4 1.61 þ 4 1.57 þ 4 1.49 þ 4 1.47 þ 4 1.29 þ 4 1.14 þ 4 1.01 þ 4 8.62 þ 3 6.67 þ 3 4.13 þ 3 3.02 þ 3 2.42 þ 3 2.59 þ 3 2.12 þ 3

6.21 þ 3 6.32 þ 3 7.14 þ 3 7.60 þ 3 7.92 þ 3 8.40 þ 3 8.95 þ 3 9.42 þ 3 9.39 þ 3 9.93 þ 3 1.06 þ 4 1.13 þ 4 1.21 þ 4 1.29 þ 4 1.35 þ 4 1.36 þ 4 1.27 þ 4 1.32 þ 4 1.37 þ 4 1.39 þ 4 1.39 þ 4 1.44 þ 4 1.50 þ 4 1.51 þ 4 1.50 þ 4 1.40 þ 4 1.38 þ 4 1.35 þ 4 1.34 þ 4 1.31 þ 4 1.21 þ 4 1.10 þ 4 9.31 þ 3 8.17 þ 3 6.44 þ 3

4.52 þ 3 4.66 þ 3 5.16 þ 3 5.45 þ 3 5.64 þ 3 5.92 þ 3 6.26 þ 3 6.58 þ 3 6.60 þ 3 7.38 þ 3 7.83 þ 3 8.29 þ 3 8.86 þ 3 9.49 þ 3 9.94 þ 3 1.01 þ 4 9.96 þ 3 1.04 þ 4 1.09 þ 4 1.12 þ 4 1.12 þ 4 1.15 þ 4 1.12 þ 4 1.16 þ 4 1.17 þ 4 1.21 þ 4 1.23 þ 4 1.26 þ 4 1.33 þ 4 1.23 þ 4 1.16 þ 4 1.17 þ 4 1.17 þ 4 1.15 þ 4 1.11 þ 4

3.13 þ 3 3.26 þ 3 3.57 þ 3 3.77 þ 3 3.89 þ 3 4.07 þ 3 4.27 þ 3 4.49 þ 3 4.53 þ 3 4.99 þ 3 5.26 þ 3 5.56 þ 3 5.94 þ 3 6.36 þ 3 6.67 þ 3 6.86 þ 3 7.20 þ 3 7.54 þ 3 7.90 þ 3 8.18 þ 3 8.36 þ 3 8.36 þ 3 8.83 þ 3 9.23 þ 3 9.30 þ 3 9.55 þ 3 8.93 þ 3 9.23 þ 3 9.68 þ 3 1.02 þ 4 1.02 þ 4 1.05 þ 4 1.04 þ 4 1.01 þ 4 8.83 þ 3

1.16 þ 4 4.21 þ 3 2.45 þ 3 2.59 þ 3 2.68 þ 3 2.79 þ 3 2.93 þ 3 3.08 þ 3 3.12 þ 3 3.39 þ 3 3.56 þ 3 3.76 þ 3 4.00 þ 3 4.27 þ 3 4.47 þ 3 4.62 þ 3 4.86 þ 3 5.11 þ 3 5.36 þ 3 5.57 þ 3 5.74 þ 3 6.09 þ 3 6.43 þ 3 6.74 þ 3 6.95 þ 3 7.24 þ 3 7.10 þ 3 7.35 þ 3 7.72 þ 3 8.14 þ 3 8.26 þ 3 7.52 þ 3 7.75 þ 3 7.88 þ 3 7.85 þ 3

7.51 þ 3 7.89 þ 3 8.81 þ 3 9.49 þ 3 8.29 þ 3 2.51 þ 3 2.01 þ 3 2.12 þ 3 2.17 þ 3 2.34 þ 3 2.47 þ 3 2.61 þ 3 2.77 þ 3 2.96 þ 3 3.10 þ 3 3.22 þ 3 3.38 þ 3 3.53 þ 3 3.70 þ 3 3.87 þ 3 4.02 þ 3 4.27 þ 3 4.51 þ 3 4.68 þ 3 4.77 þ 3 5.13 þ 3 5.34 þ 3 5.55 þ 3 5.82 þ 3 5.70 þ 3 5.72 þ 3 5.99 þ 3 6.22 þ 3 6.35 þ 3 6.22 þ 3

6.36 þ 3 6.56 þ 3 7.12 þ 3 6.10 þ 3 6.19 þ 3 6.59 þ 3 6.94 þ 3 7.32 þ 3 4.57 þ 3 1.61 þ 3 1.73 þ 3 1.87 þ 3 2.01 þ 3 2.14 þ 3 2.32 þ 3 2.42 þ 3 2.54 þ 3 2.45 þ 3 2.57 þ 3 2.69 þ 3 2.92 þ 3 3.11 þ 3 3.28 þ 3 3.28 þ 3 3.31 þ 3 3.88 þ 3 4.04 þ 3 4.20 þ 3 4.42 þ 3 4.57 þ 3 4.46 þ 3 4.57 þ 3 4.68 þ 3 4.49 þ 3 4.64 þ 3

5.21 þ 3 5.22 þ 3 5.43 þ 3 5.26 þ 3 5.47 þ 3 5.73 þ 3 5.84 þ 3 4.88 þ 3 5.01 þ 3 5.49 þ 3 5.68 þ 3 5.70 þ 3 2.84 þ 3 1.56 þ 3 1.47 þ 3 1.54 þ 3 1.62 þ 3 1.77 þ 3 1.86 þ 3 1.95 þ 3 1.99 þ 3 2.12 þ 3 2.24 þ 3 2.29 þ 3 2.36 þ 3 2.50 þ 3 2.61 þ 3 2.72 þ 3 2.88 þ 3 3.04 þ 3 3.04 þ 3 3.18 þ 3 3.30 þ 3 3.40 þ 3 3.48 þ 3

Continued

Appendix XI

Emitter wavelength (A˚) energy (keV)

Absorber

Atomic number 91 92 93 94

Element

B 6.67 þ 1a 1.83 7 1

C 4.48 þ 1 2.77 7 1

N 3.16 þ 1 3.92 7 1

O 2.36 þ 1 5.25 7 1

F 1.83 þ 1 6.77 7 1

Ne 1.46 þ 1 8.49 7 1

Na 1.19 þ 1 1.04 þ 0

Mg 9.89 þ 0 1.25 þ 0

Al 8.34 þ 0 1.49 þ 0

Pa U Np Pu

2.34 þ 3 2.57 þ 3 2.93 þ 3 3.73 þ 3

2.38 þ 3 2.30 þ 3 2.41 þ 3 2.39 þ 3

3.44 þ 3 2.70 þ 3 2.62 þ 3 2.72 þ 3

1.13 þ 4 1.09 þ 4 9.65 þ 3 6.59 þ 3

9.45 þ 3 9.39 þ 3 9.74 þ 3 1.01 þ 4

8.35 þ 3 8.36 þ 3 8.59 þ 3 8.24 þ 3

5.96 þ 3 6.03 þ 3 6.34 þ 3 6.66 þ 3

4.97 þ 3 4.96 þ 3 5.21 þ 3 5.40 þ 3

3.71 þ 3 3.76 þ 3 3.74 þ 3 3.89 þ 3

a Notation as in Appendix VI, Table 4. Source: From Veigele, 1974.

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APPENDIX XII: CORRESPONDENCE BETWEEN OLD SIEGBAHN AND NEW IUPAC NOTATION X-RAY DIAGRAM LINES Siegbahn

IUPAC

Siegbahn

IUPAC

Ka1 Ka2 Kb1 KbI2

K-L3 K-L2 K-M3 K-N3

La1 La2 Lb1 Lb2

L3-M5 L3-M4 L2-M4 L3-N5

Lg1 Lg2 Lg3 Lg4

L2-N4 L1-N2 L1-N3 L1-O3

KbII2 Kb3 KbI4 KbII4 Kb4x KbI5 KbII5

K-N2 K-M2 K-N5 K-N4 K-N4 K-M5 K-M4

Lb3 Lb4 Lb5 Lb6 Lb7 Lb07 Lb9 Lb10 Lb15 Lb17

L1-M3 L1-M2 L3-O4,5 L3-N1 L3-O1 L3-N6,7 L1-M5 L1-M4 L3-N4 L2-M3

Lg04 Lg5 Lg6 Lg8 Lg08 LZ Ll Ls Lt Lu Lv Ma1 Ma2 Mb Mg Mz

L1-O2 L2-N1 L2-O4 L2-O1 L2-N6(7) L2-M1 L3-M1 L3-M3 L3-M2 L3-N6,7 L2-N6(7) M5-N7 M5-N6 M4-N6 M3-N5 M4,5-N2,3

Source: From Jenkins et al., 1991.

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Siegbahn

IUPAC

REFERENCES Arndt UW, Brown PJ, Colliex C, Cowley JM, Creagh DC, Dolling G, Fink M, Freund AK, Hilderbrandt R, Howie A, Hubbell JH, Gjønnes, Lander G, Lynch DF, McAuley WJ, Pynn R, Ross AW, Rowe JM, Sears VF, Spence JCH, Steeds JW, Valvoda V, Willis BTM, Wilson AJC, Zvyagin BB. In: Wilson AJC, ed. International Tables for Crystallography. Vol. C. Mathematical, Physical and Chemical Tables. Dordrecht: Kluwer Academic Publishers, 1992, p 162. Auger P. J Phys 6:205, 1925. Bambynek W, Crasemann B, Fink RW, Freund HU, Mark H, Swift CD, Price RE, Venugopala Rao P. Rev Mod Phys 44:716, 1972; erratum: 46:853, 1974. Behrens P. Trends Anal Chem 11:218, 1992a. Behrens P. Trends Anal Chem 11:237, 1992b. Bethe HA. In: Annalen der Physik, Band 5. Leipzig: Verlag von Johann Ambrosius Barth, 1930, p 325. Bethe HA, Ashkin J. Experimental Nuclear Physics, Vol. 1. New York: Wiley, 1953, p 252. Birks LS. Electron Probe Microanalysis. New York: Wiley–Interscience, 1971a, p 51. Birks LS. Electron Probe Microanalysis. New York: Wiley–Interscience, 1971b, p 172. Burhop EHS. The Auger Effect and Other Radiationless Transitions, New York: Cambridge University Press, 1952. Burr A. In: Robinson JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 24. Campbell JI, Cookson JA. Nucl Instrum Methods Phys Res B3:185, 1984. Clark GL. Applied X-Rays, New York: McGraw-Hill, 1955, p 144. Clark GL. In: Clark GL, ed. Encyclopedia of X-Rays and Gamma Rays. New York: Reinhold Publishing, 1963, p 1124. Compton AH. Phys Rev 21:15, 1923a. Compton AH. Phys Rev 22:409, 1923b. Compton AH, Allison SK. X-Rays in Theory and Experiment. New York: Van Nostrand, 1935, p 537. Condon EU. In: Condon EU, Odishaw H, eds. Handbook of Physics, Part 7. New York: McGrawHill, 1958, p 7. Davis B, Mitchell DP. Phys Rev 32:331, 1928. Deslattes RD. Acta Crystallogr A25:89, 1969. Dirac PAM. The Principles of Quantum Mechanics. 4th ed. Oxford: Clarendon Press=Oxford University Press, 1958, p 136. Dyson NA. X-rays in Atomic and Nuclear Physics. London: Longman Group, 1973, p 42. Evans RD. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. XXXIV. Berlin: Springer-Verlag, 1958, p 218. Evans RD. In: Gray DE, ed. American Institute of Physics Handbook. New York: McGraw-Hill, 1963, p 8. Fink RW. In: Robinson, JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 219. Goldstein JI, Newbury DE, Echlin P, Joy DC, Fiori C, Lifshin E. Scanning Electron Microscopy and X-ray Microanalysis. New York: Plenum Press, 1981, p 105. Green M, Cosslett VE. Proc Phys Soc 78:1206, 1961. Hanke W, Wernisch J, Po¨hn C. X-ray Spectrom 14(1):43, 1985. Hasnain SS, Ed. X-ray Absorption Fine Structure, Proceedings of the Conference Held in York, August 1990. Chichester: Ellis Horwood, 1991. Heinrich KFJ. Electron Beam X-Ray Microanalysis. New York: Van Nostrand–Reinhold, l98l, p 234. Heitler W. The Quantum Theory of Radiation. 3rd ed. London: Oxford University Press, 1954, p 207.

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Hirsch PB. In: Thewlis J, ed. Encyclopaedic Dictionary of Physics. Oxford: Pergamon Press, 1962, p 800. Hubbell JH. Photon Cross Sections, Attenuation Coefficients, and Energy Absorption Coefficients from 10 keV to 100 GeV. Washington, DC: National Bureau of Standards, l969. Hubbell JH. Int J Appl Rad Isot 33:1269, 1982. Hubbell JH. Proceedings of the Workshop on New Directions in Soft X-Ray Photoabsorption, Asilomar Conference Center, Pacific Grove, CA, 1984. Hubbell JH. Report NISTIR-89-4144, U.S. Department of Commerce, National Institute of Standards and Technology, August 1989. Hubbell JH, McMaster WH, Kerr Del Grande N, Mallett JH. In: Ibers JA, Hamilton WC, eds. International Tables for X-Ray Crystallography, Vol. 4. Birmingham, UK: Kynoch Press, 1974, p 47. Hubbell JH, Veigele VJ, Briggs EA, Brown RT, Cromer DT, Howerton RJ. J Phys Chem Ref Data 4(3):471, 1975. Hubbell JH, Trehan PN, Singh N, Chand B, Mehta D, Garg ML, Garg RR, Singh S, Puri S. J Phys Chem Ref Data 23(2):339, 1994. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. New York: Marcel Dekker, 1981. Jenkins R, Manne R, Robin J, Senemaud C. Pure Appl Chem 63(5):735, 1991. Kallithrakas-Kontos N. Spectrochim Acta Part B51:1655, 1996. Kawai J, Gohshi Y. Spectrochim Acta 41B:265, 1986. Kawai J, Nakajima K, Gohshi Y. Spectrochim Acta 48B:1281, 1993. Kenney RW. In: Besanc¸on RM, ed. Encyclopedia of Physics. New York: Reinhold Publishing, 1966, p 86. Klein O, Nishina Y. Z Phys 52:853. 1929. Koningsberger DC, Prins R, eds. X-ray Absorption. Chemical Analysis Vol. 92, New York: Wiley, 1988. Kramers HA. Philos Mag 46:836, 1923. Krause MO. J Phys Chem Ref Data 8(2):307, 1979. Krause MO, Oliver JH. J Phys Chem Ref Data 8(2):329, 1979. Krishnan KS. Nature 122:961, 1928. Kulenkampff H. Ann Phys 69:548, 1922. Kulenkampff H. Ann Phys 87:597, 1928. Kulenkampff H. Handbuch der Physik, Bd. 23. Berlin: Geiger-Scheel 1933, p 433. Lagarde P. Nucl Instrum Methods 208:621, 1983. Lee PA, Pendry JB. Phys Rev B11:2795, 1975. Lifshin L. Adv X-ray Anal 19:113, 1976. Love G, Scott VD, Scanning 4:111, 1981. Love G, Cox MG, Scott VD. J Phys D 11:7, 1978. Markowicz AA, Van Grieken RE. Anal Chem 56:2049, 1984. Markowicz AA, Storms HM, Van Grieken RE. Anal Chem 57:2885, 1985. Markowicz AA, Storms H, Van Grieken R. X-ray Spectrom 15:131, 1986. McGuire EJ. Phys Rev A9: 1840, 1974. McMaster WH, Kerr Del Grande N, Mallett JH, Hubbell JH. Compilation of X-Ray Cross Sections. Lawrence Radiation Laboratory (Livermore) Report UCRL-50174, Sec. II, Rev. 1, University of California, 1969. Merzbacher E, Lewis HW. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 34. Berlin: SpringerVerlag, 1958, p 166. Mizuno Y, Ohmura Y. J Phys Soc Japan 22:445, 1967. Moseley HGJ. Philos Mag 27:703, 1914. Norrish K, Tao GY. X-ray Spectrom 22:410, 1993. Orlic I, Loh KK, Sow CH, Tang SM, Thong P. Nucl Instrum Methods Phys Res B74:352, 1993.

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Pirenne MN. The Diffraction of X-Rays and Electrons by Free Molecules. New York: Cambridge University Press, 1946. Poehn C, Wernisch J, Hanke W. X-ray Spectrom 14(3):120, 1985. Rao-Sahib TS, Wittry DB. Proceedings of the 6th International Conference on X-Ray Optics and Microanalysis. Tokyo: University of Tokyo Press, 1972, p 131. Rebohle L, Lehnert U, Zschornack G. X-ray Spectrom 25:295, 1996. Sandstro¨m AE. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 30. Berlin: Springer-Verlag, 1957, p 78. Sauter F. In: Annalen der Physik, Band 9. Leipzig: Verlag von Johann Ambrosius Barth, 1931, p 217. Sayers DE, Lytle FW, Stern EA. Adv X-ray Anal 13:248, 1970. Scherzer O. Ann Phys 13:137. 1932. Schreiber TP, Wims AM. X-ray Spectrom 11:42, 1982. Sherry WM, Vander Sande JB. X-ray Spectrom 6:154, 1977. Smith DGW, Reed SJB. X-ray Spectrom 10:198, 1981. Sparks CJ Jr. Phys Rev Lett 33:262, 1974. Sparks CJ Jr. In: Ramaseshan S, Abrahams SC, Hodgkin D, eds. Proc. Inter-Congress Conference on Anomalous Scattering held in Madrid, Spain, 22–26 April l974. Copenhagen: Munksgaard, 1975, p 175. Statham PJ. X-ray Spectrom 5:154, 1976. Statham PJ. Proc Annu Conf Microbeam Anal Soc 14:247, 1979. Stephenson ST. In: Flu¨gge S, ed. Encyclopedia of Physics, Vol. 30. Berlin, Springer-Verlag, 1957, p 337. Stobbe M. In: Annalen der Physik, Band 7. Leipzig: Verlag von Johann Ambrosius Barth, 1930, p 661. Stoev KN, Dlouhy JF. X-ray Anal 37:697, 1994. Storm E, Israel HI. Nucl Data Tables A7:565, 1970. Suzuki T. J Phys Soc Japan 21:2087, 1966. Suzuki T, Nagasawa H. J Phys Soc Japan 39:1579, 1975. Suzuki T, Kishimoto T, Kaji T, Suzuki T. J Phys Soc Japan 29:730, 1970. Szalo´ki I. X-ray Spectrom 25:21, 1996. Tertian R, Claisse F. Principles of Quantitative X-Ray Fluorescence Analysis. London: Heyden, 1982. Tertian R, Broll N. X-ray Spectrom 13:134, 1984. Veigele WJ. In: Robinson JW, ed. Handbook of Spectroscopy, Vol. I. Cleveland, OH: CRC Press, 1974, p 155. Ware NG, Reed SJB. J Phys E 6:286, 1973. Wentzel G. Z Phys 27:257, 1924. Wernisch J. X-ray Spectrom 14:109, 1985. Wernisch J, Po¨hn C, Hanke W, Ebel H. X-ray Spectrom 13:180, 1984. West RC, ed. Handbook of Chemistry and Physics. 63rd ed. Cleveland, OH: Chemical Rubber Co, 1982–83, p E-l88.

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2 Wavelength-Dispersive X-ray Fluorescence Jozef A. Helsen Catholic University of Leuven, Leuven, Belgium

Andrzej Kuczumow Lublin Catholic University, Lublin, Poland

I.

INTRODUCTION

Moseley’s law, formulated soon after the discovery of x-rays by Ro¨ntgen (Ro¨ntgen, 1896), represented the direct and definite onset of the use of x-ray spectrometry in chemistry. The first successful period was completed in the beginning of the 1920s by creating order in the periodic table of the elements and by the discovery of the missing elements (Weebs and Leicester, 1967). X-ray fluorescence (XRF) spectrometry has some unique features so that it became an irreplaceable tool for the analyst. Not may techniques had such a brilliant start to their analytical careers! gThe first commercial instrument became available around 1940 and was derived from the x-ray goniometer of a diffraction instrument. A reasonable estimation of the number of wavelength-dispersive (WDXRF) instruments working today is 15000, of which some 20% are multichannel spectrometers. In addition, about 3000 instruments working in this mode are attached to the electron microprobes, mainly in Japan. Some 3000 energy-dispersive (EDXRF) instruments are working independently, and some 1500–2000 others are attached to electron and other [e.g., proton (Uda et al., 1987, Folkmann and Frederiksen, 1990)] microprobes; a few hundred instruments are connected to synchrotrons as excitation sources. About 300 instruments are working now in total reflection (TRXRF) and several in grazing emission modes (Klockenka¨mper, 1997). The average price of a new WDXPF instrument in 1998 is of the order of magnitude of $250,000, $100,000 for EDXRF, and $200,000 for TRXRF (Claes et al., 1997) and this is not expected to rise much in the coming years. The calculation of the total money involved in the x-ray instrument market made hereafter was based on the 50% amortization estimate per instrument. We consider in our account only these parts of Particle-induced x-ray emission (PIXE), electron microscope, and synchrotron arrangements which can be treated as x-ray attachments. An auxiliary market was created also around the main equipment business, covering the spare parts, recent supplements, and service connected with the maintenance. Then, the commercial value of the instruments running today may be estimated at some US$4 billion (Markowicz and Van Grieken, 1986; Van Grieken et al., 1986). An important body of scientific literature relates to x-ray spectrometry Copyright © 2002 Marcel Dekker, Inc.

(Markowicz and Van Grieken, 1986; van Grieken et al., 1986; Jenkins, 1984; de Vries, 1986; Klockenka¨mper, 1987; Kikkert, 1983; Potts and Webb, 1992; To¨ro¨k et al., 1996, To¨ro¨k et al., 1998; Szalo´ki et al., 2000; Brill, 1996; Ellis et al., 1997 and Quisefit, 1996). Literature about essentially diffraction and scattering problems can also be very useful for the researchers working in the field of XRF (Fewster, 1996). About 10% of the WDXRF instruments are applied in geological research and prospecting (sequential and simultaneous spectrometers). Most of the simultaneous instruments are installed in industry, and the sequential instruments are distributed over industry, analytical laboratories, and research institutes at universities and public services. X-ray fluorescence spectrometry is unique in element analysis of inanimate matter: In qualitative analysis, it has unrivaled selectivity for all elements between beryllium and uranium and also for transuranium elements and an extremely wide dynamic range in quantitative analysis (ppm to 100%). Although the plasma emission spectrometry [Direct Current Plasma (DCP) or Inductively Coupled Plasma (ICP)], atomic absorption spectrometry (AAS), neutron activation analysis (NAA), and different kinds of mass spectrometry (MS) have lower or much lower detection limits, the wide dynamic range remains a unique feature of XRF (over five decades). Moreover, it possesses the scarce virtue among spectral methods—the upper useful limit reaches the concentration of several tens of percent of the total concentration for the main component. The detection limits at the ppm level without preconcentration and the limits of precision and accuracy that can be obtained (connected, in turn, to sample homogeneity and counting statistics) determine the boundaries of its application. Earlier books describing the techniques are mentioned in the reference list (Compton and Allison, 1935; Jenkins and De Vries, 1967; Birks, 1969; Dyson, 1973; Jenkins, 1974; Bertin, 1975; Tertian and Claisse, 1982; Lachance and Claisse, 1995, Jenkins et al., 1995; Buhrke et al., 1998—the latter is a kind of a guide for sample preparation in XRF and XRD, but is cited here, because the problem is very important and nontrivial). The purpose here is to describe the basic principles that allow wavelength dispersion of x-rays and to describe all the components of a spectrometer. The phenomenon of x-ray fluorescence is only a small part of a much wider problem of the interaction of charged particles or photons with matter (Feldman and Mayer, 1986). Different kinds of secondary particles are then emitted (Fig. 1a). Various combinations of exciting=emitted particles are possible, giving birth to different analytical techniques, with the combination ‘‘photons as exciting particles=emitted photons’’ as a base for the x-ray fluorescence. The angles the trajectories of the exciting or emitted particles create with the surface of the sample define normal XRF and PIXE methods and total reflection versions: TXRF (Chapter 9) and TRPIXE (Vis and van Langevelde, 1991; van Kan and Vis, 1996a, 1996b, 1997). In addition, for the x-ray fluorescence, a grazingemission method can be mentioned (GEXRF) (Sasaki, 1990; Becker et al., 1983; de Bokx and Urbach, 1995; Urbach and de Bokx, 1996; Tsuji et al., 1995; Pe´rez and Sa´nchez, 1997) (Fig. 1b). Insofar as the fundamental principles are concerned, we may recall the law of Moseley (Moseley, 1913, 1914), which is the basis of the qualitative application of XRF: pffiffiffi n¼

rffiffiffi c ¼ kðZ
sÞ l

ð1Þ

where n is the frequency (in Hz), c is the speed of light (in m=s), l is the wavelength (in m), k is the constant for a given series, and s is the screening constant. The relationship Copyright © 2002 Marcel Dekker, Inc.

Figure 1 Phenomena accompanying the interaction of x-rays with matter: (a) different applications for analytical purposes (abbreviations of methods coupled with the use of particular phenomena are mentioned; (b) different versions according to the incidence and emergence angles (I—total reflection XRF or PIXE; II—grazing emission XRF; III—grazing incidence–grazing emission XRF).

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between the wavelength of the lines belonging to the same spectral series and the atomic number is illustrated by Figure 2. For high-Z elements, the XRF spectra become more complex and full exploitation for qualitative analysis of mixtures is not always simple. Practical unraveling of spectra is dealt with in Chapter 4. The intensity of any spectral line is proportional to the number of atoms emitting photons of energies attributed to this line. However, a simple linear proportionality is not the rule. From the concepts of mass attenuation coefficients introduced in Chapter 1, it should be clear that the intensity of an analyte line of one species is attenuated by atoms of the same species and by any other atoms present in the matrix. Matrix refers to all elements present in a sample except the analyte element. Attenuation by absorption represents a first complication compromising the proportionality. If the matrix contains elements with absorption edges of slightly lower energy than the energy of the characteristic line of the analyte, strong absorption can occur. On the one hand, this results in further attenuation of the analyte line and, on the other hand, in enhancement of the spectral line related to the said absorption edge of the other matrix element. This phenomenon is known as secondary fluorescence. The process may be repeated with respect to all matrix elements with an absorption edge with a lower energy than the energy of a fluorescence line emitted by another matrix element. Secondary and higher-order fluorescence is the major complication deteriorating the simple proportionality between intensity and concentration. In Figure 3, three typical kinds of relationship between the relative intensity (measured intensity divided by the intensity obtained for the pure element) and concentration (expressed as weight fraction) are represented. Curve I is obtained when the analyte line undergoes absorption only. If enhancement occurs, the analyte line intensity is higher than expected from primary excitation only and the curve is situated above the diagonal. A similar curve, however, can be obtained when the mass attenuation coefficient of the matrix is lower than the related coefficient of the element for its own radiation, as shown in the case for Fe in FeO (curve II). In a special case, the mutual relation between attenuation

Figure 2

Representation of Moseley’s law of K and L spectral lines.

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Figure 3 Absorption phenomena in a heavy matrix (curve I), in a light matrix (curve II), and in a neural matrix (curve III).

and enhancement coefficients may be balanced such that both effects cancel precisely and then the diagonal is approximated (curve III). The effects just discussed are all energy (wavelength) dependent. This means that any calculations converting intensities into concentration should include the attenuation and higher-order fluorescence effects and also integrate over all energies present in the exciting primary beam above the value of the absorption edge and over all fluorescence lines. This is an ab initio approach for conversion of intensities into concentration implemented by several authors. We mention here the articles by Gillam and Heal (1952), Sherman (1954, 1955), Shiraiwa and Fujino (1966, 1967), who in the 1950s and the 1960s proposed suitable equations. Sparks (1976) and Li-Xing (1984) implemented small corrections and refinements concerning the details of these expressions, and de Boer (1990) and de Boer and Brouwer (1990) gave solutions to some difficult exponential integrals related to the enhancement. Important contributions to the numerical side of quantitative XRF are due to Fernandez (1989), Fernandez and Molinari (1990), Rousseau et al. (1996), and Mantler (1984). The first two authors prepared commercial, customer-oriented versions of the numerical programs XRFPc and CiROU, CiLAC, CiLT, and CiREG, respectively, and Mantler paved the way for the analyses of multilayers. It should be emphasized that Fernandez solved the transport equations for x-ray beams, the very general and powerful but difficult tool for the description of particle beams. Likewise, Gardner and Hawthorne (1975) obtained similar results by Monte Carlo simulation of x-ray excitation. A specific approach was presented by Dane et al. (1996). These authors solved the problem of quantification in situations where either an incomplete data set is available or where a reasonable assumption about composition and=or depth profile cannot be done. Then, the idea of processing whole sets of solutions (populations or strings or generations) has been introduced, instead of processing the particular concentration values in consecutive iterations. In the fundamental parameter methods, the incidence and takeoff angles are the important parameters. As a rule, they are kept constant during the traditional run of the analysis. However, there have been significant efforts to use the angle-resolved version Copyright © 2002 Marcel Dekker, Inc.

of x-ray XRF spectrometers for depth profiling of the samples (Gries, 1992; Schmitt et al., 1994). For a complete treatment of equations deriving the line intensities in x-ray fluorescence spectrum from the elemental sample composition, the reader referred is to Chapter 5.

II.

FUNDAMENTALS OF WAVELENGTH DISPERSION

The crystal monochromator is the heart of a wavelength-dispersive spectrometer. Wavelength dispersion of electromagnetic radiation in the x-ray region cannot be performed, as a rule, by normal gratings but only by diffraction on crystals or, for the long-wavelength regions, on multilayers. We briefly explain the principles because the construction features of the monochromator are directly derived from them. Consider a monochromatic beam of x-rays of wavelength l with their electrical vectors of equal amplitude in phase along any point of the direction of propagation. Assume further that the beam is parallel and is incident on a crystal at an angle W between a given crystal plane (and all the planes parallel to this first) and the incident beam direction. The beam is scattered and diffracted rays of equal l result but interfering constructively only in those directions for which the phase relationship is conserved. This happens at an angle W for scattered rays 1 and 2 (Fig. 4), for which the path difference ABC of ray 2 with ray 1 is equal to an integral number n of wavelengths. From Figure 4, it is clear that AB þ BC ¼ d sin W þ d sin W ¼ nl or, according to Bragg and Bragg (1913), who first formulated this relation, written as nl ¼ 2d sin W

ð2Þ

where n denotes the number of wavelength differences between the rays scattered by the adjacent planes. Following the recommendation of the IUPAC (Freiser and Nancollas, 1987), n is the order of diffraction. If n ¼ 1, the difference is one wavelength and the diffraction is said to be of first order. If n ¼ 2, the difference is two wavelengths and the diffraction is second order, and so on. All x-rays emitted at angles different than W cancel because they are out of phase and destructive interference occurs. From Figure 4, one can note that in the Bragg scattering,

Figure 4

Derivation of the Braggs’ law. ABC is the path difference.

Copyright © 2002 Marcel Dekker, Inc.

the incident and exit angles are equal and, in this sense, there is some analogy to mirror reflection in the classical optics. However, diffraction is by no means equal to reflection in classical optics, because the diffraction is a volume not a surface process. It is less efficient (with a great loss of intensity) and is performed only under particular angular conditions; that is, according to Bragg’s law. Full analogy between the reflection of x-rays and the reflection of optical rays happens only at grazing incident angles (i.e., smaller than the critical Snell’s angle). This is intrinsically a consequence of the refraction index, which is slightly smaller than 1 for x-rays. There are continuous efforts to construct a real mirror on the basis of multilayers, which would efficiently reflect x-rays under large incident angles, at least for soft x-rays (Kearney et al., 1991). Thus, the static condition for obtaining diffraction of a monochromatic x-ray beam in some direction in the volume surrounding the analyzing crystal is given. What happens in the case of a polychromatic beam of x-rays? For a crystal, one set of planes is selected (for different reasons) and d is constant. If only first-order diffraction is considered and constructive interference must be realized for all l present in the incident beam, then W is the only variable: l ¼ ðconstantÞðsin WÞ

ð2aÞ

The signal arriving from the diffraction angle W is detected by a detector placed on a goniometer arm. The detector rotates around an axis through the macroscopic plane of the analyzing crystal. For a source at a fixed position, the detector rotates over an angle 2W. The wavelength is calculated from constant ¼ 2d and sin W. Note that this holds only for the first-order diffraction. If the second-order diffraction is used, the wavelength is equal to half of that value. The maximum wavelength lmax that can be diffracted in the first-order diffraction by a crystal is equal to 2d because sin W  1. Because wavelength and energy are related, one more equations must be given, allowing us to convert wavelengths into energy units: E¼

12398:5 l

ð3Þ

where the energy E is in electron volts (eV) and l is in angstroms (A˚). It is a common practice that the units used by x-ray spectroscopists are still electron volts or kiloelectron volts for energy and angstroms for wavelength. When joules and nanometers are used, as required by the international rules, the numerical value of the conversion constant becomes 1.98645610
16 (nm=J). The presence of different wavelengths of different order on the same goniometer position has a particular consequence for wavelength-dispersive spectrometry. For a given position of the goniometer (and detector), one may have a first-order wavelength, say 0.6 A˚, second-order diffraction of l ¼ 0.3 A˚, third-order diffraction of l ¼ 0.2 A˚, or values of energy 20.66, 41.32, or 61.98 keV, respectively. Figures 5 and 6, reproduced from the work of Arai (1987) demonstrate a practical situation and Table 1 reprints a fragment from x-ray tables by Cauchois and Senemaud (1978). Similar compendia of spectral lines and attenuation coefficients are easily available (Birks, 1974), as a rule covering the range of elements between lithium (Z ¼ 3) to plutonium (Z ¼ 94). Much more controversial are the values of the mass attenuation coefficients for low energies and for low-Z elements at the same time (Henke et al. 1982, 1988). If necessary, one can have access to the calculation for even heavier elements (Soff et al., 1977). It is easy to understand the practical value of such calculations for the analysis. However, it is much more difficult to estimate the accuracy of numerical calculations for transuranium elements. To sort out x-rays with Copyright © 2002 Marcel Dekker, Inc.

Figure 5 Higher-order overlapping of L series x-rays of Cr, Fe, and Ni to Ka of carbon. (From Arai, 1987.)

Figure 6

Overlapping of CKa by higher-order x-rays of Ni, Fe, and Cr. (From Arai, 1987.)

wavelengths belonging to different spectral orders, a pulse-amplitude selector or detector is needed that can make the distinction between energy levels. A detector is a device of which the principle can be most easily explained by assuming the particle character of electromagnetic radiation. This is one reason that the characteristic of the impacting photons is often expressed in terms of energy, not wavelength. Copyright © 2002 Marcel Dekker, Inc.

Table 1

Fragment of Detailed Tables of X-ray Lines

l (uX) 1251.56 1251.64(a) 1252.192 1252.58 1252.6 1252.82 1255.17 1255.43 1255.50 1256.60 1256.71 1257.1(a) 1257.12 1257.66 1258.74 1259.210 1259.8 1260.30 1260.6 1260.620(a) 1261.15(a) 1261.23 1261.956 1262.68 1263.086 1263.428(a) 1264.6 1264.99(a) 1265.0 1265.24(a) 1268.520(s) 1268.66 1268.68(a) 1270.2 1270.35(a)

l (mA˚) 1254.19 1254.27 1254.824 1255.21 1255.2 1255.45 1257.81 1258.07 1258.14 1259.24 1259.35 1259.7 1259.76 1260.30 1261.39 1261.856 1262.4 1262.95 1263.2 1263.269 1263.80 1263.88 1264.608 1265.33 1265.741 1266.083 1267.3 1267.65 1267.7 1267.90 1271.186 1271.33 1271.35 1272.9 1273.02

Element 42 58 62 73 90 73 73 32 69 75 73 71 90 73 54 68 90 74 90 58 64 73 42 90 42 58 74 70 91 69 92 90 68 74 64

Mo Ce Sm Ta Th Ta Ta Ge Tm Re Ta Lu Th Ta Xe Er Th W Th Ce Gd Ta Mo Th Mo Ce W Yb Pa Tm U Th Er W Gd

Transition

Notation usuelle

Ordre

KMIV KMIV KLII abs. LIII LINIV LIIIOIV,V LIIINVI,VII KLII KLI LIMII LIIIOIII LIINI abs. LII LIIIOII KLII KLIII LIIPII,III LIMIII LIIPI KMIII KMIV,V LIIIOI KMIII LIIOIV KMII KMII LIIINIII LIINIV LIINIV LINIII LIINI LINIII LIOIV,V LIIMV KMIII

KbII5 KbII5 Ka2 — — Lb5 Lb7 Ka2 — Lb4 — Lg4 — — Ka2 Ka1 — Lb3 — Kb1 Kb5 Lb7 Kb1 Lg6 Kb3 Kb3 — Lg1 Lg1 Lg3 Lg5 Lg3 — — Kb1

2 4 4 1 2 1 1 1 5 1 1 1 2 1 3 5 2 1 2 4 5 1 2 2 2 4 1 1 2 1 2 2 1 1 5

E (keV)

n=R

ðn=RÞ1=2

9.8776

725.99

26.9441

9.8758 9.8573 9.8552

725.85 724.49 724.34

26.9416 26.9163 26.9135

9.8460 9.8452 9.842

723.66 723.60 723.4

26.9010 26.8998 26.8956

9.8377

723.05

26.8897

9.8171

721.54

26.8615

9.8099

721.01

26.8516

9.784 9.7807

719.1 718.86

26.8158 26.8116

9.7788

718.72

26.8090

9.7523 9.741

716.77 715.9

26.7726 26.7566

Source: From Cauchois and Senemaud, 1978.

The differences between wavelength-dispersive (WD) and energy-dispersive (ED) XRF are due not only to the different detectors but also to other factors: 1. The brightness of a WD spectrometer is very low, the attenuation in a crystal being responsible for an important part of losses. This problem may be overcome by the use of radiation sources of significant intensity, the synchrotron being the best known but bound to the availability of this large facility. 2. The crystal is only the dispersive device, not the detecting device. The situation is different in EDXRF, for which detectors play a double role: as the dispersive device and as the detector at the same time. Copyright © 2002 Marcel Dekker, Inc.

3.

4.

Because Bragg’s law is geometric in character, the angular conditions for the collimation of primary and secondary beams are very severe for WDXRF (8– 25 msr), not the same as for EDXRF (150 msr) (Wollman et al., 1997). This restriction on the geometrical efficiency still makes the overall photon collection efficiency of a wavelength-dispersive spectrometer worse. An attractive aspect of EDXRF is the simultaneous collection of the whole spectrum, whereas a typical WDXRF device is exclusively sequential. However, the maximum count rate for an EDXRF instrument is 30 kcps for the whole spectrum, which severely limits the total number of accumulated counts and, consequently, limits the precision (counting statistics). Similarly, the limitation on the total count rate leaves a small margin for trace element analysis. From the assumption, the trace element, if present in a sample, can emit only a very small part of the total radiation emitted by a whole sample (and there is also the background, participating in the total amount of counts).

Simultaneous WD instruments are composed of a series of individual crystal spectrometer (channels) operating simultaneously, but the number of channels is limited. A comparison of wavelength- and energy-dispersive versions of x-ray spectrometers is performed in a recent publication by Brill (1996).

III.

LAYOUT OF A SPECTROMETER

The main parts of a spectrometer can be represented in the simplest form by Figure 7. The analyzing crystal is the central point of the wavelength-dispersive instrument. On the lefthand side of the crystal, we find (1) the source of excitation, (2) the filters and devices for shaping the exciting radiation (collimators and masks), and (3) the sample. On the righthand side, we find (4) devices for shaping the diffracted beam (collimators) and (5) the detector. Signals from the detector are fed into the electronic circuitry where they are shaped to be processed by the computer software for data analysis. This arrangement can be reduced or made more complicated according to the demand. However, all possible reductions lead to less flexible devices, and complication does not necessarily enhance the quality of the instrument.

Figure 7 A wavelength-dispersive spectrometer: FC ¼ proportional flow counter; SD ¼ scintillation detector. (Reprinted with permission of Siemens AG.)

Copyright © 2002 Marcel Dekker, Inc.

A.

Sources

A variety of radiation sources, emitting either charged particles or g- or x-rays of sufficient energy, are used for excitation of some or all elements of the periodic table and some or most of the spectral lines of analytical interest. Other chapters deal with excitation by protons [proton-induced or sometimes particle-induced x-ray emission (PIXE) (Chapter 12)], by electrons [electron microprobe (Chapter 13)], or by x-rays emitted from secondary targets or from a synchrotron (Chapter 8). Excitation by x-rays or soft g-rays from radioisotopes and x-rays from low-power tubes is mainly restricted to energy-dispersive spectrometers (Chapter 3). The ideal excitation source would be a tunable x-ray laser (monochromatic and intense, allowing the best choice of exciting wavelength and often selective excitation), but this cannot be expected in the near future. Nevertheless, it is worth reading some treatises about recent and future progress in the field (Nagel, 1982; Jamelot, 1995; Fill, 1995; Crasemann, 1994; London, 1993). There are serious reasons for the slow progress in x-ray laser construction. The primary reason is that the population inversion of electrons means a much larger deviation from the energetical equilibrium if done between the levels, allowing the emission of x-rays. Consequently, it demands an excessive pumping power, turning the pumped matter into a plasma. One must be aware that the x-ray laser would be even much more useful in the fields of x-ray microscopy, holography, litography, or for the research of time-resolved phenomena than as a source for XRF analysis on a routine basis. The synchrotron is another modern source of x-ray radiation, but it cannot be considered as a tabletop instrument and a device for easy, inexpensive, and routine-based applications. From a practical point of view, vacuum x-ray tubes are the overwhelming choice among other potential excitation sources. High-power tubes are the only ones dealt with in detail in this chapter; low-power tubes are discussed in Chapter 3. All modern tubes owe their existence to Coolidge’s hot-cathode x-ray tube as presented in Physical Review some 85 years ago (Coolidge, 1913). They essentially consist of a sealed glass tube containing a hot tungsten filament for the production of electrons, a cooled anode, and a beryllium window. From a variety of modifications proposed over more than three-quarters of the century, two geometries have emerged as the most suitable for all practical purposes [the end-window tube (EWT) and the side-window tube (SWT)], but now the preponderance of the EWT pushes the SWT out of the market. Perhaps, a future deeper orientation of the WDXRF toward the microprobe applications will renew the interest in SWT, which is much better in deriving parallel x-ray beams. The general requirements of the x-ray tube as a source are as follows: 1. Sufficient photon flux over a wide spectral range, with increasing emphasis on the intensity of the long-wavelength tail of the white spectrum. The actual vivid interest in low-Z element analysis will certainly activate research in this direction. 2. Good stability of the photon flux (< 0.1% at least). Long-term stability reduces the frequency of recalibration in routine analyses; short-term stability is an absolute requirement for obtaining an acceptable precision. 3. Switchable tube potential (10–100 kV), allowing the creation of the most effective excitation conditions for each element. Still, in more high-power constructions available on the market, putting a filter in the beam path is an easier solution than switching the voltage (Shimadzu XRF-1700). For sure, the selection of the spectral region for the targeted excitation by the use of a single x-ray tube will never be as good as in dedicated synchrotron beam lines with Copyright © 2002 Marcel Dekker, Inc.

4.

monochromators of the adequate quality, but the low cost of tube operation is an obvious advantage. The intensity of the analyte lines varies considerably with excitation conditions. An extreme example is given by Vrebos and Helsen (1985a, 1985b) for simulated Al–Mo alloys. Freedom from too many interfering lines from the characteristic spectrum of the tube anode.

Freedom from interfering lines is important. The scattered characteristic lines of the anode may spectroscopically interfere with analyte lines, disturbing the qualitative recognition, the peak intensity estimation for the line of interest, and the accuracy of subsequent conversion of intensity to concentration. This is found in the results obtained after correction by some algorithms as well as by fundamental parameter calculations. Although there is some kind of a remedy by the use of more efficient spectral deconvolution methods (Remond et al., 1993; 1996), it occurs at the expense of a greatly increased computational effort and depends heavily on the assumptions made a priori. An x-ray tube is characterized by its anode element (a single element or two elements as in dual-anode tubes), its input power [expressed in watts (W) or kilowatts (kW), typically between 0.2 and 5 kW for high-power tubes], the voltage range between anode and cathode (10–100 kV for a SWT; limited to 60 kV for an EWT for the majority of commercial instruments), tube current [milliampere (mA), typically up to 60 mA or sometimes to 150 mA for high-power tubes], and an open (SWT) or closed (EWT) anode cooling circuit. The photon output of the tube or, more importantly, the photon flux hitting the sample (expressed in counts per second per watt of input power) is determined by a, the incidence angle of an electron beam on the anode, the takeoff angle b (for SWT), the distance t to the beryllium window, the thickness d of the beryllium window, and the distance between window and sample, t0 (Fig. 8). The radiation output power is rather poor with respect to the input power and is of the order of 1%, making the device a very inefficient transformer of electron current to electromagnetic radiation. For an energy balance, see Bertin (1975) and some remarks by Wollman et al. (1997). The impact of the electrons creates an excitation volume from which the white and the characteristic radiation of the target escape. This phenomenon represents the first distinct difference between a SWT and an EWT. As shown in Figure 9a, the ‘‘escape’’ path of the photons in SWT geometry is, on average, less than in the EWT geometry. An immediate consequence of this characteristic is that dual anodes are used only in a SWT, where a light element (e.g., scandium) covers a heavier element (e.g., molybdenum). By switching the excitation voltage, x-rays are produced either in the upper (lighter) element layer or in the substrate (higher-Z element), resulting in two distinct tube spectra with different yields in the low- and high-wavelength regions (Fig. 9b). The popular dual-target tube arrangements are now Rh=Cr and Rh=W. The smaller the distance t, the higher the output. A decrease of t to half of the original value increases the intensity roughly by a factor of 4. However, the reduction of t in the classical setup is limited. The smaller the value of t, the more intense the bombardment by the electrons and subsequent heating of the window by the scattered electrons (Fig. 8). In a SWT, this bombardment is rather intensive because both the anode and the window are in this geometry at ground potential. In a EWT, to the contrary, the filament and the window are both at ground potential, and heating of the window is negligible. For an EWT, however, t can be reduced to a certain extent only because it faces the anode at high potential (Fig. 10). Copyright © 2002 Marcel Dekker, Inc.

Figure 8

Details of the geometry of a side-window x-ray tube and sample.

1. Alternative Con¢gurations To overcome the absorption of the low-wavelength tail of the continuum, a windowless configuration was considered. This is an obvious solution, but it requires that the whole spectrometer with sample, collimators, crystals, and flow counter be evacuated to the low pressure suitable for securing an acceptably long life for the tube filament and conducting analyses in the required range of the soft x-rays. Nordfors (1956) advocated a dual-anode tube with two anodes physically separated. They were excited by deviating the electron beam to either of the two anodes (Fig. 11). Lack of stability prevented this solution from gaining commercial interest. Probably, a modified version of this idea was introduced in the series 2000 x-ray spectrometers of Diano (dual target þ dual filament; see Section VI). Tubes with exchangeable anodes are another possibility for solving the problem of comfortable switching between different anodes. To the best of our knowledge, this solution is not commercially available because of the tedious exchange operation and difficulties in rigorous repeating of previous conditions of the tube action. Using the sample as an anode is another solution, but then the application is bound to conducting materials. In a modified form, this solution is implemented in scanning electron microprobes. Because this ‘‘anode’’ cannot be adequately cooled, the input power must be restricted to very low values, entailing low count rates and, consequently, reducing the precision that can be obtained within reasonable counting times. Another possibility is to apply an existing 18-kW rotating anode x-ray tube generator. Sufficient intensity is obtained. Part of this gain can be sacrificed and Boehme (1987) and Nichols et al. (1987) introduced the microdiffraction beam collimator into the beam Copyright © 2002 Marcel Dekker, Inc.

Figure 9 (a) Principle of excitation in a side-window tube. (b) Principle of excitation in a dualanode x-ray tube (side window); Mo, W, or Au (for A) and Sc, Rh, or Cr (for B) are commercially available anode components. The spectral feature of the tube spectrum depends on the energy of the incident electrons.

derived from such a high-power tube. This collimates the beam to about 30 mm and allows microfluorescence. Squeezing the beam to more or less the same diameter is possible when applying the polycapillary lens, even more so with the single capillary. These solutions are very restrictive for the beam intensity; thus, a reasonable device configuration includes an energy-dispersive detector instead.

2. Optimization Because the takeoff angle is a very important parameter, the surface of the anode can be formed in steps in such a way that the yield of low-wavelength radiation is optimized. As is clear from Figure 12, the step-shaped surface decreases the escape depths. In a conventional EWT, the reduction of the anode to window distance t is limited (Fig. 10). According to an old idea of Thordarson (1939), Botden et al. (1952) developed an x-ray tube with a gold anode evaporated directly on the beryllium window. The tube was used for surface radiotherapy. After many years, Phillips returned to Botden’s construction and again reduced t to zero by sputtering the anode target element directly on the beryllium window (Fig. 13). In a SWT, the anode is at ground potential, which Copyright © 2002 Marcel Dekker, Inc.

Figure 10

Principle of end-window x-ray tubes.

Figure 11 X-ray tube with two separate anodes: (1) anodes: (2) filaments; (3) cooling; (4) separating brass wall; (5) insulator. (After Norfords, 1956.)

Figure 12

X-ray tube with stepwise configuration of the anode.

considerably facilitates and simplifies the safety requirements. This geometry enhances the intensity by an order of magnitude, allowing a much lower input power: 200 W for an intensity equivalent to the earlier 3-kW EWT. As a result of the low input power, the heat produced in the target by the electron beam is easily dissipated and no supplementary water cooling is required. The tube construction is reduced and only the most basic components are left. The commercial name for this development is T3 or TTT (target transmission tube; Fig. 14c). It is used exclusively in simultaneous WDXRF instruments, especially in the new version of the PW1660 (Philips Technical Materials Simultaneous X-Ray Spectrometer System PW 1660) (see Section VI). Copyright © 2002 Marcel Dekker, Inc.

Figure 13

Anode elements deposited on the beryllium window.

3. End-WindowTube The essential parts of an EWT are displayed in Figure 14a. The ring-shaped cathode and the cooling circuit of the anode are typical features of EWT construction. The latter consists of a closed circuit filled with deionized water. Deionized water is necessary to reduce the conductivity and enhance the safety because the circuit is at anode potential! Electrons are diverted to the anode surface by electron-optics elements. Because the filament and window are at equal potential, no electrons hit the window. Heating is mainly by radiation from the anode, and an external cooling circuit is provided, which may be connected to normal water supplies because it is not in contact with a voltage source. Although the construction just described is in a common use now, it cannot be considered as an ideal source of x-rays. Recently, van Sprang and Bekkers (1998) and Kuczumow et al. (2000) have discussed the deviations from the homogeneity in the endwindow x-ray tube output. The geometry of the excitation system, as described by an angle supported on the tube axis – the sample point – the goniometer entrance axis, deviates from the formal geometry (about 83 ). The ‘‘real,’’ not formal, geometry of the device can be read out from the shift in the Compton-scatter energy. It creates the special demands for all the constructors of microprobes and milliprobes. They must avoid using the conventional end-window tube. There are also the consequences for the normal analysis: The uniformity of the samples and spinning operation are essential for getting reproducible results. Otherwise, particular spots on a surface of the sample can be irradiated in a different manner. 4. Side-WindowTube An expanded view of a SWT is shown in Figure 14b. The essential differences from an EWT are the distance between cathode and anode, an earthed anode, and a single cooling circuit for the anode only. This circuit may be directly connected to the normal water supply because it is at ground potential. The geometric arrangement of the cathode supports higher potentials of the cathode with respect to the anode. This is the basic reason that tube potentials of 100 kV are allowed. Side-window tubes with fine-focus anodes are used in many microprobe constructions. They are appreciated for their photon output, easy to transform into a parallel, collimated beam. A consequence of the regulated anode potentials is the possibility of use of dual anodes. The available dual-anode tubes are summarized later in Table 10. 5. Tubes with Rotating Anode The electrical input in an x-ray generator is high; however, a final output of about 1% of primary power is not very impressive. This must be due to the low efficiency of conversion of the electrical energy into the energy of x-ray photons. One should remember that photon losses are even much greater in further parts of the spectrometer (especially Copyright © 2002 Marcel Dekker, Inc.

Figure 14 Different types of construction of commercial x-ray tubes: (a) end-window tube (EWT) (courtesy of Siemens AG); (b) side-window tube (SWT) (courtesy of Philips Analytical); (c) target transmission tube (TTT) (courtesy of Philips Analytical); (d) tube with a rotating anode (Vekemans 2000 courtesy of B. Vekemans).

Copyright © 2002 Marcel Dekker, Inc.

Figure 14

Continued

in collimators and crystals). The desparate need for more efficient excitation sources emerges with many more subtle applications of x-rays: the microprobe constructions, (micro)diffraction experiments with organic or biochemical materials, research on the fine structure of the absorption edge, focusing of x-rays, or the application of the total reflection, for example. If one does not have access to the synchrotron, no patience to wait for the beam time or no money for paying great sums for simple analyses, then the tube with a rotating anode seems to be the best solution. Copyright © 2002 Marcel Dekker, Inc.

Figure 14

Continued

According to our knowledge, three companies, Rigaku, Siemens, and Enraf-Nonius, constructed high-power tubes with a rotating anode. The anode, in the shape of a wide cylinder placed on a narrower cylindrical rotary shaft, can rotate with a speed of 100 rps. Mo, Cu, and Ag are the materials most frequently applied for the construction of the anodes. The sides of the anode are under intensive electron bombardment, but the high heat load is dissipated by the intensive rotation coupled with cooling by water, flowing inside the anode with a rate of 8–15 L=min. The system is evacuated by a prevacuum oilrotary pump followed by a turbo-molecular pump. It enables one to work with a voltage of 60 kV and a current of 300 mA, sometimes even greater. It gives a power of the order of 20–30 kW. Recently, the most powerful generator available is the Ru-1500 by Rigaku with a 90-kW load. The application of power of the order of 20–30 kW means roughly a 10-fold increase in the intensity as compared to the conventional x-ray tubes. When coupled with a Kumakhov lens, rotating anode tubes can be used as a powerful microfocusing source. 6. Compact Flash X-Ray Sources An x-ray tube is a very conventional x-ray source, dating back to the times of Ro¨ntgen and Coolidge. The efforts described earlier aimed at a better and more efficient construction of the x-ray tube. Different versions of the tube were by no means the only solution for constructing a better and more convenient source. The most pronounced progress of the last decades is the synchrotron, but it is still far from an ideal device in terms of price and size. Much more promising for most XRF laboratories seems to be a compact flash x-ray source (Germer, 1979). Such sources are now being developed intensively, giving, for example, a flux of photons with energies between a few kiloelectron volts to 200 keV, in pulses lasting between 10 and 50 ns, with a repetition frequency up to 50 Hz. Doses are very intensive, up to several roentgens per shot (Pouvesle et al., 1996). Primary x-rays from Copyright © 2002 Marcel Dekker, Inc.

such sources can be used for the excitation of x-ray fluorescence, being especially useful for real-time monitoring of kinetics of fast processes in ballistics and in the simulation of accidents. It seems that the polycapillary x-ray focusing waveguide would be an ideal supplement to this kind of x-ray source, giving the impetus for the development of a new generation of pulse x-ray spectrometer. This device would combine the time resolution of the source with the spatial resolution of the focusing polycapillary semilens. 7. High-Voltage Generator A high-voltage generator supplying up to 60 kV for an EWT or up to 100 kV for a SWT and an output power of 3–4 kW (or much lower, 0.2 kW for TTT tubes) is required. A special generator is needed if one wants the high-power rotating anode tube. The conventional power supplies were very cumbersome because of the size of the transformer. The new generation belongs to the so-called switched power supplies. Such a power supply is basically a dc=dc converter in which a dc voltage is electronically switched at high frequencies (several kilohertz) and fed into an inductance–capacitance network. The output power is not continuously regulated but is controlled through pulse-width modulation. One possible scheme of a switched power supply is given in Figure 15. The mains are rectified (a), high-frequency switched (b), transformed (c), and rectified and smoothed (d). The output voltage is sensed (e) and compared to a reference voltage (f), and this signal monitors the pulse-width modulator (g), which, in turn, commands the switching circuit (b). In the high-frequency transformer, a ferrite core is used; this is a lightweight component. The whole setup allows the size of the generator to be reduced to a fraction of the volume of classical generators. The whole system is generally monitored by microprocessor. Power switching of the tube is done preferentially along the isowatt curve (constant input power), all under microprocessor control. The system has a number of safety switches on flow and temperature controllers of cooling water with microswitches on all panels, which when

Figure 15

Principle of switched power supply: dc=dc convertor.

Copyright © 2002 Marcel Dekker, Inc.

removed could expose the customer either to the electrical circuitry (high voltage of the tube supply or the mains voltage) or to unacceptable radiation levels. B.

X-ray Tube Spectrum

The first attempts to propose a quantitative description of the tube intensity as a function of wavelength were made by Kulenkampff (1922) and Kramers (1923). The quantitative treatment of intensities of fluorescent x-rays (obtained by polychromatic excitation) by fundamental parameter programs requires knowledge of the intensity function of the exciting beam. This explains the continuous interest in the rigorous functional or numerical description of the tube spectrum. The description of the tube spectrum is divided into two parts, the first concerning the continuum spectrum [bremsstrahlung; see Seltzer and Berger (1985)] and the second concerning the characteristic lines of the anode material superimposed on the continuum. The relative importance of these two contributions depends on the target element: For a tungsten target, the characteristic L lines constitute the minority of the emitted radiation, about 24%, and the K lines from the copper tube represent about 60% of the emitted radiation at 45 kV. The most popular Rh x-ray tube derives about 4% of the emitted x-rays as RhK and 20% as RhL photons; the rest is emitted as bremsstrahlung (45 kV). The positive side of the latter tube is its relatively good output of the low-energy photons, with a possibility of the excitation of the lines of heavier elements by RhK lines. In the continuum research, the main effort was focused on the accuracy of Kramers’ law. Some deviations of this law from experimental results were corrected by the introduction of a nonintegral exponent m in the wavelength-dependent term in Kramers’ law (Brunetto and Riveros, 1984):
 Zr l Nl / 2
1 Kramers law l l0
 m Zr l
1 Brunetto’s result Nl / 2 l l0

ð4aÞ ð4bÞ

where l0 is the wavelength at the short-wavelength (high energy) end of the continuum and r and m are constants related to the power-dependent terms: atomic number Z and energy function, respectively. The constants were determined by fitting the theoretical curves to the experimental results. Brunetto and Riveros’ result was confirmed later by Tertian and Broll (Tertian and Broll, 1984). By no means does it exhaust the solutions; Many other descriptions of bremsstrahlung spectra have also been published; to mention only the best known, there are those by Rao-Sahib and Wittry (1972), Reed and Ware (1973), Fiori et al. (1976), and Statham (1976a, 1976b). Finally, the reader can find the interesting articles by Small et al. (1987) and Trincavelli et al. (1998) summarizing the state of the art in this area. Other corrections must be added to Eqs. (4): the absorption correction [often that of Philibert; see Pella et al. (1985) and Markowicz and Van Grieken (1984)] and the exponential correction term including the absorption of the beryllium window. Furthermore, in the article by Pella et al. (1986), an interesting algorithm was proposed for taking the characteristic line intensities into account. For this purpose, these authors made use of the equation proposed by Green and Cosslett (1961). The ratio of peak to background intensity was found to be a function of overvoltage U0 and atomic number Z: Copyright © 2002 Marcel Dekker, Inc.

"
 
2 #
Nchar U0
1 a U0 ln U0
1 ¼ exp 0:5 þ d Ncon U0
1 1:17U0 þ 3:2 b þ Z4

ð5Þ

Overvoltage U0 is the ratio of the initial electron energy E0 to the excitation energy of a given shell Eex . The symbols a, b, and d are experimentally determined constants. The intrinsic merit of this formula is its dependence on the physical parameter U0 and on Z [see also the work by Murata and Shibahara (1981) with Monte Carlo estimations of the penetration depth of electrons]. Recently, a series of papers on the numerical description of the tube output has been published by Ebel et al. (1989) and Schoßmann et al. (1995, 1997). The solutions were derived from the different depth distribution functions approximating the electron penetration in anode materials. Apart from all efforts concentrating on the numerical estimation of x-ray tube emission, experimentally recorded spectra may always be used [collected in Gilfrich and Birks (1968), Gilfrich et al. (1971), Gilfrich (1974), Brown et al. (1975), and Loomis and Keith (1976)]. The same kind of calculation as for the spectral x-ray tube output is very helpful for the description of the background in wavelength-dispersive XRF analysis. The background is mainly determined by the scattered bremsstrahlung radiation from the tube. The theoretical estimation of the continuous radiation should always be helpful for solving background problems (Arai and Omote, 1988; Omote and Arai, 1989; Arai and Shoji, 1990; Arai, 1991). Sometimes, the knowledge of background can be applied for analytical aims (Kuczumow et al., 1992). In this case, a selected bremsstrahlung channel provides quantitative (but not qualitative!) information about the sample contents. A similar approach has been applied in an article by Kuczumow et al. (1995) for the extraction of the information included in the coherently scattered signal in the energydispersive mode on the capillary version of x-ray spectrometry. A similar extraction of the information from the Rayleigh signal in wavelength-dispersive XRF and from the selected bremsstrahlung channel in the electron microprobe is possible as well (Kuczumow et al., 1999). It concerns special samples (e.g., the biological ones) and the information obtained is connected with the description of the matrix. It brings additional knowledge about the density of matrix, which is sometimes of essential significance in data evaluation. The articles just mentioned described the analysis of tree rings and other similar periodic structures, where the variability of the density of the matrix was, by far, more important than the variability in the chemical composition.

C.

Collimators and Masks

The wavelength-dispersive mode of operation depends strongly on rigorous geometric constraints on goniometer construction and analyzing crystal. Ideally, only a parallel beam of fluorescence radiation should be diffracted and stray radiation should be absent. Stray radiation enhances the background. Diverging beams result in worsening of the spectral resolution. Converging beams act in the same direction, but the latter are often necessary for making microanalysis. If applied in such conditions, they demand a special numerical treatment to obtain analytical results with a given level of confidence. Useful versions of fundamental parameter calculations for converging beams are now available (Chang and Wittry, 1994). Copyright © 2002 Marcel Dekker, Inc.

Masks, which reduce the size of the fluorescent beam in front of the collimator, cut off x-rays emitted or scattered by an area larger than the sample area. Three positions for collimators in the optical path of the spectrometer are possible: (1) between the sample and the crystal, (2) between the crystal and the detector, and sometimes (3) an auxiliary position between two detectors working in tandem. Collimators are not the ideal devices because they deliver a divergent photon beam with the angle of divergence a given by
 a a ¼ arctan ð6Þ l where a is the spacing between the blades and l is the length of the blades. In addition to divergence, there is also a substantial loss of counts while passing the collimator; for example, for a radiation of about 10 keV passing by a typical 10060.25 mm Soller collimator, the transmission is close to 10
5. On the bonus side of the use of collimators, it must be said that the resolution is improved very substantially (see Fig. 16). Thus, one needs to find a compromise between better resolution and higher brightness. In the determination of light elements, for which the intensity is relatively low (because poor excitation and efficient attenuation) but relatively well-spaced spectral lines exist, the demand for intensity prevails over resolution and, whenever possible, a coarse collimator (or none at all) is usually sufficient. Further considerations on this topic are in Section III.D.2. related to multilayers. In all focusing spectrometers (with curved crystals), the use of collimators is superfluous and the role of collimators is taken over by pinholes or slits. D.

Dispersive Elements

Both the primary photon beam derived from the x-ray tube (or another source of radiation) and the secondary or fluorescence beam derived from the sample can be monochromatized. The monochromatization of the primary beam is not essential but can be made for more efficient excitation of the selected elements by photons of chosen wavelengths or for radical simplification of calculation procedures (the fundamental parameter calculation is then transformed to simple Lachance–Traill–Tertian correction). Monochromatization may be compared to the action of a narrow bandpass filter by which only a small band of the whole spectrum is transmitted. The width of the bandpass is related to the spectral resolution. A number of parameters characterizes this process.

Figure 16 Two spectra from the same sample demonstrate how an auxilliary collimator reduces line overlap when measuring heavy elements (a) without and (b) with auxillary collimator. (Courtesy of Phillips Analytical. From Phillips Materials.)

Copyright © 2002 Marcel Dekker, Inc.

The ratio DE=E, where DE denotes the energy width of transmitted radiation band and E the energy of photons to be transmitted (the mean energy in case of a wide band), is called the relative spectral resolution of the monochromator. DE is generally expressed as full width at half-maximum (FWHM) of the transmitted band (a peak in intensity–energy representation). The reciprocal of spectral resolution, E=DE, is called the resolving power. The same can be expressed in wavelength units as Dl=l by transformation through Eq. (3). Both expressions are numerically equivalent but of opposite sign. For the wavelengthdispersive spectrometer, the angular dispersion (Bragg’s dispersion) is important and is obtained by differentiation of Bragg’s law: dW n ¼ dl 2d cos W

ð7Þ

The monochromatization of the primary beam, if required, is made by the use of filters or secondary targets. The absorption edges of the elements present in the filter should be at lower wavelengths than the wavelength domain to be transmitted. Secondary targets emit their own characteristic radiation without a white background (except the scattered part of the primary beam) and with a population density of peaks depending on the atomic number of the secondary target. In both of these cases, the monochromatization is far from being perfect. A relatively simple application of double total reflection from the quartz plates was proposed with success for monochromatization in TXRF (Schwenke and Knoth, 1982). The criteria for the monochromatization of the primary beam differ substantially, depending on the application. They become very sharp in some methods of research [photoelectron spectroscopy, extended x-ray absorption fine structure (EXAFS)]. Then, the methods of monochromatization of the beam are different also, depending very much on new discoveries in x-ray and related optics. In general, the strong monochromatization of the beam is associated with a great intensity loss and can be made for the very efficient sources of x-rays as synchrotrons. The efforts for the virtual numerical monochromatization of the real exciting beam by the introduction of the so-called ‘‘equivalent wavelength’’ concept failed. A new equivalent wavelength is needed for each analytical situation (e.g., change in composition). The gain from the simplification of the spectrum is doubtful as compared with the great increase in the numerical effort. The monochromatization of the secondary beam is the key problem. Dispersion of radiation is needed for the specific detection of characteristic lines. There are, however, a number of analytical problems in which the requisite dispersion is low. In limited cases, it may be sufficient to subdivide the spectrum into two spectral windows, from which one is to be detected. In other cases, the cutting off of a relatively wide spectral window may be sufficient, as is done in the Philips MiniMate device, where only a sealed gas proportional counter is applied for the photon energy discrimination. In practice, this selection of wide spectral domains is easily performed by energy-dispersive systems and pulse-height selection, by filters or by a set of balanced filters. A spectrometer for general use over the widest range of elements, however, should be equipped with monochromators of sufficient overall resolution, as is obtained by crystal diffraction. 1. Crystals According to the orientation to the possible crystallographic planes of the crystal, different values for d are preferentially diffracting and, coupled with it, a different resolution results Copyright © 2002 Marcel Dekker, Inc.

for a given wavelength. A few values has been calculated for LiF in different orientations and these are listed in Table 2. From the dispersion, it can be expected that the Ka line of manganese can be separated from the chromium Kb by LiF (220), not by LiF (200). The values in Table 2 are calculated for ideal crystals and a parallel beam. For real crystals, the dispersion and resolving power are less favorable; some real values for a complete spectrometer are provided later. In Table 3, a list is presented for crystals currently available for spectrometers: lithium fluoride (LiF), silicon (Si), germanium (Ge), pyrolytic graphite (PG), indium antimonide (InSb), pentaerythritol (PE or PET), ethylenediamine-d-tartrate (EDDT), ammonium dihydrogen phosphate (ADP), thallium hydrogen phosphate (TlAP), and multilayers with their commercial designations. Important characteristics other than the interplanar distances codetermine the ultimate usefulness of an analyzing crystal: spectral resolution, mosaicity, reflectivity, stability, the thermal expansion coefficient, and the spectral range. a. Resolution The spectral resolution DE=E of the crystals oscillates about the value 10
2, with few exceptions. The data for the resolution of different crystals are included later in Figure 26a, a summarizing figure on resolution [see also Kuczumow and Helsen (1989)]. The spectral resolution for the crystals is, generally speaking, better than that of other dispersive devices, such as detectors or filters, but the difference with respect to the energy-dispersive spectrometers becomes less favorable for the short wavelengths. The progress in superconducting tunnel junctions detectors (Finkbeiner et al., 1991; De Korte, 1992; Le Grand et al., 1993) and microcalorimeter detectors (Lessyna et al., 1993; McCammon et al., 1993; Silver et al., 1996) during the last decade shows that these devices would have even better spectral resolution in the medium energy range (i.e., above 4–5 keV) than the crystals (see Fig. 26b). However, it is a question of compromise among price, necessity to work at liquid-helium temperatures, the very value of spectral resolution necessary for a given analysis, and the loss of intensity in crystals on one hand and the poor count rates allowed in the newest detectors on the other hand, which decides what kind of resolving device will be the future choice for commercial applications. b. Mosaicity and Reflectivity Real crystals have all kinds of imperfections, and mosaicity is one of them. Mosaicity refers to the existence of ‘‘blocks’’ within the crystal with sizes of the order of magnitude of 100 A˚, which have slightly different orientations and lead to widening the diffracted peaks. This happens to widely differing degress: topaz, EDDT, ADP, and gypsum exhibit low mosaicity, quartz and LiF a little higher mosaicity, and PET a significantly higher

Table 2 LiF: d Spacings for Different Orientations and Angular Dispersion Crystal

2d (A˚)

Dispersion

LiF LiF LiF LiF

1.652 1.800 2.848 4.027

— — 0.5154 0.2902

(422) (420) (220) (200)

Note: Calculated for W angles corresponding to MnKa (2.102 A˚) and n ¼ 1.

Copyright © 2002 Marcel Dekker, Inc.

Table 3

Currently Available Crystals

Crystal

2d (nm)

Element range

LiF(420)

0.180

Ni–U

LiF(220)a LiF(200)a

0.2848 0.4027

V–U K–U

Si(111) Ge(111)a,b

0.6271 0.6532

P, S, Cl P, S, Cl

PG(002)a InSb(111)a,b PET(002)b

0.6708 0.784 0.8742

P, S, Cl Si Al–Cl

EDDT(020)

0.8808

Al–Cl

ADP(101) TIAP(100) PbSt LTC

1.064 2.575 10.0 15.6

Mg O–Mg F, C Up to Be

PX-1a PX-3a PX-4a OVO 55 OVO 100 OVO 160 OVO H300 PX6

4.93 19.5 12.2 5.5 10.0 16.0 30.0

O–Mg B C Mg, Na, F C, O B, C Be, B

a

Remarks High resolution, special applications for heavy-element K lines High resolution, good intensity General purpose, wide range, best diffracted intensity Supresses even orders Supresses even orders, good for intermediate- and low-Z elements Good intensity, poor resolution Good intensity, low spectral contamination, soft Lower intensity than PET, problems with stability Still lower intensity Especially F, Na, Mg, poisonous Good resolution, low intensity Longest wavelengths known for semicrystals’ structures Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity Low resolution, good intensity One of the largest 2d available up to now

Also available as curved in simultaneous instruments. Also available as transversely curved for sequential instruments.

b

mosaicity. A result is a slight angular widening of the diffracted peak on one hand, but on the other hand, the intensity of the peak is generally enhanced. Widening is smaller than widening due to other geometric factors of the whole spectrometer and the loss can be tolerated while compensated by the increase in intensity. Some mechanical treatments enhance the mosaicity, and LiF crystals are easily improved in that way. The angular intensity distribution of a diffracted peak, the rocking curve, can be determined by doublecrystal spectrometer. In Figure 17, the peak height of the represented curve obtained by a high-resolution spectrometer is determined by atomic scatter factors and the space group of the analyzing crystal, whereas the width at half-maximum is determined by the mosaicity of the analyzing crystal and the divergence of the incident and diffracted beams. The surface under the curve is the integral reflection or, when the curve is normalized to the intensity of the incident beam, the integral reflection coefficient. This coefficient is very much dependent on mosaicity. Abraded LiF has a 10-fold increase in the reflection coefficient compared to freshly cleaved LiF (3610
5 to 4610
4 rad), measured by using CuKa radiation (Birks, 1969). LiF crystals have also the great advantage of low absorption by their constituting atoms. Topaz and quartz, otherwise good crystals, have Copyright © 2002 Marcel Dekker, Inc.

Figure 17 Rocking curve that would be obtained from a real crystal-diffracting parallel monochromatic radiation (rocking curves are actually measured on double-crystal spectrometers): P ¼ the peak diffraction coefficient; R ¼ the integral reflection coefficient. (From Birks, 1969.)

poor reflectivity properties, whereas PET is very good in this respect. Reflection constants as a function of wavelength are given in Figure 18. c.

Stability and Temperature

The mechanical stability of most crystals is satisfactory, but there are exceptions. Gypsum can effloresce (especially in a high vacuum); PET has a tendency to change phases on aging

Figure18 Single-crystal integral reflection coefficients of graphite, LiF (200), LiF (220), and KAP. [From Gilfrich et al., 1971). Reprinted by permission of Analytical Chemistry. Copyright # American Chemical Society.]

Copyright © 2002 Marcel Dekker, Inc.

and it is soft so that it is easily damaged when manipulated. The temperature inside the measuring chamber of a spectrometer is kept as constant as necessary (< 1 or < 0.01 C if chemical peak shifts have to be measured). In these circumstances, damage of the organic crystals by high temperature is not likely to occur. When very precise measurements of peak position are to be made for speciation purposes, the thermal-expansion coefficient is important. PET is very disadvantageous in this respect; topaz is in the opposite situation. It should be noted that the characteristic spectral lines of the crystal can be superimposed on the spectrum, making it obscure or even unfit to use, especially when it suggests the presence of some elements in a false way. Crystals of excellent resolving features, Be and Ge (111), have their own lines in the soft x-ray region. d. Spectral Range The useful spectral range covered by one crystal is limited. The maximum wavelength is of trigonometric origin and is imposed by Bragg’s law, namely by the condition that sin W < 1 and lmax < 2d. In practice, the goniometer is scanned for values of 2W varying from a few degrees to about 150 . However, for higher values of 2W, an angular dispersion widens the peak profiles. On the contrary, in the low-2W range, only a small fraction of x-rays emitted by the sample is intercepted unless the analyzing crystal is very long. The surface of the crystal projected on a plane perpendicular to the x-ray beam may be smaller than the width of the beam emerging from the collimator. Thus, 2W values as well as the macroscopic dimension determine the useful spectral range of a crystal. For x-rays of low energies, the interplanar distances of real crystals become too small. Some substances, namely salts of heavy metals with organic acids with long chains (otherwise the soaps of heavy metals), can take over the role of dispersive structures if their organic chains are regularly arranged. There are special techniques for making such quasicrystals. The individual layers are called Langmuir–Blodgett films. The ends of hydrophobic chains of adjacent layers join each other, and the heavy metal ends are connected to the next heavy-atom ends from the next molecules. Many features of such films depend on the length of the chain. A great number of similar structures with a wide variety of interplanar distances 2d (up to 156 A˚ in the case of lead melissate) have been synthesized. Langmuir–Blodgett structures have different negative properties: for example, they are soft and not very stable, strongly hydrophobic, and of low reflectivity. Some of them, like thallium adipate, are highly poisonous substances. e. Curved Crystals and Other Focusing Systems Not only are flat crystals used in the goniometer. Some focusing systems, such as the Johann, Johansson, and Cauchois arrangements, may be used (Fig. 19). Then, the crystal must be curved, with the curvature radius equal to the diameter of the Rowland circle and, at the same time, to the Bragg curvature, being half that of the Rowland circle. Threedimensional crystals can sometimes meet both of these conditions. The focusing arrangements are extremely useful in all cases in which the sample area is small, losses of intensity are prohibitive, and scanning over large angles is not necessary. In these arrangements, x-rays are collected in one point by focusing on the detector. An obvious virtue of such an arrangement is found in the analysis of small sample areas and by a using focused exciting beam, as in electron or proton microprobes. In such situations, the curved crystal geometry allows the collection of x-rays from a relatively large solid angle of diverging x-rays to a small spot (Fig. 20). A consequence is that the collimators normally placed between the sample and the crystal, and the crystal and the detector must be replaced by slits and pinholes. The focusing geometry is a very economical arrangement for saving intensity (the brightness of such a device is estimated to be up to two orders of magnitude greater than Copyright © 2002 Marcel Dekker, Inc.

Figure19 Curved crystal x-ray arrangements in (a) Johann, (b) Cauchois (in transmission), and (c) Johansson version. S ¼ entrance slit of radiation; D ¼ exit slit of diffracted radiation; d ¼ crystal lattice spacing.

Figure 20

Focusing of x-rays by a logarithmically curved crystal. (Courtesy of Siemens AG.)

that of conventional flat geometry). The curved crystal geometry is applied in simultaneous instruments in which scanning is not used and an optimized arrangement is chosen for each wavelength (element) implemented. Then, logarithmically curved crystals are used. In a new version of the sequential instrument (Venus), Philips used the reverted channels taken from the simultaneous instrument. Crystals applied in that construction are curved as in a simultaneous device and their position is additionally adjusted by a screw to trap as much intensity as possible. In general, during the last two decades, great progress took place in the planning and construction of the optical elements based on curved reflecting or diffracting surfaces of crystals and multilayers; most of the developments were in the synchrotron version of x-ray spectroscopy (Ice and Sparks, 1984). The curved crystal version of the x-ray microprobe is not the only known solution. The progress in x-ray optics during last decades enabled the easy concentration of x-rays while being transmitted through a single capillary (Rindby, 1986; Carpenter 1989; Thiel et al., 1989) or even through a semilens based on the bundle of capillaries (Kumakhov and Komarov, 1990; Yiming et al., 1994; Dagabov et al., 1995; Xunliang et al., 1995; X-Ray Optical Systems Inc., 1997, 1998). This first possibility, due to the great loss of intensity resulting from a poor interception of the primary beam by the narrow inlet of a single capillary, is limited to the energy-dispersive version of x-ray spectrometry (Attaelmanan et al., 1994). The second version is more interesting from the point of view of WDXRF. A properly curved capillary bundle can serve as a kind of semifocusing lens with a well-defined Copyright © 2002 Marcel Dekker, Inc.

Figure 21 Cross section of a typical multicapillary ‘‘lens’’: (a) focusing configuration, from wide spot (divergent source) to another spot (semifocus); (b) configuration transforming the divergent into the parallel beam. (Courtesy of X-Ray Optical Systems Inc., Albany, NY. and personally from Dr. J. P. Bly.)

curvature and moderately-defined focus (Fig. 21). The intensity of the beam leaving the ‘‘lens’’ is, of course, lower than the intensity derived from the x-ray tube, but still sufficient for many applications. Moreover, the primary collimator is removed in this system and the gain from this removal is partially balancing the loss resulting from the bundle construction and operation. One can estimate the focal distance f of such a lens as df ¼ d0 þ 2 f ycr

ð8Þ

where d is the spot size, d0 is the diameter of the capillary end, and ycr is the critical angle of the total reflection. As a rule, the polycapillary array is introduced to concentrate the primary beam, but it can also be used for the collection of secondary x-rays outgoing from the sample, before reaching the curved crystals in nonconventional arrangements. 2. Multilayers The limitation of XRF analysis in the region of soft x-rays is one of the most serious limitations of crystal monochromators, although none of the ‘‘first principles’’ constituting the background of XRF method puts this limitation explicitly. Moseley’s law determines the rules for qualitative analysis of any element except hydrogen and helium, and the Shiraiwa and Fujino equations govern quantitative analysis. Irresolvable problems occurred until the late eighties because of troublesome absorption, lack of an adequate dispersive device, and bad detection while passing to longer wavelengths. The concept of multilayers or, strictly speaking, layered synthetic microstructures (LSMs) or multilayer interference mirrors (MIMs) solved at least part of the problems. Two approaches to the idea of a suitable analyzing device in the soft x-ray domain have been applied from the very beginning (Underwood and Barbee, 1981). The first resulted from consideration of Bragg’s equation [Eq. (2)]. This indicates that if any crystal of a longer interplanar distance is found, it will be possible to disperse longer wavelengths in the soft x-ray range. The potential candidates for long-period crystals or quasicrystals, such as clay minerals of the chlorite (14 A˚) or illite–montmorillonite (25–30 A˚) type, are not sufficiently ordered in this respect (Weaver and Pollard, 1973). Unfortunately, natural crystals, irrespective of their different crystallographic planes, offer only limited possibilities in this spectral range; specifically, their reflectivity is very low. The class of structures called Langmuir–Blodgett films was more promising, but the reflectivity and thermal, mechanical, and chemical stabilities were rather poor. Copyright © 2002 Marcel Dekker, Inc.

A correct solution resulted from generalization of Bragg’s law, namely that diffraction also occurs in media consisting of layers of different refraction coefficients—in other words, in sites of different periodically changing electron densities. Such structures may be formed in an artificial way. They may be treated as crystallike substances ordered along the c axis (i.e., spacing d or 2d), but with uncontrolled arrangement within the ab plane. The structure in the ab plane can be amorphous, provided that the optical and mechanical properties are uniform in these directions. The first effort of synthesis was made in 1940, but the Au–Cu multilayer of Du Mond and Youtz (1940) did not survive more than a few days because of deterioration by interdiffusion. The problems of thermal, radiative, diffusional, and chemical instabilities of multilayers have recently been solved. Significant progress in this field has taken place in the last three decades, hence some years before the application of multilayers in WDXRF. The second approach was the result of an increasing interest in x-ray optics. As a branch of physics, x-ray optics is nearly as old as the discovery by Ro¨ntgen, but because of the specific properties of x-rays, its progress has been slow. All existing substances exhibit complex refractive indices: m ¼ 1
d
ib

ð9Þ

where m is the refraction index, d is the so-called unit decreament, a real number, and the imaginary term b is related to absorption. The refraction indices in the x-ray region are all below but very close to 1. This makes all efforts of x-ray focusing by lenses in a traditional sense an impractical task. Franks (1977) gives an example of a lens with a curvature radius of 10 mm built of a material with d ¼ 5  10
5 . The focal length of this lens is 100 m. In such a situation, only glancing, grazing, and diffracting features of materials have been of practical interest for a long time. Otherwise, the grazing optics conditions for x-rays are extremely severe. However, the need for mirrors, monochromators, and focusing devices for the x-ray region is increasing rapidly in the fields of synchrotron radiation, plasma research and diagnostics, x-ray microscopy, x-ray fluorescence analysis, diffraction, thermonuclear fusion, x-ray laser, x-ray astronomy, and even x-ray waveguides. The invention of multilayers defined this time as a complex set of particular layers consisting of materials of periodically different refractive indices gave great impetus to progress in this branch of optics. Some important properties of the materials used to produce the multilayers are discussed in the next section. a. Nature of Materials In principle, three kinds of material are necessary for the synthesis of multilayers. First, there must be a support (sometimes called a substrate), such as a piece of flat or curved silica layer, with an ideally polished surface. This condition is very important because the roughness of the surface is translated to the deposited layers, correctable almost exclusively when one of the layers consists of amorphous materials such as carbon. After years of progress in the field, when the roughness of the interfaces have probably reached the limit of about 2 A˚, the most important contribution in it is from the deviations of the substrate from flatness (Kortright, 1996). Two sources of relatively inexpensive and precisely polished supports were found: silicon wafers as used in microelectronics (curved) and glass or fused-silica optical elements for low-scatter mirrors (flat or curved). The additional polishing is the last step in the preparation of the substrate for further manufacturing of the multilayer. An even more complicated role for the substrate is described by Nicolosi et al. (1986) in whole coal analysis. The authors determined the contents of Copyright © 2002 Marcel Dekker, Inc.

C, N, O, and F using multilayered microstructure PX2 and the contents of P, S, K, and Ca by taking advantage of diffracting properties of the supporting wafer of silicon (111) on which the multilayer was settled. Heavy metals (Hf, Pb, W, Mo, Pt, Ir, Ru, Rh, Os, Ni, Co, Fe, and Cr) and light elements or compounds [Al, B, B4C (Ovonic Synthetic Materials Co., 1987), BN, Be, C, Si, SiO2, SiC, Ti and Sc—the last with the high reflection] may be used as materials for alternating high and low optical densities, at the same time being strong and weak scatterers, respectively. Sometimes, materials from the first group are called absorbers, and those belonging to the second group are called spacers. The reactive materials and substances with a low melting point are excluded from the list. Both components of the multilayer should not share a common crystalographic structure, to avoid the epitaxial growth. The selection rule for the materials can easily be understood from Figure 22 for boron. The mass attenuation coefficients for the boron K line, as presented versus the atomic number of the absorber, have several minima. One falls in the region of boron (weak self-absorption) and carbon, and the next very deep minimum is in the region of niobium=molybdenum. It puts the constraints on the materials, which can be used for the determination of boron. The first, a low-density material, can be carbon or even better B4C; the second, a high-density material, should be niobium or molybdenum. Indeed, Philips made the structure PX3, composed of Mo=B4C, with a period of 195 A˚, adviced for the determination of boron. However, when a new multilayer should be synthesized

Figure 22 Mass attenuation coefficient for the BKa line drawn against the possible absorbers in the periodic table. Arrows show the deep minima in coefficient value. (Courtesy of Bruno Vrebos’ inspiration.)

Copyright © 2002 Marcel Dekker, Inc.

with high reflectivity in mind, then the attention is put, instead, on a pair of materials with a small value of the scattering Debye–Wallner factor [see Eq. (10)]. Here, the W=B4C tandem is a very good one. As a rule, the low-electron-density compounds are selected from the elements with K absorption edges above the energy of spectral lines to be measured. In that sense, boron and carbon are selected for multilayers of spacing above 50 A˚ and beryllium and carbon for spacings above 120 A˚. b. Smoothness This parameter is of great importance on each level, starting from the supporting substrate to the last upper layer. All rough places, uneven layers, disturbances in interfaces, interdiffusion, and chemical interaction regions or defects, act in a manner analogous to the thermal motions in classical diffraction analysis. During synthesis, the extent of roughness can vary in a different manner: In some cases, it tends to smoothness, in others to further growth. The use of amorphous materials to form some layers favors smoothing. The greater interplanar distances are, the more pronounced the roughness is. On the other hand, the smoothness probably has its limits, even in the case of multilayers with relatively small interplanar distances, with the Debye– Wallner parameter s tending to reach a value of about 2 A˚. It probably results from the smoothness level obtained during the substrate production. Finally, it puts some domination of the Debye–Wallner parameter over other possible factors influencing the quality of multilayers in the region where the interplanar distance is not very long (say, a few tens of angstroms). c. Reflectivity and Thickness of a Multilayer Classical dispersive crystals exhibit low reflectivity in the soft x-ray range. The situation was greatly improved in multilayer reflectors in which a single layer at normal incidence angles exhibits a reflection coefficient of the order of 10
4 rad. This coefficient is, however, proportional to the square of the electrical field amplitude (i.e., to a value of the order of 10
2 for the amplitude reflection coefficient). Thus, a multilayer consisting of 100 single layers may provide almost the maximum reflectivity if the absorption can be neglected (Underwood and Atwood, 1984; Underwood, 1986, http:==www-cxro.lbl.gov==multilayer=survey.html, 1998). For long wavelengths, the results are not so impressive; for example, in the very important region of the so-called water window (284–530 eV), the reflection coefficient was lower than 3% (Kozhevnikov et al., 1994). The region is of essential significance for the construction of an x-ray microscope based on zone plates for biological research (Kirz et al., 1995). These results are in accordance with the calculations by Rosenbluth concerning numbers of layers necessary for ‘‘saturation’’ of reflectivity [data cited by Barbee (1986)]. Peak reflectivities of some commercially available multilayers [e.g., Philips materials PX1, PX2 and PX3 (Nicolosi et al., 1986; Van Eenbergen and Volbert, 1987) and the OVONYX line of multilayers [(Ovonic Synthetic Materials Co., 1987)] are often compared to classical pseudocrystal dispersive structures, like lead octodecanate and thallium acid phthalate. The predominance of the multilayer structures over classical structures is striking, and spectral resolution is the only parameter in which the classic crystals are better (Van Eenbergen and Volbert, 1987; Henke et al., 1986). Still, when using multilayers, one must remember the weak points inherently coupled with those devices: great refraction effects (Martins and Urch, 1992; Luck et al., 1992), deviating the effective 2d value of the structure from the real value by as much as 5%, and poorly shaped background with ‘‘ghost’’ lines in some cases. Copyright © 2002 Marcel Dekker, Inc.

d.

Quality of Layers and Boundaries

The structure of the multilayer ought to exhibit geometric features. This condition is very strict. Thus, flat layers or figured structures must be uniform, the latter with controlled curvature, without defects but with clearly defined optical parameters. All deviations from an ideal model, easy to estimate by the reflectivity characteristics, are, as a rule, coupled by the use of the Debye–Waller factor into the effective roughness parameter seff, including both thickness errors and imperfect boundaries (Spiller and Rosenbluth, 1986). The Debye–Waller factor DW is the Gaussian-type term describing the reduction of reflected amplitude at a boundary: "
 # 2ps cos a 2 DW ¼ exp
2 ð10Þ l where s is the width of a smooth transition layer and a is the propagation angle of electromagnetic radiation, measured from the normal to the boundary between a medium with one refractive index to another. The question arises, however, whether the errors result from a rapid or from a mild change of the optical features on the boundaries. Any roughness of a given layer decreases its reflectivity (Payne and Clemens, 1993), but a random distribution of roughness may bear higher reflectivities of the higher-order peaks. Moreover, many other physical factors can influence the uniformity and expected layer characteristics: Both the support and the particular monolayers can have different slopes, densities are subject to fluctuations, the elements can diffuse, or some chemical elements can mutually react (e.g., W with C or Mo with Si) (Rosen et al., 1993). It should be emphasized that despite of so many obstacles, the magnitude of the roughness in such a totally artificial structure can be kept in the range of an atom radius (Barbee, 1986). The numerical description of multilayers is based on the matrix method (a matrix expresses the features of a layer or interlayer region, and for the characteristics of a multilayer, the product of matrices is written) or on the recursive use of layer–reflection– transmission characteristics. Deviations of empirical results (mainly reflectivity measurements) from the model predictions are associated with roughness. In this case, seff, the roughness parameter, results from the best fit of empirical data to the model. The selection of an adequate method of synthesis and subsequent control of production and quality is a key question. Many methods were checked, and the aim of all these processes was the deposition in which one atom follows another. The physical vapor condensation, chemical vapor deposition, electrochemical covering, and sequential adsorption of films are the most promising methods. However, for the time being, the first method prevails. Two basic versions of physical vapor deposition are used—with thermal and sputter sources. In the thermal source technique, the covering materials are evaporated by electron beam or laser evaporation. The vapor source and samples are in a fixed position. The time of evaporation is a parameter controlled by shutters or by pulsed laser irradiation. An in situ monitoring system was used as a feedback system for controlling the shutter. Spiller (1981) invented a system for thermal deposition. He used a soft x-ray reflectometer (a source plus a detector), placed in an evaporation chamber to register the reflectivity of successive layers. The open loop is set according to the indications of maximum reflectivity for a given layer. It is possible by this method to obtain a multilayer with an area up to 25 cm2. The rates of deposition vary in the range of 2–100 A˚=s and must be carefully controlled because the successive layers must be commensurate. Sometimes, especially Copyright © 2002 Marcel Dekker, Inc.

when multilayers of large surface are produced, a special motion of the substrate in the evaporation chamber may be necessary to guarantee the same evaporation rate for all points of the surface. Other difficulties arise from this type of deposition: the high vacuum required in the evaporation chamber (< 10
2 Pa), radiation heating of the deposited film, and the problems with thermal sources (alloys may segregate during melting). The sputter source systems (Barbee, 1981) belong to another processing method. The glow discharge, magnetron, electron cyclotron, ion beam, and triode devices may serve as sputter sources. The plasma discharge is always the first step of the process. The plasma flux is directed by the electrical potential difference to the cathode surface, where the ions and atoms dislodge the maternal atoms. The secondary atoms and ions establish the proper flux (Sigmund, 1981), which passes to the substrate to deposit a layer on it. Samples are placed on a special rotatory table, appearing at fixed intervals under the secondary flux courses. The rotation speed, precisely controlled, is the regulating factor for a designated thickness of the layer. The purity of secondary fluxes and their energies also influence the quality of the multilayer. The main errors in this method result from the ion and atom inclusion into layer (there are high-energy tails in the fluxes) and from heating caused by flux energy deposition in thin layers. However, substrate rotation may overcome many of these defects. Sputter rates range from 1 to 20 A˚=s; the covered surfaces may be quite large. Today, the magnetron sputtering is the most common method, with the best results obtained as far as the reflectivity of the products is concerned. After presenting two basic methods of layer formation, some attention must be paid to methods of control. As mentioned earlier, reflectivity estimations are of fundamental importance in multilayer quality determination, both experimentally and in theory (Spiller, 1981). A comparison of reflectivity predicted and obtained in an experiment is instructive, allowing, at least, understanding the source of the roughness (Henke et al., 1986; Luck et al., 1992). Different peak, absolute, and integrated reflectivities were measured with the use of both laboratory x-ray sources and synchrotron facilities. These results confirmed the layered structures of the synthesized materials; as a rule, reflectivity was smaller than predicted, but the results show that the optical parameters estimated until now (Henke et al., 1982, 1988) are reasonably correct, with the shorter the wavelength concerned, the better. The methods just discussed only allow the deduction of an indirect image of the multilayer structure. Other methods give direct insight into the structure. Thus, the diffraction data may be very useful, but they require large sample volumes. Progress in capillary version of the diffractometer (Bilderback et al., 1994), allowing the observation of the samples as small as 5610
3 mm3, may bring a solution to the structural problems. Very small samples are needed for obtaining the structural information by electron microscopy, especially in combination with a special technique of sample preparation, socalled microcleavage (Lepeˆtre et al., 1986). Wedge-shaped strips of materials are placed in the electron beam in such a manner that the planes of particular layers are parallel to the beam direction. Both the transmission image and the diffraction pattern can be obtained. The in-depth and lateral variations, the roughness, and the visual image of the structure are in the field of observation. High-resolution electron microscopy can give even better results, and electron energy loss spectroscopy (EELS) may provide the complementary chemical information on the subsequent layers (Lepeˆtre et al., 1986). Requirements for the application of multilayers in XRF may be summarized as follows: They should satisfy Bragg’s law for the assumed wavelength domain within permissible goniometer angles, have a high reflectivity and a good spectral resolution, Copyright © 2002 Marcel Dekker, Inc.

suppress diffraction peaks of higher order, and do not absorb soft radiation too intensively. Flat or curved multilayers can be used. Dedicated designs are described in the literature for the application of curved multilayer optics (Van Eenbergen and Volbert, 1987; Gilfrich et al., 1982). The geometric requirements imposed on the curvature of multilayers for XRF are modest in comparison with their applications in other fields of x-ray optics. The increasing spectral distance between K lines of the neighboring elements when going to lower-Z elements may require a dedicated multilayer for each element. Sometimes, great differences in the mass attenuation coefficients for the lines of neighbouring elements of interest exist for a given material—attenuator. Then, one can construct a multilayer efficiently diffracting the radiation of one element while deeply suppressing it for other elements. The dedicated multilayers are the preferable choice for the simultaneous WDXRF instrument, with an optimized channel for each element. Multilayers can be also applied in a special version of WDXRF analysis, called grazing emission x-ray analysis (de Bokx and Urbach, 1995). Curved multilayers are also used in electron microprobes (Ovonic Synthetic Materials Co., 1987). From what has just been described, it may be concluded that in order to analyze a useful range of soft x-rays, at least three different multilayers have to be selected. Such combinations are currently available commercially (e.g., for example from Osmic, Philips, Siemens, Rigaku, and Shimadzu) (Ovonic Synthetic Materials Co., 1987; Van Eenbergen and Volbert, 1987; D. K. G. de Boer, personal communication, 1998): PX1 PX3 PX4 PX5 PX6

2d ¼ 49:3 A˚ 2d ¼ 195 A˚ 2d ¼ 122 A˚ 2d ¼ 112 A˚ 2d ¼ 300 A˚

oxygen to magnesium boron and possibly for beryllium (Nicolosi et al., 1987) carbon nitrogen beryllium

and from the Ovonic Synthetic Materials Company (1987): OV-040A

2d ¼ 40 A˚

fluorine to silicon

with multilayers changing 2d spacing every 20 A˚ up to OV-140B

2d ¼ 140 A˚

carbon to oxygen

and also the MoB4C structure: OV-H series

2d ¼ 244 A˚

beryllium to boron

up to OV-300H

2d ¼ 300 A˚

Probably, materials with even greater interplanar distances can also be supplied. However, recent scientific (except the XRF field) interest in multilayers is centered, instead, more on multilayers with periods of 10–20 A˚ for their excellent reflectivity or for their polarizing abilities (Scha¨fers et al., 1998). These materials are described in the current literature as completely thermally stable; thus, they do not need temperature stabilization when used in x-ray spectrometers (Nicolosi et al., 1986; Barbee, 1986; Van Eenbergen and Volbert, 1987). Some sensitivity to the damaging influence of intense radiation [above 0.15 J=cm
2 (Kohler et al., 1985)], confirmed later in an article by Kortright et al. (1991) and by a comprehensive report by MacGowan et al. (1993), may limit the application of the multilayer for monochromatization of dense flux of x-rays, which usually happens in synchrotron, plasma, Copyright © 2002 Marcel Dekker, Inc.

or laser research. Perhaps, the more important conclusion from the latter study is that, after the impact of high power on multilayers, the amorphous carbon layer transforms partially into a graphite form. It improves the reflectivity but greatly deteriorates the stability and changes the interplanar distances. The phenomenon of ultrafast graphitization ( 40 ps) can, in the future, be applied for the construction of x-ray switches. These data can be compared with the results cited in a report by Barbee (1986). He claims that during the sputtering process of multilayer synthesis, very high temporary doses of energy can be deposited in the layers (power density ffi 105 W=cm3=s1), which involves a significant increase in local temperatures (50–180 C). The relative spectral resolution of dispersive devices is often expressed in the form of the important parameter DE=E or Dl=l, which is the basis for their choice in XRF applications. Unfortunately, this is the weakest point of the multilayer. In a commercially available specimen, the relative spectral resolution changes from about 0.025 to over 0.1 while passing from structures with 2d ¼ 40 A˚ to those with 2d ¼ 244 A˚, respectively, for different measured lines (Ovonic Synthetic Materials Co., 1987), and for the structure with 2d ¼ 75 A˚, it changes from 0.025 to 0.04 in the wavelength range 7–75 A˚ (Henke et al., 1988), which is about three times worse (Ovonic Synthetic Materials Co., 1987) or even more (Henke et al., 1986) than for a good classical pseudocrystal. The comparison of spectra of the same sample by using multilayers with different spacings, as in Figure 23, is very instructive (Van Eerbergen and Volbert, 1987). Relatively broad internal bandpasses of multilayers diminish the resolution of the whole spectrographic device, seriously increasing the detection limits. This is of special importance to the analysis of the L or M series, and avoids the efforts to search the chemical shifts in x-ray spectra [although see Habulibaz et al. (1996), where chemical shift is investigated for the SiL spectrum, diffracted on a multilayer with an interplanar distance 300 A˚]. The relatively wide bandpass of the multilayer can be used in a constructive way. As proven by Vrebos (personal communication, 1997; see Fig. 24), the imminent width of the BKa peak is so great that the contribution from the secondary beam collimation to the total width of the signal is negligible. In that situation, the analyst can remove the fine or medium collimation or

Figure 23 Comparison of sensitivity and resolution for layered synthetic microstructures with 2d values of 5, 12, and 16 nm from bottom to top, respectively. Increased sensitivity together with decreased resolution on the drop in 2d is evident. (From Van Eenbergen and Volbert, 1987.)

Copyright © 2002 Marcel Dekker, Inc.

Figure 24 Comparison of the spectra of the boron K line, obtained by a multilayer PX-3 with a fine, medium, and coarse collimator. (Courtesy of Bruno Vrebos and Philips Analytical.)

even the collimation completely. The gain is double. It results from the greater intensity given by the multilayer as such, as well as from the lack of the collimator, which is always one of the most intensity-restrictive parts of the WD spectrometer. This kind of work with a limited collimation is possible only in the region of soft x-rays, where the population of lines is rare and in the absence of the very soft L and M lines of heavier elements. The suppression of higher-order diffraction peaks gives better spectral purity and may at least partially compensate for the poorer resolution of multilayers. The isolation of the weak MgKa line from the dominant CaKa line is possible by the use of PX1 structure for the analysis of cement (Fig. 25). This is an extreme example of the superiority of multilayers over a classical crystal in particular cases (Nicolosi et al., 1986). Another great advantage of multilayers is their great reflectivity. It should be added that the peak reflectivity is 3–10 times greater than that of corresponding classical crystals. Multilayers were frequently used in the routine determination of light elements. Many examples can be cited, including carbon (Nicolosi et al., 1986, 1987; Van Eenbergen and Volbert, 1987; Philips Application Note No. 745) and boron determination (Van Eenbergen and Volbert, 1987; Nicolosi et al., 1987; Adamson et al., 1991; Philips Application Note No. 737 and No. 803; Uhlig and Mu¨ller, 1991; van Sprang and Bekkers, 1998), and beryllium detection (Nicolosi et al., 1987; Siemens Lab Report X-ray Analysis, 1994; Anonymous article, The Rigaku J., 1997). In the analysis, advocated by Rigaku, the application of the high-power 4-kW generator was the additional factor, enhancing the detection power for beryllium determination. The detection limit for that element was found to be in the region of a few tenths of a percent. Determinations of light elements were made using both flat optics sequential spectrometers and curved optics in simultaneous instruments (Van Eenbergen and Volbert, 1987). After overcoming the technical difficulties, the results obtained from such analyses may seem quite trivial (Nicolosi et al., 1987; Bonvin et al., 1995). However, not always are things going in that usual way. Van Sprang and Bekkers, (1998) emphasized the number of problems, coupled with light-eleCopyright © 2002 Marcel Dekker, Inc.

Figure 25

2W scans for Mg in cement using TlAP and PX1. (From Nicolosi et al., 1986.)

ment determination. For the analysis of boron in glass, only the combination of methods gave good results in a whole range of the detectable concentrations: The fundamental parameter method using the fluorescent signals for the low B concentrations and the use of the incoherently scattered radiation as a measure of the light-element fraction above 4%. One should note some special features of x-ray spectrometry in the soft x-ray range: The type of analysis changes from bulk to surface measurements; the influence of the chemical state and the subsequent wavelength shifts on the analysis can be significant; the wavelength of soft x-rays is comparable to the surface roughness; and the problem of the surface quality becomes extremely important. Considering the so-called total information depth of x-rays [see Sec.IV.G.1, Eq. (19)], it can be calculated that in the case of iron determination in a steel sample, the effective energy of exciting photons being 20 keV, the secondary x-rays arriving at the detector emerge from depths up to 62 mm. A carbon signal, emitted by a coal sample excited in the same manner, emerges from depths of only 8.3 mm. The problem looks even more dramatic if carbon is determined in a steel sample: In this case, signals emerge from depths as small as 0.45 mm! This means that the analytical volume for iron is about 140 times greater than that for carbon and the information on both constituents really refers to totally differing volumes of the steel sample. Moreover, the method of excitation is also critical in this case: The excitation of light elements is most efficient if made with relatively soft x-rays, the energy of which does not substantially exceed the value of the absorption edge. Excitation by use of hard radiation is not profitable in this case, as photons are mainly stopped inside the sample. The most restricting stage is the very short escape path of the characteristic photons for low-Z elements. The effect of the chemical state can be observed during the analysis of some elements using soft x-rays. The wavelengths of La1,2 lines of chromium and manganese are 0.573 and 0.637 A˚, respectively, and should be analyzed in the region otherwise reserved for the Copyright © 2002 Marcel Dekker, Inc.

K series of oxygen and fluorine, hence by the use of the PX-1, PX-4 or OV-120B pseudocrystals. The question is whether the limited spectral resolution of multilayers is sufficient to show such subtle effects as chemical shifts and to separate the mentioned lines of metals from the disturbing lines of oxygen or fluorine if present in the sample. The wavelengths of characteristic x-rays in the K series are longer than atomic radii starting from Ga (GaKa ¼ 1.337 A˚; atomic radius of Ga ¼ 1.26 A˚) and passing to the lighter elements. Radiation with a wavelength comparable to the dimensions of an obstacle always result in surface problems, with scattering and diffraction at the head. One should take into account that even the most perfectly polished sample has a surface roughness of the order of at least one atomic radius and that the best layers in multilayer structures have a roughness, expressed as the effective roughness seff, on the order of one to several atomic radii (Barbee, 1986; Kortright, 1996). These latter remarks are not intended to weaken interest in multilayer optics and its application in XRF, but only to evoke a special awareness of potential problems. 3. Spectral Resolution Figure 26a shows the features of commonly used dispersive devise (Heinrich, 1981; Caciuffo et al., 1987; Plotnikow and Pszenicznyj, 1973; Fitzgerald and Gantzel, 1971; Burkhalter and Campbell, 1967; Gohshi et al., 1982a, 1982b; Potts et al., 1985; Sparks, 1980; Salem and Lee, 1976; Bent, 1970; Bandas et al., 1978; Gilfrich, 1987). The width of the bandpass DE (expressed as FWHM whenever possible) is plotted versus the energy of radiation E; the parameter is relative to the spectral resolution DE=E. A line of given value of DE=E is called the isoresolving line. These lines divide the whole figure into two zones of different resolutions (or bandpasses). Note that both the typical dispersing devices and typical detectors are collected in Figure 26a. Only a few detecting devices with no resolving power (the reciprocal of the relative spectral resolution) at all exist (e.g., Geiger-Mu¨ller counter), and no detector exists with perfect, infinite resolving power. Real dispersive devices exhibit very contrasting features, with the parameter DE=E varying from almost 1 (scintillator counter or semiconductor photocathode) to systems with DE=E as small as 10
4 [Ge (111) crystals and two- or three-crystal spectrometers]. Even better dispersive devices are needed, for example, for x-ray inelastic scattering spectroscopy in which relative energy resolution near 10
7 is wanted. Indeed, we can cite here some exemplary articles announcing the achievement of similar values: 1:14  10
7 at a level of 14:4 keV (Chumakov et al., 1996) or 2  10
8 for 25:7 keV (Verbeni et al., 1996). XRF spectrometry has rather modest demands for resolution. We see that for introducing the characteristics of the devices with spectral resolution on a level of 10
8 in our diagram, the scale of Figure 26a should be prolonged by approximately four orders of magnitude at the bottom! Another interesting conclusion can be drawn from Figure 26a. Nearly all dispersive structures exhibit variable relative spectral resolution, but only KAP, topaz, and Ge (111) crystals and, to some extent, also multilayer structures can be considered as isoresolving systems. The microcalorimeters show rather unusual behavior, compared with other detectors. They have more or less the same absolute spectral resolution in the whole useful range (2–20 keV). As we can see from the shape of the DK and DL curves, nature does not demand isoresolving behavior from the dispersive structures. Rather, these structures, with resolution curves parallel to the curves mentioned, would be optimal for spectrometric aims: The Si–Li detector seems to have the better features. Generally speaking, a bandpass range of 10
1–10
3 is considered analytically useful. Copyright © 2002 Marcel Dekker, Inc.

Figure 26 (a) Spectral resolution of different devices: DZ ¼ spectral difference between analogous lines of adjacent elements; DW ¼ width of spectral line; Ka1 7 Ka2 ¼ difference between Ka1 and Ka2 lines; (b) the imminent features of x-ray spectra extracted from (a) for better characterization of the analytical demands; (c) characteristics of new microcalorimeter detectors as compared with typical Si–Li and LiF (200) dispersive structures.

Systems with bandpasses below 10
3 are used only in spectrophysical work. For making a proper choice of the dispersive structure for study of chemical shifts, one has to take into account the curves DW and Ka1 7 Ka2 (or even La1 7 La2) from Figure 26b. Figure 26b shows the spectral characteristics, extracted from Figure 26a, as a better demonstration of the immanent features of x-ray spectra. Depending on what matters in analysis, the adequate choice of the resolving system should be made. The regions occupied in the diagram by the most common [SiLi and crystal (e.g., LiF)] or most promising (microcalorimeter) detectors are shown in Figure 26c for easier comparison with data from Figure 26b. Figure 27 represents some example of the parallel analysis of the same sample, based on spectra from the electron microprobe acquired by the WDXRF system (TLAP), EDXRF system with Si(Li), and EDXRF system with the Copyright © 2002 Marcel Dekker, Inc.

Figure 26

Continued

microcalorimeter. For the time being, some preponderance of the LiF detector is observed for light element determinations (below 4 keV) (Wollman et al., 1997), but even this was endangered when Wollman announced the energy resolution of 2.0þ=
0.1 eV at 1.5 keV level (Wollman et al., 2000). Spectral resolution is only one parameter determining the selection of a spectrometer. Other parameters, however, such as speed of measurement, count rate capability, geometrical collection efficiency, and ease of operation may affect the choice of the most suitable instrument. E. The Goniometer The goniometer is basically a very simple construction and is the analog of a diffractometer as used in x-ray diffraction (the crystallographic research preceded the x-ray fluorescence method). The essential features are shown in Figure 7. The sample is positioned in front of the tube window. To account for possible heterogeneities, the sample is Copyright © 2002 Marcel Dekker, Inc.

Figure 26

Continued

spun about its axis at 30 rotations per minute. The analyzer crystal is positioned in the center of the diffractometer circle, rotates around an axis perpendicular to the plane of the drawing, and passes through the macroscopic plane of the crystal. The detector moves along the diffractometer circle supported by the goniometer arm. As imposed by the Bragg condition, the crystal rotates over an angle 2W, realizing equal angles of incidence and diffraction with respect to the crystal plane. The crystal is diffracting the incident fluorescent beam of the opening angle determined by the slit width and length of the collimator. Rotation of crystal and detector are either mechanically coupled through gears with a ratio of 1 : 2 or mechanically decoupled but moved by separate computer-controlled stepping motors, but both arrangements are still implemented (cf. Tables 8–10 in Sec. VI). The idea of mechanical decoupling and optical position control was first introduced by Applied Research Laboratories (see Sec. VI). This technique offers a number of advantages: Alignment of crystals is no longer necessary. They can be positioned on a known spectral line and the offsets are memorized by the computer. Copyright © 2002 Marcel Dekker, Inc.

Figure 27 (a) NIST microcalorimeter spectrum of AlGaAs (solid line) compared to parallelly determined EDS spectrum [with commercial Si(Li) detector]; (b) WDS spectrum of the same sample (with TLAP applied as a crystal), spectrum transformed from wavelength into energy representation. Observe still a little better performance of the WDS spectrometer for the considered energy region 1–1.6 keV. (From Wollman et al., 1997.)

The detectors can be juxtaposed on the goniometer arm and brought into position as required by the program. Tandem arrangement of flow and scintillation is no longer necessary. Slewing and scanning speed range is virtually unlimited. Optical position control (Moire´ fringes or optical encoder) allows very high angular precision and accuracy. Low-energy radiation is strongly attenuated by air. The Ka radiation of a pure copper sample (sample  2’’, measured on a Philips 1410 sequential spectrometer, SWT with chromium anode, 45 kV and 60 mA) produces 67800, 33200, and 2660 cps, respectively, in vacuum and 48000, 16800, and 890 cps, respectively, in air. The results mentioned are not easily repeated on recent computer-monitored instruments (e.g., Philips 2400) which do not routinely allow working under air at normal pressure. To decrease the absorption of photons, all wide-range spectrometers operate in a controlled atmosphere (vacuum or helium). The vacuum cabinet contains the entire spectrometer, except for some instruments such as the scintillator detector. Attenuation of the high-energy radiation is low, and to reduce the size of the vacuum chamber, the scintillation detector was and still is placed by some producers outside the chamber. A long rectangular window of a mechanically strong low-density polymer (Mylar) along the diffractometer circle allows the high-energy photons to reach the detector. High angular precision and accuracy is required for positioning the goniometer arm; therefore, severe mechanical constraints are imposed on its construction. Typical is an Copyright © 2002 Marcel Dekker, Inc.

angular reproducibility of 0.001 or less and a mechanical resolution of a few thousands of a degree or less (cf. Sec. VI). In most modern designs, 2W and W axes are decoupled. Scanning (step size can be software selected) or measuring at preprogrammed positions (e.g., element line, background left and right of the line for many elements) is possible. Typical angular scan speeds (2W) are in the range from 0.1 to 180 =min and slewing speeds can be up to 4800 =min. The slewing speed is the maximum rotation speed of the goniometer arm for passing from one position to another without recording. The 2W range covered is from 4 to 152 , with slight differences between manufacturers. Because of the high slewing speeds and full automation, the analysis time is severely reduced compared to the older intsruments and, when not too many elements have to measured, is often not prohibitively long. Jenkins et al. (1984) and Croke and Nicolosi (1987) announced a dual-channel sequential spectrometer. A twin primary collimator was installed consisting of an upper, fine collimator and a lower, coarse collimator. The beam is ‘‘divided’’ into two parts, each striking its own crystal. Two crystals are mounted close together but with a 15 inclination difference. The diffracted beams are detected by two detectors 30 apart. This fixed angle between both detectors, however, limits the benefits of a dual-channel instrument. Two alternative constructions should be discussed here. Simultaneous or multichannel instruments are equipped with monochromators set by the manufacturer (or by the customer) at a fixed position for the detection of a single element. Usually, curved crystal optics are used, and the whole setup is optimized for that specific element with an appropriate curved crystal, entrance and exit slits, and detector. The crystals are logarithmically or logarithmically and cylindrically curved as explained in Sec. III.D.1.e, see also Figure 20. To give some more flexibility to multichannel instruments, one or two channels can be equipped with a scanner. The optics are either focusing (e.g., Bruker) or nonfocusing (e.g., Philips). The flat crystal arrangement is principally a reduced size of the normal monochromator. The focusing arrangement is complicated by the fact that crystal and detector must be moved to focus the diffracted beam on the detector at any position. The scan range of 2W for the Bruker scanner is 30 –120 and 10 –100 for the Philips scanner. Full details of typical construction differences are given in Sec. VI on instrumentation. Two hybrid, difficult-to-classify spectrometers are discussed in this section: MDX1000 (Oxford) and the Venus 100 (Philips). The so-called ‘‘flexi-channel’’ of MDX1000 is an energy-dispersive spectrometer. The Venus 100 has only fixed channels, but they are sequentially presented to the fluorescent beam. Moreover, manufacturers are focusing more and more on dedicated hyphenated instruments combining x-ray fluorescence with diffraction measurements, such is an ARL 8600S and Philips CubiX. F.

Detectors

The objective of a detector is the transformation of photon energy into an electrical pulse. Pulses are counted over a period of time, and the count rate, expressed in counts per second or any other unit of time, is a measure of the intensity of the detected x-ray beam. From the theoretical point of view, the interaction of photons with matter is explained by the particle character of electromagnetic radiation, not by its wave nature. Four main classes of detectors may be distinguished for use in x-ray spectrometry according to the medium responsible for the energy transformation: gas detectors, scintillation detectors, semiconductor detectors, and recently introduced detectors supported on superconductive elements (tunnel junctions and microcalorimeters). They differ in the efficiency of detecting photons of a given energy, determining the range of energy or wavelength for which they Copyright © 2002 Marcel Dekker, Inc.

are suited. The width of distribution of pulse amplitudes as a result of impact of photons of the same energy determines the spectral resolution. Until recently, the best performance in energy discrimination has been obtained with semiconductor detectors and the full width at half-maximum (FWHM) is for the typical XRF domain in the range of 130 eV. This allows electronic pulse-height selection discriminating the energy of the impacting photons (at the expense of count rate, a maximum of 30 kcps). In WD spectrometers, pulse-height discrimination is required for removal of noise of low amplitude and of higher-order peaks of other elements interfering with the analytical line. It is a disadvantage typical for diffraction spectrometers. High-resolution detectors are not required in WD spectrometers because the interfering photons differ so substantially in energy that moderate discrimination is sufficient (a few hundred electron volts). High-resolution detectors constitute the heart of energy-dispersive devices and are discussed extensively in Chapter 3. Now, it is obvious that new types of device, superconductive junction detectors and microcalorimeters, exhibit energy discrimination an order of magnitude better. Of course, in the future they might be the obvious choice in applications demanding better spectral resolution, but the availability of the superconductive detectors is still a problem, as well as the maximum count rates they can manage and their routine maintenance and price. Only gas-filled and scintillation detectors will be discussed here as the devices supplementing crystal dispersive operation. Some attention will be paid also to the microcalorimeter counters, because the stormy period in their development and because of the prospects for the construction of the future generation of XRF (see Sec. VII). 1. Gas-Filled Detectors In this type of detector, the energy-exchange process occurs between photons and gas atoms or molecules in a strong electric field. When a photon strikes an atom, there is a given probability that the quantum of energy hn of the photon is imparted completely to an orbital electron of the atom. As a result of this gain in energy, the electron emerges from the atom with a kinetic energy of Ekin ¼ hn
W (Einstein’s photoelectric equation). W is the work function of the electron for leaving its orbital and is a specific constant of the atom. In the case of argon, hn

Ar
! Arþ þ e
The electron imparts its kinetic energy to other atoms, creating a series of electron–cation pairs on its path through the gas in a number equal to Ekin divided by the energy needed to expel an electron. Values for different gaseous and solid detector media are included in

Table 4

Effective Ionization Potentials (eV)

He Ne Ar Kr Xe NaI (Tl) Si (Li) Ge

27.8 27.4 26.4 22.8 20.8 50 3.8 3

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Table 4. Subsequently, the cations and electrons are accelerated toward the cathode and anode of the detector by the gradient of the electrical field between both electrodes, forming the avalanches of new ion pairs on their way to the electrodes. The discharge at the electrodes gives rise to a current pulse in the external electrical circuit. For a range of electrode potentials, the current pulse is proportional to the energy of the photon, and in spectrometers, the gas-filled detectors are exclusively used in this range. This is why these detectors are called proportional counters. Above this potential, in the Geiger–Mu¨ller plateau (Fig. 28), any photon entering the detector gives rise to a pulse of maximum amplitude without relation to the initial photon energy. The ratio of the number of ion pairs discharged at the electrodes to the ion pairs originally formed is called the amplification factor, generally denoted A. It is a complex exponential function of ionization potential of the gas, anode potential, radii of detector chamber and anode wire, and the mean free path of the electron. It implies that stability of applied potential and gas pressure is very important for reducing statistical errors on pulse amplitude. The basic parts of a gas-filled detector are a metal rectangular housing as cathode, a thin wire passing through the central axis as an anode isolated from the cathode, a filling gas, and a window either on one side or on two opposite sides of the housing. This gas flows through or is enclosed in the housing (Fig. 29). The diameter of the anode wire should be small. Realistic diameters are in the range of 40–80 mm. a. Flow-Proportional Counters These detectors cover the widest wavelength range and are used in sequential spectrometers exclusively for long wavelength ( > 2 A˚) and also in simultaneous instruments. The windows are made of thin foils of Mylar or polyester of 0.6, 1.5, 2 or 6 mm or polypropylene 1, 2, or 6 mm thick. On the inner side, windows are coated with gold or aluminum to create a homogeneous electrical field inside the detector. Because of these very thin windows, there is a leak of the filling gas that is compensated by a

Figure 28

Region of proportionality for gas detectors.

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Figure 29

Construction of proportional counters: (a) flow counter; (b) sealed counter.

constant flow of gas through the counter (Fig. 29a). This is called a flow-proportional counter. The gas is Ar with CH4 at a concentration of 10%, and the flow is about 0.5 dm3=h. For detectors in simultaneous instruments, helium is also used, or 88% He þ 12% CO2. For very specific applications, other combinations of gases may be useful. The gas is stabilized for pressure, flow, and temperature. This is of significance because a 1% change in gas pressure or temperature introduces a much more serious 6–7% change in pulse amplitude, which can discredit the results of the analysis (Short, 1991). The amplification factor in the proportional range is between 102 and 106. The current pulse is conducted over a resistor by which it is converted into a voltage pulse that is amplified, shaped, possibly discriminated in height, and counted. The count rate is a maximum of 26106 cps. The operating voltages range from 1000 to 3000 V. b. Sealed Proportional Detector Basically, the construction is the same as in the flow-proportional detector but the windows are thicker and no constant flow compensation for a leak is needed (Fig. 29b). The gases used are Ne, Kr, and Xe with Al, Be, mica, Mylar, or polypropylene windows. According to the element for which the radiation is to be detected, the most suitable combination of crystal, gas, and windows is chosen; for example, a PET crystal with a Ne detector with Al (for the determination of magnesium!) or a LiF (100) crystal with an Ar or Kr detector and a Be window for the detection many of medium-Z elements. If the filament is 80 mm thick, it is stiff and easy to install, has a long lifetime, and is easily exchanged. Because the proportional counter is often used in tandem with a scintillation counter, two windows are present on both sides of the main body. The absorption Copyright © 2002 Marcel Dekker, Inc.

probability of photons decreases with photon energy; the high-energy photons pass to the scintillation counter through the proportional counter without being essentially absorbed. The sealed detectors are applied in simultaneous instruments on their own and are not used in tandem with a scintillation counter. There is no need for a tandem arrangement because each channel is optimized for one element with the best-suited combination of detector and crystal. c. Pulse-Height Distribution In the detector, a photon generates a number of ion pairs proportional to the pulse amplitude. The formation of ion pairs is subject to statistical fluctuations, which means that the number of pairs formed oscillates around the most probable value. A record of pulses and their amplitudes as they emerge from the detector as a function of time is represented in Figure 30. When these pulses are collected according to amplitude, a distribution emerges as shown on the right side of Figure 30. The width of the distribution is measured at half-maximum and is designated FWHM. This determines the energy resolution of the detector. It is small for semiconductor detectors, such as Si(Li) (about 130–150 eV for the 5.9-keV MnKa lines from a standard 55Fe source; see Chapter 3), about 900 eV for gas proportional counters, and 3500 eV for a scintillation counter, all measured for the same photon energy. Note that this distribution differs from a distribution in intensity for a normal spectral line (intensity versus wavelength) because of natural and instrumental line broadening. d. Escape Peak As mentioned in the beginning of this section, a very probable method of energy exchange between a photon and an atom of the counter gas is the removal of an outer-shell electron. The probability of removal of inner-shell electrons is not zero, however, and this phenomenon happens in a number of interactions, in which case the specific K or L radiation is emitted. Because the wavelength of this radiation is on the long-wavelength side of the absorption edge, the mass attenuation coefficient is low and this characteristic photon has a good chance of escaping from the detector. The remainder of the photon energy is transferred to the electron of which the kinetic energy is generating a number of ion pairs but necessarily in a smaller number as an electron having received the whole photon energy. Thus, a number of pulses is created, distributed around an amplitude proportional to the energy of this electron. The absorption edge for ArK is 3203 eV; the energy of the FeKa line is 6403 eV. Pulses centered around 6403 7 3203 ¼ 3200 eV are generated, and

Figure 30 Amplitude–time record of impulses from the detector (left) and transformation of it into a pulse-height distribution (right).

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about 3200=26 ¼ 123 ion pairs are formed, 26 eV being the energy needed per creation of electron–ion pair. These pulses are present in the pulse-amplitude distribution as a second maximum on the low-energy side of the iron peak position. For Ne, Kr, or Xe, photon energies greater than 0.87, 14.32, or 35.58 keV, respectively, are needed to excite the K radiation. For lower energies, only L radiation is excited, resulting in an escape peak closer to the main pulse distribution. Due to the constant distance of escape peaks in relation to their ‘‘parent’’ fluorescence peaks and high intensity in some cases, they can be used sometimes instead of the originals. e.

Pulse-Height Discrimination

The energy resolution of a proportional counter, a few hundred electron volts at least, is not sufficient for use in energy-dispersive spectrometers (but see the simplified MiniMate construction by Philips). The spectral line separation in wavelength-dispersive spectrometers is done by the crystal monochromator. As is obvious from Bragg’s law, however, higher-order lines of other elements and the continuum radiation of the tube may be present in the same spectral window (angular position) of the crystal monochromator. In spectrometry in the ultraviolet or visible region, the higher-order diffractions of the gratings are removed by filters. This is not easily done with x-rays. For the same angle 2W, a second-order line has a substantially different wavelength or energy and is far enough apart to give two distinct pulse-amplitude distributions at the exit of the counter. For x-rays, electronic pulse-height selection or discrimination fulfills the role of filters in the optical region. The pulse-height analysis constitutes a necessary attribute to the wavelength-dispersive systems. Within an interval of 0.1 around 2W ¼ 37 for LiF (200) as analyzer crystal, we find AuLa1 in first order, E ¼ 1:2764 keV, ThLg8 in second order E ¼ 0:6390 keV, SnKb2 in third order, E ¼ 0:4259 keV, and LaKb2 in fourth order, E ¼ 0:3201 keV. The energy difference with the second-order line is 0.6374 keV and the energy resolution of a proportional counter is about 0.3 keV. The electronic circuits of all commercial instruments are able to discriminate between the pulses of both lines and, consequently, also the other higher-order lines. 2. Scintillation Detectors The energy exchange occurs in this type of detector in a medium of higher density and with high-Z elements in the matrix, namely a thallium-doped sodium iodide [NaI(Tl)], and as can be expected, this detector is only efficient for high-energy photons (< 2 A˚ or > 6 keV). The outer orbital electron for an iodide ion requires about 30 eV to be knocked out. Hence, the originally ejected electron is imparted with almost all the initial energy of the photon. The ejected electron dissipates its energy by promoting valence-band electrons to an excited state 3 eV above ground level, an energy emitted on deexcitation as a photon of 3 eV or l ¼ 410 nm. The intensity of the emitted light pulse is proportional to the number of electrons excited by the x-ray photon (ffi hv=3 eV). The pulses are detected by a photomultiplier in which the light pulse produces a few photoelectrons from the cathode material (e.g., indium antimonide). These electrons are accelerated inside the vacuum tube between the cathode and the first intermediary anode, called a dynode (Fig. 31). They gain kinetic energy and generate, in turn, a higher number of electrons from the dynode. This process is most often repeated 10 times, resulting in a substantial multiplication of the original number of electrons. The multiplication depends on the potential differences between the successive dynodes (100–150 V), and the order of magnitude of this multiplication is 106, which is called the amplification factor A. In general, the photomultiplier is a front-end Copyright © 2002 Marcel Dekker, Inc.

Figure 31

Layout of a scintillation counter.

tube, and the iodide crystal is in close contact with the tube window. The crystal itself has on its outer faces a reflective coating (but transparent for the photons above 6 keV) and a coating against moisture. In the external electronic circuit, the electrons provoke a current pulse of which the amplitude is ultimately proportional to the energy of the photon captured in the crystal. The conversion of the x-ray photons into a current pulse occurs over different steps: the initial formation of a photoelectron (I
! I0 þ e
), the quantified dissipation of the kinetic energy of this electron in 3-eV steps, emission of 410-nm light photons, the production of photoelectrons from the photocathode of the photomultiplier, and the multiplication of electrons. Although the production of the number of light photons in the crystal (hn=3 eV) is higher than the number of ion pairs created in a flow counter (hv=30 eV), the subsequent conversion into electrons is inefficient and roughly 1 electron is produced for every 10 light photons! The statistical errors of each process, particularly in the photomultiplier tube, propagate into the error of the output current pulse, thus substantially widening the distribution of pulse amplitudes and, consequently, reducing the spectral resolution. The DE is approximately two to three times worse than for a flow-proportional counter (see Fig. 26a), making a scintillation counter unsuitable for ED spectrometry. However, the resolution is still sufficient for a discrimination of higher-order spectra. Hardware for pulse amplifying, shaping, and counting are much the same as for the proportional counters. Also, an escape peak is present, with its energy equal to the initial photon energy minus the energy at the iodine K absorption edge. As already discussed, in many sequential spectrometers, flow-proportional and scintillation detectors are used in tandem, coupled in different ways. In another arrangement, both detectors are shifted over 30 . A secondary collimator is placed in front of the scintillation detector but inside the vacuum chamber; the scintillation counter is outside the chamber, but close to the exit window. According to the manufacturer (Rigaku), this allows a gain in count rate of 15%. 3. Alternative Con¢gurations Some interesting modifications of the detector have been proposed in the article by Ebel et al. (1983). These authors considered the possibility of reducing count losses between Copyright © 2002 Marcel Dekker, Inc.

analyzer crystal and detector (see Fig. 32) by removing the secondary collimator and allowing the detector to collect a divergent beam and, at the same time, by using a positionsensitive wire detector (with an energy resolution of 100 eV for detected photons with energies of about 10 keV). Position-sensitive constructions are popular in many physical applications, unfortunately not in the XRF field. Potentially, it may lead to new solutions for WDXRF. Another interesting effort is the use of a scintillation gas instead of the scintillation crystal (Kikkert, 1983; Palicarpo, 1978). This is the so-called gas-scintillation proportional counter (GPSC). As is clear from Figure 26a, the resolution is substantially better than that of its ‘‘parents.’’ The production of avalanches of ion pairs by the expelled electrons is the same as described previously. The discharge currents at the electrodes are not measured, however, but, rather, the scintillation pulses in the ultraviolet part of the optical spectrum are detected by a photomultiplier as in other scintillation counters. An essential difference with conventional gas proportional counters is that the space charge during the passage of a photon is avoided. The freedom-of-charge-related phenomena enable using higher counting rates (up to 2.56104 cps; Dos Santos et al., 1993). The lifetime of a GPSC can be quite long, up to several years, if there is no gas leakage in the meantime. The ability to use counters with large windows positively differentiates this kind of construction from room-temperature solid-state detectors, such as HgI2. Their pulse amplitude is also proportional to the energy of the incident photon. The counter can be supplied with a digital rise-time discrimination analyzer to suppress the pulses

Figure 32 Geometry with position-sensitive detector: T ¼ x-ray tube; specimen with point P irradiated by primary x-rays; crystal; mp ¼ mirror plane; M ¼ mirror point of P; PSD ¼ positionsensitive detector; l ¼ distance between mirror point and detector wire. (From Ebel et al., 1983.)

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forming the background (Simo˜es et al., 1997). This results in a much better performance of the counter. 4. Calorimetric X-ray Detectors Calorimetric detectors operate in extreme conditions—at temperature of liquid helium. The essential part of the device is a low-temperature, high-Z superconducting material with a minimal heat capacity. X-rays, even single photons, are absorbed and their energy is transferred to phonons. The detector is thermally coupled to a cold bath. The increase in temperature, which results from the absorption, is measured and is proportional to the energy of the photon. The temperature is measured by a doped semiconductor thermistor, as small as possible to minimize its heat capacity. It converts the experienced temperature change into a change in resistance. The thermalization time of the detector is in the range of 1–50 ms, and the time necessary to equalize the temperature of the detector to that of the cooling device is 100–500 ms. There is an analogy to the dead time of gas-filled and gasscintillation detectors. Still, there is a possibility for that type of device to allow count rates as high as 104 cps. Now, in more modern constructions (Irvin et al., 1996), the semiconductor thermistor is substituted by a superconducting transition-edge sensor (TES) thermometer. It is a strip of superconductor material biased within the transition range from the normal to superconductive state. The greater thermal sensitivity of the device allows the use of normal metal absorbers as a first stage of the detector. Unlike semiconductors and superconductors, metals can dissipate the absorbed energy into freeelectron excitations, not phonon ones. The free-electron deexcitation process is quick and efficient. The use of TES leads to a change in the amplifier: A less noisy superconducting quantum-interference-device current amplifier is introduced. As a whole, this new combinations exhibits a surprising feature: The energy resolution of the detector, as measured for the MnKa line, is better than the thermodynamic energy fluctuations in the microcalorimeter (7.2 eV in comparison with 10.4 eV). The total spectrometer efficiency obtained up to now is reaching 80% in some low-energy regions below 5 keV and can still be improved. Among all the detectors mentioned in the present chapter, this one does not make use of the ionization phenomenon. Strictly speaking, this type of detectors can measure energy transfers even in these processes, in which no ionization takes place. It has been recently demonstrated for, e.g., particle detection and impact energy measurements in time-of-flight mass spectrometry of single biomolecules (Hilton et al., 1998). The quantum efficiency of such an x-ray detector can be very good ( 95%) and the absolute spectral resolution is 7 eV in the range 0.2–20 keV; see Figure 26b (Silver et al., 1997). The expected absolute spectral resolution can potentially be even better than 1 eV in the mentioned range (Moseley, 1984). Of course, we are fully aware that this type of detector is not intended to compete with wavelength-dispersive systems, but we must emphasize here that the time of unbeaten behavior of crystals over all other spectral-resolving devices has gone. It seems to be a good place to present the comparison between wavelength and most modern energy-dispersive detectors at their present status. 5. Dead Time and Shift of Maximum Pulse Amplitude At high count rates (from a few thousand counts per second on!), the recorded count rate deviates proportionally from the real count rate. The reason for this discrepancy is the slow decay of the anode charge. Cation sheaths screen the cathode. This reduces the Copyright © 2002 Marcel Dekker, Inc.

effective voltage between the electrodes, resulting in a lower acceleration of the electrons and, consequently, in a smaller number of ion pairs formed. Fouling of the anode wire by impurities in the counter gas has the same effect. This phenomenon shifts the maximum of the pulse-amplitude distribution to pulses from apparently lower-energy photons. Moreover, when a photon is detected, the detector is, for a short time, unable to detect the next incident photon. This period of time is called the dead time. The value of the dead time is about 1–2 ms for scintillation counters, about 0.2 ms for proportional counters, and 200 ms for a Geiger–Mu¨ller detector. To overcome this problem, one can either choose another detector (but the choice is very limited), lower the intensity of the exciting beam by masks or absorbers, change the geometry and the optical arrangement (as is done in the case of synchrotron radiation), or use electronic anticoincidence circuits or appropriate mathematical corrections. Now, after the introduction of the high-power 4-kW option to the modern spectrometers, masks can be inserted for the analysis of the major elements (e.g., in Shimadzu instruments). The mask avoids problems with too great a count rate and its introduction is probably easier than tube power change during the analysis. The relation between the real and recorded count rate is Nr ¼

Nm 1
N m td

ð11Þ

where Nr is the real count rate (count=s or cps), Nm is the measured count rate (cps), and td is the dead time (s). If we allow a given level of discrepancy, say a%, then the following equation is valid: a ¼ N r td 100
a

ð12Þ

The real count rate and dead time are thus connected by a hyperbolic relation. If the allowed value of discrepancy must be kept constant, the left-hand side of the hyperbolic equation is constant. However, this is only part of the problem. As already stated, a photon, arriving just at the end of the dead time, is registered with reduced amplitude. It is only after the socalled recovery time that a new photon is registered with normal amplitude. An improved method for calculation of the dead time was proposed by Bonetto and Riveros (1984). These authors used the well-known fact that the second-order peaks are significantly less intensive than their first-order analogs, but the ratio of the intensities should be constant. If it is not constant, the deviation results fully from dead-time losses in the first-order signal. They derived the relation N m1 ¼ B
AN m1 N m2

ð13Þ

where B ¼ Nr1 =Nr2 and A ¼ Btd . The indices 1 and 2 refer to the order of the spectrum. A summary of the constituting parts of spectrometers and the wavelength ranges in which they are used is given in Figure 33. Note that different excitation sources are used in different branches of XRF, namely radioisotopes for EDXRF and tubes for WDXRF; there are also examples of synchrotron radiation. The x-ray laser is mentioned, but its appropriate version for XRF applications does not yet exist. Copyright © 2002 Marcel Dekker, Inc.

Figure 33 Review of different elements serving the construction of a spectrometer: optimal range (crossed area), feasible range (hatched area), and possible range (open area). (Adapted from Bertin, 1975.)

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IV.

QUALITATIVE AND QUANTITATIVE ANALYSIS

In this section, we review a number of items important to both qualitative and quantitative analyses. The qualitative application of XRF spectrometry is obvious. In simple cases, quite unequivocal assignments of spectral lines can be made based on a library of Ka or La lines or, in case of doubt, when an overlap exists, Kb or other L or M lines are checked to confirm or reject a line assignment. A few more complex situations are discussed. Quantitative analysis may also be simple. In the low-concentration range, quasilinear relationships are found between intensity and concentration. The same holds for determinations of higher concentrations but in a very narrow range for samples with quite comparable matrices. Interpolation on standard curves, obtained on suitable standard samples, allows a quick and sufficiently accurate quantitative determination. Another intensity–concentration relation, valid for binary samples, is based on the relative intensities of Ka and Kb lines in the case that the absorption edge of another element is located in between these two lines. In these boundary cases, matrix effects are negligible or interbalanced but always exist. In normal practice, they compromise the conversion of intensity into concentration in such a way that more or less complicated mathematical methods must be used for obtaining any accurate quantitative determination. The approach by fundamental parameter programs or correction algorithms is dealt with in Chapter 5. Between simple qualitative work and rigorous quantitative work, however, an area of applications is situated in which knowledge of the exact concentration or the complete qualitative composition is less important than the overall aspect of a spectrum for recognizing classes of samples (i.e., pattern recognition). An introductory discussion of this aspect is given in this section. Another point of interest in qualitative as well as to quantitative work is the measurement of the background. A good estimation of the background is necessary for obtaining net count rates, but decreasing the background allows lowering the detection limits. The origin and magnitude of the background is bound to the physical processes involved in the method of excitation and, thus, indirectly to the type of spectrometer. This aspect is also treated in this section. Finally, the said disturbing phenomena associated with the excitation of fluorescence radiation, as bremsstrahlung and the scattered radiation, can be used to increase the total analytical information and they also have some interesting applications. A.

Background

The background in WDXRF has four main sources: (1) the coherent and incoherent scattering of source radiation, (2) the presence of characteristic radiation of other than sample origin (materials of sample cup masks, sample cups, collimators, spectrometer housing, for example), (3) the detector, and (4) environmental noise (i.e., resulting from the threshold natural radiation of materials and even from a cosmic origin). The background from characteristic radiation from construction materials of the spectrometer can be minimized by careful construction and geometric arrangement of the device. Scattering from sample cup masks can efficiently be reduced by masks in front of the primary collimator. This contribution is especially great when small samples are analyzed. The beam mask reduces the irradiated area and may totally eliminate scatter from sample cup masks. Higher-order scatter increases with increasing wavelength. Pulse-height selection, as explained earlier, supresses counts from higher-order diffraction, but this technique is most efficient in the long-wavelength region and is decreasingly efficient toward the short-wavelength end. The short-wavelengths contribution to the background can be reduced by Copyright © 2002 Marcel Dekker, Inc.

installing adequate filters. A 100-mm titanium foil removes the characteristic lines of chromium from a chromium anode tube, significantly improving the conditions for the detection of Cr, Mn, Fe, Co, and Cu (Tertian and Claisse, 1982). For Cr tube radiation and a LiF (200) analyzer crystal, removal of scattered chromium radiation may be necessary when detecting the neighboring first-order lines. As Table 5 shows, not very many common elements are interfering. For steel analysis, only manganese really represents a problem. Because chromium is present in prevailing concentrations in the environment (tube anode!), manganese determination is also difficult with an aluminum filter. The simplest solution in such a case is the use of a LiF (220) crystal with better resolution in this region (but also with lower intensity). If tubes with other targets are used, tables are similar to Table 5 but with even more complex interference if targets like Mo and Rh are applied. The other elements listed in Table 5 refer mostly to less common elements and less intense lines. Other higher-order lines may interfere as well if the concentration of elements, from which the lines are derived, is high. For special cases, a convenient combination of tube targets, analyzer crystals, and filters can almost always be selected. Let us look at an example for the rhodium target tube. In the determination of palladium, PdKa1 overlaps with the Compton peak of RhKb (Fig. 34). A molybdenum filter in the primary beam reduces this peak to a very small level, allowing a perfect estimation of the location of the maximum and the determination of the intensity of the palladium line. Another efficient solution for background suppression is the use of linearly polarized x-rays. An excellent example was given by Kno¨chel et al. (1983) and it is reproduced in Figure 35. The analysis of nitrogen with a small concentration of xenon was made by a synchrotron excited XRF. Some special geometries allowing the use of polarized x-rays have been proposed by Wobrauschek and Aiginger (1980, 1983, 1985, 1986) for the more conventional XRF spectrometers. For EDXRF systems, a special commercial polarizing design is available (SPECTRO A.I. Gmbh). The subject is of great significance to synchrotron-based XRF and discussed as such in Chapter 8. A careful estimation of the background is apparently the simplest way to take the background into account! A first approach, but with limited application, is measurement on blanks—simple at first glance, but the adequate blank is often a rarity. Moreover, the blank ought to have the same scattering properties. Intensive efforts for mathematical modeling of background were performed but frequently for other purposes. The most common method of background evaluation is the linear (or even nonlinear) interpolation from background countings at a suitable angle above and below the peak position (suitable means not too far away for an easy linear interpolation, outside the tailings of the peak, and not at the position of an adjacent peak). Consequently, a good

Table 5 CrKa Pm La2 La Lb2 Ba Lg5 La Lb7 Ce Lb6 Pm La1 V Kb

Interfering Lines for CrKa and CrKb with LiF 200 crystal DE (eV)

D ( 2W)

CrKb


10
30
40 30 20 20 10

0.05 0.31 0.60
0.23
0.31
0.31
0.56

Ho Pr Mn Pm Ba

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Ll Lb7 Ka Lb1 Lg4

DE (eV)

D ( 2W)

<
10
20
50 10 30

0.04 0.24 0.61
0.17
0.30

Figure 34 Spectra of Pd excited by a Rh tube and recorded with and without a primary beam molybdenum filter (0.1 mm Mo). The peaks are labeled PdKa, PdKb, RhKa, and RhKb with their respective order indices. (Courtesy of ARL.)

spectral resolution of the system is essential because only in this case are peaks very narrow and do not overlap, and the background level is easily reached in the intervals between peaks. Finally, we refer to a method based on the relationship of fluorescent and scattered radiation intensities versus the value of attenuation coefficients. No measurements of background for the estimation of net line intensities is required in this case (Bougault et al., 1977). If we cannot overcome the background problem, it would be advantageous to use the background or scattered tube lines for increasing the information about the sample. There are very interesting examples of such work and they will be shown in Sec. IV.C. Copyright © 2002 Marcel Dekker, Inc.

Figure 35 Synchrotron-excited spectra of gaseous nitrogen with a small amount of xenon taken in the direction of polarization and perpendicular to the orbit plane. Counts normalized to stored electron current are integrated over the lifetime of the detection system. DORIS was operated at 3.3 GeV and about 55 mA. (From Kno¨chel et al., 1983. Reprinted by permission of the author and Elsevier Science Publishers.)

B.

Qualitative Spectrometry

Although the number of spectral lines is very limited compared to atomic emission spectra in the ultraviolet or visible region, overlapping of lines in x-ray spectra does exist. A classic example can be found in the works of Gentry et al. (1976), Sparks et al. (1978), and Sparks (1980), and an example is reproduced in Figure 36. At energies at which K lines of elements between palladium and cesium (about 22 and up to 30 keV) can be present, additionally the tails of L lines of uranium and thorium are found, and also the theoretically calculated M lines of the superheavy elements of Z ¼ 110 [in the time which passed since 1976 this element has been discovered (Hofmann et al., 1994)] and 126 should be expected. The erroneous attribution of some lines to these hypothetical elements was the subject of extended research and long discussions (Sparks et al., 1978; Sparks, 1980). In the software of modern spectrometers, special searching blocks exist for the assignment of lines: The detected line is compared with data compiled in the program. After the initial assignment, other lines of the element are identified according to data from the library. The initial assignment is eventually rejected if not confirmed by other lines (Fig. 37). Multichannel wavelength-dispersive instruments are not suited for this type of work, except for elements implemented on the spectrometer (maximum 28 elements). Sequential instruments can cover all elements (from beryllium on). The spectra can be displayed on the monitor screen of all modern spectrometers, documented, memorized as files, and printed. Particular parts of the spectrum can be scanned occasionally with higher resolution and sensitivity (step scanning) according to the time available for the analysis and the information required. Parts of particular interest can be displayed separately with a linear intensity axis or rescaled to square root or logarithmic scales. The peaks can be identified on the display by the angle 2W, by element name, either by wavelength or energy. Copyright © 2002 Marcel Dekker, Inc.

Figure 36 Fluorescence spectrum from a monazite particle excited with 37-keV synchrotron radiation (ORNL DATA) shows an improved signal-to-background ratio over that excited with 5.7-MeV protons (FSU DATA). Data points every 20 eV in ORNL spectrum and every 61.8 eV in FSU spectrum. (From Sparks, 1980.)

Figure 37 Example of qualitative analysis by rejection. The suspected MoLa peak in the lower spectrum was rejected after comparison with the standard containing molybdenum (upper spectrum). (Courtesy of Phillips Analytical.)

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The spectrum can be scanned linearly with respect to 2W or in a more optimal way. If Bragg’s and Moseley’s laws are combined, the following relationship among Z, the atomic number, and W is found: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þs ð14Þ Z¼ k2d sin W Z is proportional to the inverse square root of sin W. Jenkins et al. (1979) proposed computer monitoring of the scan speed according to this equation. It optimizes the scan speed, which represents a high gain in time especially in the low-Z range where the angular distances between the lines are long. The newest applications of WDXRF softwares try to avoid the old-fashioned treatment of data. One such traditional manual estimation was that concerning the possible spectral interferences. Normally, the analyst has to decide which place in a spectrum is relatively empty of disturbing peaks and compromising background contributions. This job was rather time-consuming, with the wide use of spectral tables and other datasets concerning, for example, intensity ratios. The results were not always encouraging. Now, virtual synthesis of wavelength-dispersive spectra is in progress (Reed and Buckley, 1996), in the Internet demonstration version also (Buckley, 1998). The expected pure-element spectra are loaded from the memory and installed in a region of interest of the anticipated spectrum. Those partial spectra are apparently corrected for the expected concentrations of the sample constituents. From the result shown on a screen of the computer, one can easily estimate whether the region of interest is suitable for a given analysis or should be excluded. Such a procedure makes the selection of lines for the qualitative analysis much easier, as well as the estimation of the background and also the assessment of the detection limits. C.

Special Quantitative Applications

Although the main quantitative treatment of XRF has been left to Chapter 5, still there are some special and nonconventional applications which proved to have potential practical meaning. Their significance becomes obvious when, considering the excitation process, one wonders how great a part of the radiation is transformed into the scattered form and how it can supplement the essential quantification based on characteristic signals. As stated earlier, monochromatic excitation simplifies the quantitative treatment of spectral data considerably, but this case is only seldom met in WDXRF. The most common excitation source is the x-ray tube, in which a set of characteristic lines superimposed on a white spectrum is generated. For this reason, only the fundamental coefficient method can be considered as rigorous and somewhat simplified in comparison with a complete fundamental parameter method. How the correction terms can be derived in a rigorous manner for the case of monochromatic excitation was shown in an article by Kuczumow (1982). In its original formulation, the Lachance–Traill equation is (Lachance and Traill, 1966) ! X Wi ¼ R i 1 þ aij Wj ð15Þ j6¼i

with aij defined as a constant written as aij ¼

mjC1 ðEÞ þ mjC2 ðEi Þ
miC1 ðEÞ
miC2 ðEi Þ miC1 ðEÞ þ miC2 ðEi Þ

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ð15aÞ

where m denotes the mass absorption coefficients for primary or secondary photons of energy E or Ei, respectively, corrected for the incident C1 and takeoff C2 angles. It has been proved (Claisse and Quintin, 1967) that this expression is valid only for boundary conditions: monochromatic excitation and absence of enhancement. In all other conditions, a’s are not constant, as experimentally shown (Tertian and Claisse, 1982; Lachance, 1981). It is impossible to derive a more flexible algorithm with constant coefficients directly from Shiraiwa and Fujino’s equation (Kuczumow, 1982; Rousseau and Claisse, 1974). Other algorithms similar to that of Tertian (Rousseau, 1974) and Lachance COLA (Lachance, 1981) are not recognized as fundamental. An algorithm that can be considered partially fundamental is that of Claisse and Quintin (1967), but it suffers from inaccuracies in the approximations introduced. Later, Rousseau (1984, 1987) and Kuczumow and Holland (1989) gave examples of algorithms derived from fundamental assumptions. Kuczumow and Holland describe the change in a when enhancement is included (monochromatic excitation) as follows: ! X aij Wij þ Daik Wk Wi ¼ R i 1 þ ð16Þ j6¼i;k

where all a’s are defined as previously but Da is defined as Daik ¼

m0kC2 ðEi Þ
mkC2 ðEi Þ miC1 ðEÞ
miC2 ðEi Þ

ð16aÞ

The new term m0kC2 ðEi Þ was introduced in Eq. (16). Although dependent on the excitation condition E and composition, this new term has the great advantage of being dependent on the latter in a simple way. The possibility of deriving the Lachance–Traill equation directly from the Shiraiwa and Fujino equation under the conditions applicable to Eq. (16) is also well established (Kuczumow and Holland, 1989). The next interesting case concerns scattered radiation as a source of analytical algorithms. Similar to the derivation made by Kuczumow (1982), an expression on the basis of coherently scattered radiation is obtained (Kuczumow, 1988): X coh coh coh þ R a
b ð17Þ Wi ¼ Rcoh Wj i i ij ij j6¼i

where acoh ij ¼ bcoh ij ¼

mjC1 ðE0 Þ þ mjC2 ðE0 Þ
miC1 ðE0 Þ
miC2 ðE0 Þ miC1 ðE0 Þ þ miC2 ðE0 Þ coh sj ðE0 Þ scoh i ðE0 Þ

ð17aÞ ð17bÞ

and Ricoh results from the comparison of a coherently scattered line from the sample and from the pure element i and scoh is the mass coherent scatter coefficient (for the set of incident and takeoff angles, as always in the conditions of the analysis). For Comptonscattered radiation, analogous expressions are obtained: X Wi ¼ Rcom þ Rcom acom
bcom ð18Þ Wj ij i i ij j6¼i

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with mjC1 ðE0 Þ þ mjC2 ðEcom Þ
miC1 ðE0 Þ
miC2 ðEcom Þ miC1 ðE0 Þ þ miC2 ðEcom Þ scom ðE Þ 0 j ¼ com si ðE0 Þ

acom ¼ ij

ð18aÞ

bcom ij

ð18bÞ

is the ratio of the Compton peak intensity of the sample to that of the pure eleRcom i ment. Equations (17) and (18) may be called Lachance–Traill equations for scattered radiation by their general appearance as well as by the way they have been derived. These equations are less sensitive to the type of components and changes in compositions than their analogs for fluorescent radiation, which otherwise is an advantage in analyses of samples widely varying in composition. From assumption, they include only the terms for strictly defined energy values, even if the analysis is carried out with a polychromatic source! In Table 6, the coefficients a and b from Eqs. (15)–(18) are collected, calculated for the K lines of La and Sm, the main components of samples consisting of La2O3 þ Sm2O3 excited by 60-keV photons at the incidence and emergence angles equal to 80 and 70 , respectively (the example is taken from EDXRF, but it is also instructive here). When applied to nonhomogeneous samples, also the a’s are proved to be dependent not only on the composition and the energy but also on particle size for dispersions or layer thickness for piles of layers of different composition (Helsen and Vrebos, 1984a, 1984b). Kuczumow et al. (1992) have proved that parallely with the conventional analysis of binary Au–Cu alloys by fluorescent lines (Fig. 38), it is possible to get similar information using coherently or incoherently scattered tube radiation. Sometimes, the results are even better than for the direct fluorescence analysis. Still other possibilities of using the scattered lines exist. Kuczumow et al. (1995) have proved, by a proper mathematical transformation of the scattered Rayleigh signal, that for light, chemically homogeneous matrices (as biological materials, e.g., wood), this new function of scattered radiation, called the corrected scattered intensity, strictly follows the density changes in the analyzed material (Fig. 39). The use of Rayleigh signals eliminates the necessity of transmission measurements and brings important auxiliary information (e.g., about the density). Although the method just described has been applied up to now for x-ray capillary-microprobe-type measurements only, it may be used generally (Luggar and Gilboy, 1994).

Table 6

a and b Coefficients for Fluorescent and Scattered Signals La2O3

af Daf acoh bcoh acom bcom

(La2O3 Sm2O3)

0.292
0.895 0.285 1.153 0.281 1.000

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Sm2O3 (Sm2O3 La2O3)

1.009 —
0.222 0.867
0.219 1.000

Figure 38 Analytical curves in the system R=W-1 versus w
w0 for the CuKa line in Au–Cu alloys. The solid line with open circles denotes the authors’ fluorescence measurements; the solid line with crosses denotes Rayleigh measurements, and the solid line with solid square denotes Compton measurements. R is the intensity relative to the intensity measured for the pure element and W is the composition expressed as weight ratio; w represents the ratio of the mass attenuation coefficient of the sample to the sine of the takeoff angle C2 . The subscript O refers to the values for the pure element o. (Modified from Kuczumow et al., 1992.)

D.

Pattern Recognition

The spectrum of a sample of complex composition is characterized by a set of lines, documented by their position and intensity. If we imagine an n-dimensional space with the intensities on the coordinate axes, the set of intensities determines a point in this space. Such an intensity space is convertible into an n-dimensional space of concentrations, and the point in the intensity representation has its equivalent in the concentration space (Klimasara and Berry, 1987). The aim of pattern recognition is to attribute a sample to a given class. A class is in a discrete way a set of points of similar features in the intensity or concentration space or, in a continuous way, a subspace inside a complete n-dimensional space. This subspace is restrained by a closed hyperplane, which can be approximated by a number of hyperplanes of smaller dimensionality. Inherent to all experimental determinations, the determination of intensity is also bound to a certain degree of uncertainty. This implies that a point, as such, cannot be determined, only a small subspace of limited size, limited by the assumed confidence levels. Equally ill-defined for the same reasons is the boundary hyperplane surrounding the subspace occupied by a class of samples. The ‘‘thickness’’ of this ‘‘space ring’’ surrounding the subspace is determined in relation to each Copyright © 2002 Marcel Dekker, Inc.

Figure 39 Line scans from x-ray microprobe measurements that show the correspondence between transmission (dotted) and corrected scattered (solid) data, and the sequence of tree rings of Silesian Pinus silvestris. The linear scan was made using a 89-mm capillary and a step size of 89 mm. (Modified from Kuczumow et al., 1995.)

coordinate axis by the appropriate multiplicity of the projection of standard deviation on this axis. 1. Delimitation of Classes If we consider classes as separate subspaces in n-dimensional configuration space, it is obvious that each of these classes has its own boundary hyperplane or closed space ring. If these hyperplanes can be determined in an analytical way, we are able to know the boundaries of classes. In simple cases, a class can be delimited by construction of a lowdimensional hyperplane that closely approximates the real boundary. The simplest example is the trivial division of the whole configuration space on two categories; occupied space–empty space (zero–one decision). It can be done by assuming the limiting surface (‘‘wall’’) inbetween the mentioned subspaces. If classes are separated by empty spaces, the differentiation of classes is even easier. 2. Method of Nearest Neighbours Here, the classes are considered as small subsets of points. The classification of each new point relies on a comparison with its nearest neighbors. Knowledge of the distance between the sample and the nearest neighbors is essential. The accuracy of the method depends on the way the distance is determined and the degree of complexity of the boundaries between classes, although it can be simplified by imposing proper statistical weights on learning samples. The method of the nearest neighbors has further implications. Copyright © 2002 Marcel Dekker, Inc.

Once the sample is placed in relation to the nearest neighbors, which are nothing but the standards, a separate program can draw the average from the composition of standards, taking the distances from the neighbors as the statistical weights for the estimation of the influence of each standard on the final result (Klimasara and Berry, 1987). However, this way of using pattern-recognition methods is the same as very conventional method of comparison with standards! Of course, standards are no longer considered in a conventional sense as some numerical values in relation to some elements, but rather as the full patterns [compare also the genetic algorithm by Dane et al. (1996)]. 3. Reduction in Number of Dimensions All points and classes are included in the multidimensional space generated by the number of elements present in the sample; however, some variables may be less dependent on the related coordinates, and then the number of variables necessary to differentiate the sample may be reduced. The number of dimensions may also be reduced if some variables are correlated; the most obvious way of coupling the elements is by joining them in a chemical compound. 4. Pattern Grammar Approach Another version of ‘‘pattern recognition’’ is also possible. The full spectra for different standards and=or samples can be collected and treated as some analytical claims (sentences), with peaks considered to be analogs of single words, and the whole set of spectra as a constituent of a ‘‘spectral language.’’ At the same time, we have the complete chemical knowledge of the same standards, with standard composition treated as the sentences, their particular components as the single words, and such second system is the part of another, ‘‘chemical language.’’ The task of the chemist is to translate sentences from spectral to chemical language while working with the real samples. Even more, the translation from one language to another can be treated as mapping (rather linear than nonlinear!). 5. Examples Samples may be classified into groups according to their origin. A primitive version of such x-ray methods was demonstrated by Lagarde et al. (1984). They observed the characteristic features of spectra of ceramics samples from western Africa and then put the samples in order ‘‘by the naked eye’’ according to their origin. A classic reference here is the work of Kowalski et al. (1972), which was followed by many other articles on archeology. Kowalski recognized the origin of obsidian artifacts by studying a set of intensities of 10 trace elements and projected the image of this set from 10-dimensional configuration space into a 2-dimensional space (nonlinear mapping). Similar applications in geology are obvious. Lo I Yin et al. (1989a, 1989b) proposed the application of the pattern-recognition-supported XRF to the initial selection and analysis of Mars Rover geological samples. They discovered on this occasion that it is not necessary to use sophisticated detection systems. On the contrary, they applied probably the most primitive version of an x-ray spectrometer with a proportional counter as detector. The resolving power of such a detecting system belongs to the worst among the existing ones (compare Fig. 26a), but even in these conditions, the results of the sample-selection procedure are good. In such classification procedures, putting emphasis on advanced data analysis can be a substitute for using a sophisticated high-resolution spectrometer for special analyses in difficult conditions. Copyright © 2002 Marcel Dekker, Inc.

If only a limited amount of data is available, samples can only be subdivided in groups. Discrimination of the alloys 663, B-1900, and 1455 can be done on the basis of their zirconium content (Kevex Material Analyzer Mode 6500 brochure). Nickel–cobalt alloys were subdivided on the basis of a logical analysis of successive spectral responses: Is the ratio ‘‘contents of elements A=contents of element B’’ greater than some assumed level X: A=B > X (Fig. 40)? Complete analysis of the alloy was not necessary. The same is often done when samples can be compared to standards [e.g., the ‘‘framing action’’ by software (description of Siemens software)]. An important application of pattern recognition may be found in ‘‘two-stage analysis.’’ For complex samples with nonlinear dependence of intensity on concentration, the analysis can be broken down in steps with a smaller number of dimensions (lower degree of complexity) or the samples may first be subdivided in classes before applying correction algorithms. The appropriate sets of differential equations exist: they establish the relation between the value of relative intensities [Claisse–Thinh systems (Claisse and Thinh, 1979)] or between increments of intensities [Kloyber et al. (1980) and Kuczumow (1984) systems] and increments of concentrations. These increments are calculated with respect to a central standard that represents the origin of a local system of coordinates. In

Figure 40 Selective procedure of the KEVEX Material Analyzer Model 6500 based on successive analysis of the element ratios. (Courtesy of KEVEX Corp.)

Copyright © 2002 Marcel Dekker, Inc.

this local system, a certain set of differential equations is valid and is of linear character (if not, it is not profitable to use it in such a restricted space). After attributing the samples to given classes, the right set of differential equations may be used for accurate determination of composition. The origin of the local system of coordinates may, at the same time, be the central point of a given class. When the delimitation of classes becomes problematical, a point in the configuration space may be designated and a set of differential equations, valid about this point, may be determined. From this, the boundaries of classes can be found and determined by some leading vectors introduced in the central points of the classes. Yap (1987) and Yap and Tang (1984, 1985) posed the reciprocal question in relation to ‘‘pattern recognition’’: how to extract some number of differentiating features from the known classes, differing them by time and place of origin. This was used to analyze pottery samples. Another name for this method is the ‘‘training phase’’ of the analysis. E.

Analysis on the Basis of Neural Networks

The pattern-recognition method leads us to quite different levels of reasoning. The most popular recent way of working with a spectrum is to transform it numerically and show it in two-dimensional intensity–wavelength or equivalent intensity–2W or intensity–energy diagrams. Some logical artifacts result from the numerical transformation of the original sets of signals. The interesting comparison between functioning of the electronic chip in the binary system of information and the biological cell in the complex system of information was presented by Cholewa during 15 ICXOM, Antwerp (Cholewa, 1998) (see Fig. 41). At the same time, without such a numerical presentation, a common naked-eye observation allows one to compare a fresh spectrum to other similar pictures compiled in our memory. Our way of reasoning is not numerical at all, but resembles some procedures of pattern recognition. It is simply a process of attributing a sample to some class. In a happy case, one can recall the same pattern in memory and then classify the sample as a part of existing and well-described objects. One of the possible ways of reasoning similar to the ways of functioning of the human brain is by applying so-called ‘‘fuzzy logic.’’ In that type of logic, the object can belong to the proper set with some probability, not necessary equal to 1 (belongs) or 0 (does not belong). The spectrum can be recognized in this manner, with some artificial probabilistic rules set in the very beginning of the process. An instructive example is taken from an article by Otto et al. (1992). For the qualitative estimation of an x-ray spectrum, the following rule is very useful: If the Ka line is supposed to be in a spectrum, then the Kb line should be associated with it, being ‘‘about 5’’-fold weaker. Neural networks open another approach to the problem. They imitate the action of neurons joined by synapses in the human brain. The artificial network has a layer of input cells and a layer of output cells. The cells from layers just mentioned are connected, some of them directly and the majority indirectly, by the additional cells, grouped in one or more additional, hidden layers. Each middle cell collects one or more signals from previous cells, gives those signals the proper statistical weight (e.g., according to previously imposed rules), and sums them up. Next, the modified signal is sent from the middle cell into one or many cells in the next hidden layer or, finally, to the output layer. The initial information (e.g., spectrum) given to the input layer is transformed into a useful set of information (e.g., about the chemical composition) in the output layer (Otto et al., 1992; Larsen et al., 1992; Bishop, 1992; Bos and Weber, 1991; Luo et al., 1997). Otherwise, Copyright © 2002 Marcel Dekker, Inc.

Figure 41 Drawing showing the different degrees of complicacy involved in the action of (left) the electronic chip and (right) the single biological cell, also from the informatic point of view. The electronic chip includes only one bit of the binary information; the cell is an incomparably more complicated system. It is a basis for the differences in computing and brain thinking. (Courtesy of Copyright © 2002Cholewa.) Marcel Dekker, Inc. Dr. Marian

it resembles the ‘‘pattern grammar’’ method previously described. The method can be applied for the classification of very complex spectra, for clustering the datasets, for coupling the sets of spectra to the groups of compounds, and for nonlinear data transformation between sets with different dimensionalities.

F.

Detection Limits

The importance of the correct determination of detection limits has been acknowledged by many chemists, especially those involved in environmental regulation. The accepted definition is as follows: ‘‘The limit of detection is the lowest concentration level that can be determined to be statistically significant from an analytical blank’’ (Nomenclature, 1978). The limit of detection is expressed as a concentration cL or an amount qL and is derived from the smallest measure IL, I being the instrument reading. How the data are treated in the statistics is described in Chapter 5. For a thorough discussion we may refer to the article by Long and Winefordner (1983). This article also contains interesting references to articles of Kaiser, Boumans, and others, which are therefore not repeated here. For the derivation of some simple statistical dependencies, the small, almost forgotten book of Beers (1958) is instructive. Finally, we can recommend the excellent discussion of the physical origin of background sources and detection limits in PIXE and synchrotron-excited XRF given by Sparks (1980), which is also valid for WDXRF. Here, we only invoke that according to the International Union of Pure and Applied Chemistry (IUPAC) recommendation, the characteristic signal is statistically attributed to the element when it is elevated not less than 3s(s ¼ standard deviation) above the background level. It makes the confidence level equal to 99.86%. The detection limit is then calculated for the standard deviation multiplicity k ¼ 3 and is expressed as a function of intensity in a unit of time (R ¼ N=t) with t as the counting time and with the sensitivity m expressed in cps=% or cps=ppm. In Table 7 a series of values for cL are given for randomly chosen elements collected from manufactures’ documents. No general list can be included because detection limits as well as sensitivities depend heavily on matrix composition and on the experimental conditions. Instead, this is intended as an indication of order of magnitude. Each discussion of detection limits provides methods for deciding whether a signal value IU for a unknown was greater than background reading IB with a given level of confidence. Another important approach is advocated and applied by Clayton et al. (1987). They state that ‘‘traditional techniques for determining detection limits have been concerned only with providing protection against type I errors, or false positive conclusions (i.e., reporting an analyte as present when it is not). They have not considered type II errors, or false negative assertions (i.e., reporting an analyte as not present when it is).’’ In their article, they discuss a test of the null hypothesis c ¼ 0 versus alternatives that c > 0, where c denotes the true but unknown concentration of an analyte in a medium and for two models of calibration curves with known or with unknown parameters. The test is applied to chromatographic data. Their theory can certainly be applied to XRF data, but the attempt has not yet been reported.

G.

Limits of XRF

Compared to ultraviolet (UV)–visible emission spectra, x-ray spectra are ‘‘oligoline’’ ones and very convenient with respect to the resolution of the spectrometers. The UV-visible Copyright © 2002 Marcel Dekker, Inc.

Table 7

Detection Limitsa

Matrix

Element

Terephthalic acid Low-alloy steel

Al–Mg alloy

Aluminum Glass powder Phosphosilicate glass Cement

Copper alloys

Fe Co C Al Si Cr Mg Si Ti Mn Cu Mg P B B2O3 Na2O MgO Al2O3 SiO2 SO3 P2O5 Be

cL 0.15 0.18 80, 240 4, 4.5 2.1, 4.1 2, 1.7 7 5 3 2 1 10.5 1.3 1% 0.4% 36 27 22 50 24 32 0.2%

Data source Philips Philips, Siemens

Philips

Siemens Broll, 1985 Philips Siemensb

Philips

Source: Data from Philips Analytical ‘‘X-Ray Spectrometry Application Notes nos. 504, 622, 623, 737, 808, 913, 916; Siemens SRS Awendung 84=8, 86=1, 86=5, 86=6; and Broll, 1985. a ppm (unless otherwise specified) for 100 s counting time and k ¼ 3. b With use of multilayer OVO 55.

emission spectrum of iron contains some 2000 lines between 200 and 800 nm; its complete x-ray counterpart has a dozen important lines between 1.757 A˚ (Kb1) and 0.615 A˚ (Lg)! For everyday work, however, only two lines, 1.936 and 1.940 A˚ (Ka1,2) and 1.757 A˚ (Kb), play a role. The densely populated L series spectrum such as that of gold has a set of about 20 lines between 0.9 and 1.5 A˚, a spectrum that can be managed by a spectrometer with a relative spectral resolution of 10
2. The prospects for the x-ray spectrometric range should be bright, but what are the limits to this method? 1. Application Limits All elements from beryllium on can be detected and=or quantitatively determined. On the high-Z element or low-l side, the spectral resolution may be a problem for pure spectroscopic studies, but seldom or never for qualitative or quantitative applications. Because the emitted energy of photons is high and the attenuation coefficients are low, the critical depth is great and, consequently, the analyzed volume is very representative of the composition of the bulk. The critical depth is the thickness dcrit of a layer parallel to the surface, from which 99% of the intensity sensed by the detector is recruited. Copyright © 2002 Marcel Dekker, Inc.

It is a function of the takeoff angle C2 , the linear absorption coefficient mr of the matrix, and the density:  ln 1
II1t ln10
2 4:61 dcrit ¼ lim !  ð19Þ
r=sinC2

mr=sinC2

mr=sinC2 m where It is the intensity emitted by a thin foil and I 1 is the intensity emitted by an infinitely thick sample. For long wavelengths, mr becomes high while dcrit and analyzed volume become low. If the sampled volume is low, the analysis may be called a near-surface determination, not representative for the bulk (see discussion in Sec. III.D.2.d). In that sense, the results of the determination of heavy and light elements in the sample including different kinds of elements can be nonequivalent. The heavier elements are then determined in the bulk, whereas the light ones are determined on the surface only. This can lead to surprising results if the process of surface segregation is occurring. Moreover, during recent density determinations (Tsuji and Wagatsuma, 1996) of coating layers by a similar method, Grazing emission x-ray fluorescence, striking differences have been revealed between surface and bulk densities; for example, the density of a 27-nm-thick coating lead layer was equal to hardly 35% of the normal, the bulk value of the lead density (Tsuji et al., 1997). It is a true limit by the force of nature, although it may be exploited for near-surface determination of light elements. From the concept of critical depth, it is obvious that coatings can be analyzed up to certain thicknesses depending on the absorption (and enhancement!) characteristics of the coating and substrate; the overall thickness range is from 0.01 mm to a few hundred micrometers for the normal angle versions of XRF. Apart from the analysis of macrosamples, WD spectrometers may also be fit to electron microprobes, where they supplement the ED spectrometer for better spectral resolution of light elements. Curved crystal optics is generally used because of optimum efficiency (compare the quantum efficiency of a classical WD spectrometer ffi10
8, curved crystal WD ffi 10
6, and ED spectrometer ffi10
4). WD spectrometers are also needed for heavy-particle-induced emission; because of the presence of many satellite lines, good resolution is required (Watson et al., 1977; Chapter 12). The dynamic range of analysis covers about five orders of magnitude, from 10
3 to 100%. Below 10
3%, preconcentration procedures must be used. The intensity and uniformity of exciting beams delivered by conventional end-window tubes were not sufficient to allow spot analysis. However, Boehme (1987) and Nichols et al. (1987) published the results of two-dimensional mapping executed with small beams from high-power sources (rotating anode tubes), allowing lateral resolution of about 30 mm. This offsets the boundaries to a new domain: high-spatial-resolution x-ray spectrometry. Still, the possibilities of the tube-excited, slit-based WDXRF microprobe are relatively poor in comparison with an x-ray capillary-based EDXRF system or especially proton and electron microprobes (Chapters 11–13). However, for many applications, the lateral resolution mentioned earlier is quite sufficient and this technique is time-saving. Moreover, the tube can be coupled with the polycapillary device. Note, that, for example, the diameter of the spot obtained from the polycapillary lens is in a range 50–500 mm, but the careful work with such a beam can give a somewhat better spatial resolution. In Sec. VI.B.3, we will discuss the milliprobes, a new solution which seems to be the most suitable for the wavelength-dispersive mode of operation. Copyright © 2002 Marcel Dekker, Inc.

2. Limits of Precision and Accuracy An analyst is always threatened by errors in the analysis outliers, random, and systematic errors. When a given operation or measurement provokes an offset between the true and measured values, a systematic error is said to be introduced. This offset may be due to a wrong calibration of any kind or to a wrong standard. Such errors can only be detected by analyzing a sample from the same batch with other techniques. Random errors and outliers can be dealt with by statistical methods. One example of such a method is described by Plesch (1981) and implemented in the software package Spectra 310 (Siemens). A calibration with n standards is assumed to give a standard deviation. The standard deviation ss is calculated according to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðC0
Ci Þ2 i ð20Þ ss ¼ n
p where Ci is the true weight fraction and C0i the result obtained; p is the number of parameters used in the fit: p ¼ 2 for a linear and p ¼ 3 for a parabolic fit. If the concentration Ca of an element in one specimen differs considerably from C0a , the presence of an outlier is suspected. The suspected value is eliminated, and a new standard deviation is calculated. Ca is an outlier with a probability of P% according to the inequality (F-test) (S1=S2)2 > Fp. The threshold values for Fp are listed in the current statistical literature. Application of the F-test in fact checks whether a statistically significant difference exists between the standard deviation calculated with and without the suspected result. Possible causes of outliers are numerous, including all kinds of human mistake, temporary instrument failure, and heterogeneity of specimens. Recently, the numerical programs helping to segregate the data on the ‘‘true’’ ones and outliers are available. They are based on the Grubbs test (Grubbs, 1969; Rousseeuw and Leroy, 1987). A measurement of fluorescent radiation is, as in any other experimental determination, subject to minimal error. The standard deviation of a radiation measurement is equal to the square rootpof ffiffiffiffi the number of accumulated counts N. The real standard deviation is greater than N because of errors of instrumental origin. Another source of error is the conversion of intensity into concentration and, last but not least, inherent to the sample, microheterogeneity, which is difficult to foresee and to quantify and is present even when all normal precautions for careful sample preparation are taken. Heterogeneities caused by sample preparation are dealt with elsewhere. These artifacts should be absent here. Segregation, however, which is the origin of what we call microheterogeneity, is a bulk (or sometimes surface) phenomenon of crystallographic origin: solid solutions of compounds or elements that are not perfectly soluble in each other, segregating into multiphase components. The ‘‘degree’’ of heterogeneity in the composition of the segregated phase as well as in the size of the phases depends on the history of the solution (thermal, chemical, or mechanical treatment). It is not easily mastered and is difficult to quantify. Its importance for XRF analysis has been acknowledged by several authors; Examples are tin and lead in solders (Glade and Post, 1970), silicates in fused beads (Novosel-Radovic et al., 1984), and silicon in Si–Al alloys (Michaelis and Kilday, 1962). The first report we are aware of is an article by Claisse (1957) on the determination of FeS in a sulfur matrix. A systematic study has been devoted to this phenomenon by Helsen and Vrebos (1984a, 1984b, 1986a, 1986b) and Vrebos and Helsen (1983, 1985a, 1985b, 1986), both by Monte Carlo simulation on hypothetical mixtures and by measurements on samples of known segregation. It was found by simulation on a (hypothetical) dispersion of spherical particles of iron in a chromium matrix Copyright © 2002 Marcel Dekker, Inc.

that even for a particle diameter of 1 mm, an error is introduced in the relative intensity of FeKa of 6% (on the level of cFe ¼ 0.5) with respect to the perfectly homogeneous solution. The effect was proved experimentally on alloys of Al–Si, Al–Mo, and Al–Zn. The extent of the effect changes with time, as in fused beads. This was demonstrated by the determinations of Novosel-Radovic et al. (1984) on fluxed silicates and also by Monte Carlo simulations by Vrebos and Helsen (1985a, 1985b). Microsegregation imposes a true physical limit to the accuracy obtainable by XRF spectrometry on quite a substantial number of samples, including minerals and alloys. 3. High-Z Limit We already insisted on the low-Z limit in Secs. III.D.1 and III.D.2, but what about the high-energy end of the spectrum? Commercially available generators and side-window tubes allow working at high voltages up to 100 kV, bringing even the K line of bismuth into the potential analytical spectrometric range. If a tungsten target is used, however, characteristic lines represent only a minority of the integrated emitted intensity, thus, excitation is bound mostly to continuous radiation, the maximum of which lies at 50 keV. Thus, only elements with K absorption edges below this value can be efficiently excited (Eu, Z ¼ 63, and all elements below). The above remark is even more valid for the spectrometers with end-window tubes, most commonly used in present-day instruments, of which the high voltage is limited to 60 kV. Possible excitation of K lines for all the elements from the periodic table by the tubes operating in voltages up to 160 kV is bound to metal–ceramic low-power tubes, cooperating with EDXRF systems and is outside the scope of this chapter. Thus, the available voltage limitation is the first one. If we return to Figure 26b and compare the spectral resolution of the LiF spectrometer to spectral distances between analogous lines in the K series, it is obvious that spectral resolution is worse above 35 keV (for K lines of Pr, Z ¼ 59). This is a second limitation. Most of goniometers have their optimal 2W range between 10 –15 and 70 : At the higher 2W end, the angular dispersion becomes insufficient; at the lower 2W end, the intensity becomes very low. If we choose LiF (420) with the shortest interplanar distance, 2d ¼ 1.802 A˚, useful measurements can be made at 10.0 corresponding to photons of 39.6 keV (Sm, Z ¼ 62 in the first-order spectrum). This is a third limitation. Studying Figure 26 further, we note that the spectral resolution of the LiF spectrometer is below the spectral resolution of the semiconductor Si–Li detector for photons of energy exceeding 17 keV, which corresponds to MoKa, Z ¼ 42. This is the fourth and most serious limitation in the K series. For analysis in the L series, there is no such limitation because the whole energy range lies within the capabilities of wavelength-dispersive devices. However, using the new microcalorimetric detectors with extraordinary and relatively constant spectral resolution, one can anticipate the preponderance of the spectral resolution factor for wavelength-dispersive crystals only in a region below 4–5 keV, which means the Ka line of V, Z ¼ 23 (see Fig. 27). Then, the ability to make the efficient trace analysis (because of higher count rates) and the availability of the device are still the advantageous sides of application of the crystal dispersion in the future. The wide voltage range of generators should not be used without some awareness of inherent drawbacks. For example, the K absorption edge of Mo is 20 keV. Thus, each x-ray tube with characteristic lines above 20 keV and working above twice this 20 kV for the maximum of bremsstrahlung generation is quite good for the excitation of Mo. The potential, however, should not exceed this value substantially much because the maximum Copyright © 2002 Marcel Dekker, Inc.

of the white radiation would shift more and more away from the molybdenum absorption edge and even the spectral characteristics do not improve. If we divide the mass coefficient of photoelectron excitation by the sum of mass coefficients for Compton and coherent scatters (this is a kind of measure of peak-to-background ratio), we obtain the values 76 and 18 for tube-excitation potentials of 40 and 100 kV, respectively! This is a substantial loss in peak-to-background ratio while passing to higher voltages. The discussion in this section shows that the advantage of high-energy photons (generated by very high voltages in SWT) does not compensate for the drawbacks and this probably explains why most manufacturers fit EWT tubes to the instruments. During the last years, manufacturers rather resigned from high-voltage generators and tubes, even in high-power 4-kW constructions. The use of a higher current is now a preferred option. Thus, we expect that only spectrometers with a high voltage up to 60 kV will dominate the market in the future. The effective spectral range above 30–40 keV should probably be left for typical energy-dispersive detectors. In the preceding paragraphs, we tried to show that although the spectrometers are almost perfect, there are limits to precision and accuracy due to the nature of the sample and to the mathematical conversion of intensity to concentration. These uncertainties are generally greater than those introduced by instrumental parameters.

V.

CHEMICAL SHIFT AND SPECIATION

The photoelectric excitation of atoms and subsequent specific deexcitation by emission of x-rays is the basis of all qualitative and quantitative uses of XRF. The deexcitation process was considered to take place between unique excited and ground quantum states. If these states were always stable, x-ray emission lines were to be found at strictly determined wavelengths or energies. However, electron density and quantum states are influenced by the environment of the atom with consequent wavelength shifts, or so-called chemical shifts, of the emitted lines. Such techniques as nuclear magnetic resonance or Mo¨ssbauer spectroscopy collect their spectroscopic information essentially from measurement of chemical shifts. As a matter of fact, all kinds of spectral lines of electromagnetic radiation are subject to chemical shifts. All these shifts are small and detectable only if the spectral linewidths are sufficiently small and the instrument has the required spectral resolution. For x-rays, the line shift is at most a few electron volts. An angular error DW ¼ 0.02 on the determination of a peak position at l ¼ 1 A˚ corresponds to 5 eV. Moreover, small temperature changes of the analyzing crystal already shifts the position over similar values. The sensitivity of peak position to the chemical environment of the atom was observed at the very beginning of x-ray spectrometry by Lindh and Lundquist (1924). The following characteristics of lines are sensitive to changes: line position (chemical shift) (Sumbaev et al., 1968; Sumbaev, 1970; 1976), line shape (Urch and Wood, 1978), and mutual ratios of line intensities (Asada et al., 1975). A detailed summary of the effect can be found in the book by Agarwal (1979), for example. In spectrometers with limited resolution, such as different energy-dispersive systems, most spectral shifts are hardly detectable. Chemical shift research has been done on high-resolution spectrometers. Gohshi et al. (1982a, 1982b) constructed an excellent two-flat-crystal instrument for studying chemical shifts. Asada et al. (1975) and Haycock and Urch (1978) improved the use of commercially available instruments for this purpose: by Fourier transform and iterative procedures for peak position detection or by using finer collimators. When analyzing Copyright © 2002 Marcel Dekker, Inc.

crystals with large thermal expansion coefficients, such as PET, are used, a temperature variation of 1 C gives rise to peak shifts greater than the chemical shifts. Helsen and Wijnhoven (1972) solved this problem by inserting standards at regular intervals and by using statistical techniques for the determination of the peak position. Sumbaev and Coworkers (1968, 1970, 1976) made experiments with an instrument in a Cauchois arrangement. In an article by Kataria et al. (1986), it was proved that even using a common solid-state Si–Li detector with moderate energy resolution, it was possible to detect chemical effects. Similarly, Habulibaz et al. (1996) performed experiments with a multilayer with an exact 2d spacing of 300 A˚, which means with poor spectral resolution. They managed to register SiL spectra for different compounds, clearly showing the influence of the chemical environment. Theoretically, chemical shifts can be calculated by the self-consistent field approach for the four quantum states involved (Manne, 1981). Unfortunately, the calculations are largely dependent on initial assumptions, the choice of which is difficult and somewhat random and the results are far from unequivocal. The experimental results of Sumbaev seem to be more spectacular. He connected the chemical shift DEKL to Pauling ionicity, i, valency m, and the parameter p, which was determined by lifetime ratios of electrons on the levels K or L: DEKL ¼ im½ pK DEK;ðZ:Z
1Þ
pL DEL;ðZ:Z
1Þ 

ð21Þ

He achieved an even more surprising result by presenting chemical shifts per unit ionicity versus valence (Fig. 42). This dependence is linear and may serve as proof of both the importance of chemical shift and the accuracy of Pauling’s scale. The shift can be related to the character of the bond in the molecule or the crystal. During the formation of a chemical bond, a rearrangement of electrons occurs, thus the formation of ions or bonds of varying degree of ionicity, the influence of which is reflected on all electron energy levels, sometimes more on the K level than on others levels. An example may be the shift between the Ka position of metallic aluminum and aluminum in different coordination states (IV and VI) (Helsen and Wijnhoven, 1972). Asada and Co-workers (1973, 1975, 1976) also made measurements in which the resulting chemical shift was attributed to the ionic bond character. They investigated the

Figure 42 Chemical shift as a function of valence. The energy shift divided by the ionicity is plotted (solid line). The dotted curve illustrates the uncorrected values of DE. (From Sumbaev, 1970. Reprinted with permission of J. B. Adashko.)

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changes in the ratio of some spectral lines, such as those for the Kb5 and Kb1 satellite line, and they connected them to the oxidation number of the analyte element. The results are again surprisingly elegant (Fig. 43). Clear proof for the dependence between the shape of lines and the oxidation number was given for manganese (Urch and Wood, 1978) and also between the chemical shift and the coordination number for different elements (Gohshi, 1981). The shift of AlKa for coordination states 0, IV, and VI was given by Helsen and Wijnhoven (1972). The application of the determination of chemical shift to chemical speciation is obvious. In geological samples, sulfur can be distinguished according to its different oxidation states (sulfur, sulfide, sulfite, sulfate, and hyposulfate). Birks and Gilfrich (1978) made measurements for sulfur using a portable low-power WDXRF instrument. In environmental and biological samples, the oxidation state of chromium is important: Cr(III) does not pass the cell wall, but Cr(VI) does and is suspected of being carcinogenic. Iwatsuki and Fukasawa (1987) reported the determination of chemical states of arsenic, selenium, and bromine. The determination of the valence of iron in welding fumes by measuring the energy shift and the intensity ratio of Kb0 to Kb1,3 has yielded very fine results (Fig. 44) (Tanninen et al., 1985). Kataria et al. (1986) managed to measure the varying ratio between the total line intensities of Kb and Ka as due to

Figure 43 1975.)

Satellite peak intensity ratio Kb5 =Kb1 versus oxidation number. (From Asada et al.,

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Figure 44 Relative chemical shift DE and the average valence of iron in standard samples and four welding fumes. The squares indicate the statistical uncertainties: SS ¼ stainless steel; MS ¼ mild steel; MMA ¼ manual metal arc technique; MIG ¼ metal inert gas technique. (From Tanninen et al., 1985.)

different oxidation states of manganese using an instrument of very moderate resolving power. This discussion of other than elemental or quantitative information from x-ray spectra has been intentionally restricted to x-ray spectrometry. A counterpart exists in the spectrometry of the ejected electrons, but that discussion belongs to the domain of spectrometry of Auger electrons or primary ejected electrons, for which higher resolutions are available or better speciation is possible. Another and sometimes severe drawback of these techniques is the required high vacuum and surface charging for nonconducting specimens. It should also be mentioned that Raman spectrometry is possible in the x-ray domain, but this field of research is only really accessible when synchrotron radiation is available (Sparks, 1974; Eisenberger et al., 1976a, 1976b), although performance on conventional instruments is feasible (Suzuki et al., 1970). In analytical determinations by synchrotron radiation, the Raman signals are often viewed as an additional disturbance (Jaklevic et al., 1988). Not many researchers are working in the field of speciation by WDXRF; nevertheless, it is a field of increasing importance, with theoretical as well as practical interest. It gives more than simple elemental information. Moreover, the shift of lines affects the intensity measurements for quantitative analysis, and stresses the use of the right standards for specific cases. In extreme cases, it may even be compulsory for quantitative determinations to scan over the peak maximum for each determination and locate in each case the intensity at the maximum for good precision and accuracy. If ultimate accuracy should be required, correction for peak heights may be involved. The few examples given point out that for studying the chemical shift, shape, and intensity measurements of spectral lines, the use of dedicated instruments is indicated, although a certain amount of useful work can be performed on commercially available instruments. A very interesting discussion of the problems immanently connected with the analytical application of soft x-rays can be found in the work by van Sprang and Bekkers (1998). These remarks have the obvious implications for the speciation analysis. Copyright © 2002 Marcel Dekker, Inc.

VI.

INSTRUMENTATION

The physical principles of a WD spectrometer do not allow dramatic differences in the general layout of the spectrometer. A practical instrument unites a number of construction compromises for speed, versatility, resolution, intensity, cost, user friendliness, and other factors, and all combinations have their own merits and their own drawbacks. Buying an instrument is a decision that in itself is a compromise because all benefits wanted by a customer are seldom or never united within a single instrument. In this section, the main features of the most important commercially available instruments are juxtaposed without formulating a global or final judgment.

A.

Electrical and Electronic Features

In modern instruments, all or almost all functions are controlled by a computer. As the electronic configuration is subject to continuous change, no details will be given here. Mentioned here are a few aspects which were difficult to automate or, when feasible, were hardly automated in older instruments: High voltage generators: The costly electromechanical control of conventional generators is replaced by computer-controlled solid-state generators, switched or high frequency. An example of exploitation of this flexible control is the switching of the tube along the isowatt curve, modifying the high voltage but keeping the power constant, an interesting feature if different intensities of the primary x-ray beam are required for different elements in the analysis of a complex sample (major, minor elements) by a sequential spectrometer. A consequence of the mechanical decoupling of W and 2W axes lifts the necessity of manual alignment of the crystals by giving (and memorizing) a correcting offset angle to the W axis for each crystal. A tandem disposition for gas and scintillation counters is no longer a necessity. The one or the other is brought into the right position under software control. The sin W potentiometer for continuous adjustment of the manual setting of the lower level and window of the pulse-height discriminator is replaced by the multichannel analyzer and electronic and electro-optical detection of the position of the goniometer. Automatic digital gain control corrects for pulse shrinking. Dead time is automatically corrected. This feature ensures linear response in most instruments to 2 Mcps for a gas-flow-proportional counter and to 1 or more Mcps for a scintillation counter. The main remaining electromechanical parts are the motor(s) of the goniometer in the sequential spectrometer (dc or stepper) or alignment motors for the crystals in the simultaneous spectrometers, the vacuum pumps, the electrovalves for gas flow (He, N2, counter gas), cooling water and compressed air, temperature control and fans, safety switches, sample lock and changer (internal and external), crystal turret, collimators, diaphragms, attenuator, and primary beam filters. An aspect totally absent in the older instruments was remote servicing. The full computer control allows remote servicing over a modem which substantially reduced the mean service time, especially in situations, where the service engineer is not close to the user’s facility. Copyright © 2002 Marcel Dekker, Inc.

B.

Spectrometer Layout

Commercially available spectrometers are classified according to the x-ray tube used, the position of the sample, sequential or simultaneous acquisition of the fluorescent radiation, hybrid forms, and some special dedicated designs. They will be discussed in the following order: Sequential instruments with EWT or SWT, sample down or sample up Simultaneous instruments with sample surface up or down Hybrid or dedicated designs 1. Sequential Spectrometers with EWTor SWT A typical example of geometry with sample surface down (i.e., the x-ray tube positioned under the sample) is represented in Figure 45 (layout of the Siemens-Bruker SRS-3400). The merit of the sample-surface-down geometry is in the analysis of solutions, not easily feasible with a sample-surface-up geometry. The positions of primary beam filters, sample=spectrometer seal, collimator masks or diaphragms, and spectrometer optics are clearly shown. The front end of the x-ray tube and all other parts except the scintillation counter are enclosed in a vacuum chamber. The sample=spectrometer seal is inserted in order to isolate the sample lock from the goniometer during sample loading. The sample lock and goniometer are evacuated by their own vacuum pump. The aim of this particularity is the maintenance of a monochromator atmosphere of constant composition,

Figure 45 A spectrometer layout with EWT and sample-surface-down geometry. (Siemens SRS3400. Courtesy of Bruker.)

Copyright © 2002 Marcel Dekker, Inc.

which may contribute to the reproducibility of the determination of light elements and prevent the contamination of the monochromator, particularly the collimator, from contamination by powdery samples. A drawback is the x-ray absorption when light elements are to be determined. Philips recently introduced a spectrometer with the same geometry (PW 2400 or 2404; Fig. 46) with an end-window x-ray tube, called the Super Sharp Tube (SST). The suffix ‘‘super sharp’’ points to the angle of conical nose of the tube, allowing a decrease of the sample distance to the window. The duplex detector (replacing the classical tandem fpc=sc construction) is a combination of a flow counter and a sealed xenon detector (improved performance for Ka for Ti to Cu and for La to W). The scintillation counter is giving an offset angle with respect to the gas counters. By this, the scintillation counter can be enclosed in the vacuum chamber without increasing the volume of the chamber. The selection of the detector is software controlled and became possible by the decoupling of the W and 2W axes. Shown in Figure 47 is the method by which one or two additional fixed channels can be added. Fixed channel(s) reduce(s) the analysis time for routine analysis of samples with one or two elements constantly present or for samples with an internal standard. ARL

Figure 46 Spectrometer with EWT and sample-surface-down position but with the addition of a fixed channel. This channel is the similar to the ones used in simultaneous spectrometers. (Philips PW2404) (Philips Technical Materials. Sequential X-Ray Spectrometer System PW 2404. Courtesy of Philips.)

Copyright © 2002 Marcel Dekker, Inc.

Figure 47 Heart of the ARL 9400 spectrometer (EWT, sample surface down). Two such monochromators, operating simultaneously, are integrated in one spectrometer, model, ARL 8420S. (Courtesy of ARL.)

realizes the simultaneous determination of more than one element by enclosing within the vacuum chamber a complete second monochromator of their unique extremely compact design as represented in Figure 47. It is characterized by the complete absence of gears and a positioning system by counting Moire´fringes resulting from interference of two grating systems. A development of the last few years is the substitution of diaphragms by a smalldiameter aperture allowing analysis of an area as small as approximately 1 mm2 (see Sec. VI.A.3 about milliprobes). An appropriate mechanism is moving the sample, which allows element mapping. This feature is realized in the XRF-1700 or 1500 series of Shimadzu (Fig. 48). It is an instrument with sample-surface-up geometry and fitted with an attenuator, reducing the intensity by a factor of 10 (for a major element determination); the merit of this is that no tube power switching is required for major and minor element determination. Copyright © 2002 Marcel Dekker, Inc.

Figure 48 Sample-surface-up spectrometer, fitted with small-aperture collimator diaphragms and an attenuator; all parts are enclosed in the vacuum chamber (Shimadzu XRF-1700). (Courtesy of Shimadzu.)

Only one manufacturer still produces an instrument with SWT. A number of instruments of a former generation are still in use and the tubes are still available. Diano Corporation continues the production of a spectrometer with sample-surface-up geometry and SWT (cf. Table 10). The merits and disadvantages of EWT or SWT were previously discussed. However, if a good homogeneity of the beam is required, a fine-focus SWT is by far a better solution. Sample holders of slightly different design for solids and liquids, for sample surface up or down, are offered by the manufacturers. The holders for instruments with samplesurface-up geometry are fitted with a spring mechanism to press the sample against the sample mask, both for sequential and simultaneous instruments. The holders adapting insert for liquids have a shape that can be recognized automatically and will not be accepted if the spectrometer is not under a He atmosphere (or eventually switches automatically from vacuum to He). 2. Simultaneous Instruments All simultaneous instruments are equipped with end-window tubes with very similar construction features. Up to 30 monochromators may be arranged radially around the tube head, with alternating high and low takeoff angles. One or more monochromators may be replaced by programmable goniometers of the flat (a normal flat crystal goniometer of reduced size) or curved crystal type. The latter is small but limited in angular range; the former has a large angular range but is large and occupies at least two positions otherwise taken by fixed monochromators. As already mentioned, Philips offers a small, low-power target transmission tube (TTT tube) used in the spectrometer PW 1660 (Philips Technical Copyright © 2002 Marcel Dekker, Inc.

Materials. Simultaneous X-ray Spectrometer System PW 1660) or X0 CEM capable of simultaneous determination of only 14 fixed elements or 12 elements when equipped with a programmable goniometer. In most instruments the samples are placed in the surface-up position. Some instruments (e.g., Oxford and PW2600 from Philips) irradiate in the sampledown position. The monochromators can be fitted to the outside of the vacuum cabinet, in which case they need not be vacuum tight. Figure 49 shows a generally used geometry. In the ARL sequential instruments, fixed channels (up to eight) or a complete second monochromator can be added. In Philips’ sequential spectrometer (PW 2400), an optional two fixed channels may be installed, making this instrument partly simultaneous. This is made feasible by the use of the special x-ray front end tube, the model Super Sharp Tube. 3. Milliprobes In an obvious effort toward the new fields of application, two companies introduced options, which seem to be restricted up to now to the very special, laboratory-made instruments. Rigaku and Shimadzu independently constructed the analytical systems with special masks: nonmovable (Rigaku) and scanning type (Shimadzu). In both versions, the diameter of the opening is 1 mm. Coupled with a 4-kW generator and high-power tube, it can carry out the analysis of the small pieces of a sample. In the Shimadzu version,

Figure 49 Layout of a simultaneous spectrometer with sample-surface-up geometry, typical for Bruker MRS4000 or Philips 1606 and with the arrangement at two levels with different takeoff angles. (Courtesy of Bruker.)

Copyright © 2002 Marcel Dekker, Inc.

the opening is scanned before a front of the sample, thus enabling the elemental mapping of the sample. The suggested application would be for the spot composition and density tracing of the sample, as it has been checked elsewhere (Kuczumow et al., 1995; Luggar and Gilboy, 1994). The construction of the milliprobe requires special attention concerning the x-ray tube used. The conventional end-window tube, due to its output inhomogeneouity, might be used only in the Rigaku arrangement, with a constant mask and the sample moving in front of the tube. Otherwise, the relatively small output of such tubes prevents their use as a radiation source for the milliprobes. C.

Intercomparison of Instruments

Only a limited number of manufacturers produce x-ray spectrometers. Hereafter, we will discuss the instruments of seven manufacturers subdivided in three classes: sequential and simultaneous instruments and a third class that we called hybrid spectrometers, because they have characteristics belonging to both former classes. The series numbers together with the names of the manufacturer are given in Table 8. In Tables 9–11, the numbers in the top row of the third, fifth, and last column refer to the manufacturer and series number of Table 8. The reader will find the actual addresses of the manufacturer’s headquarters at the end of this subsection. 1. Sequential Instruments In Table 9, a summary is given of the main characteristics of the current instruments as a guide to the market for the interested reader and for those who are considering buying a new instrument or replacing an older instrument. a.

General Remarks and Items Common to All Instruments A question mark in the table means that the data were not available in the brochures. When nothing is marked, it means that the given item is not applicable.

Table 8

Available Wavelength-Dispersive Spectrometers Sequential

Manufacturer ARL Diano Oxford Philips

Rigaku Siemens Shimadzu

Series

Simultaneous

Table 9

Series

Hybrid

Table 10

9400 2000

1 2

8600S

1

PW2400 PW2404

3 30

PW1606 PW2600 X-CEM

2 3 4

RIX2100 RIX3100 SRS3400 XRF1700 XRF1500

4 40 5 6 60

MRS 4000

5

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Series

Table 11

MDX1000 Venus 100 PW4100=20 PW4120=20 PW4110=10

1 2 20 3

Table 9

Presentation of Sequential Instruments Manufacturer 1

Generator Power (kW) Voltage range=step (kV=step) Current range=step (mA) Switch time (s) Switch mode Stability D%/% mains Source X-ray tube: geometry, load (kW) Anode, type, Be window thickness (mm) Option other anodes Primary beam filters number Elements Sample Sample surface Sample turret: number of positions Sample=spectrometer seal (yes, no) Optics Diaphragms: number of positions Collimators: selectable (yes, no), number

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3, 30

2

4, 40

6, 60

5

3 3 ?–60=? (opt.: ?–70=?) 15–60=1

3, 4 20–60=1

3 or 4 60=?

3 or 4 10–60=1

3 or 4 60=?

?–100 ? ? 0.0001

0–80=0.1 ? ? 0.0005

10–125=1 < 2,5 Isowatt curve 0.0005

100 or 150=? ? ? 0.0005

5–150=1 Fast — 0.0001

100 or 140=? — — 0.0002

EWT, 3

SWT

EWT, 3

EWT, 3 or 4

EWT, 3 or 4

Rh, 3GN, 75

Cr, EA-75, 250

Rh, SS, 75 or 125 Rh

W, Mo, 125 3

W,Pt,W=Cr, Pt=Cr,Mo=Cr None

Others on request Rh=Cr,Rh,=C, Rh=W 5 1 or 5

Cu, Al, Fe

—

Al, Cu

Down 2

Up 4

y

Rh, AGRh66G, Rh 75 or 125 Mo, W, Au, Cr Rh=Cr, Rh=W 10

4

Zr, Ti, Al, Cu

Cu, Al

Al, Ti, Ni, Cr

Down 2

Up 6

Down 2

Up 8 or 2

n

n

n

y

n

3

—

4

6

4

5

y, 4

1

y, 3

y, 2 or 3

y, 4

3

Attenuator (yes, no) Crystal changer: number of positions Detectors: extra (yes, no) Goniometer angular range ( 2y) fpc angular range ( 2y) sc and 2 axes (coupled=decoupled) Positioning technology Continuous scanning ( 2y=min) Step scanning ( 2y=min) Slewing speed ( 2y=min) Angular accuracy ( 2y, y) Angular reproducibility ( 2y, y) Temperature stabilization ( C) Additional fixed channels (yes, no, or number) Parts included in vaccum Counting electronics Count rate (kcps): fpc sc Sealed Duplex Pulse-shift correction Dead-time correction Computer instrument monitoring peripherals Remote servicing (yes, no) Sample changer: type (x, y), positions

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n

n

n

y

n

y

9 n

4 or 6 Sealed Xe

10 n

8 n

10 n

17–152 0–115 dec

0–146 c

8 Sealed Xe, Ne, Kr 13–148 8–104 dec

8–148 5–118 ?

4–150 4–112 dec

7–148 0–118 dec

Moire´ fringe 0.25–128

? 0.1–10

Optical sector 0.006–120

? 0.1–240

Stepper motor 1–200

Stepper motor 0.01–180

0.001–1.00 4800 0.001 0.0002

0.01–? 600 0.001 ?

0.0001–2 2400 0.0025 0.0001

0.005–0.1 1000 ? 0.0005

— 1000 0.00001 0.001

0.002–1 1200 ? 0.0003

0.1

None

< 0.05

?

y

None

2

None

None

None

All

All

All

All except sc

All except sc

All

2000 1500

? ?

 0.01

0.3

2000 1000

2000 2000 4000

2000 1000

Pulse-height discrim. Automatic Full

Automatic

Automatic

Automatic Full

2000 1500 1500 3500 Graphical display Automatic, dynamic Graphical display Automatic Partly Full

Automatic Full

Automatic Full

y xy, 12–98

n x, 4 or 10

? x(?), 50, 100

y xy, 8, 58, 100, 110, 138

n ?

Automatic

y xy, 6, 24–168

Table 9

Continued Manufacturer 1

Installation Instrument dimensions: H*W*D (mm) Weight (kg) Electricity supply (V, kVA) Cooling water: bar, max. temp. ( C), flow (L=min) Compressed air: bar, flow (L=min)

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3, 30

2

4, 40

6, 60

5

920*1080*760

1400*860*1346

1092*1410*920

1400*1420*1054 1070*1890*880

450 220, 6 > 2, 20, 0.5–5

590 208–480, 10 > 1, 29, 19

580 750 200–240, 8, 5 200–220, 10 > 3.5–8, 20, 6–11 ?, < 30, 3.5–11

700 115–400, 8 4–6, —, 1.5

780 200–220, 8 or 9 > 1.5–3, 30, 3–10

—

—

4–5, ?

6–10, 20

—

None

1350*1770*1080

The analyzing range of all instruments goes from Be to U and the transuranians, except for Diano 2000, for which the range is limited to B–U. The mains power requires a stability from
10% to þ10%, exceptionally from
15% to þ10%. A single phase is required, except for Rigaku, which requires three phases. Most commonly offered by all manufacturers are rhodium anodes for x-ray tubes. Primary beam filters are used to remove interfering lines from the tube target element and=or to improve signal-to-background ratio: Al: improves peak-to-background ratio for Pb and As in light matrices; removes RhKa line and the CrK lines Cu: determination of Ru, Rh, Pd, Ag, and Cd; removal of interfering lines from Rh Fe: removes the Ka line of Rh Ti: determination of Cr, Co, Fe, Zn; improves peak-to-background ratio Zr: measurement of CdKa, removes the Ka line of Rh Most manufacturers supply other filters on request. Sample dimensions: Most instrument allow a sample size about 52 mm in diameter (Diano: 63 mm) and about 30 mm in height (Bruker: 57 mm). To cope with small heterogeneities, the samples are spun at 30 rpm (or 60 rpm for Shimadzu) on most instruments (for Diano, the spinner is only available on the sample changer with 10 positions). The collimators are quoted in most brochures as extra-coarse, coarse, medium, fine (high intensity to high resolution). Only Bruker mentions the divergence angles: 0.077 , 0.15 , 0.46 , 0.93 , 1.0 , 1.90 , and 1.54 . Crystals: All sequential spectrometers make use of flat crystals. Counters: A gas-flow-proportional counter (fpc) and a scintillation counter (sc) are standard on all instruments, except Diano, in which the sealed Xe detector is standard and a scintillation counter is an option. The gas flow (Ar þ 10% CH4) is 1–2 L=min. The electronic counters are all based on a multichannel analyzer (Shimadzu and Rigaku probably still use a pulse-height discriminator) with an automatic pulse shift and dead-time correction. Available on all spectrometers is the choice of vacuum or He as the spectrometer atmosphere, with nitrogen as option for many. All companies supply the control unit by personal computer in more or less sophisticated versions with software for monitoring the instrument, for qualitative analysis, and for a number of possibilities for conversion of intensity to concentration. All of these items are subject to change and not discussed in detail here. The conversion algorithms are discussed in detail in Chapter 5. The software systems supplied by Bruker, ARL, Rigaku, and Shimadzu are all very user friendly. Installation data are added to give the potential buyer an idea of the required space and infrastructure. The dimensions and weight of the instruments are given without an external sample changed fit. b.

Remarks About the Individual Instruments (in Alphabetical Order as in Table 8) ARL 9400: As far as we are aware, ARL was the first manufacturer (and still the only one) to use counting Moire´ fringes for its goniometer positioning. The goniometer is a complete gearless construction and it has the highest slew rates

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Table 10

Presentation of Simultaneous Instruments Manufacturer 1

Generator Power (kW) Voltage (kV), step (kV) Current (mA), step (mA) Stability D%/% mains Source X-ray tube: Load (kW) Anodes Be windows (mm) Primary beam filters positions Elements Primary beam attenuators (Yes, No) Samples Sample surface Size: diameter, height (mm) Sample changer (positions) Monochromator (fixed channels) Takeoff angle ( ) Maximum number Crystals Detectors Element range Goniometer

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2

3

4

5

3 ?–60=? ?–100=? 0.0001

3 ?–60=? ?–75=? 0.0005

3 5–60=1 (a) 5–125=1 0.0005

0.2 50 4 0.0005

3 20–60=1 5–150=1 0.0001

3 Rh, W

3 Rh, Cr, Mo

TTT* Cr

125, 75 4 (a)

125, 75 10

3 Rh, other on request 125, 75 3

Rh 66 G Rh, Mo, W, Au, Cr 125, 75

Cu, Al, Fe y, 14

Up 75, 50 11, 32, or 8 (b)

Up 51, 40 6, 72, 300

Down (b) 51, 40

? 30, 20, 11 (c) Curved Gas counter, sc

29, 44 28 (a) x

29, 44 28 Log. curved

Be–U

Be–U

y

40, ?

52, 57 8, 58, 100, 138

25, 39, 5 14

26, 24 (a) 28

Gas (Ar flow, Ne, Kr sealed)

fc, sc Be–U

Maximum number Collimators Crystals number

1 (d) 3 9

Type Scan speed ( 2y=min) Step scan ( 2y=step) Slewing speed ( y=min) Angular accuracy ( 2y) (LiF) Angular resolution ( 2y) Angular reproducibility ( ) Angular range gas counter ( 2y) sc ( 2y) Detector Element range Spectrometer environment

Flat 0.25–128 0.001–1 4800 0.015 0.001 0.0002 17–155

0.06–76 0.01 1000 ? ? 0.003 5–120

0–115

5–120

B–U Vac

B–U Vac, He, air

XRD channel (yes, no) Installation Dimensions: H*W*D (mm) Weight (kg) Water supply(bar, max.  C, L=min) Compressed air (bar, L=min)

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(a)

2 (c) 1 1 (LiF 200, 220, Ge) Flat 0.06–76 0.01 480 ? ? < 0.01 10–100

2 (d) Slits 4 (PE, InSb, Ge, LiF 200, 220) Cyl. curved 0.06–76 0.01 480 ? ? < 0.005 85–47

sc, Kr-sealed

fpc

Vac, other on request

Vac, other on request

8

Up to 4

F–U Vac, air

y

n

1350*1400*950

1804*1232*877

1000 > 3, < 18, 5

900

?

2 (a)

305–8, < 20, 6–10 4–5, ?

1190*815*1230

1800*1810*950

580

900 3–8, 20, 3 6–10, 20

Table 11

Hybrid Instruments Manufacturer 2, 20

1 Generator Voltage (kV)=current (mA)=power (W) Stability D%=%mains Max. mains variation (V) X-ray source Tube Anode External Cooling Sample Sample surface up, down Spinning (rpm) Dimension (diameter, height in mm) Sample changer Spectrometer chamber Monochromator (fixed) Element covered

Crystals Additional spectrometers Dimensions (H*W*D in mm) Weight (kg)

3

40–50=4=200

50=4=200

id.

0.007 210–250

0.0005 180–264

id. id.

Similar (a) Rh None

Target transmission Sc None

id. Cr id.

Down

Down

48, 40

26–47, 40

id. 30 id.

? Vac, He optional

Single, 10 Vac, He

Single Vac

F–U

S, S-bg., Zr, Ni, Ni-bg., Mn, Mn-bg., V, V-bg Curved None

Al, Si, Na, Mg, Ti, Mn, K, Ca, P, S, Fe id. id.

id.

1200*750*960

id.

180

Flat, curved optional Flexi-channel (EDXRF) 1090*800*800 160

of all spectrometers. In the series 9420S, two goniometers or additional fixed channels can be fitted, adaptable to a great variety of analytical situations, excellent for rapid nonroutine industrial control, reducing the analysis time by a factor of 2 with respect to a conventional sequential spectrometer. Separate vacuum pumps support the vacuum for the sample lock and the monochromator. Diano 2000: This offers dual-target tubes. The targets are push-button selected and no moving parts are present (dual filament). Two collimators are offered: a 0.032-in. spacing ‘‘beam tunnel’’ or a 0.010-in.-spacing ‘‘diffracted beam soller slit.’’ The instrument is not temperature stabilized because the 45-kg bronze bull gear goniometer has a sufficiently high thermal capacity, avoiding rapid temperature changes. It is, however, not sufficient when one wants to measure peak shifts. The crystal in the turret must be individually tuned. The W and 2W axes are coupled but with a bisector arrangement and moved by a dc motor. The standard detectors are a flow-proportional counter and sealed Xe, the scintillation counter is an option. The counting is also by a pulse-height selector. Diano offers a fixed-sample and fixed-element mode for analysis. Copyright © 2002 Marcel Dekker, Inc.

Philips PW 2400, 2404: Both types differ only in the power output of the generator. The tube power is switched along the isowatt curve (i.e., by keeping the power constant when either the voltage or current is changed). The fixed channels that can be added have as the detector either a flow-proportional counter (3000 kcps), Xe-sealed (1000 kcps), Kr-sealed (2000 kcps), Ne-sealed (2000 kcps) proportional counters, or a scintillation counter (1500 kcps). The positioning of the goniometer is done by optical disk, a sophisticated highperformance system, giving the highest angular accuracy and reproducibility. The goniometer is driven by dc motors. The external sample changer accepts samples of nonuniform size. Rigaku 2100, 3100: Both types differ only in power output of 3 and 4 kW. The collimators are either standard=high resolution or standard high resolution=ultralight elements. The attenuator in front of the monochromator crystal reduces the intensity by a factor 10. The advantage should be that it avoids power switching of the x-ray tube. The sample-surface-up geometry makes it difficult, if not impossible, to analyze solutions. A diaphragm of 1 mm diameter allows small-spot analysis. A built-in cleaning system removes the impurities of the wire in seconds in this instrument because the flow-counter efficiency deteriorates by contamination of the wire electrode. Siemens (Bruker) SRS 3400: The instrument is rather conventional and easy to service. The positioning of the decoupled arms of the goniometer is done by stepper motors. The positioning of the sample holders is very reproducible. The external sample changer accepts nonuniform samples also. An excellent software SPECTRA PLUS is supplied for qualitative, semiquantitative, and quantitative analyses. The instrument was formerly sold under the name of Siemens but is taken over by Bruker Analytical X-Ray Systems (Bruker AXS). Shimadzu XRF 1700, 1500: The difference between both instruments is that the selection of filters, diaphragms, attenuator, and crystals is automated in the 1700 model. A special feature of this instrument is the presence of diaphragms of diameter 1, 3, 10, 20, and 30 mm. The 1-mm diaphragm together with an appropriate sample positioning system allows element mapping. 2. Simultaneous Instruments with Sample Surface Up or Sample Surface Down A summary of the characteristics of the instruments capable to determine simultaneously a number of the elements is given in Table 10. The instruments are composed by a few till maximum 30 small-size high-efficiency spectrometers as discussed earlier, optimized for one particular element. The set of spectrometers is dedicated to a particular series of samples as used in, for example, the cement or steel industry. To attribute some flexibility to these instruments, one or more scanning goniometers are added. a.

Remarks and Features Common to All Instruments All manufacturers use x-ray tubes with either a 75- or 125-mm-thick Be window. The detectors offered are flow-proportional gas counter (Ar), sealed gas counters filled with Kr, Ne, or Xe, and a scintillation counter. The linearity of the photon count rate is equal or similar to the one given for sequential instruments. Temperature stabilization is either individual for each spectrometer or for the ensemble.

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Electricity power supply is single phase, 10–15 kVA. ARL offers the possibility of adding an additional x-ray diffraction device within the fluorescence spectrometer. Other manufacturers offer integrated sample transport from the spectrometer to an x-ray diffraction instrument. The CubiX XRF from Philips is a simultaneous spectrometer for up to 14 elements, similar to the X’Cem. It is integrated in the same module with an x-ray diffraction instrument; both can be juxtaposed, allowing a rapid sample transport from one instrument to another. Both devices are operated by a single personal computer. Cubix represents a modular concept for integrated analysis, including elemental and crystallographic determination. b.

Remarks About the Individual Instruments (in Alphabetical Order as in Table 8) APR 8600S 1. 2. 3. 4.

The primary beam filter takes the place of three monochromators and cannot be fit alongside an XRD channel. Eight positions for cassettes for large samples (Ø75 mm and height of 50 mm); 11 and 32 positions for small samples (Ø52 mm and height of 30 mm). Thirty monochromators can be fitted or 20 plus 1 goniometer, or 11 plus an XRD channel. The goniometer is the same as the module of the sequential instrument.

Philips PW1600: Maximum two fixed channels can be substituted by a programmable goniometer (with flat crystals and parallel beam collimation). One programmable channel occupies space as for one fixed channel. Philips PW2600 1. 2. 3.

Switching time of the generator is  2 s and the tube is switched along the isowatt curve. Particular feature of this instrument is that it allows liquid samples to be handled also. Two different goniometers can be fitted: (1) one with a flat crystal and parallel beam collimation and it takes the space of one fixed channel; (2) one with a cylindrically curved crystal (logarithmic spiral optics) and two slits and it takes the space of three fixed channels. Optional beam attenuators can be fitted, enabling all channels to handle concentrations from ppm levels up to 100%.

Philips X9 Cem: The x-ray tube is a low-power Target Transmission Tube (TTT), (Sec. IIIA). Because of the low power (200W), this instrument does not need external water cooling. Siemens MRS 4000:

High power (4 kW) and high speed instrument.

3. Hybrid Instruments The instrument from Oxford MDX1000 is classified as hybrid because it combines a WD fixed channel with what is called a Flexi channel, an EDXRF unit. The x-ray tube is similar to the target transmission tube, produced by Philips, but no other details are given in their brochures. The instrument series Venus 100 from Philips are called hybrid because Philips uses fixed channels, which are, however, switched sequentially in front of the sample. The instruments are delivered for specific purposes: The PW4110=10 is provided for the analysis of industrial minerals, particularly for the ceramic industry, and the PW4100=20 and Copyright © 2002 Marcel Dekker, Inc.

PW4120=20 for the determination of, for example, nickel, sulfur, and vanadium in petroleum products. The element range can be adapted to specific purposes. Both instruments from Oxford and Philips are simple, low-cost, but high-performance devices which can be used in harsh industrial environments; they are simple to install and are characterized by small dimensions, light weight, no water cooling, no compressed air, and no severe requirements for the stability of the mains. Hybrid instruments are presented in Table 11. 4. Instruments Dedicated to Special Applications and Supplements Two manufacturers, Siemens and Philips, supply for the semiconductor industry transformed versions of their sequential spectrometers SRS 3300 (power up to 4 kW, mapping possible) and PW 2800, respectively. The main feature is the sample area which must accept silicon disks as large as 10 in. in diameter. The sample introduction system is automated and meets the standards for working in a cleanroom environment. The manufacturers also provides as many auxiliary devices as possible. Advanced ‘‘analysis lines’’ are available from Rigaku, Philips, and Siemens, which offer a complete set of the mills, fluxers, and polishing devices for the XRF and XRD laboratory. Worth noting is that, on the other hand, during the last years, some new companies appeared on the market, offering not whole instruments as such but only some specialized parts of instrumentation. Among the most important participants in this kind of activity, the following should be mentioned: Advance Hivolt, offering x-ray generators, especially with 4 kW power X-Ray Optical Systems Inc., supplying x-ray focusing and collimating polycapillary optical devices plus designing the optical systems with x-ray semilens Osmic Inc., the main producer of multilayers Corporation Scientifique Claisse Inc., producing fluxers for sample preparation Contact with these producers can be very meaningful alternative in case of constructing new, home-made, and dedicated version of an instrument. We conclude this subsection on the intercomparison of instruments with a list of the addresses of the headquarters of the manufacturers: ARL Applied Research Laboratories S.A., En Vallaire Ouest C, case postale, CH1024 Ecublens, Switzerland Diano Corporation, 271 Salem Street, Woburn, MA 01801, USA Oxford Instruments, 19=20 Nuffield Way, Abingdon, Oxon OX14 1 TX, England Philips Analytical X-Ray BV, Lelyweg 1, 7602 EA Almelo, The Netherlands Rigaku International Corporation, 3-9-12, Matsubara-cho, Akishima-shi, Tokyo 196, Japan Shimadzu Corporation, International Marketing Division, 3, Kanda-Nishikicho 1-chome, Chiyoda-ku, Tokyo 101, Japan Siemens, A.G., now: Bruker AXS GMBH, Oestliche Rheinbrueckenstr. 50, D-76187, Karlsruhe, Germany

VII.

FUTURE PROSPECTS

The time passed since the first edition of this handbook has been one of the most stormy periods in research and applications of x-rays since their discovery! It is well known how Copyright © 2002 Marcel Dekker, Inc.

much Ro¨ntgen was confused, although totally convinced, by the impossibility to demonstrate the electromagnetic nature of x-rays through reflection, diffraction, or interference experiments! For many years, it remained impossible and further progress was very slow and tedious. Recent years have brought the real breakthrough. We know of three recent realizations of x-ray lenses: Two diffractive: 1.

2.

Zone plates (in other words, Fresnel diffracting structures), earlier restricted to the very soft x-rays. They are very important for the construction of the x-ray microscopes and for the observation of living biological objects in the so-called ‘‘water window.’’ But now, by combining the Fresnel diffraction patterns with Bragg’s phenomenon, the special Bragg–Fresnel lens was constructed, well suited for focusing hard synchrotron radiation (Aristov et al., 1986; Snigirev, 1995; Snigirev et al., 1995; Chevallier and Dhez, 1997). Snigirev refractive lens (Snigirev et al., 1996; 1998; Elleaume, 1998; Hastings, 1996; Eskildsen, et al., 1998), consisting of one or two mutually perpendicular arrays of holes drilled in low-Z dense material (Be, Al). Depending on the number and shape of the holes, they can focus hard x-ray beams,

One version using total reflection: 1.

Kumakhov polycapillary lenslike device, which might be somewhat poor in focusing capability (spot is 50–500 mm wide) but very efficient with respect to the fraction of the primary input power turned into a small spot on the analyzed surface. Interesting products were announced by X-Ray Optical Systems Inc., Advertising Materials.

In parallel, great progress was realized in the construction of energy-dispersive detectors with increasingly better spectral resolution. The most promising device seems to be the microcalorimeter detector, a device with very good spectral resolution parameters. Moreover, its spectral resolution is more or less constant over the whole spectral range covered (1–20 keV). With such a device, the spectrum deconvolution problems, so characteristic for devices with poor spectral resolution, should vanish. Potentially, it lowers the detection limits, by narrowing the energy region in which the very sharp peak is placed. Only a very restricted number of background counts is detected in that region, thus improving the peak-to-background count rate ratio. However, the detection limit depends on the maximum count rate allowed for these detectors, which is, at the moment, very restricted. Recent discussions show (Ladbury, 1998) that intensive work is going on to overcome this shortcoming in the nearest future. The aim is to get the device with an area of 40 mm2, with count rates up to 105 cps. Another progress took place in the construction of time-resolved excitation sources. The classic radiation generator for the purpose is synchrotron. Probably much more convenient, the ‘‘portable’’ and switchable version of a time-resolved x-ray source will be a compact-flash source. Furthermore, the construction of the x-ray laser—an old dream!—is also progressing, maybe slowly, but in a promising direction. Collected together, all available information about this progress justifies hope on the construction of new versions of existing instruments as well as the advent of new ones. It is worth noting that in the present chapter we considered an XRF spectrometer mainly as a device used for the determination of elemental composition. What is now discussed testifies that future instruments will give both compositional and spatial resolution (microprobe) and time resolution (kinetic probe). New kinds of lenses and normal incidence Copyright © 2002 Marcel Dekker, Inc.

mirrors give an impetus to the construction of x-ray microscopes also in the hard part of the x-ray spectrum. One can easily imagine a microscope (but one easily construct!) working in the hard x-rays regime supported by two different x-ray lenses—one acting as an ocular lens and one as an objective lens. The new version of the microprobe could then efficiently concentrate x-rays (e.g., by Kumakhov polycapillary) on the confined areas and then record the fluorescent radiation with the excellent spectral resolution of a microcalorimeter as the energy-dispersive detector with the whole spectrum completed in one operation. Such a device is now under construction in the National Institute of Standards and Technology (Wollman et al., 1997a, 1997b; Anonymous, 1997). The microcalorimeter detector, attached to a scanning electron microscope, has, for the time being, a spectral resolution of 7.2 eV  0.4 eV, calculated in the standard way for the MnKa line, or 2.0 eV for 1.5 keV (Wollman et al., 2000). Count rates are below 150 cps, but it is, by far, not the last word in this matter. A newcomer in the generation of possible applications of the classical WDXRF, rather unexpectedly brought to the awareness of scientists, was that the extended fine structure of fluorescent spectra emitted by solid samples can be observed (Hayashi et al., 1997; Kawai et al., 1998). Up to now, this field was reserved for the absorption mode, mostly in combination with synchrotron radiation. The discovery of the extended fine structure in experiments performed under normal laboratory conditions opens a new field in the study of local structure in condensed matter and makes this method available to lower-budget laboratories.

ACKNOWLEDGMENTS All manufacturers of instruments are gratefully acknowledged for supplying documents about their instruments. Mr. P. Van der Aa, Dr. B. Volbert, and Dr. ir. B. A. R. Vrebos from Philips Analytical and Mr. S. Ornigg, Dr. K. E. Mauser, and Dr. N. Broll from Siemens (Bruker) AG deserve our gratitude for invaluable advice and critical reading of the manuscript. Dr. D. de Boer is acknowledged for supplying us with his knowledge on the most recent progress in multilayer research. A. Kuczumow thanks the Catholic University of Leuven for a fellowship that enabled him to participate in the first edition of this work and the Flemish Government for a fellowship helpful in the present revised version.

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3 Energy-Dispersive X-ray Fluorescence Analysis Using X-ray Tube Excitation Andrew T. Ellis Oxford Instruments Analytical Ltd., High Wycombe, Buckinghamshire, England

I.

INTRODUCTION

The scope of this chapter is laboratory and industrial x-ray fluorescence (XRF) analysis systems in which x-ray tubes are used for excitation and energy-dispersive (ED) semiconductor detectors, as opposed to Bragg-diffraction (wavelength) dispersion devices, are used. Such ED detectors directly measure the energy of the x-rays by collecting the ionization produced in a suitable semiconductor material. The coverage of this chapter specifically excludes the following topics, which are covered in other chapters: on-line EDXRF systems (see Chapter 7), total-reflection XRF (TXRF) (see Chapter 9), polarized beam XRF (see Chapter 10), and x-ray microfluorescence (XRMF) (see Chapter 11). Early approaches to EDXRF used gas proportional counters or scintillation detectors to determine directly the energy of the x-rays. Such systems were limited in their application because of the inherently poor energy resolution, which precluded the separation of characteristic x-rays of adjacent elements in the periodic table. The limited actual energy resolution of such detectors led them to be known as nondispersive, but this drawback has been effectively countered in some benchtop instruments through the use of novel arrangements of primary and secondary beam filters (Ross, 1928; Kirkpatrick, 1939; Kirkpatrick, 1944; Field, 1993). The real breakthrough in EDXRF came in the late 1960s (Bertolini et al., 1965; Bowman et al., 1966; Elad and Nakamura, 1966; Aitken, 1968) with the arrival of solid-state semiconductor diode detectors and their associated pulse processing circuits. These detection systems were developed through the early 1970s, often in electron microscopes, to the point at which practical x-ray spectrometry with an energy resolution of 200 eV or less became possible (Frankel and Aitken, 1970; Landis et al., 1971; Heath, 1972). Although the energy-resolution capability of semiconductor detectors remained greatly inferior to that achieved by wavelength-dispersive (WD) XRF systems, the increased efficiency inherent in the energy-dispersive method compensated in many analytical applications and permitted the use of a multiplicity of experimental geometries not practical with WDXRF. A wide variety of EDXRF analytical systems based on radioisotopoe sources, x-ray tubes, charged-particle accelerators, microprobe electron beams, and synchrotron light sources have been developed in recent years.

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Within the scope of this chapter, only EDXRF systems in which x-ray tubes are used for excitation will be considered further. The material in this chapter discusses the instrumental aspects of the energy-dispersive method, with particular emphasis on the use of low-power, compact x-ray tubes in combination with semiconductor detectors. The basic components of an EDXRF instrument within the scope of this chapter are depicted schematically in Figure 1. Each of the main subsystems of such an instrument is covered in detail in the following sections.

II.

X-RAY TUBE EXCITATION SYSTEMS

In EDXRF spectrometry, there is no physical discrimination of the secondary radiation that leaves the sample and enters the detector. This means that all photons of all energies in this secondary beam interact with the detector. The detector and its associated signal processing chain have a limited capacity to process these events and this is typically in the range 1
50 kcps. As a direct consequence, EDXRF has a limited total counting capacity

Figure 1

Basic components of a tube-excited EDXRF instrument.

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for all element x-ray emissions and it is, therefore, essential that the content of the incident spectrum be optimized for its useful information. This may require that more than one acquisition be made under different excitation conditions in order to control the range of energies needing to be collected in the detection system. Alternatively, the excitation system may be optimized to increase the intrinsic peak-to-background ratio through the use of alternative geometries such as TXRF or polarization EDXRF. In both these cases, however, higher-power x-ray tubes are typically required and there is still the need to apply some type of selectivity to the excitation system. This selectivity can be achieved in TXRF using more than one tube target material—typically Mo and W. In the case of polarization EDXRF, more than one tube high voltage or polarizing element is likely to be applied. This selectivity of excitation employed in EDXRF is completely the reverse of conventional WDXRF practice. In WDXRF, the high discrimination power of the dispersion and detection process and its associated losses coupled with very simple detection and counting systems allow very high count rates to be processed for each narrow wavelength band. Thus, the detection process is highly selective and the count rates in an individual element channel with a modern WDXRF system can be well in excess of 106 CPS. Typically, the simple rule of thumb in WDXRF would be to use twice to three times the kilovoltage (kV) of the absorption edge of the highest element of interest and then to apply the maximum allowed current at that setting. In summary, we may regard the essential operational difference between EDXRF and WDXRF spectrometry as follows: WDXRF uses broad-band excitation and selective detection. EDXRF uses selective excitation and broad-band detection. There are various ways in which selective excitation is achieved in EDXRF spectrometry. The use of TXRF and polarization is covered elsewhere in this volume and will not be considered further here. The most important modes in which selective excitation is realized in EDXRF are as follows: Selection of tube anode material Variation of tube voltage (kV) Use of primary beam filters Use of secondary targets (and associated filters) The selection of tube anode material is typically a once-off decision made at the time of purchase in view of the intended application. The high purchase cost of x-ray tubes generally precludes operational changes and low-power dual or multiple (Kis-Varga, 1988) anode x-ray tubes are not commercially available. The types of x-ray excitation arrangements are shown schematically in Figure 2. The following subsections describe typical arrangements for excitation and describe how optimum excitation conditions can be determined. Earlier work on this subject may be found in the XRF literature (Sandborg and Shen, 1984; Vane and Stewart, 1980; Gedcke et al., 1977). A.

Direct and Filtered Direct Excitation

In Figure 2, the beam (1) represents the configuration used for direct excitation of the sample by the x-rays emitted from the anode. A primary x-ray beam filter can be used to modify the spectrum from the x-ray tube that is finally used to excite the elements in the sample. The optimum selection of kV and primary beam filter are critically important for obtaining the best data from an EDXRF system. As with all spectrometry methods, the Copyright © 2002 Marcel Dekker, Inc.

Figure 2 An x-ray tube and typical excitation geometries. The high-energy electron beam strikes the anode to produce x-rays. (1) Direct or direct filtered: the flux (1) is used directly to strike sample (1) or is first passed through a primary beam x-ray filter. (2) Secondary target: the primary flux (1) strikes a secondary target 2, which generates a flux of x-rays (2), which strikes the sample at 2. (3) Transmission geometry: the beam (3) is directed through the anode to the sample.

principal driver for best precision and detection limits is peak-to-background (P/B) ratio. However, as mentioned earlier, the limited total counting capacity of the EDXRF system and the analysis of multielement samples are added complications in deriving optimum excitation conditions. In general, the kV governs the sensitivity and the primary beam filter governs the background. The energy distribution of the spectrum arriving at the sample governs the effectiveness of excitation for element XRF lines. In order to excite x-ray fluorescence, it is necessary to have incident x-ray energies above the absorption edge for the element’s line series to be excited. In order to have adequate excitation, there needs to be a high intensity of x-rays with energies higher than the edge and conventional wisdom has suggested that the tube kV should be 1.5
2 times the absorption edge of interest. This overvoltage ensures that there is a substantial proportion of the x-ray tube output spectrum available for excitation of the element line(s) of interest. Figure 3 shows the unfiltered x-ray tube spectrum scattered from a thin-polymer-film sample when the x-ray tube is operated at 5, 10, and 15 kV. The observed scatter spectrum is a reasonable representation of the spectrum striking the sample and clearly shows the following: Copyright © 2002 Marcel Dekker, Inc.

Figure 3 Scattered, unfiltered excitation spectrum provided by a silver target x-ray tube operated at (1) 5, (2) 10, and (3) 15 kV.

The high-energy cutoff at the operating kV. This is also known as the Duane
Hunt limit. The bremsstrahlung ‘‘hump’’. The intense characteristic AgL lines (around 3 keV) from the tube anode. The intensity of the AgL lines increases with the kV (i.e., as the overpotential increases). The tube L lines of Ag and Rh are particularly effective for exciting the light elements in the range 1
2.5 keV. Once higher-energy lines (e.g., FeK at 6.4 keV) are sought, the L lines provide no excitation, as they are lower in energy than the absorption edge (FeK at 7.11 keV) and the bremsstrahlung hump provides the excitation. Despite the highest intensity of the bremsstrahlung being lower than AgL lines, it is the integrated excitation spectrum, higher in energy than the absorption edge, which is important. When the x-ray tube kV exceeds the energy of the K edge of the tube anode material, the characteristic K lines will begin to dominate the excitation spectrum. This is shown in Figure 6, which depicts the scattered excitation spectrum with the tube operated at 35 kV, at which the AgK characteristic lines are strongly excited. The basic sensitivity for analysis is governed by the kV applied to the x-ray tube and an example of this is seen in Figure 4. The sample was a pressed pellet and the kV was set in the range 6
16 kV using a vacuum path in the spectrometer and no primary beam filter. At each kV, the rhodium-target x-ray tube current was adjusted to give 50% deadtime. The K-series line intensities were used and the sensitivity for each element has been normalized to the value at 10 kV. From the plots, the following are clear: The Si intensity falls off as the kV increases. This is due to the increasing distance of the tube main spectrum from the SiK absorption edge and to the increased relative contribution of the other elements to the total counts in the spectrum. This latter Copyright © 2002 Marcel Dekker, Inc.

Figure 4 details).

Effect of kV on sensitivity for the major elements in a geological sample (see text for

factor is important and derives from the limited total count rate for the whole spectrum. The FeK lines are not excited until at least 8 kV is applied. The FeK absorption edge is at 7.111 keV and the maximum sensitivity for Fe is at approximately twice this value. The general rule of thumb in which the kV is set to twice the highest absorption-edge energy is a reasonable place to start. However, as more elements are sought across a wider energy range, there usually comes a point at which the ‘‘balance’’ of the spectrum is unfavorable for the analytical problem being solved. When this happens, split the spectrum into two or more areas of interest and apply a kV to best suit each of these element ranges. In any event, as the kV and the number of elements increase, the importance of using a primary beam filter and splitting the spectrum into optimized energy regions increases. B.

Primary Beam Filter Selection

A primary beam filter acts as an x-ray absorber and it is placed between the x-ray tube and the sample to modify the x-ray tube output spectrum to which the sample is exposed. In general, the kV is selected first to ensure high excitation sensitivity and then an appropriate filter is chosen. The filter acts to reduce scattered background in the region of interest and to reduce the excited intensity of lower-energy peaks. The characteristics of the filter are defined by its x-ray absorption curve, which is controlled by selecting the material and its thickness. Filters are typical thin, pure metal foils of thickness in the range 10
500 mm. Some typical filters, the range of kV used with each, and the applicable optimum element range (K lines) are shown in Table 1. Copyright © 2002 Marcel Dekker, Inc.

The x-ray tube spectrum provided by unfiltered radiation can be seen in Figure 3. The effect on the x-ray tube spectrum of applying a thin and thick aluminum filter is seen in Figure 5, where a tube voltage of 15 kV is used. The aluminum acts as a simple absorption filter, having a single absorption edge at 1.56 keV. From the plots, it is clear that the aluminum filters both completely absorb the AgL lines as a consequence of the high-mass-absorption coefficient of around 700 cm2=g. The hump at lower energy provided by the spectrum filtered though thin Al yields good P=B ratios for elements in the S to V range (2.3
5 keV). The absence of the AgL (or RhL, MoL) lines from the tube removes their spectral interference and avoids the need for these x-rays to consume counting capacity. The thick aluminum filter ‘‘pinches’’ the low-energy side of the bremsstrahlung hump, providing high P=B ratios for the elements with lines in the range 3
8 keV. The combination of kV and an absorption filter provides an energy region in which the P=B ratio is optimal. On the low-energy side of this region, the excitation is suppressed, which allows more of the system’s counting capacity to be used in the region itself. On the high-energy side of the region, the excitation from the tube is tuned to provide a high integrated intensity above the energy of the absorption edges of the element lines of interest. The use of a thick copper filter with maximum kV is another example of this type of filter. The copper filter used is generally thick enough to completely absorb the x-ray tube anode K lines, which are of high energy (AgKa is 22.1 keV). Such an arrangement provides for the effective determination of Ag and Cd using a silver anode x-ray tube. At the other end of the energy scale, a cellulose filter is a weakly absorbing filter which suppresses the AgL lines. The so-called Regenerative Monochromatizing Filter (RMF) is a particularly important filter in tube-excited EDXRF spectrometry. The filter acts on the x-ray tube anode K lines so the kV must be sufficiently high above the K absorption edge of the anode material to generate intense K characteristic emission. The same element as the anode material is used to preferentially transmit the characteristic K x-rays generated in the anode. The filter takes its benefit from the fact that an element has a low-mass-

Table 1

Typical Primary Beam Filters and Their Range of Use

Filter

Thickness

kV range

Elements

4
50

All, Na
Ca

No filter

N=A

Cellulose Thin aluminum Thick aluminum

Single sheet 25
75 mm 75
200 mm

5
10 8
12 10
20

Si
Ti S
V Ca
Cu

Thin anode element

25
75 mm

25
40

Ca
Mo

Thick anode element

100
150 mm

40
50

Cu
Mo

Copper

200
500 mm

50

>Fe

a

Comments Optimum for light elements with 4
8 kV excitationa Suppresses tube L linesa Removes tube L linesa Good for transition element metal alloys RMFb, wide range and good trace analysis RMFb, good for trace analysis of heavy-element L lines. Suppresses tube K lines

Needs to be used in conjunction with vacuum or He path to remove the air in the spectrometer that would severely attenuate the low-energy lines and result in ArK lines in the detected spectrum. b Regenerative monochromatizing filter (RMF). The same element as the anode material is used as the filter material to preferentially transmit the characteristic K x-rays generated in the anode.

Copyright © 2002 Marcel Dekker, Inc.

Figure 5 Scattered excitation spectrum provided by a silver target x-ray tube operated at 15 kV. The plots show the effect of two thicknesses of an Al primary beam filter: (1) unfiltered; (2) thin Al filter; (3) thick Al filter.

absorption coefficient for its own characteristic lines, as these lie immediately below the absorption-edge jump. In the case of Ag, the AgKa1,2 lines are at 22.1 keV and the absorption coefficient of Ag for this energy is only 14 cm2=g. The effect of two silver primary filters, of different thickness, on the spectrum from a silver anode x-ray tube operated at 35 kV is shown in Figure 6. The unfiltered 35-kV excitation (curve 1) provides equally intense AgL and AgK characteristic lines and a broad bremsstrahlung hump centered on 12
15 keV. Any element in the 5
15 keV range will have the benefit of excitation by both the AgK characteristic lines and the bremsstrahlung. However, there exists the disadvantage of sitting on scattered background derived from the radiation in the bremsstrahlung hump. With the thin Ag filter in place (curve 2), the AgL characteristic lines are completely absorbed and the bremsstrahlung hump is reduced to a low-energy tail on the AgK characteristic lines. The resulting excitation yields high P=B ratios for lines in the energy range 4
12 keV. Use of the thicker Ag filter (curve 3), reduces even further the residual bremsstrahlung below the AgK lines and provides pseudomonochromatic radiation, which yields high P=B ratios for lines in the energy range 5
15 keV. The strongly filtered AgK line excitation provides excellent sensitivity and background reduction and a clear, intense Compton-scatter peak, which is good for trace analysis where a matrix correction based on Compton scatter is used. The same can be said for anode and filter materials of Mo, Rh, and Pd, which also provide characteristic K lines in this important region of the spectrum. A specific and well-documented example of optimum selection of x-ray tube anode, excitation kV, and primary beam filter can be seen in work by Potts et al. (1986). In that work, a Co anode x-ray tube and an iron primary beam filter were used to remove the Copyright © 2002 Marcel Dekker, Inc.

Figure 6 Scattered excitation spectrum provided by a silver target x-ray tube operated at 35 kV. The plots show the effect of two thicknesses of Ag primary beam filter: (1) unfiltered; (2) thin Ag filter; (3) thick Ag filter.

CoKb line which would otherwise strongly excite the Fe that typically dominates this part of the spectrum in geological analysis. Most modern EDXRF systems provide the ability to use more than one set of conditions per analysis. Depending on the analytical needs, the kV and filter are chosen to optimize the important parts of the spectrum and the data from spectra at each of these conditions are consolidated to provide the best P=B values. In order to establish the optimum conditions and make the best use of analysis time, the best strategy is to select a typical sample and then to make measurements with the different conditions available. Use criteria such as detection limits to decide upon the optimum conditions for the required analysis problem. Often, the instrument manufacturer’s recommendations can conveniently be taken as the starting point for any investigation. C.

Secondary Target Excitation

The basic geometry of secondary target excitation is shown in Figure 2. Use of this mode of excitation for EDXRF spectrometry was first described by Jaklevic et al. (1972), who incorporated the secondary target inside a small, low-power x-ray tube. The high P=B ratios that they reported were encouraging and Porter (1973) subsequently described the first work in which the secondary target was external to the x-ray tube. The primary radiation from the x-ray tube strikes the secondary target, which then emits characteristic radiation and generates some scattered bremsstrahlung radiation. The emitted radiation is pseudomonochromatic and the degree of monochromaticity is further enhanced, as in the case of the RMF in direct excitation, with a filter between the secondary target and the sample. Filters are of the same material as the target and are Copyright © 2002 Marcel Dekker, Inc.

typically only used with the higher-Z targets (e.g., Mo, Ag, and Gd). With increasing distance below the exciting lines, the absence of bremsstrahlung lower in energy than the exciting lines results in a very rapid reduction in sensitivity. The provision of a number of secondary targets with associated filters, which can usually be selected automatically, provides a range of discrete and narrow excitation bands. This arrangement can be conveniently thought of as providing the equivalent of a number of transmission target x-ray tubes, each with different anode materials. In order to generate intense characteristic radiation from the secondary target, the tube kV should be two to three times the K absorption-edge energy of the element from which the secondary target is made. The K lines are used, as they offer the highest fluorescence yield, which means that there are lower losses than with other line series. Typical secondary targets are, in decreasing energy order of K line emission: Gd, Sn, Ag, Mo, Ge, Cu, Fe, Ti, and Al. The optimization of x-ray tube kV, secondary target, and filter was described by Spatz and Lieser (1979). The fluorescence yield decreases as the target atomic number decreases and the efficiency of x-ray production falls off rapidly below Ti. The result of this is that the lighter elements Si to K are not well excited by secondary targets and there are few suitable elements available to provide targets with K lines in this important region. One approach is to use a reduced kV and a scattering target which scatters the x-ray tube L lines and bremsstrahlung, which, as was seen for direct excitation, is effective in this energy region. A stable polymer block provides a suitable target for this arrangement, although the overall efficiency is much lower than with direct excitation. There are two other arrangements that eliminate this limitation. The first is to use a second x-ray tube whose geometry provides for direct, and perhaps direct filtered excitation. Such an arrangement adds considerable cost and complexity but does offer optimum performance. It is possible for both tubes to be run from the same high-voltage power supply. The second approach is to provide mechanical repositioning of the x-ray tube such that it can be operated either in secondary target or in direct and direct filtered mode. The positional repeatability of the mechanical arrangement needs to be excellent if this approach is adopted. The mechanical complexity, which inevitably adds cost and some possibility for unreliability, has not stopped this approach from being used in the past with some success in one commercially available instrument. One disadvantage of this approach is the time that is needed for the mechanical switching, which can be a severe penalty if a number of acquisitions are required for a complete analysis—which is usually the case. Very recently, a so-called Wide Angle Geometry was described by Yokhin (2000), in which a single, fixed x-ray tube was used to deliver direct unfiltered or filtered and secondary target excitation. This arrangement was made possible using the wide cone x-ray beam from the special x-ray tube and a proprietary x-ray beam distributor. The reduction in overall excitation efficiency when using secondary targets is usually offset by increasing the x-ray tube power available. A tube providing close geometric coupling and power in the range 100
400 W is typically used to offset the low overall efficiency of excitation. In order to efficiently excite the GdK lines, it is preferable to use an x-ray tube capable of being operated at 60 kV. A high-Z anode such as tungsten or gold provides the highest bremsstrahlung intensity and, therefore, the best excitation efficiency for the higher-Z secondary targets. The high degree of monochromatization in filtered secondary target systems provides for very high P=B ratios for elements with absorption edges immediately below the target x-ray emission lines. The background in secondary target geometry can be further reduced using 3-axis or Cartesian geometry as described by Standzenieks and Selin (1979), Christensen et al. (1980), and Bisgard et al. (1981). Copyright © 2002 Marcel Dekker, Inc.

Despite secondary target excitation gaining wide acceptance through the early 1980s, the use of direct and direct filtered excitation has remained as the most common excitation arrangement. This is, perhaps, partly due to the restricted energy range offered by each secondary target+filter combination and the general need for unfiltered direct excitation to obtain the highest sensitivity for the light elements. In the 1990s, the use of polarized EDXRF (see Chapter 10) tends to have supplanted secondary target EDXRF for the provision of high P=B ratios in some applications. In addition, improved detection systems have continued to improve the performance of instruments using direct filtered excitation. An application in which secondary target systems have a valuable advantage is when the secondary target lines are selected to be just below the absorption edge of a major element, but just above the absorption edges of trace elements. One example of this is the use of a copper secondary target for the determination of low levels of Mn and Fe in Cu alloys. In this example, the CuK lines from the target do not excite the Cu but are highly effective for exciting Fe. The detection and counting system is not dominated by CuK lines, neither is the FeKa peak interfered with by the Si escape peak from CuKa. Such applications are few in number and, in the case of the metals industry, analysts may often prefer to use WDXRF spectrometers. D.

X-ray Tubes

X-rays were first discovered over a century ago (see Michette and Pfauntsch, 1996) by Ro¨ntgen (1896a, 1896b), and since that time, they have become used in a great variety of applications. These applications encompass medical and dental x-ray radiology and imaging, industrial and baggage inspection, industrial nondestructive testing (NDT), industrial thickness gauging, and x-ray spectrometry in all its guises. In most, if not all, of these fields, the convenience of the high-voltage x-ray tube has substantial benefits in terms of convenience, safety, and availability of appropriate spectral and intensity output. The modern x-ray tube used for EDXRF analysis is based on a design introduced by Coolidge as long ago as 1913 (Coolidge, 1913). The basic design of x-ray tubes typically used in EDXRF instrumentation consists of the following: A radiation-shielded glass envelope An independent tungsten filament with current control unit A high-melting-point metallic anode A high-voltage connection to the anode A beryllium foil exit window Some means of dissipating the heat generated In order to operate, the x-ray tube requires a controllable, stable high-voltage power supply capable of providing typically 5
60 kV. In addition, there is a low-voltage power supply for providing current to heat the filament, thereby controlling the electron beam current and output intensity. X-ray tubes used for EDXRF analysis are typically operated with a power in the range of 1
400 W. For direct excitation, the power output is typically, no greater than 50 W. Reproducibility of analytical results is achieved for x-ray tube excitation by operating for a measured amount of time at a fixed emission current (dc operation). This approach depends on the reasonable assumption that the total output x-ray flux is proportional to the number of electrons striking the anode. In some types of x-ray tube, a control grid can be used to modulate or pulse the emission current. This can be used to modulate the electron beam current to ensure constant output of radiation from the tube in dc operation or it can be used to rapidly Copyright © 2002 Marcel Dekker, Inc.

switch the electron beam on and off for pulsed operation. If the pulsing is synchronized with x-ray detection, increased output countrate can be achieved by reducing pulse pileup (see Sec. IV.C). The key features of the x-ray tube used in EDXRF instruments are as follows: Easily shielded and safe to operate Compact and able to be placed close to the sample Stable output over an extended period, typically months Well-defined spectral output High-voltage controllable in small increments, typically 1 kV, throughout the operating range (typically 5
50 kV) Beam current controllable in small increments, typically 10 mA, throughout the operating range (typically 10
1000 mA). The geometry and designs typically used and the characteristics of each are described in the following sections. Further reading on the subject can be found in Skillicorn (1982, 1983). 1. Side-Window Geometry Side-window geometry is the oldest and, perhaps, the most common x-ray tube geometry used in EDXRF instruments and it is suitable for both high-kV and high-power operation. Figure 7 shows the essential details of this design. The geometric coupling of tube anode spot to sample is limited first by the distance of the spot itself from the exit window and then, potentially, by the tube encapsulation itself. Typically, these tubes are operated in the range 5
50 kV. They can dissipate 50 W with the need only for forced-air cooling in order to keep the tube external body below a temperature of, say, 55 C. It is important not to exceed the maximum temperature stated by the x-ray tube manufacturer. Higher powers (up to 100 W or 120 W) normally need some form of liquid cooling, which is typically achieved through the use of a closed-cycle liquid heat exchanger. The energy spectrum that is output from a side-window x-ray tube is governed by the tube anode material, the angle of the anode, the kV applied to it, and the exit window’s material and thickness. Tube anode materials are usually Rh, Ag, or W, although Ti, Cr, Co, Cu, Mo, Pd, or Au is also available. The selection of anode material depends on the intended application, but Rh, Ag, and Pd are generally the most popular, as they are highly compatible with tube manufacture and operation. In addition, their K and L characteristic emission lines are in regions of the energy spectrum where they offer high sensitivity with minimum spectral interference. Dual-target x-ray tubes have been used in attempts to maximize flexibility and performance. These efforts have met with limited success in EDXRF, in part due to the spectral cross-contamination that is typically experienced in the designs used to date. There has been, however, considerable success in the use of dual-target x-ray tubes for WDXRF, where the problems of spectral contamination have much less impact. Typically, the anode is cut at an angle of 20
35 from normal to the incident electron beam and the exit window is usually made from beryllium foil of thickness in the range 50
250 mm. The tube output spectrum can be measured or estimated and such information is important in analytical correction procedures, particularly those using fundamental parameter calculations for which accurate and computationally convenient tube output spectrum models are required (see Chapter 2, Sec. I.B for further details.) Copyright © 2002 Marcel Dekker, Inc.

Figure 7

Copyright © 2002 Marcel Dekker, Inc.

Side-window x-ray tube.

It should be remembered that the tube output spectrum, especially with higherpower tubes, may change with time. One way in which this can occur is through deposition of a thin tungsten film on the inside of the tube output window through evaporation of tungsten from the hot filament. Such deposition is unlikely to affect the low-power tubes typically used in EDXRF systems and, in any event, is usually taken care of through the normal restandardization process. 2. End-Window Geometry The design features of such an x-ray tube are shown in Figure 8. The same factors of operation and tube spectrum apply to this type of tube as was applied to side-window tubes. The tube spectrum is altered by the fact that the takeoff angle is normal to the anode surface and so there is minimal self-absorption by the target material. This high takeoff angle makes such a design particularly good for low-energy excitation. One of the important benefits of the end-window design is the very tight coupling of anode spot to sample that can be achieved with a ‘‘sharp-nose’’ design. The spot-to window coupling can be close with suitable electron optics (e.g., 12 mm in the Varian EG50 design). The use of a high positive voltage on the target minimizes the effect which repelled electrons would have of causing an unacceptable amount of exit-window heating. This reduced heating allows thinner beryllium windows to be used, which further improves the low-energy x-ray output. 3. TransmissionTarget Geometry The design features of such an x-ray tube are shown in Figure 9. In this type of x-ray tube, the geometry is basically end window, but in this embodiment, there is a metal foil target parallel to and inside the beryllium exit window. The incident high-voltage electron beam

Figure 8

End-window x-ray tube.

Copyright © 2002 Marcel Dekker, Inc.

Figure 9

Schematic of a transmission target x-ray tube.

strikes and enters one side of the thin-film target and the resulting radiation is heavily filtered as it passes through the foil and exits the far side of the target. This novel design was initially developed (Jaklevic et al., 1972) in order to reduce the bremsstrahlung from the x-ray tube, which would then reduce the scattered background in the spectrum and thereby improve detection limits. The initial design used a molybdenum transmission target of 0.12 mm thickness and, when operated at 42 kV and up to 25 W, was shown to give an output spectrum consisting principally of characteristic MoK radiation. The output spectrum was comparable to an 125I radioisotope source coupled to a molybdenum-foil secondary target. This type of x-ray tube was made commercially available by Watkins
Johnson (Hershyn, 1975a, 1975b). The spectrum output from this type of tube was investigated by Zulliger and Stewart (1975), who calculated and measured spectra from Cr, Mo, and W transmission target x-ray tubes. The targets were either thin (50 mm) metal foils or thin films (5
10 mm) on a beryllium substrate of 0.25
0.5 mm thickness. The thin targets mean that only low-power tubes (< 100 W typically) can be made successfully. These authors found, as predicted, a substantial reduction in bremsstrahlung compared to the characteristic lines and quoted detection limits for a number of applications. The Cr target was found, not unsurprisingly, to be best for the light elements S and Ca. Although the energy range covered was claimed to be wide, the intrinsic fixed filtration of this design makes it a less valuable general excitation source compared to a side- or end-window tube with an external, selectable filter. A novel, multiple-target transmission tube has even been described (Kis-Varga, 1988) in which the five anode materials (Fe, Cu, Ge, Mo, and Ag) were changed by a magnetic mechanism. The intrinsic filtration of the low-energy bremsstrahlung makes this type of tube much less suitable for excitation of low energy (i.e., < 2-keV) x-ray lines. Consequently, their usage is limited to specific applications where the intrinsic filtration and potential for close geometric coupling of the tube anode spot to the sample are major benefits. In fact, in one design, the Watkins
Johnson tubes did not have a target closely coupled to the exit window, which, when combined with the intrinsic lossess in the target, led to low output flux from the tubes and, consequently, reduced sensitivity. Finally, these tubes are not useful when operated at low voltages (Skillicorn, 1982) due to strong absorption in the target. This makes them much less sensitive than solid-target, side- or end-window tubes for selectively exciting the light elements in the presence of large amounts of heavier elements. Copyright © 2002 Marcel Dekker, Inc.

III.

SEMICONDUCTOR DETECTORS

In this type of detector, the total ionization produced by each x-ray photon striking the detector is converted to a voltage signal with amplitude proportional to the incident energy. Specially designed preamplification and processing electronics are employed to maintain the linearity of the voltage signal with respect to the original charge pulse. A multichannel analyzer accumulates an energy spectrum of the sequential events in a histogram memory. Because the energy analysis does not depend in any way on the diffraction or focusing of the x-rays by the detector, the geometry of the system is relatively insensitive to the placement of the detector with respect to the sample. An exception to this is in polarized excitation XRF (Chapter 10), where the orientation angle can be used to suppress scattered background. This insensitivity to geometry and ability to place the detector element close to the sample provides for a large solid angle and high geometric efficiency. In addition, the mechanism by which the ionization signal is measured is not restricted to a narrow energy region, thus allowing the simultaneous detection of x-rays over a wide dynamic range of the emission spectrum. The principal advantages of EDXRF derive from its capability for simultaneous detection, with high geometric efficiency, of characteristic x-rays from multiple elements. A.

Fabrication and Operating Principle

The energy-dispersive detecting element is based on the simple, semiconductor diode structure shown in Figure 10. The example shown is a structure typical for lithium-drifted silicon, Si(Li), detectors, although the basic elements are similar for high-purity germanium (HPGe) and other semiconductor detectors. The device is fabricated on a cylindrical wafer of high-quality semiconductor material with rectifying p or n contacts on opposing end surfaces. The bulk of the material is characterized by a very low concentration of free-charge carriers. This reduced free-carrier concentration is achieved either with the use of extremely pure material, in the case of HPGe detectors, or through the charge compensation of p-type silicon with lithium donor atoms in the case of Si(Li) detectors. A typical dimension for such a detector crystal is 10
80 mm2 active area with a thickness of 3
5 mm. In the geometry shown in Figure 10, the lithium-diffused region acts as an n contact and the metal surface barrier (typically an evaporated Au layer) serves as the p rectifying contact. When the diode is reverse-biased, any remaining free carriers are swept out of the bulk by the applied field and an active depletion region is created. In this condition, the only current that flows between electrodes on the respective contacts at each end of the diode cylinder is due to thermally generated charge carriers, which are excited above the narrow band gap of the semiconducting material. In order to minimize the noise from these thermally generated charges, the detector crystal is operated at reduced temperature, which is typically achieved using cryogenic cooling by means of a Dewar vacuum flask filled with liquid nitrogen. Other cooling strategies can be adopted (see Sec. III.C). When an x-ray photon is incident upon the active volume (i.e., depletion region) of the diode, it normally interacts by photoelectric absorption to create an inner-shell vacancy in the semiconductor material together with an energetic photoelectron. This photoelectron interacts with the atoms in the semiconductor crystal lattice to produce multiple low-energy ionization events. This process continues until the electron comes to rest at the end of a total range, which is short compared with the dimensions of the crystal. The energy associated with the inner-shell vacancy is also absorbed in the crystal, in most Copyright © 2002 Marcel Dekker, Inc.

Figure 10

Cross section of a typical Si(Li) detector showing electrode contacts and active volume.

cases following the emission of Auger electrons or multiple low-energy x-rays and subsequent re-absorption. The result of these multiple ionization process is the essentially instantaneous production of a large number of free electron
hole pairs in the sensitive volume of the detector structure. These free-charge carriers form as a cloud from which they are separated by the field gradient. The front detector electrical contact, being biased at
500 V, for example, attracts the hole and repels the electrons, the latter negative charge cloud is swept to the rear contact and converted, by means of the FET, to an amplified voltage pulse. The number of charge carriers produced is directly proportional to the energy of the x-ray photon incident on the detector. Thus, the number of charges collected results in a voltage pulse whose magnitude is, in turn, directly proportional to the energy of the detected photon. B.

Crystal Materials

The characteristics of energy-dispersive semiconductor x-ray spectrometers derive primarily from the properties of the materials used to make the device itself. Although the most common material for semiconductor detectors in EDXRF is silicon, devices based on the use of germanium or compound semiconductors such as mercury(II) iodide (HgI2), CdTe, CdZnTe, or GaAs have also been used to varying degrees. In particular, mercury (II) iodide detectors (Swierkowski et al., 1974; Slapa et al., 1976; Ponpon and Sieskind, 1996) have been used successfully in both research and commercial products for many years. Early devices with adequate performance at room temperature were flown on space missions. More recently, devices have been used in hand-held EDXRF spectrometers where the benefit of ambient temperature operation, with or without thermoelectric (Peltier) cooling of the field-effect transistor (FET), have been successfully exploited. Iwanczyk et al. (1996) compared the performance of HgI2, Si PIN photoiodide, and CdTe detectors for this latter, important application. Large arrays of up to 100 HgI2 detectors Copyright © 2002 Marcel Dekker, Inc.

have also been fabricated (Iwanczyk et al., 1995; Patt et al., 1995). Dabrowski et al. (1981) have been particularly successful in designing, fabricating, and characterizing HgI2 detectors throughout the past two decades and their work should be consulted for further details of this important type of EDXRF detector. Although high-purity germanium (HPGe) is widely used for detectors in electron microscopes as a result of its slightly better resolution than Si(Li) detectors, the use of HPGe detectors is less popular in photonexcited EDXRF spectrometry. One reason for this lack of acceptance may lie in the high escape peak intensities (see Sec. III.E.1) which can strongly interfere in trace analysis. Table 2 lists properties relevant to x-ray detection for Si, Ge, and HgI2, which are the materials most commonly employed in commercial EDXRF spectrometers. A variety of other compound semiconductor materials, such as GaAs, CdTe, and CdZnTe, have been used for applications in which specific properties are particularly beneficial. These properties include the following: Higher atomic number for increased photoelectric cross section Larger band gap for lower thermal leakage at room temperature A lower band gap corresponding to a smaller average energy for creation of hole
electron pairs (Knoll, 1979), which offers improved energy resolution. Recalling the direct relationship described earlier between photon energy and number of charge carriers and, therefore, the detectors signal resulting from detection of a particular x-ray photon, one can use the data from Table 2 in the following example for detection of CaKa1 (K
L3), which has an energy of 3.691 keV: No. electron
hole pairs produced ¼ 3691=3.86 ¼ 956 Charge on electron ¼ 1.6610
19 C Therefore, Charge from CaKa1 interaction ¼ 1.53610
16 C Assuming a feedback capacitance in the first stage amplification of 0.1 pF, the resulting output voltage pulse is 1.53 mV. From this, we can see the very small magnitude of the signals involved. This has led to very great efforts having been made in the design of detector and signal processing systems to minimize all sources of electronic noise which would otherwise interfere very strongly with the signal being sought. In addition to the conventional Si(Li) detector, Si PIN (Haselberger et al., 1996; Cesareo et al., 1996) and Si drift detectors (Lechner et al., 1996; Bertuccio et al., 1996) have also become available in recent years. The active area of these devices is generally

Table 2

Semiconductor Detector Material Properties

Detector properties Atomic number Atomic=molecular weight Density (300 K) g=cm3 Band gap (300 K) (eV) Average energy E per electron
hole pair (77 kV) (eV) Fano factor (77 K) a

At 300 K.

Copyright © 2002 Marcel Dekker, Inc.

Si

Ge

HgI2

14 28.09 2.33 1.115 3.86

32 72.60 5.33 0.665 2.96

80.53 454.45 6.3 2.13 4.15a

0.12

0.08

3
10 mm2 and the depletion thickness is of the order of 300 mm, which reduces detection efficiency (see Sec. III.F) compared to Si(Li) detectors. In the following section, the details of detector operation are discussed, with particular emphasis on those aspects that impact EDXRF spectrometry most directly. These include the detector energy resolution, detection efficiency, spectral response, and counting system throughput.

C.

Cooling Systems

The need to minimize the electronic noise of the detection system is paramount for obtaining the best energy resolution. The main source of noise in the detector itself is leakage current, which derives from the generation of charge carriers in the absence of x-rays through, for example, thermal vibrations of the detector crystal lattice. Whatever the cooling system used, the intention of it is to minimize leakage, and the lower the temperature, the less leakage will occur. The boiling point of liquid nitrogen (77 K) is a typical temperature used for the detector crystal, although the FET may require a different and often higher temperature to minimize its noise contribution and, therefore, obtain optimum detector resolution. The need for reduced operating temperature results in the need for careful design in the insulation from ambient temperatures of the detector crystal. Effective insulation is typically achieved by enclosing the detector head (crystal and FET) assembly in a evacuated cryostat. In order for the low-energy x-rays (< 2 keV) to reach the detector crystal, the vacuum is usually retained by a thin (typically 5
50 mm) beryllium entrance window (Fig. 11). In recent years, the thinnest windows (5, 8, and 12 mm) have become available with coatings which ensure that there is no leakage of He through the Be window. Any helium getting through the window would degrade the high vacuum within the cryostat and lead to increased temperature, leakage current, and deteriorating performance. The original and perhaps simplest way of cooling a detector unit is to attach the measurement head via a cold-finger assembly to a dewar vessel containing liquid nitrogen (LN). The dewar itself is typically a complex technological component if it is to provide high performance and long operational lifetime. The modern LN cryostat delivers optimum resolution and the ability to be cycled through multiple warm-up=cooldown cycles without degradation in performance. The inevitable and natural boiling-off from the dewar of LN over a period (1 L per day is typical) means that the dewar needs regular replenishment. In order to minimize the frequency of replenishing LN, a large dewar is preferable, although this desire for a long LN holding time may lead to a rather bulky system. LN volumes of 7
15 L are typical, although some large systems may use volumes up to 25 L and some small, hand-held units may use 1 L or less. In any event, the following are important: Use only liquid nitrogen: Liquid air, although delivering the required temperature range, will fractionate, yielding O2, which is an explosion hazard in the chemical laboratory. Ensure that there is no ice or dirt in the LN used. The presence of ice or dirt in the dewar can cause microphony, which can degrade energy resolution. Ice and dirt can be eliminated by pouring the LN through a funnel equipped with a wire screen (e.g., a motor fuel metal filter funnel). Observe safe LN handling procedures. Avoid overfilling and spillages as liquid nitrogen can cause embrittlement of polymers and cable insulation or result in Copyright © 2002 Marcel Dekker, Inc.

Figure 11

Cross section of a groove-type Si(Li) detector mounted in its cryostat end cap.

condensation from the atmosphere being deposited on components that are sensitive to moisture. It is essential that a Si(Li) detector be kept cold throughout the time that the bias is applied. If this is not done, then there is the possibility of damage to the detector assembly by, for example, migration of the drifted Li ions from the compensated region of the crystal. Should this happen, the detector will fail and, for this reason, it is usual for an automatic bias shutoff system to be implemented. Such systems typically use a temperature sensor or a LN level sensor to switch off the detector bias in case of the dewar warming up. Other detectors, such as HPGe or high-purity Si, for which drifting is not required, do not exhibit this particular problem but may still suffer some problems if thermally cycled repeatedly, unless the cryostat assembly is specifically designed to do this. The simplicity of the LN-filled dewar has led to its widespread use in EDXRF systems. However, the difficulty in obtaining LN in remote areas, or the need for a no-LN solution, has led to the introduction in the past decade or so of thermoelectric (Peltier effect) coolers or mechanical refrigerating units. The mechanical units may employ the Joule
Thompson (J
T) effect or some other cycle such as the Stirling cycle to obtain essentially LN temperatures at the detector head. The major benefit of this approach is that there is no difference in performance in terms of detector energy resolution from the conventional dewar LN systems, but the coolers come with penalties of increased complexity, cost, and bulk. HPGe detectors, which typically need liquid-nitrogen temperature to operate, may use these mechanical coolers, but other detectors which will function at higher temperatures, albeit with degraded energy resolution, are suitable for use with thermoelectric (Peltier effect) coolers. Copyright © 2002 Marcel Dekker, Inc.

Peltier coolers can offer a very compact cryostat=detector assembly that requires only electrical power to provide its cooling. Multistage (three to six) coolers may be used to provide a DT cooling, from ambient, of 50 C to a maximum of perhaps 120 C. In an ambient room temperature of 20 C, such coolers deliver working detector temperatures typically in the range
30 C to
90 C, which are suitable for Si PIN, HgI2, and extremely low-leakage Si(Li) detectors. Energy resolution in the range 200
350 eV (see Sec. III.D) is typically available with these systems and both Si PIN and HgI2 detectors are commercially available with Peltier cooling. More recently, Peltier-cooled Si drift detectors, which offer energy resolution better than 150 eV, have been developed (Lechner et al., 1996; Bertuccio et al., 1996) and are now commercially available.

D.

Energy Resolution

The energy resolution of the semiconductor spectrometer system determines the ability of a given system to resolve characteristic x-rays from multielement samples. It is normally defined as the full width at half-maximum (FWHM) of the pulse
height distribution measured for a monoenergetic x-ray at a specified energy. A convenient choice of x-ray energy at which resolution is quoted is the weighted mean MnKa line at 5.895 keV, as this emission line is readily available from 55Fe radioisotope sources. The MnK x-rays are of sufficiently low energy that the contribution to the FWHM of the unresolved Ka1 þ Ka2 doublet (5.898 and 5.887 keV, respectively) and the intrinsic width of the emission lines can be neglected. This is not true for higher-energy characteristic x-rays (e.g., MoK lines) for which the separation of the Ka1 þ Ka2 doublet and the intrinsic width of the emission line must be considered. Figure 12 shows a typical pulse
height spectrum of MnK x-rays taken from an 55Fe source. Typical state-of-the-art 30-mm2 Si(Li) detectors achieve a FWHM better than 140 eV at 5.9 keV, although values as low as 130 eV are achievable. However, this number is but one indicator of the quality of an EDXRF detector system, and other factors such as maximum count rate or the presence of background and artifacts may be more important in many analytical applications. Peltier-cooled Si PIN detectors offer FWHM values typically in the range 165
220 eV, whereas Si drift detectors claim < 150 eV. The interdependence of energy resolution and count rate is discussed below, although it is worth noting at this point that the user can exert a large degree of control over this by making an appropriate selection of pulse processor settings. If one neglects the natural linewidth of the x-ray lines, the instrumental energy resolution of a semiconductor detector x-ray spectrometer is a function of two independent factors. One of these is determined by the properties of the detector itself; the other is dependent on the nature of the electronic pulse processing employed. In some systems, a pulser or strobe signal may be injected into the measurement system in order to monitor the resolution of the electronic system independent of any peak broadening due to the detector itself. The measured FWHM of the x-ray line (DETotal) is the sum, in quadrature, of the contribution due solely to detector processes (DEDet) and that associated with the electronic pulse processing system (DEElec): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DETotal ¼ DE2Det þ DE2Elec ð1Þ The DEDet component is determined by the statistics of the free-charge production process occurring in the depleted volume of the diode. The average number of electron
hole pairs Copyright © 2002 Marcel Dekker, Inc.

Figure 12

Pulse
height spectrum of MnK x-rays taken with a 30-mm2 Si(Li) detector.

produced by an incident photon can be calculated as the total photon energy divided by the mean energy required to produce a single electron
hole pair (see Sec. III.B). If the fluctuation in this average were governed by Poisson statistics, the standard deviation would be rffiffiffiffi pffiffiffi E s¼ n¼ ð2Þ e In semiconductor devices, the details of the energy-loss process are such that the individual events are not strictly independent and a departure from Poisson behavior is observed. This departure is taken into account by the inclusion of the Fano factor in the expression for the detector contribution to the FWHM: rffiffiffiffiffiffiffi pffiffiffiffiffiffi FE s ¼ Fn ¼ ð3Þ e Taking sðEÞ pffiffiffiffiffiffi ¼ Fn E

ð4Þ

and, rearranging,

pffiffiffiffiffiffiffiffi DEDet ¼ 2:35 FeE

ð5Þ

where e is the average energy required to produce a free electron
hole pair, E is the energy of the photon, F is the Fano factor (see Table 2), and the factor 2.35 converts the standard deviation to FWHM for a Poisson distribution. A typical value for the detector contribution to resolution, also known as the dispersion, is 120. Examination of the values of e and F listed in Table 2 shows that for an Copyright © 2002 Marcel Dekker, Inc.

equivalent energy, the detector contribution to the resolution is 28% less for the case of Ge compared to Si. This results in better energy resolution being obtained for HPGe detectors and could be an important consideration in the choice of detector for certain applications. However, this theoretical advantage in energy resolution of Ge over Si(Li) is mitigated in many systems because of the dominance of electronic noise (e.g., from the FET). In addition, the very high GeK escape peak probability (see Sec. III.E.1) is likely to be a serious limitation to the use of HPGe detectors in most EDXRF work. HPGe detectors are widely used in electron microscopes, where energies < 10 keV are likely to be of most interest. E.

Spectrum Artifacts

The simplicity of the element line spectra encountered in XRF spectrometry is one of its major benefits when compared with other atomic spectroscopy methods. However, a few peaks can arise in the EDXRF spectrum from sources other than the elements in the sample and it is wise to be aware of their nature. Unlike basic detector efficiency and spectrum background, which are defined by the selection of detector window and the detector manufacturing process, the end user can exert some control over the effects of spectrum artifacts. From the spectra presented so far, it is clear that the bremsstrahlung continuum and the Compton (and Rayleigh)-scatter peaks arise from the interaction of the exciting radiation with the sample. These features are not regarded as artifacts. 1. Escape Peaks For an incident x-ray with energy higher than the SiK absorption edge, the detection process will involve the generation, through x-ray fluorescence of the detector material, of SiK x-rays. The vast majority of these will themselves immediately be absorbed in the detector volume and contribute to the overall charge collected for the original incident x-ray photon. This is the normal mechanism of x-ray detection, which was described earlier. There is, however, a finite probability that the SiKa x-rays produced will escape from the detector volume and not contribute to the charge collected for the original photon that was detected. The resulting energy detected will be reduced by 1.740 keV, which is the energy of the SiKa x-ray that escaped. The greatest probability of escape arises close to the front of the detector crystal from which SiK x-rays can more easily escape. X-rays detected far within the volume will still generate SiK x-rays, but they will all be absorbed before they can reach the outer surface of the crystal and escape. The probability of generating the SiKb x-ray and of it escaping is extremely low and this effect can safely be ignored. The result of the SiK escape process is a peak in the collected spectrum, which is 1.74 keV lower in energy than the parent peak. The Si escape peaks for TiKa and TiKb are clearly seen, 1.74 keV below the TiK lines, in Figure 13 (indicated by 1 and 2, respectively) (see also Table 3). In many detectors, the incomplete charge-collection tail on the lowenergy side of the parent peak will obscure the Si escape peak from TiKb. The escape probability is highest for lines closest to the SiK absorption edge at 1.838 keV and may be calculated as described by Statham (1976) and Dyson (1974). The intensity falls rapidly from around 3% of the parent peak at 2
3 keV to around 0.1% of the parent peak at 10 keV. At energies above 10 keV, the effect is negligible, but below this energy, it is important that spectrum-processing software takes account of the peak. Typically, the processing software will make a correction in which the Si escape peak is removed and the most comprehensive packages will reinstate the lost intensity to the parent peak. If such packages are available, then make use of them before carrying out Copyright © 2002 Marcel Dekker, Inc.

Figure13

Peak artifacts from a Ti spectrum collected by a Si(Li) detector (see Table 3 for details).

qualitative analysis in order to avoid classic errors such as the Si escape peaks from ZnK being identified as Co! In the case of HPGe, the escape probability for GeL lines is negligible in XRF and can be ignored. The escape of the GeK lines has, however, a very high probability, and for Se, the escape peak intensity is around 16% of the parent peak. This is due to the combination of the high detector efficiency for GeK x-rays and the much greater depth within the crystal from which the GeK lines can escape. There is further complication in that there are many GeK lines and each of these lines will result in an escape peak. The complexity and intensity of Ge escape peaks can easily be seen in the spectrum from a molybdenum oxide sample, shown in Figure 14. The principal GeK lines and their characteristic energies are as follows: GeKa: 9.876 keV GeKb1: 10.981 keV GeKb2: 11.100 keV The assignments of the numbered GeK escape peaks arising from the detection of the MoK lines are listed in Table 4.

Table 3 Peak No. 1 2 3 4 5

Artifact Peak Energy and Assignments for Figure 13 Energy (keV)

Assignment

2.8 3.2 9.0 9.4 9.9

TiKa
SiK escape TiKb
SiK escape TiKa þ TiKa sum peak TiKa þ TiKb sum peak TiKb þ TiKb sum peak

Copyright © 2002 Marcel Dekker, Inc.

Figure 14 details).

GeK escape peaks from Mo spectrum collected by HPGe detector (see Table 4 for

The energy and intensity of these GeK escape peaks interferes very strongly over a wide and important region of the spectrum. The interference is considerable from the GeK escape peaks of Nb and Mo in stainless steels and corrections for these intense escape peaks are not sufficiently accurate to retain the intrinsic performance of EDXRF for what would be a reliable and straightforward analysis using a Si(Li) detector. The severe limitation posed by the Ge escape peaks in EDXRF is also demonstrated in Figure 15 (see also Table 5) for a typical determination of minor and trace heavy metals in a low-Z matrix. In this case, the complex GeK escape peaks from the Rh backscatter peaks completely dominate the important part of the spectrum for trace analysis. In conclusion, the Si escape peak is small and easily corrected, but the number and high intensity of the GeK escape peaks severely limit the value of the HPGe detector for tube-excited EDXRF. In the case of energy-dispersive (ED) microanalysis, however,

Table 4 Peak No. 1 2 3 4 5

GeK Escape Peak Assignments and Relative Intensity for Figure 14 Energy (keV)

Assignment

% of parent

6.4 7.57 8.6 9.74 10.12

MoKa
GeKb1 MoKa
GeKa MoKb
GeKb MoKb1
GeKa MoKb2
GeKa

1.5 9.6 1.0 8.8 2.5

Copyright © 2002 Marcel Dekker, Inc.

Figure 15 Comparison of Si(Li) and HPGe detectors for tube-excited EDXRF trace analysis of low-Z sample matrix (see Table 5 for details).

the HPGe detector has benefits in that the intensity of the GeL escape peaks is negligible and the energy resolution is better than that of Si(Li) detectors. 2. Sum Peaks Sum peaks arise from a specific form of peak pileup (see Sec. IV.C) where two events from high-intensity peaks arrive in the pulse processing electronics so close together in time that the pileup inspector cannot recognize them as two events. The effect of this is for the signals to be seen as one and for them to be registered at the energy that is the sum of the two. The sum peaks of Ti are seen in Figure 13, in which the most intense peak (3) is from the sum of two TiKa events. The smallest peak (5), from the sum of two TiKb events is only visible when the parent peak is very intense. At first glance, peaks 3 and 4 may appear like a typical Ka, Kb

Table 5

GeK Escape Peak Assignments for Figure 15

Peak 1 2 3 4

Copyright © 2002 Marcel Dekker, Inc.

Assignment RhKa Compton
GeKa RhKa Rayleigh
GeKa RhKb Compton
GeKa RhKb Rayleigh
GeKa

doublet, but the peak separation is too small for K lines in the region of the spectrum in which they occur. In addition, the relative line ratio is not the same as for a characteristic K line series at this energy. Consequently, sum peaks are unlikely to be incorrectly identified as elements, but they may interfere with important lines in a particular analysis. This is particularly the case in some environmental analyses, (e.g., where high concentrations of Fe lead to intense FeK lines). This will result in FeK sum peaks in the region of the K lines of Se and Br and the L lines of toxic heavy elements such as Hg, Tl, and Pb. Some spectrum-processing software packages are able to correct for sum peaks and should be used to minimize potential errors in analysis (See Chapter 4). If it is found that such features are not available, then reducing the count rate will substantially reduce the effect of the sum peaks. In order not to lose analytical performance, however, selection of alternate excitation conditions (e.g., using a more absorbing beam filter) will ‘‘rebalance’’ the spectrum in favor of the peaks at higher energy and minimize the interference. 3. Diffraction Peaks Whenever a crystalline sample is measured in an EDXRF spectrometer, there is the possibility that the conditions for Bragg diffraction will be met. This is exacerbated by the divergent geometry of the incident and detected beam and the variety of excitation conditions available. The worst cases are where unfiltered primary radiation is used, as there are then many energies and angles which increase the probability of meeting the Bragg condition for a crystalline component in the sample. The more monochromatic the exciting beam is, the less likely it is that the Bragg condition will be met. Figure 16 shows the diffraction peak obtained from a boric acid pellet which was excited with a silver anode x-ray tube operated at 5 kV with no primary beam filter and in an approximately 90 geometry. The spectrum is overlaid with one from a sulfur pellet measured under exactly the same conditions. It can be seen that the diffraction peak is broader than a K series line at that energy and the low-energy tailing is more pronounced. There is some possibility for misidentification, but the most likely problem arises in any spectrum processing not setup to take account of this peak. In cases such as silicon wafers, where the crystal planes are highly oriented, the sample can be tilted or rotated to minimize the often sharp diffraction peaks that can be observed. Alternatively, effective use can be made of additional collimation to constrain more closely the angles that can satisfy the Bragg condition or of alternate excitation conditions, which change or reduce the incident energies available for diffraction. Figure 17 shows a comparison of the spectra taken from the same two samples under different excitation conditions. The increased kV and use of an absorption filter completely change the energy distribution of the excitation spectrum and eliminate the diffraction peak. This second set of conditions is particularly effective for lines in this part of the spectrum and effectively removes problems from diffraction peaks. Once a diffraction peak is identified as causing a problem, a change of excitation conditions is often the best measure to adopt. The use of monochromatic or polarized excitation or of secondary targets in Cartesian geometry will also eliminate diffraction peaks. 4. System Contamination Peaks Careful screening of all the components in an EDXRF spectrometer is required in order to eliminate spurious element peaks in measured spectra. Each spectrometer designer will adopt different approaches and materials for this purpose and, thus, there is a variety of Copyright © 2002 Marcel Dekker, Inc.

Figure 16 Spectrum from boric acid, overlaid with one from a sulfur pellet, showing the diffraction peak. Conditions: Ag target x-ray tube, 5 kV, no primary beam filter.

Figure 17 Spectrum from boric acid, overlaid with one from a sulfur pellet, showing removal of the diffraction peak seen in Figure 16. Conditions: Ag target x-ray tube, 15 kV, thin Al primary beam filter.

Copyright © 2002 Marcel Dekker, Inc.

potential spectrum contaminant peaks. Careful design will eliminate them, although, in some cases, the use of a simple blank correction may be required in some trace analysis. Detector components can be a cause of some system peaks. Beryllium entrance windows usually contain some trace elements that, in extreme cases such as TXRF, can be seen in blank spectra. The use of thick beryllium windows exacerbates this potential problem, but window thicknesses commonly used (8
50 mm) should pose no practical problem. The Si internal fluorescence peak and absorption edge caused by the dead layer are effects not seen in practice and need very careful characterization with specialized conditions if they are to be seen at all. The most likely source of spurious spectrum peaks is sample handling and presentation, both of which are largely in the control of the user. F.

Detection Efficiency and Entrance Windows

One of the more important advantages of semiconductor spectrometers is the absolute efficiency with which fluorescence x-rays are detected and their energies measured. This is the result of the intrinsically high photoelectric absorption efficiencies for semiconductor materials in the x-ray energy range and from the large solid angles achieved in typical EDXRF geometries. The solid angle is determined by the area of the detector and the sample
detector distance and varies with the design of the system. Typical areas are 10
80 mm2 for Si(Li) detectors. Although the additional solid angle is advantageous for many applications, one must realize that the added capacitance associated with increased active area results in an increased contribution in the electronic noise of the system. This, in turn, results in an increased energy resolution of the detection system. The intrinsic efficiency of the semiconductor device can be approximated by a simple model in which the probability of detecting an x-ray incident on the detector is assumed to be the probability of photoelectric absorption within the sensitive volume. This can be expressed as eðEÞ ¼ e
mt ð1
e
sd Þ where e(E) is the energy-dependent intrinsic efficiency of the detector, t is the thickness of any absorptive layer between the sample and the detector, m ¼ m(E ) is the mass absorption coefficient of the absorptive layer, d is the detector thickness, and s ¼ s(E ) is the photoelectric mass-absorption coefficient for the detector material. Figure 18 shows plots of the above equation for the case of a 3-mm-thick Si(Li) and a 5-mm-thick HPGe detector. The poor efficiency at low energies is assumed to be determined by the combined absorption of the 25-mm beryllium entrance window and a nominal 2-cm air pair between sample and detector. The energies of the characteristic K emission lines for several elements are also shown. The plots show the near-unity intrinsic efficiencies for both detectors over a wide range of useful x-ray energies. The Ge detector is efficient at much higher energies than Si because of the higher atomic number and subsequent larger photoelectric cross section. The simple model gives a semi-quantitative picture of the efficiency behavior of semiconductor spectrometers. However, several other factors must be considered for quantitative calibration of a fluorescence spectrometer. The concept of a thin entrance window that either absorbs or transmits an incident x-ray does not describe cases in which the secondary electrons, either photoelectric or Auger, are emitted into the active volume from the window layer. Similarly, detailed studies of the low-energy efficiency have indicated the presence of an absorbing layer on the surface of both Si and Ge devices Copyright © 2002 Marcel Dekker, Inc.

Figure 18 Calculated intrinsic photopeak efficiency for Si and Ge detectors as a function of incident photon energy. The high-energy limits are established by the photoelectric cross section of the detector material and the diode thickness. The low-energy cutoff is determined either by absorption in the thin Be window or in a nominal 2-cm air path from sample to detector or fluorescence source.

associated with the surface layer of the semiconductor material itself. This layer is comprised of an evaporated metal contact used to form the rectifying Schottky barrier and a thin surface layer (the dead layer) of inactive semiconductor material from which charge cannot be collected. The thickness of evaporated metal, normally Au, can be determined by direct measurement during the manufacturing process and is typically in the range of 10 nm. The effective ‘‘dead layer’’ of semiconductor material is a more complicated parameter to determine. Empirical studies have attempted to measure the absorption inherent in the semiconductor surface layer. Such studies established that the effective thickness is determined largely by the absorption length of the low-energy photons and the chargetransport characteristics of the associated ionization products. In the simplest model, there is competition from the rate of diffusion of the electron
hole distribution, against the gradient of the applied field. For distances near the entry contact, a part of the charge can diffuse into the contact and be lost to the signal before the electrical field can sweep it to the opposing electrode. This loss of charge can be interpreted in terms of an effective ‘‘window’’ thickness of typically 0.2 mm silicon equivalent. In addition, thin evaporated contacts are deposited on the entry surface that can absorb incident low-energy photons. Detailed experiments have indicated that the charge-collection efficiency for events near the surface depends in a complex way on low-energy x-ray and charge-transport properties (Llacer et al., 1977; Goulding, 1977). As a consequence of these entry-windowrelated effects, the efficiency for low-energy photons can be reduced. The events lost from the photopeak can then appear in a continuum background below the full-energy peak, where they reduce detectability and interfere with spectral analysis. Similarly, secondary electrons or photons originating from the photoelectric absorption events that occur initially in the active volume can escape. This results in a collected charge pulse of reduced Copyright © 2002 Marcel Dekker, Inc.

amplitude. An easily observed manifestation involves the observation of discrete peaks in the spectrum. Continuous-loss processes involving electron escape can also occur, although the probability is small. These and related mechanisms that reduce the amplitude for a given event can lower the photopeak efficiency relative to the simple model described. There have been studies in which the efficiencies of Si(Li) and Ge have been carefully measured using calibrated sources of x-rays spanning the energy region of interest (Cohen, 1980; Szoghy et al., 1981; Campbell and McGhee, 1986). These indicate that the maximum intrinsic photopeak efficiency is reduced by a few percent relative to the curves shown in Figure 18 and is slightly higher at the upper energy cut off than calculated. These results substantiate the overall validity of the simple photoelectric absorption model but demonstrate the limitations if precise results are required. For laboratory applications of EDXRF, it is not necessary to explicitly determine the photoelectric efficiency function because it is included in the overall calibration factor of the instrument. In any event, the internal efficiency of the detector itself is defined by the design and manufacture of the detector and this is out of the control of most users. The efficiency curve shown in Figure 18 illustrates that for conventional EDXRF measurements, the absorption of fluorescence x-rays in air and in the Be entry window limits the accessible energy range to photons greater than approximately 2 keV. The absorption losses in the air path can be significantly reduced by the common practice of employing a vacuum or He atmosphere. A vacuum path is preferable due to higher x-ray transmission, and if helium is used, the beryllium entrance window must be able to eliminate leakage into the cryostat with subsequent loss of its vacuum. Coated beryllium windows are commonly used nowadays to minimize helium leakage into the cryostat. The necessity of a beryllium window to maintain vacuum integrity between the cryostat enclosure and atmospheric pressure does not constitute a serious limitation for most analyses. However, there are applications in which the detection of x-rays with energies below 1 keV (e.g., for F) becomes necessary. The relatively high detection limits achieved by EDXRF systems because of the poor energy resolution (compared with WDXRF) combined with absorption by the beryllium window generally make EDXRF the second choice for these very low-energy x-ray lines. A beryllium window thickness of 8 mm is typically used for very light-element analysis, although windows as thin as 5 mm have been used. The use of extremely thin windows may lead to increasing concerns for helium integrity and equipment cost. High-strength thin windows made from low-atomic-number elements that are capable of withstanding a full 1-atm-pressure differential have become commercially available in the last decade. These include self-supporting 0.5-mm diamond polycrystalline films and 0.25-mm windows composed of a vapor-deposited amorphous material consisting of 90% boron by weight with nitrogen and oxygen for the remainder. These windows exhibit significant x-ray transmission for photons well below 1 keV and have found widespread use in x-ray microanalysis. G.

Detector Background

Effects associated with the partial collection of the photoelectric signal by the detector have a small effect on its efficiency. However, these incomplete charge-collection processes can have consequences that are far more serious for analytical performance through their effect on spectral background. The effect of incomplete charge collection of these events is to produce tailing on the low-energy side of a peak. In addition, it produces a continuum of events that appear as a shelf on the low-energy side of major peaks in the spectrum. This Copyright © 2002 Marcel Dekker, Inc.

background tailing and shelf interfere with the measurement of lower-energy fluorescence x-rays and is a significant factor in obtaining the lowest detection limits for the determination of trace quantities on the low-energy side of major peaks. Studies designed to reduce this background have indicated that in addition to fundamental x-ray and electron-energy-loss processes, a more significant background resulting from incomplete charge collection from the detector active volume is normally the dominant contribution (Goulding et al., 1972; Jaklevic and Goulding, 1972). This process is an artifact of the detector operation in which the collection of the free charge from the depleted volume is inhibited as a result of nonuniformities in the applied electrical field. These nonuniformities are typically associated with edge effects at the periphery of the cylindrical detector. The incomplete charge-collection background can be reduced either by external collimators that prevent the incident radiation from interacting in the periphery of the detector or by internal electronic collimation brought about by the use of a guard-ring structure (Goulding et al., 1972). The tailing can sometimes be improved by increasing the detector bias, but this is not an option available to the majority of users. There has been much work on characterizing the detector line shape and the effects on background and tailing of incomplete charge collection (Campbell and Wang, 1991; Campbell et al., 1997; Heckel and Scholz, 1987; Lepy et al., 1997). Such studies have led to a greater understanding of the processes involved and to peak-fitting procedures that can take account of them. However, improvements in this form of background can only be achieved through improvements in detector design and manufacturing processes. The P=B ratio of a detector is a standard measurement of detector quality. The measurement is made using a 55Fe radioisotope source to ensure that there is no other contribution to background. The measurement cannot be made using x-ray tube excitation, as it is extremely difficult to eliminate the residual bremsstrahlung continuum. The peak is taken from the most intense channel in the MnKa peak and the background is taken as the average intensity in the channels from 0.9 to 1.1 keV. A high-quality Si(Li) detector will deliver a P=B ratio in excess of 10,000:1 and have minimal low-energy tailing.

IV.

SEMICONDUCTOR DETECTOR ELECTRONICS

A.

Sources of Electronic Noise

The contribution to resolution associated with electronic noise (DEElec) is the result of random fluctuations in thermally generated leakage currents within the detector itself and in the early stages of the amplifier components. Although these processes are intrinsic to the overall measurement process, there are methods for limiting the impact on the final system resolution. Figure 19 is a schematic diagram of a typical pulse processing system employed in a semiconductor detector x-ray spectrometer. The pulse processing can be divided between the charge integration, which takes place in the preamplifier, and the voltage amplification and pulse shaping, which occur in the main amplifier (pulse processor). The function of the charge-sensitive preamplifier and subsequent amplification stages is to convert the integrated charge pulse, produced by collection of the photoelectrically induced ionization, to a voltage pulse that can be measured and stored in the multichannel pulse
height analyzer (MCA). The first stage of the process occurs in the FET, which is attached to the rear of the detector crystal. The method of charge restoration used for the FET has a fundamental effect on the electronic noise of the overall detection system and is Copyright © 2002 Marcel Dekker, Inc.

Figure 19

Typical pulse processing system used for energy-dispersive detectors.

particularly important. Early detectors used pulsed optical charge restoration, but newer systems make use of integrated solid-state devices incorporating low-noise-junction FETs to minimize electronic noise (Statham and Nashashibi, 1988; Lund et al., 1995, Lund et al., 1996). However, the pulse processing must also achieve a very important goal of amplifying the weak charge signal to a measurable level while suppressing random fluctuations in the signal amplitude produced by thermal noise. This is achieved by generating a carefully defined pulse shape in the main amplifier, which restricts the Fourier frequency components in the final signal in such a way that the signal contributions are emphasized relative to the noise fluctuations. The most common pulse shapers employed in modern semiconductor spectrometers generate output pulses that are Gaussian, triangular, or cusp shaped (Fairstein and Hahn, 1965; Kandiah et al., 1972; Landis et al., 1982). Each is capable of achieving adequate energy resolution for EDXRF analysis; the differences are due mainly to the effective time interval required for processing a pulse. Because the relative amplitude of the noise contribution is a strong function of the characteristic time constant associated with the pulse shaper, the difference between pulse shapes becomes important for applications in which high counting rates are important. A detailed analysis of the effects of various pulse-shaping options on spectrometer performance is presented by Goulding and Landis (1982). More recently, sufficiently fast, sampling analog-to-digital converters (ADCs) have become available and these have resulted in the availability of digital pulse processors (Warburton et al., 1988). These devices can implement the optimum noisereduction filters in conjunction with much higher count rates than conventional analog pulse processors. As pointed out below in the following subsection, however, it is not always desirable to operate at the minimum-noise shaping time because an unacceptably large dead time may result. Although EDXRF instruments are designed with a suitable compromise between energy resolution and count rate capabilities, an understanding of these trade-offs on the part of the user is important for optimum usage of the instrument. Copyright © 2002 Marcel Dekker, Inc.

B.

Resolution and Count Rate

There exists a great deal of literature on the nature of the detection and pulse processing systems and the great strides which have been made over the past 20 years to improve resolution and increase count rate. For many years, the complete integration of the detection and electronics systems within the instrumentation has meant that the individual choice of and matching by the EDXRF user of detector, main amplifier, ADC unit, and MCA is no longer required. Once the application is determined, the choice of system will typically revolve around achievable analytical performance. The user will typically be left with a relatively straightforward selection of the single signal processing setting providing the required resolution and count rate (see Fig. 20). There is only a single setting and, unfortunately, resolution and rate go in opposite directions when this setting is changed. In one direction, the amplifier time constant will increase, which gives better resolution (lower FWHM) at the expense of a lower count rate into the MCA memory. In the opposite direction, resolution degrades but count rate increases. Once the EDXRF system is chosen, this control is really the only way in which the user can influence resolution and count rate. It is a little like choosing a car where you select the type of vehicle most suited to your use. That vehicle then has a gearing selection system that allows you some degree of trade-off in control. Low gears give highest accuracy in maneuverability (i.e., resolution) but lowest speed (i.e., count rate). Conversely, the highest gears give a high speed (rate) at the expense of accurate maneuverability! At one end of the range, best resolution is obtained with lowest total count rate, whereas at the other, highest count rate is delivered with the worst energy resolution. When severe peak overlap is the dominant limitation to analytical performance, the best resolution should be selected. In this case, which is typical for the light elements, the

Figure 20

Plots of energy resolution (at 5.9 keV) and count rate for a Si(Li) detector.

Copyright © 2002 Marcel Dekker, Inc.

excitation conditions will need adjusting to make best use of the limited count rate available. Where peak separation is good, such as above 15 keV, or where there are no severe overlaps, select the fastest count rate. In both cases, the current is used to adjust for optimum dead time (50% typically) and the measurement time is used to control the precision. C.

Pulse Pileup

The count-rate limitations associated with a semiconductor spectrometer are an inherent property associated with the finite pulse processing time required by the electronic shaping network. When a random sequence of pulses is incident on the detection system, some of the events cannot be processed without ambiguity. To appreciate the fundamental nature of this limitation and its relationship to system performance, some elementary concepts of electronic pulse processing must be considered. Figure 21 illustrates the time sequence of pulses that occur at various stages in the pulse processing chain. Trace A shows the output of the charge-integrating preamplifier. The steps at times 1, 2, and 3 represent the charge integrals of discrete events. Traces B and C are the outputs of the fast (short shaping time) amplifier and an associated discriminator that serves as a timing marker for the individual events. The main shaping amplifier output is shown as the Gaussian pulse shapes (in trace D). For each event, a total pulse processing time (neglecting analog-to-digital conversion in the pulse
height analyzer) of td is required after the arrival of the pulse and before the system is ready to accept the next event. Although the average counting rate detected by the system can be well below the frequency defined by the reciprocal of the pulse width td, that the events are statistically uncorrelated implies that the events are not uniformly distributed in time. There then exists a probability that two pulses will occur within the same processing time interval. This is illustrated by the overlap of pulses 2 and 3 in the trace. At a low average counting rate, this pulse overlap is not a limiting factor. As the average counting rate increases, however, a point is reached at which there is a significant probability that a second event will occur before the first event has been fully analyzed. If the two events occur within a time less than the shaping time of the amplifier, the charge signals are indistinguishable and an erroneous ‘‘pileup’’ energy signal results. Modern spectrometers employ some form of rejection circuitry to eliminate these pileup events from the pulse
height spectrum. Typical systems rely on the inspection of the fast discriminator output to determine if two pulses have occurred in rapid succession. Appropriate logic is then employed to gate the output of the processor to eliminate the resultant ambiguous energy signals. This is shown in traces E and F of Figure 21, in which the logic signal causes the output to be inhibited when the pulses overlapped to produce an ambiguous pileup energy output. Because the fast shaper that generates the discriminator output has inherently poorer energy resolution relative to the slow channel, the pileup rejection circuit is limited in the minimum energy of pulses it can detect. However, in typical tube-excited EDXRF, the major fraction of pulses occur from scattered high-energy radiation and a pileup rejector system is an important feature in the system design. The pileup probability is obviously a function of the characteristic shaping time tp, which, in turn, establishes the effective dead time, td, of the system. This probability is independent of the details of the specific type of pileup rejection circuit used to eliminate the ambiguous events. The number of events that experience pileup and are consequently eliminated from the spectrum can be estimated for the case of a totally random arrival Copyright © 2002 Marcel Dekker, Inc.

Figure 21 Time sequence of pulses during the processing of a series of x-ray events; (A) the output of the preamplifier is represented as a series of voltage steps reflecting the integrated charge from the individual events; (B) the output from a fast shaping amplifier that operates with a shorter time constant than the main shaping system; (C) a fast discriminator timing pulse derived from B; (D) the output of the main shaper, with the pileup of pulses 2 and 3 indicated; (E) the dead-time gates; and (F) the final output with the ambiguous 2 and 3 pileup pulses rejected.

distribution. For a series of events randomly distributed in time with an average frequency N0, the probability P that no pileup events occur within a characteristic time td after a given event can be expressed as P ¼ N0 e
N0 td From this expression, we can calculate the fraction of events transmitted through the system. Figure 22 is a plot of nonpileup output rate versus input rate expressed in terms of a characteristic pulse processing time. The input rate for which the output rate is a maximum is seen to be the reciprocal of the shaping time. The output rate at this point is reduced by a factor of 1=e. It should be emphasized that this behavior is a fundamental consequence of random arrival statistics and a finite measurement time. However, one should be aware that it is always possible to reduce the characteristic processing time to achieve an increase in counting rate at some sacrifice in energy resolution. It is also worth noting that, by working at too high a dead Copyright © 2002 Marcel Dekker, Inc.

Figure 22 Nonpileup output rate (N0) as a function of input rate (N1) scaled as a function of a characteristic shaping time. Maximum output is 0.37 of the input rate that is equal to the reciprocal of the time constant.

time, the throughput curve inverts and the output rate begins to fall. Further increase of the dead time will eventually lead to paralysis of the system and negligible output countrate. Although different algorithms can be used to correct for events lost to pileup, there is no way to eliminate the effect through passive pulse processing (Gedcke, 1972; Hayes et al., 1978; Statham, 1977). Because different manufacturers of x-ray equipment vary in their approaches to pileup rejection, the only way to evaluate the throughput of the system is using a variable intensity source of radiation. The input rate can be determined by scaling the fast discriminator output, which is sometimes available from the instrument data system. The output rate can be simultaneously measured in the pulse
height analyser. A plot of nonpileup output rate versus input rate can then be generated and compared to the ideal case shown in Figure 22. When the excitation source can be switched off in a time interval that is short compared to the characteristic pulse processing time, it is possible to increase the average output counting rate by eliminating the effects of pileup. Such pulsed excitation systems rely on the ability to detect an event in the fast channel and to shut off the excitation before a second pileup event can occur. In this mode of operation, the output rate can equal the input rate up to the point at which the system is continuously busy. This method of pileup control has been implemented using pulsed x-ray tubes (Jaklevic et al., 1976; Stewart et al., 1976), although the increased complexity of the electron optics tends to lead to increased cost. D.

Dead-Time Correction

The presence of pulse pileup must be considered in any system designed for quantitative measurements because it causes the efficiency with which pulses are processed to be rate dependent. EDXRF analytical systems are designed to correct for this discrepancy through a variety of approaches. The most straightforward involve the use of a live-time Copyright © 2002 Marcel Dekker, Inc.

clock consisting of a gated oscillator and scaler. The oscillator clock is turned off when the system is busy processing pulses. In this way, the duration of the measurement in terms of a live-time interval is extended to correct for those periods when the system is incapable of processing a pulse. Alternative methods rely on the direct measurement of the fast discriminator pulses to keep track of those events that were missed. An additional correction is added to compensate for those intervals in which pileup occurs (Bloomfield et al., 1983; Hayes et al., 1978). A simple empirical method used to correct for dead-time losses is a measurement of the ratio of input to output counts as a function of input rate over the range of values normally encountered. Subsequent analyses require that the input rate be measured for each experiment. The live-time correction can then be applied using the previously determined response function. Because all live-time correction methods have some limitation on the range of counting rates over which they can be used, it is important that methods be devised to evaluate the precision with which such corrections are generated. A carefully prepared series of standards of varying concentrations represents a direct approach. A potentially more precise method involves the use of a single thin-film standard. Variable-mass targets are placed behind the standard to vary the total counting rate over the range of interest. If the variable mass target is chosen so that the variable intensity of scattered or fluorescence radiation does not induce fluorescence in the thin-film standard, the measured intensity of fluorescence from the standard should be independent of total counting rate. The ability of the dead-time correction system to compensate for pileup can then be empirically evaluated. All modern EDXRF systems incorporate effective dead-time correction circuitry, which is likely to be at least as accurate as experiments designed to test this standard feature.

V.

SUMMARY

The technology that makes chemical analysis with EDXRF practical is based on the use of semiconductor x-ray detectors and associated pulse processing and data acquisition systems. The present chapter attempted to explain to the analyst the basic concepts behind the operation of these components and the manner in which they influence overall system performance. The trade-offs one must make between such parameters as excitation conditions, detector resolution, count rate, and other design variables determine how effectively one can tailor a given instrument or experimental apparatus to a specific application. Furthermore, a thorough understanding of the factors that limit performance should enable one to implement experimental tests to determine the effectiveness of a particular approach and evaluate various commercial options.

REFERENCES Aitken DW. IEEE Trans Nucl Sci NS-15:10, 1968. Bertolini G, Cappelani F, Restelli G. Nucl Instrum Methods 32:86, 1965. Bertuccio G, Fasoli L, Fiorini C, Gatti E, Longoni A, Sampietro M, Hauff D, Kemmer J, Richter R. Nucl Instrum Methods 377:352, 1996. Bisgard KM, Laursen J, Nielsen BS. X-Ray Spectrom 10:17, 1981. Bloomfield DJ, Love G, Scott VD. X-Ray Spectrom 12:2, 1983.

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Bowman HR, Hyde EK, Thompson DG, Jared RC. Science 151:562, 1966. Campbell JL, McGhee PL. Nucl Instrum Methods Phys Res A248:393, 1986. Campbell JL, Wang JX. X-Ray Spectrom 20:191, 1991. Campbell JL, Maxwell JA, Papp T, White G. X-Ray Spectrom 26:223, 1997. Cesareo R, Gigante GE, Canegallo P, Castellano A, Iwanczyk JS, Dabrowski A. Nuclear Instrum Methods 380:440, 1996. Christensen LH, Rasmussen SE, Pind N, Henriksen K. Anal Chim Acta 116:7, 1980. Cohen DD. Nucl Instrum Methods 178:481, 1980. Coolidge WD. Phys Rev 2:409, 1913. Dabrowski AJ, Singh M, Huth GC, Iwanczyk JS. In: Heinrich KFJ, Newbury DE, Myklebust RL, Fiori CE, eds. Energy Dispersive X-ray Spectrometry Special Publication 304. Gaithersburg, MD: National Bureau of Standards, 1981, p 45. Dyson NA. Nucl Instrum Methods 114:131, 1974. Elad E, Nakamura M. Nucl Instrum Methods 41:161, 1966. Fairstein E, Hahn J. Nucleonics 23:50, 1965. Field KM, Anderson RJ. US Patent 08=277, 768, 1993. Frankel RS, Aitken DW. Appl Spectrosc 24:557, 1970. Gedcke DA. X-ray Spectrom 2:129, 1972. Gedcke DA, Elad E, Denee PB. X-Ray Spectrom 6:21, 1977. Goulding FS, Jaklevic JM, Jarrett BV, Landis DA. Adv X-ray Anal 15:470, 1972. Goulding FS. Nucl Instrum Methods 142:213, 1977. Goulding FS, Landis DA. IEEE Trans Nucl Sci NS-29:1125, 1982. Haselberger N, Markowicz A, Valkovic V. Appl Radiat Isot 47:455, 1996. Hayes JM, Matthews DE, Schoeller DA. Anal Chem 50:25, 1978. Heath RL. Adv X-ray Anal 15:1, 1972. Heckel J, Scholz W. X-Ray Spectrom 16:181, 1987. Hershyn W. Research=Development 26:32, 1975a. Hershyn W. Watkins Johnson Tech-Notes 2:1, 1975b. Iwanczyk JS, Patt BE, Wang YJ, Croft M, Kalman Z, Mayo W. IEEE Trans Nucl Sci NS-42:606, 1995. Iwanczyk JS, Patt BE, Wang YJ, Khusainov AK. Nucl Instrum Methods 380:186, 1996. Jaklevic JM, Giauque RD, Malone DF, Searles WL. Adv X-ray Anal 15:266, 1972. Jaklevic JM, Goulding FS. IEEE Trans Nucl Sci NS-19:384, 1972. Jaklevic JM, Landis DA, Goulding FS. Adv X-ray Anal 19:253, 1976. Kandiah K, Smith AJ, White G. IEEE Trans Nucl Sci NS-22:2058, 1972. Kirkpatrick P. Rev Sci Instrum 10:186, 1939. Kirkpatrick P. Rev Sci Instrum 15:223, 1944. Kis-Varga M. Isotopenpraxis 24:435, 1988. Knoll GF. Radiation Detectors and Measurements. Wiley, New York: 1979. Landis DA, Goulding FS, Pehl RH, Walton JT. IEEE Trans Nucl Sci NS-18:115, 1971. Landis DA, Cork CP, Madden NW, Goulding FS. IEEE Trans Nucl Sci NS-29:619, 1982. Lechner P, Eckbauer S, Hartman R, Krisch S, Hauff D, Richter R, Soltau H, Strueder L, Fiorini C. Nucl Instrum Methods 377:346, 1996. Lepy MC, Plagnard J, Stemmler P, Ban G, Beck L, Dhez P. X-Ray Spectrom 26:195, 1997. Llacer J, Haller EE, Cordi RC. IEEE Trans Nucl Sci NS-24:53, 1977. Lund JC, Olschner F, Bennett P, Rehn L. IEEE Trans Nucl Sci NS-42:820, 1995. Lund MW, Decker KW, Perkins RT, Phillips JD. Nucl Instrum Methods Phys Res A380:318, 1996. Michette A, Pfauntsch S. X-rays: The First Hundred Years. Chichester: Wiley, 1996. Patt BE, Iwanczyk JS, Szczebiot R, Maculewicz G, Wang M, Wang YJ, Hedman B, Hodgson KO, Cox AD. IEEE Trans Nucl Sci NS-42:558, 1995. Ponpon JP, Sieskind M. Nucl Instrum Methods 380:173, 1996. Porter DE. X-Ray Spectrom 2:85, 1973. Potts PJ, Webb PC, Watson JS. J Anal Atomic Spectrom 1:467, 1986.

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Ro¨ntgen WC. Science 3:227, 1896a. Ro¨ntgen WC. Science 3:726, 1896b. Ross PA. J Opt Soc Am 16:433, 1928. Sandborg A, Shen R. EDAX Editor 14:8, 1984. Skillicorn B. Adv X-ray Anal 25:49, 1982. Skillicorn B. Kevex Analyst 5:2, 1983. Slapa M, Huth GC, Seibt M, Randtke PT. IEEE Trans Nucl Sci NS-23:101, 1976. Spatz R, Lieser KH. X-Ray Spectrom 8:110, 1979. Standzenieks P, Selin E. Nucl Instrum Methods 165:63, 1979. Statham PJ. J Phys E 9:1023, 1976. Statham PJ. X-Ray Spectrom 6:95, 1977. Statham PJ, Nashashibi T. Microbeam Anal 50, 1988. Stewart JE, Zulliger HR, Drummond WE. Adv X-ray Anal 19:153, 1976. Swierkowski SP, Armantrout GA, Wichner R. IEEE Trans Nucl Sci NS-21:302, 1974. Szoghy IM, Simon J, Kish L. X-Ray Spectrom 10:168, 1981. Zulliger HR, Stewart JE. Adv X-ray Anal 18:278, 1975. Vane RA, Stewart WD. Adv X-ray Anal 23:231, 1980. Warburton WK, Darknell DA, Hubbard-Nelson B. Mater Res Soc Symp Proc 487:559, 1998. Yokhin B. Adv X-ray Anal 42:11, 2000.

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4 Spectrum Evaluation Piet Van Espen University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION

This chapter deals with (mathematical) procedures to extract relevant information from acquired x-ray spectra. Smoothing of the spectrum results in a graph that can be easier interpreted by the human observer. To determine which elements are present in a specimen, peak search methods are used. To obtain the analytically important net peak areas of the fluorescence lines, a variety of methods, ranging from simple summation to sophisticated least-squares-fitting procedures, are to the disposal of the spectroscopist. Spectrum evaluation is a crucial step in x-ray spectrometry, as much as sample preparation and quantification. As with any analytical technique, the performance of x-ray fluorescence analysis is determined by the weakest step in the process. Spectrum evaluation in energy-dispersive x-ray fluorescence analysis (EDXRF) is certainly more critical than in wavelength-dispersive spectrometry (WDXRF) because of the relatively low resolution of the solid-state detectors employed. The often-quoted inferior accuracy of EDXRF can, to a large part, be attributed to errors associated with the evaluation of these spectra. As a consequence of this, most of the published work in this field deals with ED spectrometry. Although rate meters and=or strip-chart recorders have been employed in WD spectrometry for a long time, the processing of ED spectra by means of computers has always been more evident because of their inherent digital nature. Some of the techniques to be discussed have their roots in g-ray spectrometry developed mainly in the sixties; for others (notably the spectrum-fitting procedures), EDXRF has developed its own specialized data processing methodology. The availability of inexpensive and fast personal computers together with the implementation of mature spectrum evaluation packages on these machines has brought sophisticated spectrum evaluation within the reach of each x-ray spectrometry laboratory. In this chapter, various methods for spectrum evaluation are discussed, with emphasis on energy-dispersive x-ray spectra. Most of the methods are relevant for x-ray fluorescence, particle-induced x-ray emission (PIXE), and electron beam x-ray analysis [electron probe xray microanalysis (EPXMA), scanning electron microscopy – energy dispersive x-ray analysis (SEM–EDX), and analytical electron microscopy (AEM)]. The principles of the methods and their practical use are discussed. Least-squares fitting, which is of importance not only for spectrum evaluation but also for qualification procedures, is discussed in detail in Sec. IX. Section X presents computer implementations of the main algorithms. Copyright © 2002 Marcel Dekker, Inc.

II.

FUNDAMENTAL ASPECTS

The aims of spectrum evaluation is to extract analytically relevant information from experimental spectra. Obtaining this information is not straightforward because the spectral data are always corrupted with measurement noise. A.

Amplitude and Energy Noise

In x-ray spectra, we can distinguish between amplitude and energy noise. Amplitude noise is due to the statistical nature of the counting process, in which random events (the arrival of x-ray photons at the detector) are observed during a finite time interval. For such a process, the probability of observing N counts when the ‘‘true’’ number of counts is N0 is given by the Poisson distribution (Bevington and Robinson, 1992): PðN; N0 Þ ¼

NN 0
N0 e N!

ð1Þ

The number of counts in each channel of an x-ray spectrum as well as the sum over a number of channels obey this Poisson distribution. For a Poisson random variable, the population standard deviation is equal to the square root of the true number of counts: pffiffiffiffiffiffi sN0 ¼ N0 ð2Þ The sample standard deviation, which is an estimate of the true standard deviation, therefore can be calculated as the square root of the observed number of counts: pffiffiffiffi ð3Þ sN ¼ N  sN0 The statistical nature of the counting process (Poisson statistics or counting statistics) causes the typical channel to channel fluctuations observed in x-ray spectra. That the uncertainty of the data can be calculated from the data itself [Eq. (3)] is of great importance for the spectrum evaluation methods. Energy noise, on the other hand, causes the characteristic x-ray lines of ED spectra to appear much wider than their natural linewidth of about 5
10 eV. Part of this line broadening is due to the nature of photon-to-charge conversion process in the detector, and part of it is associated with the electronic noise in the pulse amplification and processing circuit, as discussed in Chapter 3. As a result, x-ray photons with energy E, which on average correspond to a pulse height stored in channel i, from time to time gives rise to slightly higher or lower pulses, causing the x-ray events to be stored in channels above and below i, respectively. Accordingly, characteristic x-ray lines appear as relatively broad (140
250 eV), nearly Gaussian-shaped peaks in the spectrum. The peaks observed in wavelength-dispersive spectra are also wider than the natural linewidth because of imperfections in the diffraction crystal and the finite size of the collimators. B.

Information Content of a Spectrum

In the absence of these two noise contributions, spectrum evaluation would be trivial. A spectrum would consist of a well-defined continuum on which sharp characteristic lines were superimposed. The intensity of the continuum and the net x-ray line intensities could be determined without error. Any remaining peak overlap (e.g., between AsKa and PbLa, where the separation of 8 eV is less than the natural width of the AsKa line) could be dealt with in an exact manner. Copyright © 2002 Marcel Dekker, Inc.

Unfortunately, we cannot eliminate the noise in the measurements completely. It is possible, however, to reduce the noise in various ways. Amplitude noise (i.e., counting statistics) can be reduced by acquiring the spectrum for a longer period of time or by using a more intense primary beam. The effect of energy noise can be lowered by using a detector and associated electronics of good quality and by shielding the system from external sources of electronic noise. Although these suggestions may sound straightforward and not appropriate in the context of spectrum evaluation, it is important to realize that once a spectrum has been acquired, the information content remains constant. No spectrum processing procedure, no matter how sophisticated, can produce more information than present originally. It is therefore much more efficient to employ optimal experimental conditions when acquiring the data rather than to rely on mathematical techniques in an attempt to obtain information which is not present in the first place (Statham and Nashashibi, 1988). From this point of view, spectrum processing can be seen as any (mathematical) procedure that transforms the information content of a measured spectrum into a form that is more useful for our purposes (i.e., more accessible). As is indicated in Figure 1, most of the procedures that calculate this ‘‘useful’’ information require some form of additional input. Sometimes, this extra information is intuitive and not clearly defined; in other cases, additional information is used in the form of mathematical model. In this respect, not the complexity of model but rather the ability to accurately describe the physical reality is of relevance. When we use a procedure to estimate the net peak area of a characteristic line by summing the appropriate channels and interpolating the continuum left and right of the peak, explicitly some ‘‘additional information’’ is given to this ‘‘spectrum processing’’ method in the form of peak and continuum boundaries. In addition (implicitly), a certain mathematical model is assumed regarding the shape of the peak and the continuum. Provided this model is correct, this relatively simple procedure returns a correct estimate of the net peak area as good as one could obtain by a complicated fitting procedure. The important distinction between simple and more sophisticated spectrum evaluation procedures lies in the incorporated flexibility and applicability. A simple area estimation based on the integration of peak and continuum regions is not generally applicable in real-world situations in which peak overlap, curvature in the continuum, and peak tailing occur. Procedures employing more complex models are adaptable to each specific situation, yielding reliable peak estimates. Spectrum evaluation procedures should

Figure 1 Spectrum evaluation seen as an information process: spectrum processing requires additional information to extract useful information from measured spectral data.

Copyright © 2002 Marcel Dekker, Inc.

therefore be compared on the basis of the explicit and implicit assumptions that are made in the model(s) they employ. C.

Components of an X-ray Spectrum

To evaluate an x-ray spectrum correctly, it is necessary to understand all the phenomena that contribute to the final appearance of the spectrum. This includes the two main features, characteristic lines and continuum, and also a number of spectral artifacts, which become important especially in trace analysis work (Van Espen et al., 1980). 1. Characteristic Lines The characteristic radiation of a particular x-ray line has a Lorentz distribution. Peak profiles observed with a semiconductor detector are the convolution of this Lorentz distribution with nearly Gaussian detector response function, giving rise to what is known as Voigt profile (Wilkinson, 1971). Because the Lorentz width is of the order of only 10 eV for elements with atomic number below 50, whereas the width of the detector response function is of the order of 160 eV, a Gauss function is an adequate approximation of the line profile. Only for K lines of elements such as U and Th does the Lorentz contribution become significant and need to be taken into account (Gunnink, 1977). A more close inspection of the peak shape reveals some distinct tailing at the low energy side of the peak and a shelf extending to zero energy. This is mainly due to incomplete charge collection caused by detector imperfections (dead layer and regions of low electric field) as discussed in Chapter 3. The effect is most pronounced for low energy x-rays. For photons above 15 keV Compton scatter in the detector also contributes to deviation from the Gaussian shape. The distortion caused by incomplete charge collection has been described theoretically (Joy, 1985; Heckel and Scholz, 1987). Various functions have been proposed to model the real peak shape more accurately (Campbell et al., 1987). The observed emission spectrum of an element is the result of many transitions, as explained in Chapter 1. The resulting x-ray lines, including possible satellite lines, need to be considered in the evaluation of an x-ray spectrum. A more detailed discussion on the representation of the K and L spectra and the peak shape is given in Sec. VII. 2. Continuum The continuum observed in x-ray spectra results from a variety of processes. The continuum in electron-induced x-ray spectra is almost completely due to the retardation of the primary electrons (bremsstrahlung). The intensity distribution of the continuum radiation emitted by the sample is in first approximation given by Kramer’s formula (Chapter 1). Absorption in the detector windows and in the sample causes this continuous decreasing function to fall off at low energies, giving rise to the typical continuum shape observed. The attenuation of this bremsstrahlung by major elements in the sample also causes absorption edges to appear in these spectra. Continuum modeling for electron-induced x-ray spectra has been studied in detail by a number of authors (Statham, 1976a, 1976b, Smith et al., 1975; Sherry and Vander Sande, 1977; Bloomfield and Love, 1985). For particle-induced x-ray emission, a similar continuum is observed also mainly due to secondary electron bremsstrahlung. Other (nuclear) processes contribute here, making it virtually impossible to derive a real physical model for the continuum. Special absorbers placed between the sample and detector further alter the shape of the continuum. Copyright © 2002 Marcel Dekker, Inc.

In x-ray fluorescence, the main source of the continuum is the coherent and incoherent scattering of the excitation radiation by the sample. The shape can therefore become very complex and depends both on the initial shape of the excitation spectrum and on the sample composition. When white radiation is used for the excitation, the continuum is mainly radiative and absorption edges can also be observed. With quasimonoenergetic excitation (secondary target, radioisotope), the incomplete charge collection of the intense coherently and incoherently scattered peaks is responsible for most of the continuum (see Chapter 3). Also here, realistic physical models for the description of the continuum are not used. The incomplete charge collection of intense fluorescence lines in the spectrum further complicates the continuum. The cumulative effect of the incomplete charge collection of all lines causes the apparent continuum at lower energies to be significantly higher that expected on the basis of the primary continuum generating processes. 3. Escape Peaks Escape peaks result from the escape of SiK or GeK photons from the detector after photoelectric absorption of the impinging x-ray photon near the edge regions of the detector. The energy deposited in the detector by the incoming x-ray is diminished with the energy of the escaping SiK or GeK photon. Typical examples of the interference due to Si escape peaks are the interference of TiKa (4.51 keV) by the FeKa escape at 4.65 keV and the interference of FeKa by the CuKa escape. For a Si(Li) detector, the escape peaks is expected 1.742 keV (SiKa) below the parent peak. Experimentally, it is observed that the energy difference is slightly but significantly higher, 1.750 keV (Van Espen et al., 1980). Ge escape peaks are observed 9.876 (GeKa) and 10.984 keV (GeKb) below the parent peak. The width of the escape peaks is smaller than the width of the parent peak and corresponds to the spectrometer resolution at the energy of the escape peak. The escape fraction f is defined as the number of counts in the escape peak Ne divided by the number of detected counts (escape þ parent). Assuming normal incidence to the detector and escape only from the front surface, the following formula can be derived for the escape fraction (Reed and Ware, 1972):

 Ne 1 1 m m 1
K ln 1 þ I f¼ ¼ oK 1
ð4Þ r Np þ Ne 2 mI mK where mI and mK are the mass-attenuation coefficient of silicon for the impinging and the SiK radiation, respectively, oK is the K fluorescence yield, and r is the K jump ratio of silicon. Using 0.047 for the fluorescence yield and 10.8 for the jump ratio, the calculated escape fraction is in very good agreement with the experimentally determined values for impinging photons up to 15 keV (Van Espen et al., 1980). Equation (4) is also applicable for estimating the escape fraction in Ge detectors, provided that the parameters related to Si are substituted with these of Ge. Knowing the energy, width, and intensity of the escape peak, corrections for its presence can be made in a straightforward manner. 4. Pileup and Sum Peaks With modern pulse processing electronics, the pileup effects are suppressed to a large extend. With a pulse-pair resolution time of a few 100 ns or less, only true sum peaks are observed. These sum peaks are, within a few electron volts, located at their expected position, and they are slightly wider (5%) than normal peaks located at the same energy in the spectrum (Van Espen et al., 1980). The count rate of a sum peak is given by Copyright © 2002 Marcel Dekker, Inc.

N_ 11 ¼ tN_ 1 N_ 2

ð5Þ

N_ 12 ¼ 2tN_ 1 N_ 1

ð6Þ

and

with N_ 11 the count rate (counts=s) in a sum peak due to the coincidence of two x-rays with the same energy, N_ 12 the count rate of a sum peak resulting from two x-rays with different energy, and t the pulse-pair resolution time in seconds. Sum peaks are often found when a few large peaks at lower energy dominate the spectrum. Typical examples are PIXE spectra of biological and geological samples. The high count rate of the K and CaK lines produces sum peaks that are easily observed in the high-energy region of the spectrum where the continuum is low. It is important to note that the intensity of sum peaks is count-rate dependent, they can be reduced and virtually eliminated by performing the measurement with a lower primary beam intensity. A method for correcting for the contribution of sum peaks in least-squares fitting has been proposed by Johansson (1982) and is discussed further in Sec. VII. 5. Discrete Character of a Pulse
Height Spectrum Another aspect of spectral data that should be mentioned is that a pulse
height spectrum is a discrete histogram representing a continuous function. Digitization of this continuous function, especially the Gaussian peaks, into too few channels causes considerable systematic errors. If the peak contains less than 2.5 channels at the FWHM, the peak area estimate, for example, obtained by summing the channel contents is largely overestimated. This lower limit of 2.5 channels at FWHM corresponds to a spectrometer gain of 60 eV=channel for a peak width of 150 eV. In practice, 40 eV=channel or lower is recommended, otherwise peak position and width determinations and the results of spectrum fitting become unreliable. 6. Other Artifacts A number of other features might appear in an x-ray spectrum and can cause problems during the spectrum evaluation. In the K x-ray spectra of elements with atomic number between 20 and 40, one can detect a peaklike structure with a rather poorly defined maximum and a slowly declining tail (Van Espen et al., 1979a). This structure is due to the K
LL radiative Auger transition, which is an alternative decay mode of the K vacancy. The maximum is observed at the energy of the K
LL Auger electron transition energy. The intensity of the radiative Auger structure varies from approximately 1% of the Ka line for elements below Ca to 0.1% for elements above Zn. For chlorine and lower atomic number elements, the radiative Auger band overlaps with the Ka peak. In most analytical applications, this effect will not cause serious problems. The structure can be considered as part of non-Gaussian peak tail. The scattering of the excitation radiation in x-ray fluorescence is responsible for most of the continuum observed in the spectrum. When characteristic lines are present in the excitation spectrum, two peaks might be observed. The Rayleigh (coherently)-scattered peak has a position and width as expected for a normal fluorescence line. The Compton (incoherently scattered) peak is shifted to lower energies according to the well-known Compton formula and is much broader than a normal characteristic line at that energy. This broader structure, resulting from scattering over a range of angles and Doppler effect, Copyright © 2002 Marcel Dekker, Inc.

is difficult to model analytically (Van Dyck and Van Grieken, 1983). The structure is further complicated by multiple scattering phenomena in the sample (Vincze et al., 1999). Apart from these commonly encountered scattering processes, it is possible to detect xray Raman scattering (Van Espen et al., 1979b). Again, a bandlike structure is observed with an energy maximum given by the incident photon energy minus the electron-binding energy. The Raman effect is most prominently present when exciting elements with atomic number Z 7 2 to Z þ 7 with the K radiation of element Z. In this case, Raman scattering occurs on L electrons. For x-ray excitation energies between 15 and 25 keV, the Raman scattering on the K electrons of Al to Cl can also be observed. Because of its high-energy edge, the effect may appear as a peak in the spectrum with possible erroneous identification as a fluorescence line. The intensity of the Raman band increases as the incident photon energy comes closer to the binding energy of the electron. The observed intensity can amount to as much as 10% of the L fluorescence intensity for the elements Rh to Cs when excitation with MoK x-rays is used. When the excitation source is highly collimated diffraction peaks can be observed in the x-ray fluorescence spectrum of crystalline materials. It is often difficult to deal with these diffraction patterns. They can interfere with the fluorescence lines or even be misinterpreted as being fluorescence lines, giving rise to false identification of elements.

III.

SPECTRUM PROCESSING METHODS

Spectrum processing refers to mathematical techniques that alter the outlook of the spectral data. This is often done, using some digital filter, to reduce the noise, to locate peaks, or to suppress the continuum. In this section, various methods of filtering are discussed. Because of its relation to the frequency domain, the concept of Fourier transformation is introduced first. A.

Fourier Transformation, Convolution, and Deconvolution

One can think of an x-ray spectrum as consisting of a number of components with different frequencies. In the spectrum shown in Figure 2, one recognizes a nearly constant component (the continuum) as well as a component that fluctuates from channel to channel ( ¼ fast). The latter is obviously the noise due to counting statistics. Peaks, then, must have frequency components intermediate between these two. The frequency characteristics of a spectrum can be studied in the Fourier space. For any discrete function f(x), x ¼ 0, . . . , n 7 1 (e.g., a pulse
height spectrum), the discrete Fourier transform is defined as
 n
1 1 X
j2pux fðxÞ exp FðuÞ ¼ n x¼0 n n
1  1 X ux ux fðxÞ cos 2p
j sin 2p ð7Þ n x¼0 n n pffiffiffiffiffiffiffi with j ¼
1 and u ¼ 0, . . . , n 7 1. F(u) is a complex number. The real part, R(u), and the imaginary part, I(u), represent respectively the amplitude of the cosine and the sine functions that are necessary to describe the original data. The square of F(u) is called the power spectrum,

¼

jFðuÞj2 ¼ R2 ðuÞ þ I2 ðuÞ Copyright © 2002 Marcel Dekker, Inc.

ð8Þ

Figure 2 A 256-channel pulse
height spectrum (single Gaussian on a constant continuum) and the Fourier-filtered spectrum.

and gives an idea about the dominant frequencies in the spectrum. Because there are n different nonzero real and imaginary coefficients, no information is lost by the Fourier transform and the inverse transformation is always possible:
 n
1 X j2pux FðuÞ exp fðxÞ ¼ ð9Þ n u¼0 Figure 3 shows that power spectrum of the pulse
height distribution in Figure 2 (a single Gaussian on a constant continuum). The frequency (inverse channel number) is defined as u=n, with n ¼ 256 and u ¼ 0, . . . , n=2. The amplitude of the zero frequency jFð0Þj2 , which is equal to the average of the spectrum, is not shown. The dominating low frequencies originate from the continuum and from the Gaussian peak, whereas the higher frequencies are caused mainly by the counting statistics. It is clear that if we eliminate those high frequencies, we are reducing this noise. This can be done by multiplying the Fourier transform with a suitable function: GðuÞ ¼ FðuÞHðuÞ An example of such a function is a high-frequency cutoff filter:

1; u  ucrit HðuÞ ¼ 0; u > ucrit

ð10Þ

ð11Þ

which sets the real and imaginary coefficients above a frequency ucrit to zero. If we apply this filter to the Fourier transform of Figure 3 using ucrit ¼ 0.05 and then apply the inverse Fourier transformation [Eq. (9)], the result as shown by the solid line in Figure 2 is obtained. The peak shape is preserved, but most of the statistical fluctuations are eliminated. Copyright © 2002 Marcel Dekker, Inc.

Figure 3

Fourier power spectrum of the pulse
height distribution shown in Figure 2.

If we would cut off at even lower frequencies, peak distortions at the top and at the base of the peak would become more pronounced. This Fourier filtering can also be done directly in the original data space. Indeed, the convolution theorem says that multiplication in the Fourier space is equivalent to convolution in the original space: GðuÞ ¼ FðuÞHðuÞ , fðxÞ  hðxÞ ¼ gðxÞ

ð12Þ

The convolute at data point x is defined as the sum of the products of the original data and the filter centered around point x: X fðx
x0 Þhðx0 Þ ð13Þ gðxÞ ¼ fðxÞ  hðxÞ ¼ x0

h(x) is called a digital filter and is the inverse Fourier transformation of H(u). In general, this convolution or filtering of a spectrum yi with some weighing function is expressed as yi ¼

j¼m 1 X hj yiþj N j¼
m

ð14Þ

where hj are the convolution integers and N is a suitable normalization factor. The filter width is given by 2m þ 1. Fourier filtering with the purpose to reduce or eliminate some (high or low) frequency components in the spectrum can thus be implemented as a convolution of the original data with a digital filter. This convolution also alters the variance of the original data. Applying the concept of error propagation, one finds that the variance of the convoluted data is given by Copyright © 2002 Marcel Dekker, Inc.

s2y ¼ i

m 1 X h2 yiþj N2 j¼
m i

ð15Þ

when the original data follows a Poisson distribution (s2y ¼ y). Because the measured spectrum y(x) is itself a convolution of the original (x-ray emission) signal f(x), with the instrument (or detector) response function h(x), it is, in principle, possible to restore the measured signal if this response function is know. This can be accomplished by dividing the Fourier transform (FT) of the measured spectrum by Fourier transform of the (nearly Gaussian) response function, followed by the inverse Fourier transform (IFT) of the resulting quotient: ) FT IFT yðxÞ!YðuÞ YðuÞ ¼ FðuÞ ! fðxÞ ð16Þ FT HðuÞ hðxÞ!HðuÞ The fact that the detector response function changes with energy (becomes broader) and, more importantly, the presence of noise prohibits the straightforward application of this Fourier deconvolution technique. Indeed, in the presence of noise, the measured signal must be presented by yðxÞ ¼ fðxÞ  hðxÞ þ nðxÞ

ð17Þ

and its Fourier transform YðuÞ ¼ FðuÞHðuÞ þ NðuÞ

ð18Þ

YðuÞ NðuÞ ¼ FðuÞ þ HðuÞ HðuÞ

ð19Þ

or

At high frequencies, the response, H(u), goes to zero while N(u) is still significant, so that the noise is emphasized in the inverse transformation. This clearly shows that the noise (counting statistics) is the ultimate limitation for any spectrum processing and analysis method. A clear introduction to Fourier transformations related to signal processing can be found in the work of Massart et al. (1998). Algorithms for Fourier transformation and related topics are given in the work of Press et al. (1998). Detailed discussions on Fourier deconvolution can be found in many textbooks (Jansson, 1984; Brook and Wynne, 1988). Fourier deconvolution in x-ray spectrometry based on maximum a posteriori or maximum entropy principles is discussed by several authors (Schwalbe and Trussell, 1981; Nunez et al., 1988; Gertner et al., 1989). Gertner implemented this method for the analysis of real x-ray spectra and compared the results with those obtained by simple peak fitting. The problem that the deconvolution algorithms are limited to systems exhibiting translational invariance was overcome by a transformation of the spectrum, so that the resolution becomes independent of the energy. B.

Smoothing

pffiffiffi Because of the uncertainty y on each channel content yi , fictitious maxima can occur both on the continuum and on the slope of the characteristic peaks. Removal or suppression of these fluctuations can be useful during the visual inspection of spectra (e.g., for locating small peaks on a noisy continuum) and is used in most automatic peak search and Copyright © 2002 Marcel Dekker, Inc.

continuum estimation procedures. Although smoothing can be useful in qualitative analysis, its use is not recommended as part of any quantitative spectrum evaluation. Smoothing, although reducing the uncertainty in the data locally, redistributes the original channel content over the neighboring channel, thus introducing distortion in the spectrum. Accordingly, smoothing can provide a (small) improvement in the statistical precision obtainable with simple peak integration but is of no advantage when used with leastsquares-fitting procedures in which assumptions about the peak shapes are made. 1. Moving Average The most straightforward way of smoothing (any) fluctuating signal is to employ the ‘‘box-car’’ or moving-average technique. Starting from a measured spectrum y, a smoothed spectrum y* is obtained by calculating the mean channel content around each channel i: yi ¼ y
i ¼

þm X 1 yiþj 2m þ 1 j¼
m

ð20Þ

This can be seen as a convolution [Eq. (14)] with all coefficients hj ¼ 1. The smoothing effect obviously depends on the width of the filter, 2m þ 1. The operation being a simple pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi averaging, the standard deviation of the smoothed data is reduced by a factor 2m þ 1 in regions where yi is nearly constant. On the other hand, such a filter introduces a considerable amount of peak distortion. This distortion depends on the filter width-to-peak width ratio. Figure 4 shows the peak distortion effects when a moving-average filter of widths 9, 17, and 25 is applied to a peak with P full width at half-maximum (FWHM) equal to nine channels. Being a unit area filter hj =N ¼ 1 with N ¼ 2m þ 1, the total counts in the peak is not affected in an appreciable way other than by rounding errors.

Figure 4 Distortion introduced by smoothing of a peak with a moving-average filter 9, 17, and 25 points wide. The FWHM of the original peak is nine channels.

Copyright © 2002 Marcel Dekker, Inc.

Figure 5 shows the effect on the peak height and width when applying this type of filter with different sizes. The peak distortion is caused by the fact that in the calculation of yi , the content of all neighboring channels is used with equal weight. Consequently, by employing a nonuniform filter h, which places more weight on the central channels and less on the channels near the edge of the filter, smoothing can be achieved with less broadening effects. 2. Savitsky and Golay Polynomial Filters Another way of dealing with statistical fluctuations in experimental data is by drawing a best-fitting curve through the data points. This idea resulted in the development by Savitsky and Golay (1964) of a general type of smoothing filters with very interesting features. The method is based on the fact that nearly all experimental data can be modeled by a polynomial of order r, a0 þ a1 x þ a2 x2 þ    þ ar xr , when the data are confined to a sufficiently small interval. If we consider a number of data points around a central channel io , such as yi0
2 ; yi0
1 ; yi0 ; yi0 þ1 ; yi0 þ2 , a least-squares fit with the function. yðiÞ ¼ a0 þ a1 ði
i0 Þ þ a2 ði
i0 Þ2

ð21Þ

can be made. Once we have determined the coefficients aj , the value of the polynomial at the central channel i0 can be used as the smoothed value: yi ¼ yði0 Þ ¼ a0

ð22Þ

This concept is schematically illustrated in Figure 6. By moving the central channel to the right (from i0 to i0 þ 1), the next smoothed channel content can be calculated by repeating the entire procedure.

Figure 5 Percent change in peak height and width introduced by filtering with a moving-average and a Savitsky and Golay polynomial filter as a function of the filter width-to-peak FWHM ratio, (2m þ 1)/FWHM.

Copyright © 2002 Marcel Dekker, Inc.

Figure 6 Concept of polynomial smoothing. A parabola is fitted trough the points i0
3 to i0 þ3. The value of the parabola at i0 is the smoothed value ( ).



At first sight, this smoothing method would require a least-squares fit for each channel in the spectrum. However, the fact that the x values are equidistant allows us to formulate the problem in such a way that the polynomial coefficients can be expressed as simple linear combinations involving only yi values: j¼m 1 X ak ¼ Ck; j yiþk ð23Þ Nk j¼
m This means that it is possible to implement the least-squares-fitting procedure more efficiently as a convolution of the spectrum with a filter having appropriate weights. For this second-order polynomial, the coefficients are given by C0; j 3ð3m2 þ 3m
1
5j2 Þ ¼ N0 ð2m
1Þð2m þ 1Þð2m þ 3Þ

ð24Þ

Smoothing with a five-point second-degree polynomial ð2m þ 1 ¼ 5Þ thus becomes 1 yi ¼ a0 ¼ ð
3yi
2 þ 12yi
1 þ 17yi þ 12yiþ1
3yiþ2 Þ ð25Þ 35 In general, for a polynomial of degree r fitted to 2m þ 1 points, this can be written as yi ¼

j¼m 1 X Crm; j yiþj Nrm j¼
m

ð26Þ

where the convolution integers Crm;j and the normalization factors Nrm do not depend on the data to be smoothed but are a function only of the polynomial degree r and on the filter Copyright © 2002 Marcel Dekker, Inc.

half-width m. Table 1 lists the coefficients of polynomial smoothing filters with widths between 5 and 25 points. The coefficients for a second- and a third-degree polynomial are identical. In comparison with the moving-average filter of the same width, polynomial smoothing filters are less effective in removing noise but have the advantage of causing less peak distortion. The distortion effect as a function of the filter width-to-peak width ratio is given in Figure 5. When the filter becomes much wider than the peak, the smoothed spectrum features oscillations near the peak boundaries, as illustrated in Figure 7. An interesting feature of this type of filters is that they can produce not only a smoothed spectrum but also a smoothed first and second derivative of the spectrum. If we differentiate Eq. (21) and take the value at the center position,  dyðiÞ 0 y ¼ ¼ a1 ð27Þ di i¼i0  d 2 yðiÞ ¼ 2a2 ð28Þ y00 ¼ di2 i¼i0 it follows from Eq. (23) that the smoothed first and second derivative of the spectrum can also be calculated by means of suitable convolution coefficients. For instance, for the first derivative of a second-order polynomial, using five data points this becomes y0 i ¼

1 ð
2yi
2
yi
1 þ yiþ1 þ 2yiþ2 Þ 10

ð29Þ

The corresponding convolution integers for the calculation of the smoothed first and second derivative are listed in Tables 2
4. The use of the derivative spectra is illustrated in next section dealing with peak search methods. The FORTRAN implementation of the Savitsky and Golay filters is given in Sec. X. Variations on these smoothing strategies, such as the use of variable-width filters are reviewed by Yule (1967). The effect of repeated smoothing on the accuracy and precision of peak area determination is discussed by Nielson (1978). A more comprehensive treatment on polynomial smoothing can be found in Enke and Nieman (1976) and its references. Table 1 Savitsky and Golay Coefficients for Second- and Third-Degree (r ¼ 2 and r ¼ 3) Pj¼m Polynomial Smoothing Filter; yi ¼ ð1=Nrm Þ j¼
m Crm; j yiþj , Filter Width ¼ 2m þ 1 j ðCrm; j ¼ Crm;
j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

35 21 231 429 143 1105 323 2261 3059 805 5175

17 7 59 89 25 167 43 269 329 79 467

12 6 54 84 24 162 42 264 324 78 462


3 3 39 69 21 147 39 249 309 75 447


2 14 44 16 122 34 224 284 70 422


21 9 9 87 27 189 249 63 387


36 0 42 18 144 204 54 342


11
13 7 89 149 43 287


78
6 24 84 30 222


21
51 9 15 147


136
76
2 62


171
21
33


42
138


253

Copyright © 2002 Marcel Dekker, Inc.

Figure 7 Effect of the smoothing of a peak with a Savitsky and Golay polynomial filter. The FWHM of the original peak is nine channels.

3. Low Statistics Digital Filter Most smoothing techniques originate from signal processing and were initially introduced in the field of g-ray spectroscopy. Also in the PIXE community, considerable attention has been devoted to a number of aspects of spectrum processing and evaluation. Within the framework of continuum estimation (see later), a smoothing algorithm was developed that removes noise from a spectrum on a selective basis (Ryan et al., 1988). The method provides an n-point means smoothing in regions of low statistics (few counts) while avoiding spreading of the base of peaks and degradation of valleys between peaks. Table 2 Savitsky and Golay Coefficients for Second-Degree (r ¼ 2) Polynomial, First-Derivative Pj¼m Filter; yi ¼ ð1=Nrm Þ j¼
m Crm; j yiþj , Filter Width ¼ 2m þ 1 j ðCrm; j ¼ Crm;
j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

10 28 60 110 182 280 408 570 770 1012 1300

0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3

4 4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5

6 6 6 6 6 6 6

7 7 7 7 7 7

8 8 8 8 8

9 9 9 9

10 10 10

11 11

12

Copyright © 2002 Marcel Dekker, Inc.

Table 3 Savitsky and Golay Coefficients for Third-Degree (r ¼ 3) Polynomial, First-Derivative Filter; yi ¼ ð1=Nrm Þ Width ¼ 2m þ 1

Pj¼m j¼
m

Crm; j yiþj , Filter

j ðCrm; j ¼ Crm;
j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

12 252 1,188 5,148 24,024 334,152 23,256 255,816 3,634,092 197,340 1,776,060

0 0 0 0 0 0 0 0 0 0 0

8 58 126 296 832 7,506 358 2816 29,592 1,222 8,558


1 67 193 503 1,489 13,843 673 5,363 56,881 2,365 16,649


22 142 532 1,796 17,842 902 7,372 79,564 3,350 23,806


86 294 1,578 18,334 1,002 8,574 95,338 4,098 29,562


300 660 14,150 930 8,700 101,900 4,530 33,450


1,133 4,121 643 7,481 96,947 4,567 35,003


12,922 98 4,648 78,176 4,130 33,754


748
68 43,284 3,140 29,236


6,936
10,032 1,518 20,982


84,075
815 8,525


3,938
8,602


30,866

Copyright © 2002 Marcel Dekker, Inc.

Table 4 Savitsky and Golay Coefficients for P Second- and Third-Degree (r ¼ 2 and r ¼ 3) j¼m Polynomial Second-Derivative Filter; yi ¼ ð1=Nrm Þ j¼
m Crm; j yiþj , Filter Width ¼ 2m þ 1 j ðCrm; j ¼ Crm;
j Þ m

Nrm

0

1

2

3

4

5

6

7

8

9

10

11

12

2 3 4 5 6 7 8 9 10 11 12

7 42 462 429 1,001 6,188 3,876 6,783 33,649 17,710 26,910


2
4
20
10
14
56
24
30
110
44
52


1
3
17
9
13
53
23
29
107
43
51

2 0
8
6
10
44
20
26
98
40
48

5 7
1
5
29
15
21
83
35
43

28 6 2
8
8
14
62
28
36

15 11 19 1
5
35
19
27

22 52 12 6
2
8
16

91 25 19 37 5
3

40 34 82 20 12

51 133 37 29

190 56 48

77 69

92

The rather heuristic algorithm is discussed in Section X, together with the computer implementation. The effect of the filter is illustrated in Figure 8, where it is compared with the other smoothing methods discussed.

C.

Peak Search Methods

Several methods have been developed for the automatic localization of peaks in a spectrum. Nearly all methods follow a strategy where the original spectrum is transformed into a form that emphasizes the peaklike structures and reduces the continuum, followed by a decision whether these peaklike structures are statistically significant. The latter involves some adjustable parameter(s) controlling the sensitivity of the peak search. Although visual inspection of the spectrum still appears to be the best method, peak search algorithms, which are heavily used in g-ray spectrometry, may have some value in x-ray spectrometry. Their use in energy-dispersive x-ray analysis as part of an automated qualitative analysis procedure is hampered by the extreme peak overlap in these spectra. More elaborated procedures involving artificial intelligence techniques have been considered for this (Janssens and Van Espen, 1986; Janssens et al., 1988). Peak search procedures usually involve three steps: (1) transformation of the original spectrum so that continuum contribution is eliminated, peaks are readily locatable, and overlapping peaks are (partially) resolved; (2) significance test and approximate location of the peak maximum; and (3) more accurate peak position estimate in the original spectrum. The various peak search algorithms mainly differ in the choice of the transformation. Some methods use the first and second smoothed derivative of the spectrum. This method is illustrated in Figure 9. The sign change (crossing of the x axis) of the first derivative and the minimum of the second derivative are quire suitable to detect the peaks in the original spectrum. Other methods employ some form of correlation technique, which is basically the convolution of the original spectrum with a filter that approximates the shape of the peak and, therefore, emphasizes the peak. If a zero-area correlator (filter) is used, the continuum is at the same time effectively suppressed. The simplest and most effective correlators belong to the group of zero-area rectangular filters. These filters have a central window with Copyright © 2002 Marcel Dekker, Inc.

Figure 8 Effect of various smoothing methods applied to part of a x-ray spectrum: (A) original spectrum, (B) nine-point Savitsky and Golay filter, (C) nine-point moving average, (D) low-noise statistical filter.

constant and positive coefficients and two side lobes with constant and negative coefficients. Convoluting an x-ray spectrum with this kind of filter yields spectra in which the continuum is removed and peaks are easily locatable. They are similar to inverted second-derivative spectra. An important representative of this group of filters is the ‘‘top-hat’’ filter, which has a central window with an odd number of channels w and two side windows each n channels wide. The value of the filter coefficients follows from the zero-area constraint: 8 1 <
2n ;
n
w2  k <
w2 1 ð30Þ hk ¼ ;
w2  k  þ w2 : w1
2n ; þ w2 < k  w2 þ n The filtered spectrum is obtained by the convolution of the original spectrum with this filter: yi ¼

k¼nþw=2 X

hk yiþk

ð31Þ

k¼
n
w=2

The effect of this filter on a typical spectrum is shown in Figure 10. The variance of the filtered spectrum is obtained by simple error propagation: s2y ¼

k¼nþw=2 X

i

h2k yiþk

k¼
n
w=2

Copyright © 2002 Marcel Dekker, Inc.

ð32Þ

Figure 9 Peak search using first and second derivatives: (A) doublet, peak width (s) four channels, separation eight channels, on a constant continuum. (B) First and (C) second smoothed derivative using a five-point Savitsky and Golay filter.

If yi is significantly different from zero, a peak structure is found and the top of the peak can approximately be located by searching for the maximum. Thus, i corresponds to the position of a peak maximum in the original spectrum if yi > rsyi Copyright © 2002 Marcel Dekker, Inc.

ð33Þ

Figure 10 Result of applying a top-hat filter. Dotted line: typical X-ray spectrum; solid line: filtered spectrum.

and yi
1  yi > yiþ1

ð34Þ

In Figure 11, the positive part of the filtered spectrum (w ¼ 9 and n ¼ 5) and the decision line ðrsy Þ for r ¼ 1 and 4 are displayed. If required, other peak features can be obtained from the filtered spectrum: The distance between the two local minima is a measure of the width of the peak and the height at the maximum is related to the net peak area. Because the width and heights of the peaks in the filtered spectrum strongly depend on the dimensions of the filter, it is important that its dimensions are matched to the width of the peaks in the original spectrum. From considerations of peak detectability (signal-tonoise ratio) and resolution (peak broadening), it follows that the optimum width of the positive window w is equal to the FWHM of the peaks (Robertson et al., 1972). The width of the negative side windows should be chosen as large as the curvature of the continuum allows. A reasonable compromise between sensitivity to peak shapes and rejection of continuum is reached when n equals FWHM=2 to FWHM=3. Typical values for the sensitivity factor r are between 2 and 4. Higher values result in the loss of small peaks; lower values will cause continuum fluctuations to be interpreted as peaks. Other zero-area rectangular filters, variations to the top-hat filter, are also in use, such as the square-wave filter with typical coefficient sequence
1,
1, 2, 2,
1,
1 (Philips and Marlow 1976; McCullagh, 1982) and the symmetric square-wave filter with coefficients
1, 1, 1,
1 (Op De Beeck and Hoste 1975). A detailed account of the performance of this filter is given by Op De Beeck and Hoste (1975). A method using a Gaussian correlator function is discussed by Black (1969). Copyright © 2002 Marcel Dekker, Inc.

Figure 11 Peak search using the positive part of the top-hat filtered spectrum and the decision level for one and four times the standard deviation. The original spectrum is shown at the bottom.

Once the peak top is approximately located in the filtered spectrum, a more precise maximum can be found by fitting a parabola over a few channels around the peak. For a well-defined peak on a low continuum (or after continuum subtraction), the channel content near the top of the peak can be approximated by a Gaussian: "

ðx
mÞ2 yi  h exp
2s2

# ð35Þ

The logarithm of the data then is a simple polynomial
ln yi ¼

m2 ln h
2 2s

 þ

m m2 2 x
x 2s2 s2

ð36Þ

If we fit ln yi with a polynomial a0 þ a1 x þ a2 x2 , where x represents the channel number, the position of the peak m is obtained from m¼


a1 2a2

Copyright © 2002 Marcel Dekker, Inc.

ð37Þ

with an accuracy of 0.1 channel or better if the continuum contribution is small and if the peak is interference free [i.e., if Eq. (35) accurately describes the data]. An estimate of the peak’s width and height is obtained at the same time: rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 1 FWHM ¼ 2 2 ln 2s ¼ 2:3548
ð38Þ 2a2   a2 h ¼ exp a0 þ 1 ð39Þ 4a2 To obtain a reliable estimate of the parameters, only the channels in the FWHM or, at most, within the FWTM region of the peak must be included in the fit. As a somewhat simpler and faster alternative, one can find an estimate of the peak maximum by fitting the parabola over the three top channels of the peak. If i is the peak maximum found in the filtered spectrum, a better estimate of the maximum in the original spectrum is found by m¼iþ

1 yi
1
yiþ1 2 yi
1 þ yiþ1
2yi

ð40Þ

This method might be preferred for small peaks when the continuum cannot be disregarded. A FORTRAN implementation of a peak search algorithm is given in Section X.

IV.

CONTINUUM ESTIMATION METHODS

Except for some special quantification procedures (e.g., the peak-to-background method in electron microscopy), the relevant analytical information is found in the net peak areas and continuum is considered a nuisance. There are, in principle, three ways to deal with the continuum: (1) the continuum can be suppressed or eliminated by a suitable filter; (2) the continuum can be estimated and subtracted from the spectrum prior to the estimation of the net peak areas; and (3) the continuum can be estimated simultaneously with the other features in the spectrum. The first approach is discussed in Section VI, where the continuum is removed from spectra by applying a top-hat filter followed by linear leastsquares fit of the spectrum with a number of (also filtered) reference spectra. Least-squares fit (linear or nonlinear) with analytical functions (Sec. VII) allows the simultaneous estimation of continuum and peaks, providing a suitable mathematical function can be found for the continuum. In this section, we discuss a number of procedures that aim to estimate the continuum independently of the other features in the spectrum. Once estimated, this continuum can be subtracted from the original spectrum and all methods for further processing, ranging from simple peak integration to least-squares fitting can be applied. Any continuum estimation procedure must fulfill two important requirements. First, the method must be able to reliably estimate the continuum in all kinds of situations (e.g., small isolated peaks on a high continuum as well as in the proximity of a matrix line). Second, to permit processing of a large number of spectra, the method needs to be nearly free of user-adjustable parameters. Although a number of useful continuum estimation procedures has been developed, it must be realized that their accuracy in estimating the continuum is not optimal. In one way or another, they rely on the difference in frequency response of the continuum compared to other structures such as peaks, the former mainly consisting of low Copyright © 2002 Marcel Dekker, Inc.

frequencies (slowly varying). Because the peaks also exhibit low frequencies at the peak boundaries, it is difficult to control the method in such a way that it correctly discriminates between peaks and continuum. This results in either a small underestimation or overestimation of the continuum, introducing potentially large relative errors for small peaks. In this respect, the fitting of the continuum with analytical functions may provide more optimal results (Vekemans et al., 1994). A considerable advantage of the methods discussed here is that they do not assume an explicit mathematical model of the continuum. Constructing a detailed and accurate analytical model for the continuum based in physical theory is nearly impossible except for some simple geometry and particular excitation conditions. Most often, some polynomial type of function must be chosen when fitting a portion of the spectrum with analytical functions. A.

Peak Stripping

These methods are based on the removal of rapidly varying structures in a spectrum by comparing the channel content yi with the channel content of its neighbors. Clayton et al. (1987) proposed a method which compares the content of channel i with the mean of its two direct neighbors: mi ¼

yi
1 þ yiþ1 2

ð41Þ

If yi is smaller than mi, the content of channel i is replaced by the mean mi. If this transformation is executed once of all channels, one can observe a slight reduction in the height of the peaks while the rest of the spectrum remains virtually unchanged. By repeating this procedure, the peaks are gradually ‘‘stripped’’ from the spectrum. Because the method tends to connect local minima, it is very sensitive to local fluctuations in the continuum due to counting statistics. This makes smoothing of the spectrum, as discussed in the previous section, prior to the stripping process mandatory. Depending on the width of the peaks after typically 1000 cycles, the stripping converges and a more or less smooth continuum remains. To reduce the number of iterations, it might be advantageous to perform a log or pffiffiffiffi square root transformation to the data prior to the stripping: y0i ¼ logðyi þ 1Þ or y0i ¼ yi . After the stripping, the continuum shape is obtained by applying the inverse transformation. A major disadvantage of this method is that after a number of cycles, the bases of partially overlapping peaks are transformed into broad ‘‘humps,’’ which take much longer to remove than isolated peaks. The method was originally applied to PIXE spectra but proves to be generally applicable for pulse
height spectra. In Figure 12, this method is applied to estimate the continuum of an x-ray spectrum in the region between 1.6 and 13.0 keV. The spectrum results from a 200-mg=cm2 pellet of NIST SRM Bovine Liver sample excited with the white spectrum of an Rh-anode x-ray tube filtered through a thin Rh filter (Tracor Spectrace 5000). Because of the white tube spectrum, a considerable continuum intensity was observed, increasing quite steeply in the region above 10 keV. To calculate the continuum, the following algorithm was used: (1) the square root of the original spectrum was taken; (2) these data were smoothed with a 10-point Savitsky and Golay filter; (3) a number of iterations were performed applying Eq. (41) over the region of interest; (4) the square of each data point was taken (back transformation) to obtain the final continuum shape. In Figure 12, the continuum after 10, 100, and, finally, 500 iterations is shown. As a generalization of the above-discussed method, the average of two channels a distance w away from i can be used: Copyright © 2002 Marcel Dekker, Inc.

Figure 12 Continuum estimate after 10, 100, and 500 iterations obtained with simple iterative peak stripping.

mi ¼

yi
w þ yiþw 2

ð42Þ

Ryan et al. (1988) proposed using twice the FWHM of the spectrometer at channel i as the value for w. They reported that only 24 passes were required to produce acceptable continuum shapes pffiffiffi in PIXE spectra. During the last eight cycles, w is progressively reduced by the factor 2 to obtain a smooth continuum. To compress the dynamic range of the spectrum, a double-log transformation of the spectrum, log[log( yi+1)+1], before the actual stripping was proposed. In combination with the low statistics digital filter, this procedure is called the SNIP algorithm (Statistical Nonlinear Iterative Peak clipping). A variant of this procedure is implemented in the procedure ‘‘SNIPBG’’ given in Sec. X. Instead of the double logarithmic, we employed a square root transformation, and a Savitsky and Golay smoothing is performed on the square root data. The width w is kept constant over the entire spectrum. The value of w is also used as the width of the smoothing filter. Using this implementation, the continuum of the above-discussed spectrum is calculated and represented in Figure 13. The width was set 11 to channels approximately corresponding to the FWHM of the peaks in the center of the spectrum and 24 iterations were done. Apart from delivering a smoother continuum with smaller ‘‘humps,’’ this method executes much faster than the original method proposed by Clayton. B.

Continuum Estimation Using Orthogonal Polynomials

Another interesting continuum estimation procedure was introduced by Steenstrup (1981), who applied the method to energy-dispersive x-ray diffraction spectra. The spectrum is fitted using orthogonal polynomials, and the weights of the least-squares fit are iteratively adjusted so that only channels belonging to the continuum are included in the fit. Copyright © 2002 Marcel Dekker, Inc.

Figure13 Continuum estimate using statistical nonlinear iterative peak clipping (SNIP algorithm) with 24 iterations.

The method is generally applicable to pulse
height spectra and can be implemented as an algorithm that needs little or no control parameters. The continuum is described by a set of polynomials up to degree m: m X yðiÞ ¼ cj Pj ðxi Þ ð43Þ j¼0

where Pj(xi) is an orthogonal polynomial of degree j. As an example of m ¼ 3, the function becomes yðiÞ ¼ c0 þ c1 ðxi
a0 Þ þ c2 ððx1
a1 Þðxi
a0 Þ
b1 Þ

ð44Þ

The least-squares estimates of the parameters cj are given by cj ¼

n X wi yi Pj ðxi Þ gj i¼1

ð45Þ

where wi are the weights of the fit. A detailed discussion on orthogonal polynomial fitting can be found in Sec. IX. Because the polynomial terms Pj ðxi Þ are orthogonal, no matrix inversion is required to obtain the results. Orthogonal polynomials of a much higher degree can be fitted to the experimental data without running into problems with illconditioned normal equations and oscillating terms. The goal of this continuum estimation method is to fit the continuum with the above-described orthogonal polynomial of degree m and to interpolate under the peaks. This can be achieved by careful manual selection of only those data pairs ðxi ; yi Þ that belong to the continuum. The more elegant approach proposed by Steenstrup consisted of using all channels and automatically adjusting the weights in such a way that the continuum contributions are emphasized. If yðiÞ is a polynomial approximation of the continuum of degree m, then yi  yðiÞ if i is a continuum channel, otherwise yi > yðiÞ. Copyright © 2002 Marcel Dekker, Inc.

A better approximation to the continuum can then be found by choosing small weights for the data points where yi > yðiÞ and repeating the fit. The following weighting scheme is proposed by Steenstrup: pffiffiffiffiffiffiffiffi 1 if yi  yðiÞ þ r yðiÞ wi ¼ ð46Þ yðiÞ pffiffiffiffiffiffiffiffi 1 wi ¼ if y > yðiÞ þ r yðiÞ ð47Þ i ½ yðiÞ
yi 2 where r is an adjustable parameter. Typically, r ¼ 2 is used. If yi p is ffiffiffiffiffiffiffi normally distributed ffi (which is approximately the case for yi > 30 counts), yi  yðiÞ þ 2 yðiÞ holds for 97.7% of the channels containing only continuum. A too high value of r will cause the inclusion of the tails of the peaks into the fit. With the new weights, a new polynomial fit of degree m is performed. The process is stopped when the new polynomial coefficients cj are within one standard deviation from the previous ones. We have also obtained good results by setting the weights effectively to zero when the channel content is statistically above the fitted continuum [Eq. (47)]. The method can be made even more unsupervised by including a procedure to automatically select the best degree of the polynomial. This can be done by fitting (as described earlier) successive polynomials with higher degrees and testing the significance of each additional polynomial coefficient. If the coefficient Cmþ1 is statistically not significant different from zero, jCmþ1 j < 2sCmþ1

ð48Þ

a polynomial of degree m is retained. Figure 14 shows the spectrum of bovine liver sample (same as Fig. 12) and the fitted continuum using a fourth-, fifth-, and sixth-degree orthogonal polynomial. A value of 1.5 was chosen for r [Eqs. (46) and (47)] and the weights were set to zero for the channels not belonging to the continuum; wi ¼ 0 in Eq. (47). Table 5 gives the data on the fits. The final number of continuum channels retained, the number of weight adjustments performed, and the value and standard deviation of the highest polynomial coefficient are shown. The total number of channels in the fitting region was 571. Because the value of the seventh degree polynomial coefficient is insignificant [Eq. (48)], it is concluded that a sixth-degree orthogonal polynomial is needed to model the continuum. Section X lists the FORTRAN code that implements the complete procedure to estimate the continuum of a given spectrum by an mth-degree orthogonal polynomial. V.

SIMPLE NET PEAK AREA DETERMINATION

In both in WDXRF and EDXRF, the concentration of the analyte is proportional to the number of counts under the characteristic x-ray peak corrected for the continuum. At constant resolution, this proportionality also exits for the net peak height. In EDXRF, preference is given to the peak area. In WDXRF, the acquisition of the entire peak profile is very time-consuming and the count rate is usually measured only at the peak maximum. A.

Peak-Area Determination in EDXRF

The most straightforward method to obtain the net area of an isolated peak in a spectrum consists of interpolating the continuum under the peak and summing the continuumcorrected channel contents in a window over the peak, as shown in Figure 15. Copyright © 2002 Marcel Dekker, Inc.

Figure 14 Continuum estimate obtained by fitting a fourth-, fifth-, and sixth-degree orthogonal polynomial.

The net peak area, Np , of an isolated peak on a continuum is given by NP ¼

iP1 X

½ yi
yB ðiÞ ¼

iP2

X

yi


i

X

yB ðiÞ ¼ NT
NB

ð49Þ

i

where NT and NB are the total number of counts of the spectrum and the continuum in the integration window iP1
iP2 . The uncertainty of the estimated net peak area due to counting statistics is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sNP ¼ NT þ NB ð50Þ The continuum yB ðiÞ is calculated by interpolation, assuming a straight line (see Fig. 15): yB ðiÞ ¼ YBL þ ðYBR
YBL Þ

i
iBL iBR
iBL

ð51Þ

Table 5 Data on the Fit of the Continuum of the Spectrum Shown in Figure 14 Using Orthogonal Polynomials Degree of polynomial 4 5 6 7

No. of channels used

No. of weight adjustments

Highest degree coefficient and standard deviation

126 199 267 266

7 16 6 6

(3.98±0.20)10
6 (1.88±0.09)10
7 (7.66±0.28)10
10 (3.66±2.03)10
13

Copyright © 2002 Marcel Dekker, Inc.

Figure 15

Determination of the net peak area by interpolating the continuum.

YBL and YBR are the values of the continuum at the channels iBL and iBR , left and right from the peaks, respectively. These values are best estimated by averaging over a number of channels: YBL ¼ YBR ¼

iBL1 1 X NBL yi ¼ nBL i¼iBL2 nBL

1

iBR1 X

nBR

i¼iBR2

yi ¼

NBR nBR

ð52Þ ð53Þ

The number of channels in the continuum windows are nBL ¼ iBL2
iBL1 þ 1 and nBR ¼ iBR2
iBR1 þ 1. The center position of the continuum windows (not necessarily an integer number!) used in Eq. (51) are iBL ¼ ðiBL1 þ iBL2 Þ=2 and iBR ¼ ðiBR1 þ iBR2 Þ=2. If both continuum windows have equal width, nBL ¼ nBR ¼ nB =2, and are positioned symmetrically with respect to the peak window ðiP
iBL ¼ iBR
iP Þ a much simpler expression is obtained for the net peak area: nP ð54Þ NP ¼ NT
ðNBL þ NBR Þ nB where NBL and NBR are the total counts in the left and right continuum windows, respectively, each nB =2 channels wide, and nP equals the number of channels in the peak window. Applying the principle of error propagation, the uncertainty in the net peak area is then given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 nP ðNBL þ NBR Þ ð55Þ sNP ¼ N T þ nB Copyright © 2002 Marcel Dekker, Inc.

From this equation, it can be seen that, in principle, the continuum should be estimated using as many channels as possible ðnB Þ to minimize random errors due to counting statistics. In practice, the width will be limited by curvature of the continuum and by the presence of other peaks. Most often, the total width of the continuum ðnB ¼ nBL þ nBR Þ is taken equal to or slightly larger than the width of the peak window. The optimum width of the peak window to minimize counting statistics depends on the peak-to-continuum ratio (Jenkins et al., 1981). For low peak-to-continuum ratios, the peak height being only a fraction of the continuum height, the optimum width of the integration window is 1.17 times the FWHM of the peak. If the ratio of the peak height to the continuum height is larger than 1, a slightly wider window is optimal, although, in practice, the improvement in precision is negligible. The method does not deliver the total net peak area, only a fraction of it. Integrating over 1.176FWHM (from
1:378s to þ1:378s) covers only 83% of the peak. To cover 99% of the peak, the window should be 2.196FWHM (±2.579s). If position or width changes are observed from one spectrum to the other, care should be taken to define the windows in such a way that cover the same part of the spectrum. The wider the peak window, the less sensitive the method for peak shift. Although such a peak-area-determination method seems naively simple, if correctly used it provides results that are as accurate and precise as the most sophisticated procedures. The premises for use are that the peak window should be known to be free from interferences, that there should be no peaks in the continuum windows, and that the continuum should be linear over the extent of the windows. A peak search procedure can, in principle, be used to setup the windows automatically. However, its practical use is limited by complexity of the EDXRF spectra. Moreover, the use of such automated procedures is hazardous because no measure can be given for the presence or absence of systematic errors. Because of this restrictions, a simple peak integration method cannot be used as a general tool for spectrum processing. In a limited number of applications, good results can be obtained. An evaluation of various peak integration methods is given by Hertogen et al. (1973). B.

Net Count Rate Determination in WDXRF

In WDXRF, the count rate at the 2Y angle of the peak maximum corrected for the continuum is used as analytical signal. The continuum is estimated at a 2Y position on the left or right side of the peak. Continuum interpolation as described in the previous section is also possible. If NP is the number of counts accumulated during a time interval tP at the top of the peak and NB is the number of the continuum counts accumulated during time tB , then the net count rate, I, is given by I ¼ IP
IB ¼

NP NB
tP tB

and the uncertainty in the net count rate due to counting statistics is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NP NB sI ¼ þ 2 t2P tB

ð56Þ

ð57Þ

Various counting strategies can be considered and the effect on the precision can be estimated using Eq. (57) (Bertin, 1970). In a ‘‘optimum fixed time’’ strategy, the minimum Copyright © 2002 Marcel Dekker, Inc.

uncertainty is obtained when, for a total measurement time t ¼ tP þ tB ; tP and tB are chosen in such a way that their ratio is equal to the square root of the peak-to-continuum ratio: rffiffiffiffiffi tP IP ð58Þ ¼ tB IB Under, these conditions, the uncertainty in the net intensity is given by pffiffiffiffiffi pffiffiffiffiffi IP þ IB sI ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tP þ tB

VI.

ð59Þ

LEAST-SQUARES FITTING USING REFERENCE SPECTRA

In this section, two techniques based on linear least squares are discussed. The filter-fit methods makes use of library spetra, measured or calculated spectra of pure compounds, that are used to describe the spectra of complex samples. The other method is based on partial least-squares (PLS) regression, a multivariate calibration technique. In this case, no spectrum evaluation in the strict sense is performed, but, rather, relations between the concentrations of the compounds in the samples and the entire spectrum are established. In this way, quantitative analysis is possible without obtaining net peak areas of the characteristic lines. A.

Filter-Fit Method

1. Theory If a measured spectrum of an unknown sample can be described as a linear combination of spectra of pure elements constituting the sample, then the following mathematical model can be written: ¼ ymod i

m X

aj xji

ð60Þ

j¼1

the content of channel i in the model spectrum and xji the content of channel i in with ymod i the jth reference spectrum. The coefficients aj are a measure of the contribution of pure reference spectra to the unknown spectrum and can be used for quantitative analysis. The values of the coefficients aj are obtained via multiple linear least-squares fitting, minimizing the sum of the weighted squared differences between the measured spectrum and the model: " #2 n2 n2 m X X X 1 1 2 2 w ¼ ½ yi
yðiÞ ¼ yi
aj xji ð61Þ s2 s2 i¼n1 i i¼n1 i j¼1 where yi and si are the channel content and the uncertainty of the measured spectrum, respectively, and n1 and n2 are the limits of the fitting region. A detailed discussion of the least-squares-fitting method is given in Sec. IX. The assumption of linear additivity [Eq. (60)] holds normally reasonable well for the characteristic lines in the spectrum, but not for the continuum. To apply this technique, the continuum can be removed from the unknown spectrum and from the reference spectra using one of the procedures described in Sec. IV before the actual least-squares fit. Copyright © 2002 Marcel Dekker, Inc.

Another, frequently used approach is to apply a digital filter to both unknown and reference spectra. This variant is known as the filter-fit method (Schamber, 1977; Statham, 1978; McCarthy and Schamber, 1981) and is discussed in some detail below. By the discrete convolution of a spectrum with top-hat filter [Eqs. (30) and (31)], the low-frequency component (i.e. the slowly varying continuum) is effectively suppressed as discussed in Sec. III. Apart from removing the slowly varying continuum, a rather severe distortion of the peaks is also introduced. If we apply this filter to both the unknown spectrum and the reference spectra, the nonadditive continuum is removed and the same type of peak distortion will be introduced in all spectra, allowing us to apply the method of multiple linear least-squares fitting to the filtered spectra. Equation (61) then becomes " #2 n2 m X X 1 w2 ¼ y0
aj x0ji s0 2 i j¼1 i¼n1 i

ð62Þ

Where y0i and x0ji are the filtered unknown and reference spectra, respectively, s0i 2 is the variance of y0i given by X s02 h2k yiþk ð63Þ i ¼ k

The least-squares estimates of the contribution of each reference spectrum is then given by (see Sec. IX) aj ¼

m X

a
1 jk bk ;

j ¼ 1; . . . ; m

ð64Þ

k¼1

with n2 X 1 0 0 yx 02 i ji s i¼n1 i n2 X 1 0 0 ajk ¼ x x s0 2 ki ji i¼n1 i

bj ¼

ð65Þ ð66Þ

The uncertainty in each coefficient aj is directly estimated from the error matrix: s2aj ¼ a
1 jj

ð67Þ

Schamber (1977) suggested the following equation for the uncertainties, taking into account the effect of the filter: s2aj ¼

nw
1 a n þ w jj

ð68Þ

Where w is the width of the central positive part of the filter and n is the width of the negative wings. A measure of the goodness of fit is available through the reduced w2 value: w2n ¼

1 w2 ðn2
n1 þ 1Þ
m

ð69Þ

which is the w2 value of Eq. (62) divided by the number of points in the fit minus the number of reference spectra. A value close to 1 means a good fit, indicating that the reference spectra are capable of adequately describing the unknown spectrum. Copyright © 2002 Marcel Dekker, Inc.

Most of the merits and the disadvantages of the filter-fit method can be deduced directly from the mathematical derivation given in the preceding paragraphs. The most interesting aspect of the filter-fit method is that it does not require any mathematical model for the continuum and that, at least in principle, the shape of the peaks in the unknown spectrum are exactly represented by the reference spectra. Reference spectra should be acquired with good counting statistics, at least better than the unknown spectrum, because the least-squares method assumes that there is no error in the independent variables x0ji . Reference spectra can be obtained from single-element standards. Only the portion of the spectrum that contains peaks needs to be retained as reference in the fit. Multielement standards can be used if the peaks of each element are well separated. The reference spectra must provide an accurate model of the peak structure present in the unknown spectrum. This requires that reference and unknown spectra are acquired under strictly identical spectrometer conditions. Changes in resolution and, especially, energy shifts can cause large systematic errors. The magnitude of this error depends on the degree of peak overlap. Peak shifts of more than a few electron volts should be avoided, which is readily possible with modern detector electronics. If shifts are observed over long periods of operations of the spectrometer, careful recalibration of the spectrometer is required or, better, the reference spectra should be acquired again. Also, peak shift and peak broadening due to differences in count rate between standards and unknown must be avoided. Differential absorption is another problem that might influence the accuracy of the model. Because of the difference in x-ray attenuation in the reference and the unknown, the Kb to Ka ratios might be different in the two spectra. This becomes especially problematic if the Kb line is above and the Ka line below an absorption edge of a major element of the unknown sample. The magnitude of the error depends on the peak overlap. Careful selection of the samples to produce the reference spectra is therefore required. The procedure requires that a reference spectrum be included for each element present in the unknown. The method allows no mechanism to deal with sum peaks. Apart from removing the continuum, the filter also has some smoothing effect on the spectrum and causes the peak structure to be spread out over more channels. This is equivalent to fitting a spectrum with a somewhat lower resolution than originally acquired. Therefore, the precision and detection limits attainable with the filter-fit method are slightly worse than optimal. The width of the filter is important in this respect. Schamber (1977) suggests taking the width of the top of the filter equal to the FWHM resolution of the spectrometer, u ¼ FWHM. The width of the wings can be taken as v ¼ u=2. 2. Application The calculation procedure is quite simple and requires the following steps. The top-hat filter is applied to the unknown spectrum and the m reference spectra [Eqs. (30) and (31)], and the modified uncertainties are calculated using Eq. (63). Next, the vector b of length m and the m  m square matrix a are formed using Eqs. (65) and (66), summing over the part of the spectrum one wants to analyze (n1
n2 ). Only the relevant part, such as the Ka or the Ka plus the Kb, of the reference spectra needs to be retained; the rest of the filtered spectrum can be set to zero. After calculating the inverse matrix a
1 , the contribution of each reference to the unknown and its uncertainty are calculated using Eqs. (64) and (67) or (68). A computer implementation of the filter-fit method is given in Sec. X. The method was used to analyze part of an x-ray spectrum of a polished NIST SRM 1103 brass sample (Fig. 16A). The measurements were carried out using a Mo x-ray tube and a Zr secondary target and filter assembly. Spectra of pure metals (Fe, Ni, Cu, Zn, and Pb) were acquired Copyright © 2002 Marcel Dekker, Inc.

Figure 16 (A) Part of the x-ray spectrum of NIST SRM 1103 brass sample. (B) Top-hat-filtered spectrum and result of fit using reference spectra.

under identical experimental conditions. A top-hat filter of width w ¼ 5 was used. Table 6 shows how the spectra were divided in regions of interest to produce the reference spectra. Because considerable x-ray attenuation is present in brass, separate references were created for the Ka and Kb of Cu and Zn. This was not done for Fe, Ni, and Pb because these elements are only present as minor constituents in the brass sample. Figure 16B shows the filtered brass spectrum and the resulting fit using the seven (filtered) reference spectra. The region below CuKa is expanded 100 times, and the region above ZnKa is expanded 10 times. As can be seen, the agreement between the filtered brass spectrum and the fit is very good. The reduced w2 value is 8.5. This high value is probably due to small peaks shifts in the reference spectra compared to brass spectrum. Table 7 compares the results of the filter fit with the results obtained by nonlinear least-squares fitting using analytical functions (see Sec. VII). Although the w2 value of the nonlinear fit is slightly better (2.7), one observes an excellent agreement between the two methods for the analytical important data (i.e., intensity ratios). The uncertainties for small peaks are slightly higher with filter-fit method, as explained previously. The filter-fit method is fast and relatively easy to implement. It can produce reliable results when the spectrometer calibration can be kept constant within a few electronvolts and suitable standards for each element present in the sample are available. The method Copyright © 2002 Marcel Dekker, Inc.

Table 6 Data on the Reference Spectra and the Unknown Spectrum Used in the Filter-Fit Procedure (Fig. 16) Spectrum

Region of interest (keV)

Pure Fe Pure Ni Pure Cu

4.25
8.12 4.73
9.57 5.70
8.46 8.46
9.81 6.59
9.09 9.09
10.30 8.36
13.68 5.22
11.26

Pure Zn Pure Pb SRM 1103

Used as FeKa þ Kb reference NiKa þ Kb reference CuKa reference CuKb reference ZnKa reference ZnKb reference PbLa reference Unknown

Table 7 Comparison of Spectrum Evaluation Results Using Filter-Fit Method and Nonlinear Least-Squares Fitting; Ratios of the Intensity from the SRM 1130 Standard to the Pure Element Given Ratio of intensity in SRM 1104 to pure element Element Fe Ni Cu Zn Pb

Filter fit

Nonlinear fit

% Diff

0.0053±0.0002 0.0019±0.0002 0.546±0.001 0.390±0.001 0.0310±0.0006

0.0047±0.0001 0.00201±0.00008 0.551±0.001 0.3912±0.0008 0.0308±0.0003

11
6
0.9
0.31 0.65

performs well when one has to deal with a difficult to model continuum. If information on trace elements and major elements is required (very large peaks next to very small ones), the method might not be optimal. This filter-fit method is frequently used to process x-ray spectra obtained with electron microscopes (SEM-EDX), often in combination with a ZAF or Phi-Rho-Z correction procedure. B.

Partial Least-Squares Regression

Spectrum evaluation as discussed in most of this chapter aims to obtain the net intensity of the fluorescence lines. These net peak intensities are then used to determine analyte concentrations using one of the many empirical, semiempirical, or fundamental approaches detailed in various other chapters. In this, sub-section, we discuss an approach that is relatively new for the XRF community and is based on multivariate calibration. We will concentrate our discussion on partial least squares (PLS), a chemometrical technique that is extensively used in infrared and relates spectrometric methods. The basic idea of the method is to find a (multivariate linear) model that directly relates the spectral data to the analyte concentrations, thus avoiding the explicit evaluation of the spectrum in terms of the net peak areas. The method involves a calibration step where a large number of standards are used to build and validate the model and the actual analysis step where the model is applied to spectra of unknown samples. Copyright © 2002 Marcel Dekker, Inc.

1. Theory Multivariate spectroscopic calibration attempts to predicts properties of samples (concentration of analytes) based on measured spectral data via the following relation: Y ¼ XB þ F

ð70Þ

The Ynm matrix holds concentration of m analytes in the n samples. The Xnp matrix represents the spectral data, with n measured spectral data each having p x-ray intensities (channels) and Fnm is the matrix the residuals, part of the Y matrix not explained by the model. The regression coefficients B can be calculated in several ways. The most straightforward approach is the use of multiple linear regression (see Sec. IX). The leastsquares solution is given then given by B ¼ ðX0 XÞ
1 X0 Y

ð71Þ

When the number of x variables exceeds the number of samples ( p > n) and=or when there is a high degree of correlation between the x variables, the least-squares solution becomes unstable or cannot be obtained due to the fact that the covariance matrix X0 X cannot be inverted. This is exactly the fact when we attempt to apply multivariate calibration in XRF. The number of channels in the spectrum (1024 or more) largely exceeds the number of samples and the intensities of neighboring channels in each peak are very strongly correlated. To overcome this problem, the method of singular value decomposition can be used. The X matrix is decomposed in a number of linearly independent variables, know as principal components. In principle, there are as many principal components as there are original variables (channels), but only the most significant [i.e., having the highest variance (or eigenvalue)] are retained. Using the matrix of this smaller number of principal components scores, rather than the entire X matrix in Eq. (70), is known as principal components regression (PCR). The disadvantage of PCR is that the selection of the principal components is based on how much variance they explain of the X matrix. The first few principal components may have little relation with the concentrations that need to be predicted by the model. Partial least-squares regression (PLSR) is a variant of PCR that largely overcomes this problem. The methods uses two outer relations and one inner relation. The outer relations describe the decomposition of the X and Y matrices: X ¼ TP0 þ E ¼ Y ¼ UQ0 þ F ¼

A X a¼1 A X

ta p0a þ E

ð72Þ

ua q0a þ F

ð73Þ

a¼1

TnA and UnA are the score matrices, the values of the AðA pÞ latent variables. P0Ap and Q0Am are the loading matrices describing the relation between the latent variables (T and U) and the original variables (X and Y). The number of latent variables A in the model is of crucial importance and its optimum value must be found by cross-validation. The matrices E and F contain the residuals, part of the original spectral data and the concentration data, respectively, not accounted for when using A latent variables. The inner relation is written as: ua ¼ ba ta ;

a ¼ 1; . . . ; A

ð74Þ

from which the regression coefficients B can be obtained. This operation can be seen as a leastsquares fit between the X block and Y block scores. The final PLS model can thus be written as Copyright © 2002 Marcel Dekker, Inc.

Y ¼ TBQ0 þ F 

ð75Þ

A graphical representation of the PLS model is given in Figure 17. In the normal PLS algorithm Y is a vector of concentrations for one element and a separate model is build for each element. If all Y-variables are predicted simultaneously, as in the case of Equation (70), the PLS2 algorithm is used. This method performs better when the concentrations in the samples are highly correlated. The quality of the calibration model can be judged by the root mean square error of prediction (RMSEP): " #1=2 n 1X 2 RMSEP ¼ ð^ yi
yi Þ ð76Þ n i¼1 where y^i is the concentration in sample i predicted by the model yi is the true concentration in the standard. To determine the optimum number of latent variables, the RMSEP is calculated using PLS models with different numbers of latent variables A. The RMSEP values are plotted against A and the value where a minimum or a plateau is reached is taken. For small datasets, the calculation of the RMSEP is done using cross-validation. When a large number of standards are available, they are split in a training set (approximately two-thirds of the samples) and a prediction set (approximately one-third of the samples). The training set is used to build the PLS model and the RMSEP is calculated based on the concentrations of the prediction set. Alternatively, for smaller calibration sets, leave-one-out cross-validation can be used. Each sample is excluded once from the dataset and predicted by the model built with the remaining samples. This is repeated until all samples have been excluded once. Geladi and Kowalski (1986) published a tutorial on PLS and its relation to other regression techniques. A standard work on the PLS method is the book by Martens and

Figure 17

Graphical representation of the PLS algorithm.

Copyright © 2002 Marcel Dekker, Inc.

Naes (Martens and Naes, 1989). Theoretical and statistical aspects of the method can be found in chemometrics literature (Manne, 1987; Lorber et al., 1987; Pratar and De Jong, 1997; Hoskuldsson, 1988). 2. Application The PLS method is illustrated with the analysis of aqueous solution containing Ni, Cu, and As with concentration in the ranges 18
50, 0.5
45, and 5
20 g=L, respectively, whereas Zn, Fe, and Sb were present in more or less constant amounts. Spectra were acquired for 1000 s from 5-mL solution using an EDXRF spectrometer equipped with a Rh x-ray tube operating at 30 kV and a Si(Li) detector having 160 eV resolution at MnKa. From the 22 samples, 10 were used to build the PLS mode and the remaining were used as the prediction set. Table 8 gives the composition of the samples in the calibration set. Figure 18 shows part of the spectrum between 5 and 14 keV of sample number 9. The CuKa line is considerably interfered by the NiKb line. Absorption effects can be expected because the NiK absorption edge (8.33 keV) is just above the CuKa line (8.04 keV) and the AsK lines can cause secondary fluorescence of Cu and Ni. A PLS 1 model is build for Cu. The X matrix consists of 451 x-ray intensities (variables) between 5 and 14 keV (channels 250 to 700) of the 10 samples. The Y matrix contains the Cu concentration of those samples. In Figure 19, the RMSEP based on leave-one-out cross-validation is plotted as function of the number of latent variables. A minimum error of 0.57 g=L is obtained for three latent variables so that the PLS model with three latent variable is retained. Figure 20 compares the true Cu concentration with the predicted concentration for the calibration set. The PLS model predicts very accurately the Cu concentrations in the range from 1 to 40 g=L. The so-called regression coefficients of the PLS model show which variables (channels) are used in the model. They are plotted in Figure 21. As could be expected, the Cu concentrations are predicted from the channel content corresponding to the CuK lines. The influence of absorption and enhancement results in a small negative contribution from the Ni peak and a small positive contribution from the As peaks, respectively. The PLS model thus handles both the problem of spectral interference and matrix effects. The Cu concentration predicted by the PLS model for the test set is given in Table 9. Except for the two samples with the highest concentrations, the Cu concentration

Table 8

Composition of Samples Used to Build the PLS Model Concentration (g=L)

Sample No. 1 2 3 4 5 6 7 8 9 10

Ni

Cu

Zn

18.0 18.1 18.5 18.1 53.6 53.8 57.3 56.9 51.4 17.9

6.7 5.0 3.5 8.4 1.8 0.8 1.1 1.5 17.7 41.5

6.1 6.0 6.1 5.7 20.1 19.8 19.3 20.0 20.9 5.3

Copyright © 2002 Marcel Dekker, Inc.

Figure 18

X-ray spectrum of sample number 9 in the calibration set used to build the PLS model.

Figure 19 RMSEP values versus the number of latent variables for the prediction of the Cu concentration.

in the ‘‘unknown’’ samples is very well estimated. The predicted of the test set is generally somewhat worse that the prediction of the calibration set. To build an accurate model, a large number of standards spanning the concentration range of interest for each element is required. This is certainly the major drawback of PLS for its application in XRF, Copyright © 2002 Marcel Dekker, Inc.

Figure 20 Comparison of ‘‘true’’ and predicted Cu concentrations for the samples in the PLS calibration set.

Figure 21 Regression coefficients of the PLS model for Cu, showing which variables (channels in the x-ray spectrum) are used to predict Cu.

Copyright © 2002 Marcel Dekker, Inc.

Table 9

Comparison of ‘‘True’’ and Predicted Cu Concentration Using the PLS Model Cu concentration (g=L)

Sample No. 11 12 13 14 15 16 17 18 19 20 21 22

‘‘True’’

Predicted

Difference

6.30 7.20 6.30 7.50 6.30 0.90 1.00 1.20 1.80 2.50 17.60 44.70

6.38 7.34 6.32 7.40 6.09 1.27 0.75 1.55 2.11 2.86 16.51 40.59

0.08 0.14 0.02
0.10
0.21 0.37
0.25 0.35 0.31 0.36
1.09
4.11

especially when solid samples are considered. This problem can to some extend be overcome by building a calibration set via Monte Carlo simulation. Just as with the filterfit method, standards and unknowns need to be measured under strictly identical spectrometer conditions. Changes in gain or resolution will cause systematic errors in the calculated concentrations. Swerts and Van Espen (1993) demonstrated the use of PLS for the determination of S in graphite using a Rh x-ray tube excitation XRF equipped with a Si(Li)detector. Because of diffraction effects, least-squares fitting of the spectra was nearly impossible. Using PLS, the sulfur content could be determined in a concentration range 2
60%, with an accuracy of better than 5% relative standard deviation. Urba´nski and Kowalska applied the PLS method to the determination of Sr and Fe in powdered rock samples and to the determination of the S and ash content in coal using a radioisotope XRF system equipped with a low-resolution gas proportional counter. They also demonstrated the usefulness of this method for the determination of the thickness and composition of Sn-Pb and Ni-Fe layered structures (Urba´nski and Kowalska, 1995). Molt and Schramm (1987) compared principal components regression (PCR), PLS for the determination of S, exhibiting strong interference from Mo, in aqueous and solid samples. The results were also compared with quantitative analysis using the method developed by Lucas-Tooth and Price (1961). Equally good results were obtained with all three methods. Similar results were obtained by Lemberge and Van Espen for the determination of Ni, Cu, and As in liquid samples. They demonstrated that taking the square root of the data improves the PLS model and that the PLS method extracts information from the scattered excitation radiation to describe the matrix effects (Lemberge and Van Espen, 1999).

VII.

LEAST-SQUARES FITTING USING ANALYTICAL FUNCTIONS

A widely used and certainly the most flexible procedure for evaluating complex x-ray spectra is based on least-squares fitting of the spectral data with an analytical function. The method is conceptually simple, but not trivial to implement and use. Copyright © 2002 Marcel Dekker, Inc.

A.

Concept

In this method, an algebraic function, including analytical importantly parameters such as the net areas of the fluorescence lines, is used as a model for the measured spectrum. The object function (w2) is defined as the weighted sum of squares of the differences between this model y(i) and the measured spectrum yi over a region of the spectrum: n2 X 1 w2 ¼ ½ yi
yði; a1 ; . . . ; am Þ2 ð77Þ 2 s i i¼n1 where s2i is the variance of data point i, normally taken as s2i ¼ yi , and aj are the parameters of the model. The optimum values of the parameters are those for which w2 is minimal. They can be found by setting the partial derivatives of w2 with respect to the parameters to zero: @w2 ¼ 0; @aj

j ¼ 1; . . . ; m

ð78Þ

If the model is linear in all the parameters aj, these equations result in set of m linear equations in the m unknowns aj, which can be solved algebraically. This is known as linear least-squares fitting. If the model is nonlinear in one or more of its parameters, a direct solution is not possible and the optimum value of the parameters must be found iteratively. An initial value is given to the parameters and they are varied in some way until a minimum for w2 is obtained. The latter is equivalent to searching for a minimum in the m þ 1-dimensional w2 hypersurface. This is known as nonlinear least-squares fitting. The selection of a suitable minimization algorithm is very important because it determines to a large extent the performance of the method. A detailed discussion of linear and nonlinear least-squares fitting is given in Sec. IX. The most difficult problem to solve when applying this least-squares procedure is the construction an analytical function that accurately describes the observed spectrum. The model must be capable of describing accurately the spectral data in the fitting region. This requires an appropriate model for the continuum, the characteristic lines of the elements and for all other features present in the spectrum such as absorption edges, escape, and sum peaks. Although the response function of the energy-dispersive detector is, to a very good approximation, Gaussian, deviation from the Gaussian shape needs to be taken into account. Failure to construct an accurate model will result in systematic errors, which under certain conditions may lead to gross positive or negative errors in the estimated peak areas. On the other hand, the fitting function should remain simple, with as few parameters as possible. Especially for the nonlinear fitting, the location of the w2 minimum becomes problematic when a large number of parameters is involved. In general, the fitting model consists of two parts: X yP ðiÞ ð79Þ yðiÞ ¼ yB ðiÞ þ P

where y(i) is calculated content of channel i and the first part describes the continuum and the second part the contributions of all peaklike structures. Because the fitting functions for both linear and nonlinear least-squares fitting have many features in common, we treat the detailed description of the fitting function for the most general case of nonlinear least squares. Moreover, if the least-squares fit is done using the Marquardt algorithm, the linear least-squares fit is computationally a particular case of the nonlinear least-squares fit. Programs based on this algorithm can perform Copyright © 2002 Marcel Dekker, Inc.

linear and nonlinear fitting using the same computer code. A large part of the discussion given here is based on the computer code AXIL, developed by the author for spectrum fitting of photon-, electron-, and particle-induced x-ray spectra (Van Espen et al., 1977a, 1977b, 1979b, 1986). B.

Description of the Continuum

To model the continuum, various analytical expressions are in use, depending on the excitation conditions and on the width of the fitting region. Except for electron microscopy, it is virtually impossible to construct an acceptable physical model that describes the continuum, mainly because of the large number of processes that contribute to it. For this reason, very often some type of polynomial expression is used. 1. Linear Polynomial A linear polynomial of the type yB ðiÞ ¼ a0 þ a1 ðEi
E0 Þ þ a2 ðEi
E0 Þ2 þ    þ ak ðEi
E0 Þk

ð80Þ

is useful to describe the continuum over a region 2
3 keV wide. Wider regions often exhibit too much curvature to be described by this type of polynominal. In Eq. (80), Ei is the energy (in keV) of channel i [see Eq. (84)] and E0 is a suitable reference energy, often the middle of the fitting region. Expressing the polynomial as a functin of (Ei 7 E0) rather than as a function of the channel number is done for computational reasons; (Ei 7 E0)3 is, at most, of the order of 103, whereas i3 can be as high as 109. Most computer programs that implement a polynomial model for the continuum allow the user to specify the degree of the polynomial; k ¼ 0, 1, and 2 produce respectively a constant, a straight line, and a parabolic continuum. Values of k larger than 4 are rarely useful because such high-degree polynomials tend to have physical nonrealistic oscillations. Equation (80) is linear in the fitting parameters a0, . . . , ak, so that this function can be used in linear as well as in nonlinear least-squares fitting. 2. Exponential Polynomial A linear polynomial cannot be used to fit the continuum over the entire spectrum or to fit regions of high positive or negative curvature. Higher curvature can be modeled by functions of the type yB ðiÞ ¼ a0 exp½a1 ðEi
E0 Þ þ a1 ðEi
E0 Þ2 þ    þ ak ðEi
E0 Þk 

ð81Þ

where k is the degree of the exponential polynomial. A value of k as high as 6 might be required for an accurate description of a continuum from 2 to 16 keV. This function is nonlinear in the fitting parameters a1, . . . , ak and requires a nonlinear least-squares procedure and some initial guess of these parameters. Initial values for these nonlinear parameters can be determined by first estimating the shape of the continuum using one of the procedures described in Sec. IV followed by a linear fit of the logarithm of this continuum. These initial guesses are then further refined in the nonlinear fitting procedure. 3. Bremsstrahlung Continuum The exponential polynomial is not suitable for describing the shape of the continuum observed in electron- and particle-induced x-ray spectra, mainly because of the high Copyright © 2002 Marcel Dekker, Inc.

curvature at the low-energy region of the spectrum. The continuum results from photons emitted from the sample by the retardation of fast electrons. The slope of the emitted continuum is essentially an exponentially decreasing function according to Kramer’s formula. At low energies, the emitted photons are strongly absorbed by the detector windows and by the sample. A suitable function to describe such a radiative continuum is an exponential polynominal multiplied by the absorption characteristics of the spectrometer: yB ðiÞ ¼ a0 exp½a1 ðEi
E0 Þ þ    þ ak ðEi
E0 Þk Ta ðEi Þ

ð82Þ

A detailed discussion of the function Ta(E) is given on page 288. To be physically correct, the absorption term must be convoluted with the detector response function, because the sharp edges due to absorption by detector windows (Si or Au) or elements present in the sample are smeared out by the finite resolution of the detector. 4. Continuum Removal An alternative to an algebraic description of the continuum is to estimate the continuum first using one of the procedures outlined in Sec. IV and to substract this continuum from the measured spectrum before the actual least-squares fitting. To correctly implement the least-squares fit after subtraction of the continuum, the weights 1=s2i [Eq. (77)] must be adjusted. If y0i represents the spectral data after subtraction of the continuum, y0i ¼ yi
yB ðiÞ, the variance of y0i is given by s0i 2 ¼ s2i þ s2yB ðiÞ . A reasonable approximation for s2yB ðiÞ is yB ðiÞ itself, so that the total variance becomes s0i 2 ¼ yi þ yB ðiÞ. If this adjustment of the weights is not made, the uncertainty in the net peak areas are understimated, especially for small peaks on a high continuum. It is rather difficult for an inexperienced user to select the appropriate continuum model for a given spectrum. The following might serve as a general guideline. For fitting regions 2
3 keV wide, a linear polynomial continuum is often adequate. To fit large regions of XRF spectra, the exponential polynomial provides the most accurate results, with k typically, between 4 and 6. The same holds for the bremsstrahlung continuum of SEMEDX and PIXE spectra. The simplest method from the users’ point of view is the continuum stripping, but this method does not provide optimum results. A slight underestimation or overestimation might occur, resulting in large relative errors in the area determination of small peaks (Vekemans et al., 1995). C.

Description of Fluorescence Lines

Because the response function of most solid-state detectors is predominantly Gaussian, all mathematical expressions used to describe the fluorescence lines involve this function. When dealing with the K lines of high-atomic-number elements, such as Pb or U, the influence of the natural line shape becomes appreciable and the use of the more complicated Voigt profile is required (Wilkinsin, 1971; Gunnink, 1977; Pessara and Debertin, 1981). 1. Single Gaussian A Gaussian peak is characterized by three parameters: the position, width, and height or area. It is desirable to describe the peak in terms of its area rather than its height because the area is directly related to the number of x-ray photons detected, whereas the height depends on the spectrometer resolution. The first approximation to the profile of a single peak is then given by Copyright © 2002 Marcel Dekker, Inc.

" # A ðxi
mÞ2 pffiffiffiffiffiffi exp
2s2 s 2p

ð83Þ

where A is the peak area (counts), s is the width of the Gaussian expressed in channels, and m isp the location of the peak maximum. The often used FWHM is related to s by the ffiffiffiffiffiffiffiffiffiffi ffi factor 2 2 ln 2 or FWHM ¼ 2.35s. In Eq. (83), the peak area is a linear parameter; the width and position are nonlinear parameters. This implies that a nonlinear least-squares procedure is required to find optimum values for the latter two parameters. Using a linear least-squares method assumes that the position and width of the peak are know with high accuracy from calibration. To describe part of a measured spectrum, the fitting function must contain a number of such functions, one for each peak. For 10 elements and 2 peaks (Ka and Kb) per element, we would need to optimize 60 parameters. It is highly unlikely that such a nonlinear leastsquares fit will terminate successfully at the global minimum. To overcome this problem, the fitting function can be written in a different way as shown in the next subsection. 2. Energy and Resolution Calibration Function The first obvious step is to drop the idea of optimizing the position and width of each peak independently. In x-ray spectrometry, the energies of the fluorescence lines are know with an accuracy of 1 eV or better. The pattern of peaks observed in the spectrum is directly related to elements present in the sample. Based on those elements, we can predict all x-ray lines that constitute the spectrum and their energies. The peak function [Eq. (83)] is therefore written in terms of energy rather than channel number. Defining ZERO as the energy of channel 0 and expressing the spectrum GAIN in electronvolts=channel, the energy of channel i is given by EðiÞ ¼ ZERO þ ðGAINÞi and the Gaussian peak can be written as " # GAIN ½Ej
EðiÞ2 Gði; Ej Þ ¼ pffiffiffiffiffiffi exp
2s2 s 2p with Ej the energy (in eV) of the x-ray line and s the peak width given by
 NOISE 2 2 s ¼ þ3:58ðFANOÞEj 2:3548

ð84Þ

ð85Þ

ð86Þ

In this equation, NOISE is the electronic contribution to the peak width (typically 80
100 eV FWHM) with the factor 2.3548 to convert to s units, FANO is the Fano factor ( 0.114), and 3.85 pffiffiffiffiffi ffi is the energy required to produce an electron
hole pair in silicon. The term GAIN=s 2p in Eq. (85) is required to normalize the Gaussian so that the sum over all channels is unity. For linear least-squares fitting, ZERO, GAIN, NOISE, and FANO are physically meaningful constants. In the case of nonlinear least squares, they are parameters to be refined during the fitting. The advantage of optimizing the energy and resolution calibration parameters rather than the position and width of each peak is a vast reduction of the dimensionality of the problem. The nonlinear fit of 10 peaks would now involve 14 parameters compared to 30. Even more importantly, all information available in the spectrum is used to estimate ZERO, GAIN, NOISE, and FANO and thus the positions and the widths Copyright © 2002 Marcel Dekker, Inc.

of all peaks. Imagine a small, severely overlapping doublet with a well-defined peak on both sides of this doublet. These two peaks will contribute most to the determination of the four calibration parameters, virtually fixing the position and the width of the two peaks in the doublet. As a consequence, their areas can be determined much more accurately. Referring to our discussion on information content in Sec. II, we did not obtain this extra performance for nothing. We have supplied extra information: the energy of the peaks and the two calibration relations [Eqs. (84) and (86)]. Fitting with Eq. (85) requires that the extra information we supply is indeed correct. With modern electronics, the linearity of the energy calibration [Eq. (84)] holds very well in regions above 2 keV. Fitting the entire spectrum, including the low-energy region, might require more complex energy calibration functions. To fit PIXE spectra from 1 to 30 keV, Maenhaut and Vandenhaut (Maenhaut and Vandenhaut, 1986) suggested the following function: i ¼ C1 þ C2 E þ C3 expð
C4 EÞ. The relation between the square of the peak width and the energy [Eq. (86)] is based on theoretical considerations. The relation holds very well if the doublet splitting of the x-ray lines is taken into account. The Ka1
Ka2 separation increases from a negligible value for Ca (3.5 eV) to nearly 100 eV for Mo. The observed peak shape of a K line is actually an envelope of two peaks. This envelope can be represented rather well by a single Gaussian, but failing to take this doublet splitting into account (i.e., fitting with a single Gaussian where doublets are required) will result in peak widths that do not obey Eq. (86). To illustrate this, the observed width of a number of Ka lines as function of the x-ray energy is presented in Figure 22. The dotted line represents the width of the Ka doublet fitted as one peak. The solid (straight) line shows the width of the individual lines in the doublet. 3. Response Function for an Element A second modification to the fitting function that will reduce the number of fitting parameters is modeling an entire element, rather than single peaks. A number of lines can be considered as logically belonging together, such the Ka1 and Ka2 of the above-mentioned doublets or all K lines of an element. This group can be fitted with one area parameter A representing the total number of counts of all the lines in the group. The spectrum of an element can then be represented by yP ðiÞ ¼ A

NP X

Rj Gði; Ej Þ

ð87Þ

j¼1

where G are the Gaussians for the various lines with energy Ej and Rj the relative inP Rj ¼ 1. tensities of the lines. The summation runs over all lines in the group (NP) with The transition probabilities of all lines originating from a vacancy in the same (sub) shell (K, LI, LII, . . .) are constants, independent of the excitation. However, the relative intensities depend on the absorption in the sample and in the detector windows. To take this into account, the x-ray attenuation must be included in Eq. (87). The relative intensity ratios are obtained by multiplying the transition probabilities with an absorption correction term: Rj Ta ðEnj Þ R0j ¼ P j Rj Ta ðEnj Þ

ð88Þ

Contributions from various subgroups (i.e., between K and L, between LI and LII, etc.) depend on the type of excitation (photons, electrons, protons) and on the excitation energy. General values cannot be given and must be determined for the particular excitation Copyright © 2002 Marcel Dekker, Inc.

Figure 22

FWHM of various Ka lines, fitted as single peak and as a Ka1
Ka2 doublet.

if one wants to combine lines of different subgroups. Transition probabilities of lines in various groups, determined experimentally and calculated from first principles, can be found in the literature (McCrary et al., 1971; Salem et al., 1970; Salem and Wimmer, 1970; Scofield 1970, 1974a, 1974b). In Figure 23, the fit of a tungsten L line spectrum using 24 transitions from the 3 L subshells is shown. The relative intensities of the L lines within each subshell were taken from the literature (Scofield, 1974b). The fit was thus done with one peak-area parameter for the L1, the L2 and the L3 sublevels. Fitting elements rather than individual peaks enhances the capability of the method to resolve overlapping peaks. The area of the CrKa peak, interfered by a VKb peak, can be obtained with higher precision (lower standard deviation) because the area of the VKb peak is related to the area of the (interference free) VKa peak. Again, we have introduced more a priori knowledge into our model. If this information (the relative intensity ratio) is not correct, we will introduce systematic errors (e.g., the CrKa peak area, although having a low standard deviation, will not be correct). In practice, there will be a trade-off between the gain in accuracy and the gain in precision. Errors in the values of the transition probabilities and in the value of the absorption correction term are sufficiently small so that small to moderate high peaks (up to 105 counts peak area) can be fitted as one group. 4. Modified Gaussians When fitting very large peaks with a Gaussian, the deviation from the pure Gaussian shape becomes significant. In Figure 24, a MnK spectrum with 107 counts in the MnKa peak is shown. One observes a tailing on the low-energy side of the peaks and a continuum that is Copyright © 2002 Marcel Dekker, Inc.

Figure 23 Fit of a complex L line spectrum of tungsten. In total, 24 transitions, divided over the 3 L subshells, are required for the description on the spectrum.

Figure 24 MnK line spectrum with very good counting statistics. The difference from the Gaussian response is obtained by subtracting all Gaussian peaks. From this, the peak-shape correction is calculated.

Copyright © 2002 Marcel Dekker, Inc.

higher at the low energy side of the Ka peak than at the high energy side of the Kb peak. The observed peak shape is partially caused by nonideal behavior of the detector. For lowenergy lines ( < 10 keV), incomplete charge collection and other detector artifacts play an important role. For higher energetic lines, Compton scattering in the detector also contributes. Another part of the deviation from the Gaussian shape is attributed to phenomena taking place in the sample itself. X-ray satellite transitions such as KLM radiative Auger transitions on the low-energy side of the Ka and peak and KMM transitions on the low-energy side of the Kb contribute to the overall shape of the spectrum. In the literature, considerable attention has been given to the explanation and to the accurate mathematical description of the observed peak shape (McNelles and Compbell, 1975; Van Espen et al., 1977b; Wielopolski and Gardner, 1979; Campbell et al., 1985, 1987, 1997, 1998, Gardner et al., 1986; Yacout et al., 1986; Campbell, 1996). Failure to account for the deviation from the Gaussian peak shape causes a number of problems when fitting x-ray spectra. Small peaks sitting on the tail of large ones (e.g., NiKa in front of CuKa) cannot be fitted accurately, resulting in large systematic errors for the small peak. Because the least-squares method seeks to minimize the difference between the observed spectrum and the fitted function, the tail might become filled up with peaks of elements that are not really present. Also, the continuum over the entire range of the spectrum becomes difficult to describe. This is illustrated in Figure 25a, where a spectrum

Figure 25 Fit of the spectrum of a brass sample (NIST SRM 1106) (a) fitted with a simple Gaussians and (b) fitted with Gaussians including tail and step functions.

Copyright © 2002 Marcel Dekker, Inc.

of a brass sample (NIST SRM 1106) is fitted with simple Gaussians and a linear continuum. An unrealistically high area for the NiKa peak is obtained in this way. A number of analytical functions have been proposed to account for the true line shape. Nearly all of them include a flat shelf and an exponential tail, both convoluted with the Gaussian response function. Similar functions are also used to describe the peak shape of g-ray spectra (Phillips and Marlow, 1976). To account for the deviation from the Gaussian peak shape, the Gauss function G(i, Ej) in Eq. (85) can be replaced by Pði; Ejk Þ ¼ Gði; Ejk Þ þ fS Sði; Ejk Þ þ fT Tði; Ejk Þ

ð89Þ

where G(i, Ej) is the Gaussian part given by Eq. (85) and S and T are the step and tail functions, respectively,   GAIN EðiÞ
Ejk pffiffiffi ð90Þ Sði; Ejk Þ ¼ erfc 2Ejk s 2     GAIN EðiÞ
Ejk EðiÞ
Ejk 1 pffiffiffi þ pffiffiffi exp Tði; Ejk Þ ¼ ð91Þ erfc 2 2gs exp½
1=ð2g Þ gs g 2 s 2 In these equations, s represents spectrometer resolution [Eq. (86)] and g is the broadening of the exponential tail. The parameters fS and fT in Eq. (89) describe the fraction of the photons that arrive in the step and the tail, respectively. The complement of the error function used to convolute the step and the tail is defined as Z u 2 2 e
t dt ð92Þ erfcðuÞ ¼ 1
erfðuÞ ¼ 1
pffiffiffi p 0 and can be calculated via series expansion (Press et al., 1988). Figure 25b shows a fit of a brass spectrum when these step and tail functions are included. In this case, a much more realistic area for the NiKa is obtained and the continuum is described more correctly. Campbell et al. (1987) used a Gaussian, a short-tail, and a long-tail exponential to fit the peak in energy-dispersive spectra obtained from Ka1 and La1 lines selected by Bragg reflection from a curved crystal. In this way, the influence of the doublet structure and the satellite lines is eliminated. Excellent fits with reduces chi-square values between 1.02 and 1.16 were obtained for peaks having 106 counts. Fitting real x-ray spectra with modified Gaussians as given by Eq. (89) increases dramatically the number of parameters that need to be optimized ( fS, fT, g) for each peak. The phenomena contributing to the deviation from the Gaussian shape depend on the energy of the detected x-ray line, so that these parameters can be expressed as smooth functions of energy (Wielopolski and Gardner, 1979; Yacout et al., 1986). These relations can be established using pure element spectra with very good counting statistics. An additional problem is that the tailing of the Kb peaks tend to be larger due to the more intense KMM radiative transition, thus requiring a separate relation for Ka and Kb tails. A further modification of the fitting function involves the use of a Voigt profile to account for the natural line width of the x-ray lines. This becomes important when fitting K lines of higher-Z elements (Ba and above) where the natural linewidth becomes considerable ( > 20 eV) compared to the detector resolution. The peak profile, Gði; Ej Þ in Eq. (89) must then be replaced by a Voigt profile (the convolution of a Lorenzian and a Gaussian):
 GAIN EðiÞ
Ejk aL pffiffiffiffiffiffi K pffiffiffi ; pffiffiffi ð93Þ s 2p s 2 2s 2 with K the real part of the complex error function and aL the natural width of the x-ray line: Copyright © 2002 Marcel Dekker, Inc.

Kðx; yÞ ¼ Rebe
z erfcð
izÞc; 2

z ¼ x þ iy

ð94Þ

Voigt profiles can be calculated using numerical approximations (Schreier, 1992). The imaginary part of the complex error function can be used to calculate the derivatives necessary in the least-squares-fitting algorithm. An alternative procedure to describe the peak shape is used in the AXIL program (Van Espen et al., 1977b). The deviation from the Gaussian shape is stored as a table on numerical values, representing the difference of the observed shape and the pure Gaussian. The table extends from zero energy up to the high-energy side of the Kb peak and is normalized to the area of the Ka peak. The deviation is obtained from pure element spectra having very good counting statistics. (Ka 107 counts). Preferably, thin films are used to keep the continuum as low as possible and to avoid absorption effects. The area, position, and width of all peaks in the spectrum are determined by fitting Gaussians on a constant continuum over the full width at tenth maximum (FWTM) of the peaks. The Gaussian contributions are then stripped from the spectrum. The resulting non-Gaussian part as shown in Figure 24 is further smoothed to reduce the channel to channel fluctuations and subsequently used as a numerical peak-shape correction. The fitting function for an element is then given by ( ) NP X 0 0 Rj Gði; Ej Þ ð95Þ yP ðiÞ ¼ A RKa ½Gði; EKa Þ þ CðiÞ þ j¼2

where CðiÞ is the numerical peak-shape correction at channel i. Values in the table are interpolated to account for the difference between the energy scale of the correction and the actual energy calibration of the spectrum. Similar to the parameters of the nonGaussian analytical functions, the shape of the numeric correction seems to vary slowly from one element to another. This allows us to interpolate the peak-shape correction for all elements from a limited set of experimentally determined corrections. A major disadvantage of this method is that it is quite difficult and laborious to obtain good experimental peak-shape corrections. Although they are, in principle, detector dependent, experience has proven that the same set of corrections can be used for different detectors with reasonable success, proving the fundamental nature of the observed non-Gaussian shape. Another disadvantage is that the peak-shape correction for the Kß becomes underestimated if strong differential absorption takes place because the peak-shape correction is only related to the area of the Ka peak. Also, it is nearly impossible to apply this method to the description of L-line spectra. A mayor advantage however is the computational simplicity of the method and the fact that no extra parameters are required in the model. 5. Absorption Correction The absorption correction term Ta ðEÞ, used in the Eqs. (82) and (88), includes the x-ray attenuation in all layers and windows between the sample surface and the active area of the detector. For high energetic photons also, the transparency of the detector crystal needs to be taken into account. In x-ray fluorescence, the attenuation in the sample, causing additional changes in the relative intensities, can also be considered, providing the sample composition is known. The total correction term is thus composed of a number of contributions: Ta ðEÞ ¼ TDet þ TPath þ TSample The detector contribution for a Si(Li) detector is given by Copyright © 2002 Marcel Dekker, Inc.

ð96Þ

TDet ðEÞ ¼ e
mBe ðrdÞBe e
mAu ðrdÞAu e
mSi ðrdÞSi ð1
e
mSi ðrDÞSi Þ

ð97Þ

where m, r, and d are the mass-attenuation coefficient, the density, and the thickness of the Be window, the gold contact layer, and the silicon dead layer. In the last term, D is the thickness of the detector crystal. Any absorption in the path between the sample and the detector can be modeled in a similar way. For an air path, the absorption is given by TPath ðEÞ ¼ e
mair ðrdÞair

ð98Þ

The mass-attenuation coefficient of air can be calculated assuming a composition of 79.8% N2, 19.9% O2, and 0.3% Ar. The density of air in standard conditions is 1.261073 g=cm3. In PIXE setups, it is common practice to use additional absorbers between the sample and the detector, with one absorber sometimes having a hole (funny filter). The absorption behavior of such a structure can be modeled by TPath ðEÞ ¼ e
mf ðrdÞf ½h þ ð1
hÞe
mff ðrdÞff 

ð99Þ

where the first term accounts for the absorption in (all) solid filters and the second term accounts for the absorption of the filter having a hole. The fraction of the detector solid angle subtended by the hole is denoted by h. The sample absorption correction in the case of x-ray fluorescence, assuming excitation with an (equivalent) energy E0 , is given by Tsample ðEÞ ¼

1
e
ws ðrdÞs ws ðrdÞs

ð100Þ

and the sample attenuation coefficient is given by ws ¼

ms ðEÞ ms ðE0 Þ þ sin y1 sin y2

ð101Þ

This sample attenuation coefficient can only be calculated if the weight fractions of the constituting elements are known, which is often not the case because the aim of the spectrum evaluation is to obtain the net peak areas from which the concentrations are to be calculated. Although x-ray attenuation in solid samples might become very large, it is important to realize that not the absolute values of the absorption corrections but their ratio is of importance in Eq. (88). This ratio changes less dramatically especially because the energy difference of the lines involved is small. For an infinitely thin Fe sample, the Kb=Ka intensity ratio is 0.134. In an infinitely thick Fe matrix, the absorption correction term Ta ðEKb Þ and Ta ðEKa Þ are respectively 161 and 134, assuming MoKa excitation, causing the Kb=Ka ratio to change to 0.161. Therefore, a rough estimate of the sample composition is often sufficient. Van Dyck and Van Grieken (1983) demonstrated the integration of spectrum analysis with quantitative analysis. The sample absorption term is recalculated based on the estimated sample composition, and the spectrum fitting is repeated. 6. Sum and Escape Peaks As indicated in Sec. II, Si escape peaks can be modeled by a Gaussian with an energy 1.750 keV below the parent peak. The area relative to the area of the parent peak can be calculated from the escape fraction [Eq. (4)]: Copyright © 2002 Marcel Dekker, Inc.

Z¼

Ne f ¼ Np 1
f

ð102Þ

Including the escape peaks, the description of the fluorescence of an element becomes yP ðiÞ ¼ A

NP X

R0j ½Gði; Ej Þ þ ZGðði; Ej
1:750Þ

ð103Þ

J¼1

Also, various polynomial type functions expressing the escape ratio as a function of the energy of the parent peak are in use. The coefficients of the function can be determined by least-squares fitting from experimental escape ratios. For spectra obtained with a Ge detector, one needs to account in a similar way for both the GeKa and GeKb escape peak for elements above arsenic. The incorporation of the sum peaks in the fitting model is more complex. The method discussed below was first implemented by Johansson in the HEX program (Johansson, 1982). Initially, the spectrum is fitted without considering pileup peaks. The peaks are then sorted according to their height and the n largest peaks are retained. Peaks that differs less than 50 eV are combined to one peak. Using Eqs. (5) and (6), the relative intensities of all possible nðn þ 1Þ=2 pileup peaks and their energies are calculated and the m most intense are retained. Knowing the relative intensities and the energies of these m pileup peaks, they can be included in the fitting model as one ‘‘pileup element.’’ In the next iteration, the total peak area A of this pileup element is obtained. The construction of the pileup element can be repeated during the next iterations as more reliable peak areas become available. In Figure 26, part of an PIXE spectrum is shown fitted with and without sum peaks included. D.

Special Aspects of Least-Squares Fitting

1. Constraints When using nonlinear least-squares fitting, it can be advantages to impose limits on the fitting parameters to eliminate physically meaningless results. Some illustration of what can happen if too much freedom is given to the fitting parameters is provided by Statham (1978). The incorporation of the energy and resolution calibration functions [Eqs. (84) and (86)] in the fitting model already places a severe constraint on the fit, avoiding the situation in which two peaks that are very close together swapping their positions or becoming one broader peak. The fitting parameter can be effectively constrained by defining the real physically meaningful parameter Pj [e.g., the GAIN in Eq. (84)] as an arctangent function of the fitting parameter aj (McCarthy and Schamber, 1981; Nullens et al., 1979): 2Lj arctan aj ð104Þ Pj ¼ P0j þ p where P0j is the expected value (initial guess) of the parameter Pj , and Lj defines the range. As a result of this transformation, the parameter Pj will always be in the range P0j  Lj . Apart from significantly adding to the mathematical complexity, such transformation has the disadvantage that the w2 minimum, although lying within the selected interval, cannot be reached because the path toward the minimum passes a ‘‘forbidden’’ region. Also, this constraint makes no distinction between the more probable values of the parameter lying near the center of the interval and the unlikely values at the limits. Copyright © 2002 Marcel Dekker, Inc.

Figure 26 Fit of part of a PIXE spectrum having very high count rates for Fe and Zn. (Top: without considering sum peaks; bottom: sum peaks included in the fitting model.)

An alternative approach, proposed by Nullens et al. (1979), relies on the modification of the curvature of the w2 surface. The fitting parameters, such as ZERO, GAIN, NOISE, and FANO, are considered as random variables with an expected value equal to the initial guess a0j and having an uncertainty Daj . They can be included in the expression of w2 , just as the observed data points yi  si : X 1 X 1 w2 ¼ ½ yi
yðiÞ2 þ ½a0
aj 2 ð105Þ 2 2 j s Þ ðDa i j j i Using this expression in the Marquardt nonlinear least-squares-fitting algorithm results in modified equations for the diagonal terms of the a matrix and for the b vector (see Sec. IX): X 1
@y0 ðiÞ2 1 þ ð106Þ ajj ¼ 2 @a si j ðDaj Þ2 i and bj ¼

X 1 @y0 ðiÞ a0j
aj ½ y
y ðiÞ þ i 0 @aj s2i ðDaj Þ2 i

ð107Þ

If no significant peaks are present in the fitting interval, the derivatives of the fitting function with respect to the position and width of the peaks are zero. Therefore, the Copyright © 2002 Marcel Dekker, Inc.

second term in bj will dominate, causing the parameter estimate in the next iteration to be such that aj tends toward a0j (the initial value). If well-defined peaks are present, however, the second term in b will be negligibly small compared to the first term and the method acts as if no constraints were present. Another way to look at this is by considering the curvature of the w2 surface near the minimum. Figure 27 shows a cross section through the w2 surface along the peak position parameter. The variation of the w2 values, with and without constraints, are shown for a large (10,000 counts) and a small (100 counts) peak. The true peak position is at channel 100 and the FWHM of the peak is eight channels. The constraint, Da, on the peak position is one channel. From Figure 27, it is evident that the w2 minimum is much better defined for a small peak when the constraint is applied, whereas this has no influence for a large peak. This method has been implemented in the AXIL program to constraint the energy and resolution calibration parameters. 2. Weighting the Fit The weights wi used in the least-square methods [Eq. (77)] are defined in terms of the ‘‘true’’ (population) standard deviation of the data yi , which can be obtained directly from the data itself: wi ¼

1 1  2 yi si

ð108Þ

This approximation is valid for moderate to good statistics. When regions of the spectrum with very bad statistics (low channel contents) are fitted, estimating the weight from the measured channel content causes a systematic bias leading to an underestimate of the peak areas. To overcome this problem, Phillips suggested estimating the weights from the

Figure 27 Effect of the use of a constraint on the shape of the w2 response surface. (Left: marginal effect for a large peak; right: important contribution for a small peak.)

Copyright © 2002 Marcel Dekker, Inc.

average of three channels (Phillips, 1978). In the AXIL program, the weights are initially based on the measured channel content, but when the overall w2 value of the fit falls below 3, the weights are calculated from the fitted channel content. This is based on the idea that when the calculated spectrum approached the measured spectrum, the calculated channel contents are a better estimate of the ‘‘true’’ channel content than the measured values. The effect of the weighting on the fit is considerable, as can be seen in Figure 28. E.

Examples

To illustrate the working of the nonlinear least-squares-fitting method, an artificial spectrum with CuK and ZnK lines is fitted with four Gaussian peaks on a constant continuum. Using the Marquardt algorithm, the position, width, and area of each peak are determined. The fitting function thus is " # 4 X ði
ajþ1 Þ2 aj exp
ð109Þ yðiÞ ¼ a0 i þ 2a2jþ2 j¼1 with i the channel number (independent variable) and aj the parameters to be determined, 13 in total. In Table 10, the values of the parameters to generate the spectrum (true values), the initial guesses for the nonlinear parameters, and the final fitted values are given. The initial guesses were deliberately taken rather far away from the true values. Figure 29 (top) shows the fitted spectrum after the first and second iterations. During the second iteration, the Marquardt algorithm evolved into a gradient search, drastically changing the position and width of the peaks. Even after five iterations, the calculated spectrum deviates considerably from the ‘‘measured’’ spectrum, as can be seen from Figure 29 (bottom).

Figure 28 The effect of weighing the least-squares fit is shown on part of an PIXE spectrum with a very small number of counts per channel.

Copyright © 2002 Marcel Dekker, Inc.

Table 10 True Value of Spectrum Parameters Used to Generate the Artificial Spectrum in Figure 29, Initial Guesses and Fitted Values of the Peak Area, Position, and Width Obtained by Nonlinear Least-Squares Fitting Parameter Area (counts) CuKa CuKb ZnKa ZnKb Position (channel number) CuKa CuKb ZnKa ZnKb Width (channels) CuKa CuKb ZnKa ZnKb

True value 100,000 13,400 30,000 4,106 402.05 445.25 431.55 478.60 3.913 4.033 3.995 4.123

Initial guess

Fitted value

0 0 0 0

100,134±321 13,163±169 30,092±213 4138±83

395 450 435 485

402.03±0.01 445.35±0.06 431.59±0.04 478.68±0.09

3 3 3 3

3.91±0.01 3.99±0.05 4.02±0.03 4.06±0.08

Finally, after 10 iterations, a perfect match between them is obtained, with a reduced w2 value of 0.96. From Table 10, it is evident that the fit was quite successful, with all peak areas, positions, and widths estimated correctly within the calculated standard deviation. One observes that the uncertainties in the peak areas are approximately equal to the square root of the peak area and that the position and the width of the peaks are estimated very precisely (within 0.01 channel or 0.2 eV), especially for the larger peaks. By observing how the iteration procedure changes the peak width and position parameters, one can imagine that something might go wrong. Especially if the spectrum is more complex, chances are high that the iteration stops in a false minimum or even drifts away completely. In both cases, physically incorrect parameter estimates will be obtained. In practice, it is course possible to give much better initial estimates for the peak position and width parameters than used in this example. In a next example, a complex spectrum (geological reference material JG1, excited with MoK x-rays from a secondary target system) is evaluated using nonlinear leastsquares fitting. In Figure 30, the spectrum and the fit are shown together with the residuals of the fit (see p. 298). Due to the large number of overlapping lines, the method used in the first example (fitting the position and width of each peak independently) is not applicable in this case. For the description of the spectrum from 1 to 18 keV, the fluorescence lines of 21 elements were used. Au, Hg, Pb, and Th were treated each as one L group (L1 þ L2 þ L3 ) and Al, Si, Ti, V, Cr, Mn, Ni, Cu, Zn, Ga, Al, Br, Sr, Rb, Y as one K group (Ka þ Kb); K, Ca, and Fe were fitted with individual Ka and Kb peaks. The coherently scattered MoKa radiation was fitted with a Gaussian and the incoherent scattered MoKa radiation is fitted with a Voigt profile. Including escape and sum peaks, this amounts to well over 100 peak profiles. Step and tail functions [Eq. (89)] were included for all peaks, using expressions to relate the step and tail heights and the tail width to the energy of the peak. The continuum is described by an exponential function [Eq. (81)] with six parameters. The least-squares fit thus performed required the refinement of 37 parameters Copyright © 2002 Marcel Dekker, Inc.

Figure 29 Artificial CuK and ZnK line spectrum fitted with a nonlinear least-squares procedures to optimize peak area, position, and width. (Top: Fitted spectrum after first and second iteration; bottom: after fifth and final iterations.)

(4 calibration parameters, 27 peak areas, 6 continuum parameters, 3 step and tail parameters, and 1 Voigt parameter). The minimum w2 value of 1.47 is obtained after 16 iterations. The residuals indicate an overall good fit with most of the residuals in the 73 to þ3 interval, without any systematic patterns. Its interesting to note that the fitted continuum is well below the base of the peaks, especially in the region from channel 200 to 600. The continuum describes correctly the small scattered bremsstrahlung contribution from the x-ray tube above 12 keV (channel 600) and the Compton scattering in the detector at the low-energy side. Most of the apparent continuum in this secondary-target EDXRF spectrum is due to incomplete charge collection and tailing phenomena of the scattered Mo excitation radiation and the fluorescence lines. F.

Evaluation of Fitting Results

In order to understand and appreciate the capabilities and limitations of least-squares fitting, either using library spectra, linear, or nonlinear analytical functions, it is useful to study the effect of random and systematic errors in some detail. Random errors are associated with the uncertainty si of the channel content yi . As will be seen, these uncertainties influence the precision of the net peak areas and determine the ultimate resolving power of the least-squares method. Systematic errors, on the other hand, Copyright © 2002 Marcel Dekker, Inc.

Figure 30 Example of the fit of a complex x-ray spectrum (thin film deposit of geological reference material JG1 excited with a Mo secondary-target system). The residuals are plotted as an indicator of the quality of the fit.

are caused by discrepancies between the fitting model and the observed data and cause inaccuracies in the net peak areas. 1. Error Estimate Section IX explains how the least-squares-fitting method (linear as well as nonlinear) allows the estimation of the uncertainties in the fitted parameters. These uncertainties result from the propagation of the statistical fluctuations in the spectral data into the parameters. Intuitively, one could come to the conclusion that the standard deviation of the peak area should be close to the square root of the peak area. This is indeed the case for a large, interference-free peak on a low continuum, but if the peak is sitting on a high continuum and=or is seriously overlapped by another peak, the uncertainty in the estimated peak area will be much larger. A properly implemented least-squares method not only correctly estimates the net peak areas but also their uncertainty, taking into account the continuum and the degree of peak overlap, providing, of course, that the fitting model is capable to describe the measured spectrum. The closer together the peaks are, the higher the uncertainties in the two peak areas will become. (Theoretically, in the limit of complete overlap, the uncertainty will be infinite and the area of the two peaks will have complete erratic values, but their sum will still represent correctly the total net peak area of the two peaks; in practice, the curvature matrix a will be singular so that the matrix inversion fails.) Copyright © 2002 Marcel Dekker, Inc.

The uncertainties in the net peak areas can be used to decide if a peak is indeed present in the spectrum and to calculate the detection limit. A peak area is statistically not significant different from zero if its value is in the range ±3s. Any value above 3s gives clear evidence that the peak is present; any value below 73s would indicate that there is something wrong with the model because truly negative peak areas are physically meaningless. Because the uncertainty in the net peak area includes the influence of continuum and peak overlap, it can be used to calculate the a posteriori detection limits of the elements (peaks) present in the spectrum. Three situations can occur: 1. Estimated peak area > 36 standard deviation ? report: area±standard deviation 2. 736 standard deviation  estimated area  36 standard deviation ? report: detection limit equal to 36 standard deviation 3. Area < 736 standard deviation ? revise the fitting model 2. Criteria for Quality of Fit From the definition of the least-squares method, it follows that the w2 value [Eq. (75)] estimates how well the model describes the data. The reduced w2 value, obtained by dividing w2 by the number of degrees of freedom; 1 1 w2 w2n ¼ w2 ¼ n n
m

ð110Þ

has an expected value of 1 for a ‘‘perfect’’ fit. The number of degrees of freedom equals the number of data points (n) minus the number of parameters (m) estimated during the fit. Because w2 is also a random variable, the observed value will often be slightly larger or smaller than 1. Actually, w2 follows (approximately) a chi-square distribution and the 90% confidence interval is given by w2n;0:95  w2n  w2n;0:05

ð111Þ

where w2n;0:95 and w2n;0:05 are the tabulated w2 values at the 95% and 5% confidence intervals, respectively. For n ¼ 20, the confidence interval is 0.543
1.571, and for n ¼ 200, it is 0.841
1.170. An observed value of w2n in this interval indicates that the model describes the experimental data within the statistical uncertainty of the data. In other words, all remaining differences between the data and the fit can be attributed to the noise fluctuations in the channel contents and are statistically not significant. When fitting complex spectra with good statistics, much higher chi-square values will be obtained due to small imperfections in the continuum or peak model. This does not mean that the estimated peak areas are not useful any more, but high reduced w2 values ( > 3) might indicate that the fitting model needs improvement. The reduced chi-square value as defined in Eq. (110) gives an estimate of the overall fit quality over the entire fitting region. Locally, in some part of the spectrum, the fit might actually be worse than indicated by this value. It is therefore useful to define the chi-square value for each peak separately: w2P ¼

m 1 X 1 ½yi
yðiÞ2 n1
n2 i¼n2 s2i

Copyright © 2002 Marcel Dekker, Inc.

ð112Þ

where n1 and n2 are the boundaries of the peak at FWTM and n2
n1 approximates the number of degrees of freedom. High values of w2P indicate that the peak is fitted poorly and the resulting peak area should be used with caution. In this case ðw2P > 1Þ, it is advisable to give a conservative estimate for the uncertainty in the net peak area, by multiplying the calculated uncertainty with the square root of the w2 value: qffiffiffiffiffi sA0 ¼ sA w2P ð113Þ Although the w2 values gives an indication of the goodness of fit, visual inspection of the fit is highly recommended. Because of the large dynamic range of the data, a plot of the spectrum and the fit on a linear scale nearly always give the impression of a perfect fit. A plot of the logarithm or the square root of the data is more appropriate. The best method is to plot the residuals of the fit as in done in Figure 30. The residuals are defined as ri ¼

yi
yðiÞ si

ð114Þ

It is the sum of the squares of these residuals that were minimized by the least-squares method. Values in excess of þ3 or 73 and especially the presence of a pattern in the residuals then indicate poorly fitted regions. G.

Available Computer Codes

In the literature, a number of computer programs for spectrum evaluation based on the least-squares method are reported. Without attempting to be complete, the main characteristics of a number of programs are summarized in the following paragraphs. Wa¨tjen made a compilation of the characteristics of eight computer packages for PIXE analysis (Wa¨tjen, 1987). An intercomparison of five computer program for the analysis of PIXE spectra revealed a very good internal consistency among the five programs (Campbell et al., 1987). PIXE spectra of biological, environmental, and geological samples were used and their complexity put high demands of the spectrum analysis procedures. The following programs were tested: AXIL, University of Gent, Belgium; HEX, University of Lund, Sweden; SESAM-X (Marburg, FRG), the Guelph program; PIXAN, Lucas Heights. It was concluded that the most serious disagreement occurred for small peaks on the low-energy tails of very large peaks, pointing to a need to a more accurate description of the tail functions. Also, very good agreement between the linear (SESAM-X) and the nonlinear least-squares approach was observed. The Los Alamos PIXE Data Reduction software (Duffy et al., 1987) contains three components. The K and L relative x-ray intensities of the elements making up the sample are computed taking into account the detector and sample absorption. Using Gaussian peak shape, the energy and resolution calibrations of the spectrum are calculated. With the relative peak areas and the calibration functions obtained is this way, the spectrum is fitted using a Gaussian peak shape and a polynomial continuum. Escape and pileup peaks can be included. A linear least-squares fit is done with the relative elemental concentrations and the polynomial continuum coefficient as unknowns. The continuum and the relative concentrations are constrained to non-negative values and all elements having x-ray lines in the spectrum interval considered are included. The PIXASE computer package (Zolnai and Szabo´, 1988) performs spectrum analysis using nonlinear least-squares fitting. Elements are represented by groups of lines with Copyright © 2002 Marcel Dekker, Inc.

fixed, absorption-corrected, relative intensities and including escape peaks. Each peak is modeled by a Gaussian with the addition of a exponential tail and an error function as step. The square of the peak width is a first-order polynomial of the peak energy and the position is a second-order polynomial of the peak energy. The continuum is described as the sum of an exponential polynomial and two simple exponentials. Pileup effects are treated as one pileup element. The nonlinear least-squares fitting is done using a simple grid search technique. The search space of each parameter is limited by user-supplied minimum and maximum values. Fitting a large series of similar spectra linear least-squares fitting using a library of calculated spectrum components can be done. Bombelka et al. (1987) described an PIXE analysis program based on linear least squares. The peak shape includes a Gaussian low-energy tail function to account for the incomplete charge collection and the escape peak. The position and the square of the width of the peaks are given by first-order linear function of the energy. The fluorescence lines of an element are modeled as a sum of those peak shapes with relative intensity ratios corrected for absorption in the detector windows and absorbers. The continuum is composed of a fourth-order exponential polynomial multiplied by the x-ray attenuation term (bremsstrahlung continuum) and a second-order linear polynomial. Pileup is taken into account as a pileup element. The energy and resolution calibration parameters are obtained from selected peaks in the spectrum. The parameters of the exponential continuum are calculated from continuum spectra. The parameters obtained by the linear leastsquares fit are the amplitude parameters of each element and of one pileup element, the amplitude parameter of the bremsstrahlung continuum, and the linear polynomial continuum parameters. The computer implementation for PIXE is called SESAMX and is highly interactive with graphical representation of the spectral data. Other computer program for tube-excited (Breschinsky et al., 1979) and synchrotron radiation- (Petersen et al., 1986) excited XRF were developed based on this code. SAMPO-X (Aarnio and Lauranto, 1989) is intended for the analysis of electron induced x-ray spectra and is based on the well-known SAMPO code original developed for g-ray spectroscopy (Routti and Prussin, 1969). A Gaussian with two exponential tails, as in the original SAMPO program, is used to represent the peaks. The height and the position parameters are obtained by the nonlinear least-squares fit. The peak width and the tail parameters are obtained from shape-calibration tables by interpolation. The continuum is modeled by the semi-empirical electron bremsstrahlung intensity function proposed by Pella et al. (1985). The thickness of the detector beryllium window and the atomic number of the sample, which occur in this formula, are adjusted by the leastsquares fit. The program also includes an element identification based on the energy and intensity of the fitted peaks and a standard ZAF matrix correction. Jensen and Pind (1985) described a program for the analysis of energy-dispersive xray spectra. The program uses a sum of Gaussians, one for each fluorescence line. The continuum is subtracted first using a linear, parabolic, or exponential function fitted from peak-free regions in the spectrum. The peak width is obtained from a calibration function which expressed the log of the peak width as a linear function of the peak position. The width calibration is done using nonoverlapping peaks in a calibration spectrum or in the spectrum to analyze. Peak positions are determined using a peak search method or entered by the operator with the aid of a graphical display of the spectrum. The peak heights are then determined using linear least-squares fitting. The computer code developed at the technical University of Graz (Marageter et al., 1984a) is primary intended for the evaluation of x-ray fluorescence spectra. A Gaussian response function with an low-energy tail is used to describe the peaks. The square of the Copyright © 2002 Marcel Dekker, Inc.

peak width is a linear function of the peak energy. A straight-line equation relates the peak position to the energy of the peaks. A parabola is used to describe the continuum, whereas absorption edges are modeled by a complementary error function. The fitting parameters are the peak heights, the three continuum parameters, the height of the absorption edges, and the two energy calibration parameters. Nonlinear least-squares fitting is done with the Marquardt algorithm using a tangent transformation to constrain the fitting parameters to physical meaningful values (Nullens et al., 1979). Provision for escape peaks and Auger peaks is also made (Marageter et al., 1984b). The AXIL program (Van Espen et al., 1977a) was originally developed for the analysis of x-ray fluorescence spectra and later modified to allow the evaluation of electronand particle-induced x-ray spectra (Van Espen et al., 1979b, 1986). It uses the Marquardt algorithm for nonlinear least-squares fitting with a modified (constrained) chi-square function. Linear, exponential, and bremsstrahlung polynomials can be used to model the continuum as well as continuum stripping. X-ray lines are described by Gaussian functions with an optional numerical peak-shape correction. Escape and sum peaks can be included in the model. The peak position and the square of the peak width is related to the x-ray energy by linear functions. Provision is made to correct for the absorption in detector windows, filters, and the sample. In a later version, an orthogonal polynomial continuum model was added, as well as step and tail functions to describe the deviation from the Gaussian peak shape (Vekemans et al., 1994). The code was implemented as a Windows application. An example of the screen output of this program is given in Figure 31.

VIII.

METHODS BASED ON THE MONTE CARLO TECHNIQUE

Monte Carlo techniques for the simulation of x-ray spectra are becoming more and more popular, particularly because of the fast computers available today. These simulated spectra are useful for studying the behavior and performance of various spectrum processing methods. The Monte Carlo technique also can be used in quantitative analysis procedures, as will be discussed in Sec. VIII. B. A.

Simulation of X-ray Spectra

During the development and test phase of a spectrum processing method, it is often advantageous to use computer-simulated spectra. For these spectra, such features as position, width, and area of peaks are exactly known in advance. They can be generated to any desired complexity. In order to make any real use of them, the simulated spectra must possess the same channel-to-channel variation according to a Poisson distribution as experimental spectra. A simple and adequate procedure consist of first calculating, over the channel range of interest, the ideal spectrum y0 using, for example, a polynomial for the continuum and a series of Gaussians as given by Eq. (109). More complex functions, including a more physically realistic model for the continuum and tailed peaks, can be used if desired. The next step is to add or subtract some number of counts from each channel content so as to obtain data with the desired counting statistical noise. In other words, the ‘‘true’’ content y0i needs to be converted into a random variable, yi , so that is obeys a Poisson distribution [Eq. (1)] with N0 ¼ y0 . Poisson-distributed random variables can be generated by various computer algorithms. An example (Press, 1988) is given in Sec. X. For y0 > 30 the Poisson-distributed Copyright © 2002 Marcel Dekker, Inc.

Figure 31

Screen capture of the spectrum analysis program WinAxil, showing the fitted spectrum and the results obtained.

Copyright © 2002 Marcel Dekker, Inc.

random variable can be approximated by the much easier to calculate normal-distributed random variable. The probability to observe y counts in a channel, assuming a normal distribution, is given by ðy
mÞ2 1 ð115Þ PðyÞ ¼ pffiffiffiffiffiffi e 2s2 s 2p with m ¼ s2 ¼ y0 . For large y0 the normal distribution is a very good approximation of the Poisson distribution. Even for small channel contents this approximation is quite satisfactory. The probability of observing 6 counts, assuming the true value is 4, is 0.10 according to a Poisson distribution, whereas it is 0.12 according to a normal distribution. The problem of adding counting statistics to the calculated spectrum is thus reduced to calculating a normal-distributed random variable y with mean value m ¼ y0 and variance s2 ¼ y0 . Starting from a uniformly distributed random variable U in the interval (0, 1), which can be generated by a pseudo-random number generator, normal distributed random variables with zero mean and unit variance can be obtained with the BoxMuller method (Press, 1988). n1 ¼ 2Ui
1 n2 ¼ 2Uiþ1
1 r¼

n21

þ

ð116Þ

n22

if r < 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ln r=r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2 ¼ n2
2 ln r=r

z1 ¼ n 1

n1 and n2 and r are calculated from two uniformly distributed random numbers Ui and Uiþ1 . If r is less than 1, two normally distributed random numbers z1 and z2 can be calculated. If r 1; n1 ; n2 ; and r are recalculated using a new set of uniform random numbers. The normally distributed random number y, with mean m ¼ y0 and variance s2 ¼ y0 , is then obtained by simple scaling: pffiffiffiffiffi y ¼ m þ zs ¼ y0 þ z y0 ð117Þ Applying this to all channels will produce the desired counting statistics. Since z is normally distributed with mean 0 and unit variance, z can be negative as well as positive. The count p rate in each channel will thus be increased or decreased randomly and proportional to y0 . The final step in the computer simulation is to convert the real numbers that were used during the calculation to integers. Another interesting procedure is to generate artificial spectra from parent spectra. The parent spectrum is a spectrum acquired for a very long time so that it exhibits extremely good counting statistics (high channel content). A large number of child spectra with lower and varying counting statistics can be generated by the procedure explained below. This method is useful to study the effect of counting statistics on spectrumprocessing algorithms (Ryan, 1988). From the parent spectrum yi , which might be first smoothed to reduce the noise even further providing not too much distortion is introduced, the normalized cumulative distribution function Yj is calculated Pj yi ð118Þ Yj ¼ Pi¼0 n i¼0 yi Copyright © 2002 Marcel Dekker, Inc.

n being the number of channels in the spectrum and Yj in the interval (0
1). To generate the child spectrum we select N times a channel i according to the equation i ¼ Y
1 ðUÞ

ð119Þ

where Y
1 is the inverse cumulative distribution and U is a uniformly distributed random number. Each time channel i is selected, one count is added to that channel. Since N is the total number of counts in the child spectrum, the counting statistics can be controlled by varying N. However, Y
1 cannot be expressed as an analytical function, but for each random number U we select the channel i for which Yi  U < Yiþ1 . Figure 32 shows an EDXRF spectrum from a 0.187 mg=cm2 pellet of IAEA Animal Blood reference material acquired for 1000 s with a Tracor Spectrace 5000 instrument, which is used as a parent spectrum. The cumulative distribution function and a child spectrum simulated with N ¼ 36104 are also shown. The total number of counts in the original spectrum is 36106. The child spectrum is equivalent to a spectrum that would have been acquired for 10 s. Some computer routines useful for simulation experiments are given in Sec. X.

Figure 32 Simulation of a ‘‘child’’ spectrum from a ‘‘parent.’’ (Top: original spectrum; middle: the cumulative distribution; bottom: generated child.)

Copyright © 2002 Marcel Dekker, Inc.

B.

Spectrum Evaluation Using Monte Carlo Techniques

Another interesting application of the Monte Carlo technique is the simulation of the complete response of an x-ray fluorescence setup. Given an excitation spectrum and the excitation-detection geometry the interactions of the primary photons with the sample are simulated and the events giving rise to detectable phenomena are registered. With such a Monte Carlo simulation, the intensity of the characteristic lines and the scattered excitation spectrum are estimated, taking primary and secondary effects (absorption, enhancement, and single and multiple scattering) into account. The obtained spectrum is one as seen by an ideal detector with infinite resolution. This spectrum can then be convolution with the response function of a real detector to mimic a typical observed pulse-height spectrum. Apart from a detailed Monte Carlo simulation code, this technique requires detailed knowledge of the excitation spectrum and a very accurate detector response function. Doster and Gardner were among the first to apply this technique to simulate the complete spectral response of an EDXRF system (Doster and Gardner, 1982a; Doster and Gardner, 1982b; Yacout and Dunn, 1987). More recently, Janssens and coworkers developed highly efficient computer codes for the simulation of conventional and synchrotron energy-dispersive x-ray fluorescence systems, allowing such conditions as polarized radiation and heterogeneous samples to be taken into consideration (Janssens et al., 1993; Vincze et al., 1993; Vincze et al., 1995a, 1995b; Vincze et al., 1999). Figure 33 shows the results of the application of this Monte Carlo simulation for a NIST SRM 1155 steel sample excited with a filtered Rh-anode x-ray tube. At the top the simulated spectrum as seen by a perfect detector is shown, at the bottom the spectrum after convolution with a suitable detector response function is compared with a real measured spectrum of this sample. Except for some sum peaks that occur in the measured spectrum, an excellent agreement between the simulated and measured fluorescence lines, scattered peaks, and continuum is obtained. These simulated x-ray spectra can be used in various ways in quantitative analysis. Using the complete spectral response of spectrometer, one can try to find a sample composition that minimizes (in the least-squares sense) the difference between the simulated and the measured spectrum. The analysis involves the following steps: (1) simulation of the X-ray intensities over the expected composition range of the unknowns; (2) convolution with the detector response function to obtain spectra; (3) construction of a w2 map (weighted sum of squares of differences between simulated spectra and experimental spectrum) as a function of the sample composition; (4) interpolation of w2 for the composition corresponding to the minimum. An interesting aspect of this method is that all the information present in the spectrum (characteristic lines, scattered radiation, continuum) is considered. Doster and Garnder demonstrated an analytical accuracy of the order of 2% absolute for the analysis of Cr-Fe-Ni alloys with a 109Cd radioisotope system (Doster and Gardner, 1982a). Based on this work, Yacout and Dunn (Yacout and Dunn, 1987) demonstrated the use of the inverse Monte Carlo method, which requires in principle only one simulation to analyze a set of similar samples. Another method is called Monte Carlo
library least-squares analysis (Verghese et al., 1988). Starting from an initial guess of the composition of an unknown sample, a spectrum is simulated taking into account all the interactions in the sample. During the simulation one keeps track of the response of each element to construct library spectra. After the simulation these library spectra are used to obtain the elemental concentrations by linear least-squares fitting (see Sec. VI). In the concentration in the unknown samples differ too much from the initial assumed concentration, the simulation is repeated. Copyright © 2002 Marcel Dekker, Inc.

Figure 33 Monte Carlo simulation of the spectral data from an NIST SRM 1155 sample excited with a Rh-anode x-ray tube. (Top: simulated spectrum as seen by an ideal detector; bottom: simulated spectrum after convolution with the detector response function and comparison with measured spectrum.)

In contrast to the normal library least-squares method, this method has the advantage that the library spectra are simulated for a composition close to the composition of the spectrum to analyzed, rather than measured from standards. This eliminates the necessity of applying the top-hat filter and problems related to changes in Kb=Ka ratios, and again the method combines spectrum evaluation with quantitative analysis. Finally the ability to simulate x-ray spectra that agree very well with real measured spectra opens the possibility to used them as ‘‘standards’’ for the quantitative analysis based on partial least-squares regression (see Sec. VI.B). Indeed, as this method only functions correctly if the PLS model is built using a large number of standards covering the entire concentration domain, it seems advantageous to use simulated spectra for this. All the inter-element interactions can be accounted for by the simulation and only a few real standards are required to scale the simulated spectra. Copyright © 2002 Marcel Dekker, Inc.

IX. THE LEAST-SQUARES-FITTING METHOD The aim of the least-squares method is to obtain ‘‘optimal’’ values for the parameters of a function that models the dependence of experimental data. The method has its roots in statistics but is also considered part of numerical analysis. The least-squares parameter estimate, also known as curve fitting, plays an important role in experimental science. In x-ray fluorescence, it is used in many calibration procedures and it forms the basis of a series of spectrum analysis techniques. In this section, an overview of the least-squares method with emphasis on spectrum analysis is given. Based on the type of fitting function, one makes a distinction between linear and nonlinear least-squares fitting because one requires numerical techniques of different complexity to solve the problem in the two cases. The linear least-squares method deals with the fitting of functions that are linear in the parameters to be estimated. For this problem, a direct algebra solution exists. If the fitting function is not linear in one or more of the parameters, one uses nonlinear least-squares fitting and the solution can only be found iteratively. A group of linear functions of general interest are the polynomials, the straight line being the simplest case. The special case of orthogonal polynomials will also be considered. If more than one independent variable ðx1i ; x2i ; . . . ; xmi Þ is associated with each measurement of the dependent variable yi , one speaks of multivariate regression. Spectrum analysis using the library function (e.g., filter-fit method) belongs to this category. If analytical function (e.g., Gaussians) are fitted to a spectrum, the method of linear or nonlinear least square is used, depending on whether nonlinear parameters, such as the peak position and width, are determined. A.

Linear-Least Squares

Considered the problem of fitting experimental data with the following linear function: y ¼ a1 X1 þ a2 X2 þ    þ am Xm

ð120Þ

This function covers all linear least-squares problems. If m ¼ 2; X1 ¼ 1, and X2 ¼ x, the straight-line equation y ¼ a1 þ a2 x is obtained. For m > 2 and Xk ¼ xk
1 , Eq. (120) is a polynomial y ¼ a1 þ a2 x þ a3 x2 þ    þ am xm
1 to be fitted to the experimental data points fxi ; yi ; si g i ¼ 1; . . . ; n. If X1 ; . . . ; Xm represents different independent variables, the case of multiple linear regression is dealt with. Because of this generality, we will discuss the linear least-squares method based on Eq. (120) in detail. Assume a set of n experimental data points: fx1i ; x2i ; . . . ; xmi ; yi ; si g;

i ¼ 1; . . . ; n

ð121Þ

with xki the value of the kth independent variables Xk in measurement i, assumed to be known without error, and yi the value of the dependent variable measured with uncertainly si . The optimum set of parameters a1 ; . . . ; am that gives a least-squares fit of Eq. (120) to these experimental data are those values of a1 ; . . . ; am that minimize the chisquare function: w2 ¼

n X 1 ðyi
a1 X1
a2 X2
  
am Xm Þ2 2 s i i¼1

ð122Þ

The minimum is found by setting the partial derivatives of w2 with respect to the parameters to zero: Copyright © 2002 Marcel Dekker, Inc.

n X @w2 1 ¼
2 ðyi
a1 X1
a2 X2
  
am Xm ÞXk ¼ 0; 2 @ak s i i¼1

k ¼ 1; . . . ; m

ð123Þ

Dropping the weights 1=s2i temporarily for clarity, we obtain a set of m simultaneous equations in the m unknown ak : X X X X yi X 1 ¼ a1 X1 X1 þ a2 X2 X1 þ    þ am Xm X1 X X X X yi X 2 ¼ a1 X1 X2 þ a2 X2 X2 þ    þ am Xm X2 ð124Þ .. . X X X X yi X m ¼ a1 X 1 X m þ a2 X2 Xm þ    þ am Xm Xm where the summations run over all experimental data points i. These equations are known as the normal equations. The solution—the values of ak —can easily be found using matrix algebra. Because two (column) matrices are equal if their corresponding elements are equal, the set of equations can be in matrix form as X X 3 2 X 3 2X a1 X1 X1 þ a2 X 2 X 1 þ    þ am Xm X1 yi X1 X X X X 7 6 7 6 6 6 yi X2 7 X1 X2 þ a2 X 2 X 2 þ    þ am Xm X2 7 7 6 a1 7 6 ð125Þ 7¼6 7 6 .. .. 6 7 6 7 4X . 5 4 X 5 . X X yi Xm a1 X 1 X m þ a2 X2 Xm þ    þ am Xm Xm The right-hand column matrix can be written as the product of a square matrix a and a column matrix a: X X 3 2 X 3 2X X1 X1 yi X1 X2 X1    Xm X1 2 a1 3 X X 7 6 X 76 7 6X 6 6 yi X2 7 X1 X2 X2 X2    Xm X2 7 a2 7 7 6 76 6 6 . 7 ¼ ð126Þ 7 7 6 6 6 . .. 7 6 74 .. 7 6 5 5 4X 5 4 X .. . X X a2 yi Xm X1 Xm Xm Xm X2 Xm    or b ¼ aa

ð127Þ

This equation can be solved for a by premultiplying both sides of the equation with the inverse matrix a
1 , a
1 b ¼ a
1 aa ¼ Ia

ð128Þ

or, I being the identity matrix, a ¼ a
1 b Introducing the weights again, the elements of the matrices are given by n X 1 bj ¼ y X ; j ¼ 1; . . . ; m 2 i j s i i¼1 n X 1 ajk ¼ Xk Xj ; j ¼ 1; . . . ; m; k ¼ 1; . . . ; m s2i i¼1 and Copyright © 2002 Marcel Dekker, Inc.

ð129Þ

ð130Þ ð131Þ

aj ¼

m X

a
1 jk bk ;

j ¼ 1; . . . ; m

ð132Þ

k¼1
1 where a
1 jk are the elements of the inverse of the matrix a . The uncertainty in the estimate of aj is due to the uncertainty of each measurement multiplied by the effect that measurement has on aj :
2 n X @aj s2aj ¼ s2i ð133Þ @yi i¼1

Because a
1 jk is independent of yi , the partial derive is simply m @aj 1X ¼ 2 a
1 Xk ðiÞ @yi si k¼1 jk

After some rearrangements, it can be shown that " # m X m n X X 1 2
1
1 saj ¼ ajk ajl Xk Xl s2i i¼1 k¼1 l¼1 the term in the brackets being akl and XX
1
1 s2aj ¼ a
1 jk ajl akl ¼ ajj k

ð134Þ

ð135Þ

ð136Þ

l

This results in the simple statement that the variance (square of uncertainty) of a fitted parameter aj is given by the diagonal element j of the inverse matrix a
1 . The off-diagonal elements are the covariances. For this reason, a
1 is often called the error matrix. Similarly, a is called the curvature matrix because the elements are a measure of the curvature of the w2 hypersurface in the m-dimensional parameter space. It can easily be shown that X 1 1 @ 2 w2 ¼ Xk Xj ¼ ajk 2 @aj @ak s2i i

ð137Þ

If the uncertainties in the data points si are unknown and are the same for all data points ðsi ¼ sÞ, these equations can still be used by setting the weights wi ¼ 1=s2i to 1. Assuming the fitting model is correct, s can be estimated from the data: 1 X s2i ¼ s2  s2 ¼ ðyi
a1 X1
a2 X2
  
am Xm Þ2 ð138Þ n
m i The uncertainties in the parameters are then given by s2aj ¼ s2 a
1 jj

ð139Þ

If the uncertainties in the data points are known, the reduced w2 value can be calculated as a measure of the goodness of fit: 1 X 1 1 w2 w2n ¼ ðyi
a1 X1
a2 X2
  
am Xm Þ2 ¼ ð140Þ n
m i s2i n
m The expected value of w2n is 1.0, but due to the random nature of the experimental data values slightly smaller or greater than 1 will be observed even for a ‘‘perfect’’ fit. w2n follows a chisquare distribution with n
m degrees of freedom, and a 90% confidence interval is defined by Copyright © 2002 Marcel Dekker, Inc.

w2 ðn; P ¼ 0:95Þ  w2n  w2 ðn; P ¼ 0:05Þ

ð141Þ

where w2 ðn; PÞ is the (tabulated) critical value of the w2 distribution for n degrees of freedom at a confidence level P. Observed w2n values outside this interval indicate a deviation between the fit and the data that cannot be attribute to random statistical fluctuations. B.

Least-Squares Fitting Using Orthogonal Polynomials

A special group of linear functions are orthogonal polynomials. Orthogonality means that the polynomial terms are uncorrelated, and this has some distinct advantages. Let Pj ðxi Þ be an orthogonal polynomial of degrees j; a function can then be constructed as a sum of these orthogonal polynomials of successive higher degree: m X yðiÞ ¼ Cj Pj ðxi Þ ð142Þ j¼0

The least-squares estimates of the coefficient Cj are determined by minimizing the weighted sum of squares: n X wi ðyi
yðiÞÞ2 ð143Þ w2 ¼ i¼1

which results in a set of m þ 1 normal equations in the m þ 1 unknown. Because the Pj ðxi Þ are a set of orthogonal polynomials, they have the property that n X wi Pj ðxi ÞPk ðxi Þ ¼ gk djk ð144Þ i¼1

with gk a normalization constant and djk ¼ 0 for j 6¼ k. Because of this property, the matrix of the normal equations is diagonal and the polynomial coefficients are directly obtained from n X wi yi Pj ðxi Þ ð145Þ Cj ¼ gj i¼1 and the variance of the coefficient is given by 1 s2cj ¼ gj

ð146Þ

Another advantage of the use of orthogonal polynomials is that the addition of one extra term Cmþ1 Pmþ1 does not change the values of the already determined coefficients C0 ; . . . ; Cm . Further, if the yi are independent, then also the Cj are independent; that is, the variance
covariance matrix is also diagonal. As a result, much higher-degree orthogonal polynomials can be fitted compared to ordinary polynomials without running into problems with ill-conditioned normal equations and oscillating terms. Orthogonal polynomials can constructed by following recurrence relation: Pjþ1 ðxi Þ ¼ ðxi
aj ÞPj ðxi Þ
bj Pj
1 ðxi Þ

j ¼ 0; . . . ; m
1

aj and bj are constants independent of yi given by Pn wi xi ½Pj ðxi Þ2 aj ¼ i¼1 ; j ¼ 0; . . . ; m gj Copyright © 2002 Marcel Dekker, Inc.

ð147Þ

ð148Þ

Pn bj ¼

i¼1

wi xi Pj ðxi ÞPj
1 ðxi Þ ; gj
1

j ¼ 0; . . . ; m

ð149Þ

Further, the normalization factor is given by gj ¼

n X

wi ½Pj ðxi Þ2

ð150Þ

i¼1

and b0 ¼ 0 and

P0 ðxi Þ ¼ 1

ð151Þ

Thus, an example of a first-order orthogonal polynomial is C0 P0 þ C1 P1 ¼ C0 þ C1 ðxi
a0 Þ

ð152Þ

with a0 ¼

n X

w i xi

.X

ð153Þ

wi

i¼1

C.

Nonlinear Least Squares

In this part, we consider the fitting of a function that is nonlinear in one or more fitting parameters. Examples of such functions are a decay curve, yðxÞ ¼ a1 expða2 xÞ

ð154Þ

or a Gaussian on a linear background, ðx
a4 Þ2 yðxÞ ¼ a1 þ a2 x þ a3 exp
2a25

! ð155Þ

Equation (154) is nonlinear in a2 ; Eq. (155) is nonlinear in the parameters a4 and a5 . Equation (154) is representative for a group of functions for which linear least-squares fitting can be applied after suitable transformation. Fitting with Eq. (155) implies the application of truly nonlinear least-squares fitting with iterative optimization of the fitting parameters. 1. Transformation to Linear Functions Taking the logarithm of Eq. (154), ln y ¼ ln a1 þ a2 x

ð156Þ

and defining y0 ¼ ln y; and a01 ¼ ln a1 , a linear (straight line) fitting function is obtained, y0 ¼ a01 þ a2 x

ð157Þ

and the method discussed earlier can be applied, but not without making the following important remark. We have transformed our original data yi to y0i ¼ ln yi . Consequently, also the variance s02 i has been changed according to the general error propagation formula:
0 2 @y s0i 2 ¼ s2i ð158Þ @y Copyright © 2002 Marcel Dekker, Inc.

or in this particular case, s0i 2 ¼

s2i yi

ð159Þ

Thus, even if all original data points had the same uncertainty ðsi ¼ sÞ and unweighed linear least-squares fitting could have been used, after the transformation a weighted linear least-squares fit is required. The results of the fit are the parameters and the associated uncertainties of the transformed equation, and to obtain the original parameter, we must perform a back transformation with the appropriate error propagation, 0

a1 ¼ e a 1 s2a1 ¼ s2a0

1




@a1 @a01

ð160Þ

2 ¼ a1 s2a0

1

ð161Þ

2. General Nonlinear Least Squares For the general case of least-squares fitting with a function that is nonlinear in one or more of its fitting parameters, no direct solution exists. Still, we can define the object function w2 : X 1 ½y
yðxi ; aÞ2 ð162Þ w2 ¼ 2 i s i i whose minimum will be reached when the partial derivative with respect to the parameters are zero; however, this will result in a set of m equations for which no general solution exists. The other approach to the problem is then to consider w2 as a continuous function of the parameters aj (i.e., w2 will take a certain value for each set of values of the parameters aj for a given dataset fxi ; yi ; s2i gÞ. w2 thus forms a hypersurface in the m-dimensional space, formed by the fitting parameter aj . This surface must be searched to locate the minimum of w2 . Once found, the corresponding coordinate values of the axes are the optimum values of the fitting parameters. The problem of nonlinear least-squares fitting is thus reduced to the problem of finding the minimum of a function in an m-dimensional space. Any algorithm that performs this task should operate according to the following: 1. Given some initial set of values for the parameters aini evaluate w2 : w2old ¼ w2 ðaini Þ 2. Find a new set of values anew such that w2new < w2old . 3. Test the minimum of w2 value: if else

w2new is the (true) minimum accept anew as the optimum values of the fit w2old ¼ w2new and repeat Step 2.

From the scheme, the iterative nature of the nonlinear least-squares fitting methods becomes evident. Moreover, it shows some other important aspects of the method: Initial values are required to start the search, we need a procedure to obtain a new set of parameters which preferably are such that w2 is decreasing, and we need to be sure that the true minimum, not some local minimum, is finally reached. A variety of algorithms has been proposed, ranging from brute-force mapping procedures dividing the m-dimensional parameter space in small cells and evaluating w2 in each point, to more subtle simplex search procedures (Fiori et al., 1981). The most Copyright © 2002 Marcel Dekker, Inc.

important group of algorithms is nevertheless based on the evaluation of the curvature matrix. The gradient method and the first-order expansion will be discussed briefly, as they form the basis of the most widely used Leverberg
Marquardt algorithm (Marquardt, 1963; Bevington and Robinson, 1992; Press et al., 1988). a. The Gradient Method Having a fitting function y ¼ yðx; aÞ and w2 defined as a function of the m parameters aj , n X 1 w2 ¼ w2 ðaÞ ¼ ½y
yðxi ; aÞ2 ð163Þ 2 i s i i¼1 the gradient of w2 in the m-dimensional parameter space is given by Hw2 ¼

X @w2 j @aj j

ð164Þ

where j is the unit vector along the axis j and the components of the gradient are given by X 1 @w2 @y ¼
2 ½yi
yðxi ; aÞ 2 @aj @aj si i

ð165Þ

It is convenient to define bj ¼


1 @w2 2 @aj

ð166Þ

The gradient gives the direction in which w2 increases most rapidly. A method to locate the minimum can thus be developed on this basis. Given the current set of parameters aj , a new set of parameters a0j is calculated (for all j simultaneously): a0j ¼ aj þ Daj bj

ð167Þ

which follows the direction of steepest descent and guarantees a decrease of w2 (at least if the appropriate step sizes Daj are taken). The gradient method works quite well away from the minimum, but near the minimum, the gradient becomes very small (at the minimum, even zero). Fortunately, the method discussed next behaves in the opposite way. b. First-Order Expansion If we write the fitting function yðxi ; aÞ as a first-order Taylor expansion of the parameters aj around y0 , yðx; aÞ ¼ y0 ðx; aÞ þ

X @y0 ðx; aÞ daj @aj j

ð168Þ

we obtain an (approximation) to the fitting function which is linear in the parameter increments daj . y0 ðx; aÞ is the value of the fitting function for some initial set of parameter a. Using this function, we can now express w2 as " #2 X 1 X @y0 ðxi ; aÞ 2 w ¼ yi
y0 ðxi ; aÞ
daj ð169Þ @aj s2i j i and we can use the method of linear least squares to find the parameters daj so that w2 will be minimal. We are thus fitting the difference y0i ¼ yi
y0 ðxi ; aÞ with the derivatives as Copyright © 2002 Marcel Dekker, Inc.

variables and the increments daj as unknowns. With reference to the section on linear least-squares fitting [Eq. (122)], Xj ¼

@y0 ðxi Þ @aj

ð170Þ

and [Eq. (130) and (131)] n X 1 @y0 ðxi Þ ½y
y0 ðxi Þ 2 i @aj s i i¼1 n X 1 @y0 ðxi Þ @y0 ðxi Þ ajk ¼ @ak s2i @aj i¼1

bj ¼

ð171Þ ð172Þ

defining a set of m normal equations in the unknowns daj , b ¼ ada

ð173Þ

with solution daj ¼

m X

a
1 jk bk

ð174Þ

k¼1

It is not very difficult to prove that bj ¼


1 @w20 2 @ak

ð175Þ

(i.e., the component of the gradient of w2 at the point of expansion) and ajk 

1 @ 2 w20 2 @aj dak

ð176Þ

Thus, ajk in Eq. (172) is the first-order approximation to the curvature matrix whose inverse is the error matrix. The first-order expansion of the fitting function is closely related to the first-order Taylor expansion of the w2 hypersurface itself: w2 ¼ w20 þ

X @w2 0 daj @a j j

ð177Þ

where w20 is the w2 function at the point of expansion: w20 ¼

n X 1 ½y
y0 ðxi ; aÞ2 2 i s i i¼1

ð178Þ

At the minimum, the partial derivation of w2 with respect to the parameter ak will be zero: @w2 @w20 X @ 2 w20 ¼ þ dak ¼ 0 @ak @ak @aj @ak j

ð179Þ

This results in a set of equations in the parameters dak : m X @w20 @w20 ¼
dak @ak @aj @ak j¼1

Copyright © 2002 Marcel Dekker, Inc.

ð180Þ

bk ¼

X

ajk dak

ð181Þ

which is the same set of equations, except that in the expansion of the fitting function, only a first-order approximation of the curvature matrix is used. Because near the minimum the first-order expansion of the w2 surface is a good approximation, we can conclude that also the first-order expansion of the fitting function (which is computationally more elegant because only derivatives of functions and not of w2 are required) will yield parameter increments daj which will direct us toward the minimum. For each linear parameter in the fitting function, the first-order expansion of the function in this parameter is exact and the calculated increment daj will be such that the new value aj þ daj is optimum (for the given set of nonlinear parameters which might not be at their optimum value yet). c. The Marquardt Algorithm Based on the observation that away from the minimum the gradient method is effective and near the minimum the first-order expansion is useful, Marquardt developed an algorithm that combines both methods using a scaling factor l that moves the algorithm either in the direction of the gradient search or into the direction of first-order expansion (Marquardt, 1963). The diagonal terms of the curvature matrix are modified as follows:

ajk ð1 þ lÞ; j ¼ k 0 ajk ¼ ð182Þ j 6¼ k ajk ; where ajk is given by Eq. (172) and the matrix equation to be solved for the increments daj is X a0jk dak ð183Þ bj ¼ k

When l is very large (l 1), the diagonal elements of a dominate and Eq. (183) reduces to bj  a0jj dak or dak 

1 1 @w2 b  j a0jj a0jj @ak

ð184Þ ð185Þ

which is the gradient, scaled by a factor a0jj . On the other hand, for small values of l (l 1), the solution is very close to first-order expansion. The algorithm proceeds as follows: 1. 2. 3. 4.

Given some initial values of the parameters aj , evaluate w2 ¼ w2 ðaÞ and initialize l ¼ 0.0001. Compute b and a matrices using Eqs. (171) and (172). Modify the diagonal elements a0jj ¼ ajj þ l and compute da. If w2 ða þ daÞ w2 ðaÞ increase l by a factor of 10 and repeat Step 3; If w2 ða þ daÞ < w2 ðaÞ decrease l by a factor of 10 accept new parameters estimates a a þ da and repeat Step 2.

The algorithm thus performs two loops: the inner loop incrementing l until w2 starts to decrease and the outer loop calculating successively better approximations to the optimum values of the parameters. The outer loop can be stopped when w2 decreases by a negligible absolute or relative amount. Copyright © 2002 Marcel Dekker, Inc.

Once the minimum is reached, the diagonal elements are an estimate of the uncertainty in the fitting parameters just as in the case of linear least squares: s2aj ¼ a
1 jj

ð186Þ

which is equal to ða0jj Þ
1 providing the scaling factor l is much smaller than 1. In Sec. X, a number of computer programs for linear and nonlinear least-squares fitting are given. Further information can be found in many textbook (Press et al., 1988). The book by Bevington and Robinson (1992) contains a very clear and practical discussion of the least-squares method.

X.

COMPUTER IMPLEMENTATION OF VARIOUS ALGORITHMS

In this section, a number of computer routines related to spectrum evaluation are listed. The calculation routines are written in FORTRAN. Some example programs, calling this FORTRAN routines, are written in C. The programs were tested using Microsoft FORTRAN version 4.0A and C version 5.1. Most of the routines are written for clarity rather than optimized for speed or minimum space requirement.

A.

Smoothing

1. Savitsky and Golay Polynomial Smoothing The subroutine SGSMTH calculates a smoothed spectrum using a second-degree polynomial filter (see Sec. III.B.2). Input:

Output

Y NCHAN ICH1, ICH2 IWID S

Original spectrum Number of channels in the spectrum First and last channel number to be smoothed Width of the filter (2m þ 1), IWID < 42 Smoothed spectrum, only defined between ICH1 and ICH2

SUBROUTINE SGSMTH (Y, S, NCHAN, ICH1, ICH2, IWID) INTEGER*2 NCHAN, ICH1, ICH2, IWID REAL*4 Y(0:NCHAN
1), S(0:NCHAN
1) REAL C(
20:20) C - - Calculate filter coefficients IW ¼ MIN(IWID, 41) M ¼ IW=2 SUM ¼ FLOAT( (2*M 7 1)*(2*M þ 1)*(2*M þ 3) ) DO 10 J ¼
M, M C(J) ¼ FLOAT( 3*(3*M*M þ 3*M 7 1 7 5*J*J) ) 10 CONTINUE C - - Convolute spectrum with filter JCH1 ¼ MAX( ICH1, M ) JCH2 ¼ MIN( ICH2, NCHAN 7 1 7 M ) DO 30 I ¼ JCH1, JCH2 S(I) ¼ 0.

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20 30

DO 20 J ¼ 7 M, M S(I) ¼ S(I) þ C(J)*Y(I þ J) CONTINUE S(I) ¼ S(I)=SUM CONTINUE RETURN END

2. Low Statistics Digital Filter The subroutine LOWSFIL smooths a spectrum using the low statistics digital filter algorithm (see Sec. III.B.3). Input:

Output:

Y NCHAN ICH1, ICH2 IFWHM S

Original spectrum Number of channels First and last channel to be smoothed FWHM in channels of a peak in the middle of the smoothing region Smoothed spectrum, only defined between ICH1 and ICH2

SUBROUTINE LOWSFIL (Y, S, NCHAN, ICH1, ICH2, IFWHM) INTEGER*2 NCHAN, ICH1, ICH2, IFWHM REAL Y(0:NCHAN 7 1), S(0:NCHAN 7 1) LOGICAL NOKSLOPE, STOOHIGH REAL AFACT, FFACT, MFACT, RFACT PARAMETER (AFACT ¼ 75., FFACT ¼ 1.5, MFACT ¼ 10., RFACT ¼ 1.3) C - - Adjust smoothing region IW ¼ NINT( FFACT*IFWHM ) JCH1 ¼ MAX(ICH1 7 IW, IW) JCH2 ¼ MIN(ICH2 þ IW, NCHAN 7 1 7 IW) DO 100 I ¼ JCH1, JCH2 IW ¼ NINT( FFACT*IFWHM ) SUML ¼ 0. SUMR ¼ 0. DO 20 J ¼ I 7 IW, I 7 1 SUML ¼ SUML þ Y(J) 20 CONTINUE DO 30 J ¼ I þ 1, I þ IW SUMR ¼ SUMR þ Y(J) 30 CONTINUE C - - Adjust window 50 CONTINUE SUMT ¼ SUML þ Y(I) þ SUMR IF( SUMT .GT. MFACT ) THEN SLOPE ¼ (SUMR þ 1.)=(SUML þ 1.) NOKSLOPE ¼ SLOPE.GT.RFACT .OR. SLOPE .LT. 1.=RFACT STOOHIGH ¼ SUMT .GT. AFACT*SQRT(Y(I)) IF( NOKSLOPE .OR. STOOHIGH .AND. IW .GT. 1 ) THEN SUML ¼ SUML 7 Y(I 7 IW) SUMR ¼ SUMR 7 Y(I 7 IW) IW ¼ IW 7 1 GOTO 50

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C -100 C --

110

120

ENDIF ENDIF Smoothed value S(I) ¼ SUMT=FLOAT (2*IW þ 1) CONTINUE Copy data points that could not be smoothed DO 110 I ¼ ICH1, JCH1 7 1 S(I) ¼ Y(I) CONTINUE DO 120 I ¼ JCH2 þ 1, ICH2 S(I) ¼ Y(I) CONTINUE RETURN END

For each channel i in the spectrum, two windows, one on each side of the channel of width f6FWHM(Ei ) channels are considered. In both windows, the channel contents are summed, yielding a left sum L and a right sum R. Both windows are subsequently reduced in width until either the total sum S ¼ L þ yi þ R falls below some constant minimum M or until two conditions are met: pffiffiffiffi 1. S is less than a cutoff value N ¼ A yi , with A a constant. 2. The slope ðR þ 1Þ=ðL þ 1Þ lies between 1=r and r, with r a constant. The minimum constant M sets the base degree of smoothing in a region of vanishing counts. The first condition ensures that smoothing is confined to the low statistics region of the spectrum; the second condition avoids the incorporation of the edges of the peaks in the averaging. When the above conditions are satisfied, the average S=(2f6FWHM þ 1) is adopted as a smoothed channel count. The following parameters were found to yield good results when treating PIXE spectra: f ¼ 1.5, A ¼ 75, M ¼ 10, and r ¼ 1.3.

B.

Peak Search

The subroutine LOCPEAKS locates peaks in a spectrum using positive part of tophyhat filter (see Sec III.C). Input:

Output:

Y NCHAN R IWID MAXP NPEAK IPOS

Spectrum Number of channels in the spectrum Peak search sensitivity factor, typical 2 to 4 Width of the filter, approx. equal to the FWHM of the peaks Maximum number of peaks to locate (size of array IPOS) Number of peaks found Peak positions (channel number)

SUBROUTINE LOCPEAKS (Y, NCHAN, IWID, R, IPOS, NPEAKS, MAXP) INTEGER*2 NCHAN, IWID, NPEAKS, MAXP INTEGER*2 IPOS (MAXP) REAL Y (NCHAN), R C - - Width of filter (number of channels in the top)

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C

C --

C --

20

22 C --

C C

C

100

must be odd, and at least 3 NP ¼ MAX (IWID=2)*2 þ 1, 3) NPEAKS ¼ 0 Calculate half width and start and stop channel N ¼ NP=2 I1 ¼ NP I2 ¼ NCHAN-NP Initialize running sums I ¼ I1 TOTAL ¼ 0. TOP ¼ 0. DO 20 K ¼ 7N*2, NP TOTAL ¼ TOTAL þ Y(I1 þ K) CONTINUE DO 22 K ¼ 7N, N TOP ¼ TOP þ Y(I1 þ K) CONTINUE Loop over all channels LASTPOS ¼ 0 SENS ¼ R*R FI ¼ 0. FNEXT ¼ 0. SNEXT ¼ 0. DO 100 I ¼ I1 þ 1, I2 TOP ¼ TOP 7 Y(I-N-1) þ Y(I þ N) TOTAL ¼ TOTAL 7 Y(I-NP) þ Y(I þ NP) FPREV ¼ FI FI ¼ FNEXT SI ¼ SNEXT FNEXT ¼ TOP þ TOP 7 TOTAL SNEXT ¼ TOTAL Significant? IF( FI.GT.0. .AND. (FI*FI.GT.SENS*SI) ) THEN Find maximum IF( FI .GT. FPREV .AND. FI .GT. FNEXT ) THEN IF( FPREV.GT.0. .AND. FNEXT.GT.0. ) THEN and store (ch# is array index
1 and FI refers to I
1) NEWPOS ¼ I
2 IF( NEWPOS.GT.LASTPOS þ 2 ) THEN NPEAKS ¼ NPEAKS þ 1 IPOS(NPEAKS) ¼ NEWPOS LASTPOS ¼ NEWPOS IF( NPEAKS .EQ. MAXP ) RETURN ENDIF ENDIF ENDIF ENDIF CONTINUE RETURN END

The routine is optimized for speed and requires no other arrays than the spectrum and a table to store the peak maxima found. This is achieved by using a variant of the top-hat Copyright © 2002 Marcel Dekker, Inc.

filter; that is, for a filter width of 5, the coefficients are 71, 71, þ1, þ1, þ1, þ1, þ1, 71, 71, 71. The next point in the filtered spectrum can be calculated from the current by subtracting and adding and only the value of the previous, the current, and the next points in the filtered spectrum are retained. This makes the routine quite cryptic, but it works very fast and is reliable. C.

Continuum Estimation

1. Peak Stripping The subroutine SNIPBG, a variant of the SNIP algorithm, calculates the continuum via peak stripping (see Sec. IV.A). INPUT:

Output: Comment:

Y NCHAN ICH1,ICH2 FWHM NITER YBACK Uses subroutine

Spectrum Number of channels in the spectrum First and last channels of region to calculate the continuum Width parameter for smoothing and stripping algorithm, set it to average FWHM of peaks in the spectrum, typical value 8.0 Number of iterations of SNIP algorithm, typical 24 Calculated continuum in the region ICH1
ICH2 SGSMTH

SUBROUTINE SNIPBG (Y, YBACK, NCHAN, ICH1, ICH2, FWHM, NITER) INTEGER*2 NCHAN, ICH1, ICH2, NITER REAL*4 Y(0:NCHAN-1), YBACK(0:NCHAN-1), FWHM PARAMETER (SQRT2 ¼ 1.4142, NREDUC ¼ 8) C - - Smooth spectrum IW ¼ NINT (FWHM) I1 ¼ MAX (ICH1 7 IW, 0) I2 ¼ MIN (ICH2 þ IW, NCHAN-1) CALL SGSMTH (Y, YBACK, NCHAN, I1, I2, IW) C - - Square root transformation over required spectrum region DO 10 I ¼ I1, I2 YBACK(I) ¼ SQRT (MAX(YBACK(I), 0.)) 10 CONTINUE C - - Peak stripping REDFAC ¼ 1. DO 30 N ¼ 1, NITER C.. Set width, reduce width for last NREDUC iterations IF( N .GT. NITER-NREDUC ) REDFAC ¼ REDFAC=SQRT2 IW ¼ NINT (REDFAC*FWHM) DO 20 I ¼ ICH1, ICH2 I1 ¼ MAX (I 7 IW, 0) I2 ¼ MIN (I þ IW, NCHAN-1) YBACK(I) ¼ MIN(YBACK(I), 0.5* (YBACK(I1) þ YBACK(I2))) 20 CONTINUE 30 CONTINUE C - - Back transformation DO 40 I ¼ ICH1, ICH2 YBACK(I) ¼ YBACK(I)*YBACK(I)

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40

CONTINUE RETURN END

2. Orthogonal Polynomial Continuum The subroutine OPOLBAC fits the continuum of a pulse-height spectrum using an orthogonal polynomial. Continuum channels are selected by adjusting the weight of the fit (see Sec. IV.C). Input:

Output:

Workspace:

NPTS X Y R YBACK W A,B C SC FAILED

Number of data points (channels) Array of channel numbers Array of spectrum Adjustable parameter [Eqs. (51) and (52)], typical value 2 Array of fitted continuum Array of weights Coefficients of the orthogonal polynomial Fitted parameters of the orthogonal polynomial Uncertainties of C Logical variable TRUE if no convergence after MAXADJ weight adjustments RCHISQ Reduced chi-square value of the fitted continuum WORK1, WORK2 of size NPTS

The routine calls ADJWEIG to adjust the weights. Further, the subroutine ORTPOL is used to fit the polynomial. The iteration (adjustment of weights) stops when all coefficients cj change less than one standard deviation or when the maximum number of iterations is reached.

SUBROUTINE OPOLBAC (NPTS, X, Y, W, YBACK, WORK1, WORK2, > NDEGR, A, B, C, COLD, SC, FAILED, RCHISQ, R) INTEGER NPTS, NDEGR REAL*4 X(NPTS),Y(NPTS),W(NPTS),YBACK(NPTS) REAL*4 WORK1 (NPTS), WORK2 (NPTS) REAL*4 A (NDEGR), B(NDEGR), C(NDEGR), COLD(NDEGR), SC(NDEGR) REAL*4 RCHISQ, R LOGICAL*2 FAILED PARAMETER (MAXADJ ¼ 20) LOGICAL*2 NEXT C -- Initialize DO 10 J ¼ 1, NDEGR COLD(J) ¼ 0. 10 CONTINUE DO 20 I ¼ 1, NPTS YBACK(I) ¼ 0. 20 CONTINUE C - - Main iteration loop DO 100 K ¼ 1, MAXADJ C .. Calculate weights CALL ADJWEIG (NPTS, Y, W, YBACK, R, NBPNTS)

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C .. Fit orthogonal polynomial CALL ORTPOL (NPTS, X, Y, W, YBACK, WORK1, WORK2, > NDEGR, A, B, C, SC, SUMSQ) RCHISQ ¼ SUMSQ=FLOAT (NBPNTS 7 NDEGR) S ¼ SQRT (RCHISQ) C .. Test if further adjustments of weights is required NEXT ¼ .FALSE. DO 30 J ¼ 1, NDEGR SC(J) ¼ S * SC(J) IF( ABS(COLD(J)-C(J)) .GT. SC(J) ) NEXT ¼ .TRUE. COLD(J) ¼ C(J) 30 CONTINUE C .. convergence IF( .NOT.NEXT ) THEN FAILED ¼ .FALSE. RETURN ENDIF 100 CONTINUE C - - No convergence after MAXADJ iterations FAILED ¼ .TRUE. RETURN END SUBROUTINE ADJWEIG (NPTS, Y, W, YFIT, R, NBPNTS) C * * Adjust weights to emphasize the continuum INTEGER NPTS, NBPNTS REAL*4 Y(NPTS), W(NPTS), YFIT(NPTS), R NBPNTS ¼ 0 C - - Loop over all data points DO 10 I ¼ 1, NPTS IF (YFIT(I) .GT. 0.) THEN IF (Y(I).LE. YFIT(I) þ R*SQRT (YFIT(I))) THEN C .. Point is considered as continuum W(I) ¼ 1./YFIT(I) NBPNTS ¼ NBPNTS þ 1 ELSE C .. Point is NOT considered as continuum. W(I) ¼ 1./(Y(I) 7 YFIT(I))**2 ENDIF ELSE C .. Continuum <¼ 0, weight based original data C (initial condition) W(I) ¼ 1./MAX(Y(I),1.) NBPNTS ¼ NBPNTS þ 1 ENDIF 10 CONTINUE RETURN END

D.

Filter-Fit method

The C program FILFIT is a test implementation of the filter-fit method (see Sec. VI). This program simply coordinates all input and output, allocates the required memory, and calls two FORTRAN routines that do the actual work. The subroutine TOPHAT returns the Copyright © 2002 Marcel Dekker, Inc.

convolute of the spectrum with the top-hat filter or the weights (the inverse of the variance of the filtered spectrum). The general-purpose subroutine LINREG is called to perform the multiple linear least-squares fit. The output includes the reduced w2 value, the parameters of the fit aj (which is an estimate of the ratio of the intensity in the analyzed spectrum to the intensity in the standard for the considered X-ray lines), and their standard deviation. The routine ‘‘GETSPEC’’ reads the spectral data and must be supplied by the user. #include #include #include #include

<stdio.h> <malloc.h> <math.h>

void fortran TOPHAT( ); void fortran LINREG( ); float spec[2048]; main( ) { int nchan, first_ch_fit, last_ch_fit, width, ierr; int i, first_ch_ref, last_ch_ref, ref, num_ref, num_points; int filter_mode ¼ 0, weight_mode ¼ 1, ioff; float meas_time, ref_meas_time, *scale_fac; float *x, *xp, *y, *w, *yfit, *a, *sa, chi; double *beta, *alpha; char filename [64]; // input width of tophat filter scanf(‘‘%hd’’, &width); // input spectrum to fit and fitting region scanf(‘‘%s’’, filename); scanf(‘‘%hd %hd’’, &first_ch_fit, &last_ch_fit); nchan ¼ Getspec(spec, filename, &meas_time); num_points ¼ last_ch_fit 7 first_ch_fit þ 1; // filter spectrum and store in y[ ] y ¼ (float *)calloc(num_points, sizeof(float)); TOPHAT(spec, y, &nchan, &first_ch_fit, &last_ch_fit, &width, &filter_mode); //calculate weights of fit and save in w[ ] w ¼ (float *)calloc(num_points, sizeof(float)); TOPHAT(spec, w, &nchan, &first_ch_fit, &last_ch_fit, &width, &weight_mode); //read reference spectra, filter and store in x[ ] scanf(‘‘%hd’’, &num_ref); scale_fac ¼ (float *)calloc(num_ref, sizeof(float)); x ¼ (float *)calloc(num_points*num_ref, sizeof(float)); for(ref ¼ 0; ref < num_ref; refþþ) { scanf(‘‘%s’’, filename); nchan ¼ Getspec(spec, filename, &ref_meas_time); scale_fac[ref] ¼ ref_meas_time/meas_time; scanf(‘‘%hd %hd’’, &first_ch_ref, &last_ch_ref); if (first_ch_ref < first_ch_fit) first_ch_ref ¼ first_ch_fit; if(last_ch_ref > last_ch_fit) last_ch_ref ¼ last_ch-fit; ioff ¼ ref*num_points þ first_ch_ref 7 first_ch_fit; xp ¼ x þ ioff; TOPHAT(spec, xp, &nchan, &first_ch_ref, &last_ch_ref, &width, &filter_mode); }

Copyright © 2002 Marcel Dekker, Inc.

// perform least squares fit yfit ¼ (float *)calloc(num_points, sizeof(float)); a ¼ (float *)calloc(num_ref, sizeof(float)); sa ¼ (float *)calloc(num_ref, sizeof(float)); beta ¼ (double *)calloc(num_ref, sizeof(double)); alpha ¼ (double *)calloc(num_ref* (num_ref þ 1)/2, sizeof(double)); LINREG (y,w,x &num_points, &num_ref, &num_points, &num_ref, yfit, a, sa, &chi, &ierr, beta, alpha); if( ierr ¼ ¼ 0 ) { printf(‘‘Filter fit: Chi-square ¼ %f\n’’, chi); printf(‘‘Standard Int. in analyse spectrum/Int. in standard\n’’); for(i ¼ 0; i < num_ref; i þ þ ) printf(‘‘ %hd %f n%f\n’’, i þ 1, a[i]*scale_fac[i], sa[i]*sacle_fac[i]); for(i ¼ 0; i < num_points; i þ þ ){ printf(‘‘%4hd %7.0f %9.2f’’, first_ch_fit þ i, y[i],yfit[i]); for (ref ¼ 0;ref < num_ref; ref þ þ ) printf(‘‘%7.0f’’, x[ref * num_points þ i]); printf(‘‘\n’’); } } }

SUBROUTINE TOPHAT (IN, OUT, NCHAN, IFRST, ILAST, IWIDTH, MODE) INTEGER*2 NCHAN, IFRST, ILAST, IWIDTH, MODE REAL*4 IN (NCHAN), OUT(1) C ** Tophat filter of width IWIDTH, MODE ¼ 0 calculate filtered specturm, C MODE ! ¼ 0 calculate weights (1/variance of filtered spectrum) C - - Calculate filter constants. IW ¼ IWIDTH IF(MOD(IW,2).EQ.0)IW ¼ IW þ 1 FPOS ¼ 1./FLOAT(IW) KPOS ¼ IW/2 IV ¼ IW/2 FNEG ¼ 7 1./FLOAT(2*IV) KNEG1 ¼ IW/2 þ 1 KNEG2 ¼ IW/2 þ IV N¼0 C - - Loop over all requested channels. DO 30 I ¼ IFRST þ 1, ILAST þ 1 C . . Central positive part, YPOS ¼ 0. DO 10 K ¼ 7 KPOS, KPOS IK ¼ MIN(MAX(I þ K,1),NCHAN) YPOS ¼ YPOS þ IN(IK) 10 CONTINUE C . . Left and right negative part, YNEG ¼ 0. DO 20 K ¼ KNEG1, KNEG2 IK ¼ MIN(MAX(I 7 K,1),NCHAN) YNEG ¼ YNEG þ IN(IK) IK ¼ MIN(MAX(I þ K,1),NCHAN) YNEG ¼ YNEG þ IN(IK)

Copyright © 2002 Marcel Dekker, Inc.

20

30

E.

CONTINUE N¼Nþ1 IF(MODE.EQ.0) THEN OUT(N) ¼ FPOS * YPOS þ FNEG * YNEG ELSE VAR ¼ FPOS*FPOS*YPOS þ FNEG*FNEG*YNEG OUT(N) ¼ 1./MAX(VAR,1.) ENDIF CONTINUE RETURN END

Fitting Using Analytical Functions

The C program NLRFIT is an example implementation of the nonlinear spectrum fitting using an analytical function (see Sec. VII). The program only coordinates input and output. The actual fitting is done using the Marquardt algorithm with the FORTRAN subroutine MARQFIT. The fitting function consists of a polynomial continuum width NB terms and NP Gaussians. The continuum parameters and the area, position, and with of each Gaussian are optimized during the fit. The fitting function is calculated using the routine FITFUNC. The derivatives of the fitting function with respect to the parameters are calculated by the routine DERFUNC. // Program NLRFIT #include <stdio.h> #include <malloc.h> #include #include <math.h> #define MAX_PERKS 10 #define MAX_CHAN 1024 void fortran MARQFIT(); float spec [MAX_CHAN]; // Fortran common block structure COMMON/FITFUN/NB, NP struct common_block {short NB, NP;}; extern struct common_block fortran FITFUN; main( ) { char specfile[64]; int nchan, first_ch_fit, last_ch_fit, nb, np; int i, j, n, num_points, num_param, ierr, max_iter; float ini_pos[MAX_PEAKS], ini_wid[MAX_PEAKS]; float *x, *xp, *y, *w, *yfit, *a, *sa, chi, lamda, crit_dif; float *b, *beta, *deriv, *alpha; double *work; // Input of parameters and spectral data scanf(‘‘%s’’, specfile); scanf(‘‘%hd %hd %hd %f’’, &first_ch_fit, &last_ch_fit, &max_iter, &crit_dif); scanf(‘‘%hd %hd’’, &np, &nb); for(i ¼ 0; i < np; i þ þ ) scanf(‘‘%f %f’’, &ini_pos[i], &ini_wid[i]);

Copyright © 2002 Marcel Dekker, Inc.

nchan ¼ GetSpec(spec, specfile); num_points ¼ last_ch_fit 7 first_ch_fit þ 1; num_param ¼ nb þ 3*np; //Allocate memory for y[ ], w[ ], x[ ] y ¼ (float *)calloc(num_points, sizeof(float)); w ¼ (float *)calloc(num_points, sizeof(float)); x ¼ (float *)calloc(num_points, sizeof(float)); // Store independent variable (spectrum), weights and dep. var (channel #) for(i ¼ first_ch_fit, n ¼ 0; i < ¼ last_ch_fit; i þ þ , n þ þ ) { y[n] ¼ spec[i]; w[n] ¼ (spec[i] > 0.) ? 1./spec[i] : 1.; x[n] ¼ (float)i; } // allocate memory for other arrays required yfit ¼ (float *)calloc(num_points, sizeof(float)); a ¼ (float *)calloc(num_param, sizeof(float)); sa ¼ (float *)calloc(num_param, sizeof(float)); b ¼ (float *)calloc(num_param, sizeof(float)); beta ¼ (float *)calloc(num_param, sizeof(float)); deriv ¼ (float *)calloc(num_param, sizeof(float)); alpha ¼ (float *)calloc(num_param*(num_param þ 1)/2, sizeof(float)); work ¼ (double *)calloc(num_param*(num_param þ 1)/2, sizeof(double)); // initialize, all linear parameters to zero, peak position and width // to their initial guesses lamda ¼ 0.001; for(i ¼ 0; i < np; i þ þ ){ a[nb þ np þ i] ¼ ini_pos[i]; a[nb þ 2*np þ i] ¼ ini_wid[i]; } // perform least squares fit FITFUN.NP ¼ np; FITFUN.NB ¼ nb; MARQFIT(&ierr, &chi, &lamda, &crit_dit, &max_iter, x, y, w, yfit, &num_points, a, sa, &num_param, b, beta, deriv, alpha, work); if( ierr ¼ ¼ 0 ) { printf(‘‘\nNon-linear fit: Chi-square ¼ %f\n’’, chi); printf(‘‘Polynomial continuum parameters\n’’); for(i ¼ 0; i < nb; i þ þ ) printf(‘‘%hd %f n %f\n’’, i þ 1, a[i], sa[i]); printf(‘‘Peak parameters Area Position Width\n’’); for(i ¼ 0; i < np; i þ þ ){ printf(‘‘%hd %10.0f n % 7 10.0f’’, i þ 1, a[nb þ i], sa[nb þ i]); printf(‘‘%10.3f n % 7 10.3f’’, a[nb þ np þ i], sa[nb þ np þ i]); printf(‘‘%10.3f n % 7 10.3f\n’’, a[nb þ 2*np þ i], sa[nb þ 2*np þ i]); } for (i ¼ 0; i < num_points; i þ þ ) printf(‘‘%4hd %7.0f %9.2f\n’’, first_ch_fit þ i, y[i], yfit[i]); } }

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SUBROUTINE FITFUNC(X, YFIT, NPTS, A, NTERMS ) REAL*4 X(NPTS), YFIT(NPTS), A(NTERMS) COMMON /FITFUN/ NB, NP C ** Fitting function, polynomial continuum and NP gaussians C with position, width and area as parameters PARAMETER(SQR2PI ¼ 2.50663) C - - Loop over all channels DO 100 I ¼ 1, NPTS C . . continuum YFIT(I) ¼ A(1) DO 20 J ¼ 2, NB YFIT(I) YFIT(I) þ A(J) * X(I)**(J 7 1) 20 CONTINUE C . . Peaks DO 30 K ¼ 1, NP AREA ¼ A(NB þ K) POS ¼ A(NB þ NP þ K) SWID ¼ A(NB þ 2*NP þ K) Z ¼ ((POS 7 X(I))/SWID)**2 IF( Z.LT.50.) THEN G ¼ EXP( 7 Z/2.)/SWID/SQR2PI YFIT(I) ¼ YFIT(I) þ AREA*G ENDIF 30 CONTINUE 100 CONTINUE RETURN END SUBROUTINE DERFUNC(X, NPTS, A, NTERMS, DERIV, I) REAL*4 X(NPTS), A(NTERMS), DERIV(NTERMS) COMMON/FITFUN/NB, NP C ** Derivatives of fitting function: polynomial continuum and NP Gaussians C with postion, width and area as parameters PARAMETER (SQR2PI ¼ 2.50663) C -- Derivatives of function with respect to the continuum parameters DERIV(1) ¼ 1. DO 10 J ¼ 2, NB DERIV(J) ¼ X(I)**(J 7 1) 10 CONTINUE C -- Derivatives of function with respect to the peak parameters DO 30 K ¼ 1, NP AREA ¼ A(NB þ K) POS ¼ A(NB þ NP þ K) SWID ¼ A(NB þ 2*NP þ K) Z ¼ ((POS 7 X(I))/SWID)**2 IF(Z.LT.50.) THEN G ¼ EXP( 7 Z/2.)/SWID/SQR2PI C . . Peak area DERIV(NB þ K) ¼ G C . . Peak position DERIV(NB þ NP þ K) ¼ 7 AREA*G*(POS 7 X(I))/SWID/SWID C . . Peak width DERIV(NB þ 2*NP þ K) ¼ AREA*G*(Z 7 1.)/SWID ELSE

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DERIV(NB þ K) ¼ 0. DERIV(NB þ NP þ K) ¼ 0. DERIV(NB þ 2*NP þ K) ¼ 0. ENDIF CONTINUE RETURN END

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F.

Monte Carlo Methods

1. Uniform Random-Number Generator The function URAND is a FORTRAN function returning uniform distributed random numbers in the interval 0  U < 0. The random-number generator is based on Knuth’s ‘‘subtractive’’ method (Press et al., 1988) (see Sec. VIII. A) Input: Output:

ISEED URAND

Set to any negative number to initialize the random generator Uniform random number in the interval 0  URAND < 1

REAL*4 FUNCTION URAND (ISEED) INTEGER*2 ISEED REAL*4 UTABLE(56) REAL*4 UBIT, USEED PARAMETER (UBIG ¼ 4000000., USEED ¼ 1618033.) SAVE I1, I2, UTABLE, INIT C -- Initialize table IF (ISEED.LT.0 .OR. INIT.EQ.0) THEN U ¼ USEED þ FLOAT(ISEED) U ¼ MOD(U,UBIG) UTABLE(55) ¼ U UTMP ¼ 1. DO 10 I ¼ 1, 54 II ¼ MOD(I*21, 55) UTABLE(II) ¼ UTMP UTMP ¼ U 7 UTMP IF(UTMP.LT.0.) UTMP ¼ UTMP þ UBIG U ¼ UTABLE(II) 10 CONTINUE DO 30 K ¼ 1, 4 DO 20 I ¼ 1, 55 UTABLE(I) ¼ UTABLE(I) 7 UTABLE(1 þ MOD(I þ 30,55)) IF(UTABLE(I).LT.0) UTABLE(I) ¼ UTABLE(I) þ UBIG 20 CONTINUE 30 CONTINUE I1 ¼ 0 I2 ¼ 31 ISEED ¼ 1 INIT ¼ 1 ENDIF C -- Get next ‘‘random’’ number I1 ¼ I1 þ 1 IF(I1.EQ.56) I1 ¼ 1

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I2 ¼ I2 þ 1 IF(I2.EQ.56) I2 ¼ 1 U ¼ UTABLE(I1) 7 UTABLE(I2) IF(U.LT.0.)U ¼ U þ UBIG UTABLE(I1) ¼ U URAND ¼ U/UBIG RETURN END

2. Normal Distributed Random Deviate The function NRAND returns normal distributed random number with zero mean and unit variance, using the Box
Muller method (see Sec. VIII. B). Input: Output:

ISEED NRAND

Set to any negative number to initialize the random sequence Normal distributed random deviate with zero mean and unit variance.

REAL*4 FUNCTION NRAND (ISEED) INTEGER*2 ISEED SAVE NEXT, FAC, V1, V2 IF (NEXT.EQ.0 .OR. ISEED.LT.0) THEN 10 CONTINUE V1 ¼ 2. * URAND(ISEED) 7 1. V2 ¼ 2. * URAND(ISEED) 7 1. R ¼ V1*V1 þ V2*V2 IF(R.G.E.1. .OR. R. EQ. 0.) GOTO 10 FAC ¼ SQRT( 7 2. * LOG(R)/R) NRAND ¼ V1*FAC NEXT ¼ 1 ELSE NRAND ¼ V2*FAC NEXT ¼ 0 ENDIF RETURN END

3. Poisson Distributed Random Deviate The function ‘‘PRAND’’ can be used to produce approximately Poisson distributed random deviates. For small numbers ( < 20), the direct method is used; for larger numbers, the Poisson distribution is approximated by the Normal distribution. Input: Output:

Y ISEED PRAND

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(population) mean of deviate Set to any negative number to initialize the random sequence Poisson distributed random deviate with mean Y

REAL *4 FUNCTION PRAND(Y, ISEED) INTEGER*2 ISEED REAL*4 Y REAL*4 NRAND, URAND IF(Y.LT.20.) THEN C -- Use direct method G ¼ EXP( 7 Y) PRAND ¼ 7 1. T ¼ 1. 10 CONTINUE PRAND ¼ PRAND þ 1. T ¼ T * URAND(ISEED) IF(T.GT.G) GOTO 10 ELSE C -- Approximate by normal distribution PRAND ¼ Y þ SQRT(Y) * NRAND(ISEED) ENDIF RETURN END

G.

Least-Squares Procedures

1. Linear Regression Subroutine LINREG is a general-purpose (multiple) linear regression routine (see Sec. IX. A). Input:

Output:

Workspace:

Y W X N M NMAX,MMAX YFIT A SA CHI IERR BETA

Array of dependent variable Array of weights 1/s2i ) Matrix of independent variables Number of data points Number of independent variables (columns of X) Size of X matrix Array of fitted Y values Estimated least-squares parameters Standard deviation of A Chi-square value Error condition,
1 if fit failed (singular matrix) of size M

SUBROUTINE LINREG (Y, W, X, N, M, NMAX, MMAX, > YFIT, A, SA, CHI, IERR, BETA, ALPHA) INTEGER*2 N, M, NMAX, MMAX, IERR REAL*4 Y(N), W(N), YFIT(N), A(M), SA(M), CHI REAL*4 X(NMAX, MMAX) REAL*8 BETA(M), ALPHA(1) c Accumulate BETA and ALPHA matrices JK ¼ 0 DO 10 J ¼ 1, M BETA(J) ¼ 0.0D0 DO 2 I ¼ 1, N

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BETA(J) ¼ BETA(J) þ W(I)*Y(I)*X(I, J) CONTINUE DO 6 K ¼ 1, J JK ¼ JK þ 1 ALPHA(JK) ¼ 0.0D0 DO 4 I ¼ 1, N ALPHA(JK) ¼ ALPHA(JK) þ W(I)*X(I,K)*X(I,J) 4 CONTINUE 6 CONTINUE 10 CONTINUE Invert ALPHA matrix CALL LMINV (ALPHA, M, IERR) IF(IERR .EQ. 7 1) THEN RETURN ENDIF Calculate fitting parameters A DO 20 J ¼ 1, M A(J) ¼ 0. JJ ¼ J*(J 7 1)/2 DO 12 K ¼ 1, J JK ¼ K þ JJ A(J) ¼ A(J) þ ALPHA(JK)*BETA(K) 12 CONTINUE DO 14 K ¼ J+1, m JK ¼ J þ K* (K 7 1)/2 A(J) ¼ A(J) þ ALPHA(JK)*BETA(K) 14 CONTINUE 20 CONTINUE Calculate uncertainties in the parameters DO 30 J ¼ 1, M JJ ¼ J*(J þ 1)/2 SA(J) ¼ DSQRT(ALPHA(JJ)) 30 CONTINUE Calculate fitted values and Chi-square CHI ¼ 0. DO 40 I ¼ 1, N YFIT(I) ¼ 0. DO 32 J ¼ 1, M YFIT(I) ¼ YFIT(I) þ A(J)*(I, J) 32 CONTINUE CHI ¼ CHI þ W(I)*(YFIT(I) 7 Y(I))**2 40 CONTINUE CHI ¼ CHI/FLOAT(N 7 M) RETURN END 2

c

c

c

c

2. Orthogonal Polynomial Regression Subroutine ORTPOL fits an orthogonal polynomial to a set of data points [xi, yi, wi] (see Sec. IX.B).

Copyright © 2002 Marcel Dekker, Inc.

Input:

Output:

Workspace:

NPTS X Y W NDEGR A, B C SC SUMSQ PJ, PJMIN

Number of data points Array of independent variable Array of dependent variable Array of weights (wi ¼ 1s2i ) Degree of orthogonal polynomial to be fitted Parameters of the orthogonal polynomials Fitted orthogonal polynomial coefficients Standard deviation of C Chi-square value of size NPTS

SUBROUTINE ORTPOL (NPTS, X, Y, W, YFIT, PJ, PJMIN, NDEGR, A, B, C, SC, > SUMSQ) INTEGER NPTS, NDEGR REAL*4 X(NPTS), Y(NPTS), W(NPTS), YFIT(NPTS) REAL*4 PJ(NPTS), PJMIN(NPTS) REAL*4 A(NDEGR), B(NDEGR), C(NDEGR), SC(NDEGR), SUMSQ C - - Initialize DO 10 I ¼ 1, NPTS PJ(I) ¼ 1. PJMIN(I) ¼ 0. YFIT(I) ¼ 0. 10 CONTINUE GAMJMIN ¼ 1. C - - Loop over all polynomial terms DO 100 J ¼ 1, NDEGR C .. Accumulate normalization factor, A and B constants for term j GAMJ ¼ 0. A(J) ¼ 0. B(J) ¼ 0. DO 20 I ¼ 1, NPTS GAMJ ¼ GAMJ þ W(I)*PJ(I)*PJ(I) A(J) ¼ A(J) þ W(I)*X(I)*PJ(I)*PJ(I) B(J) ¼ A(J) þ W(I)*X(I)*PJ(I)*PJMIN(I) 20 CONTINUE A(J) ¼ A(J) / GAMJ B(J) ¼ B(J) / GAMJMIN C .. Least squares estimate of coefficient C C(J) ¼ 0. DO 30 I ¼ 1, NPTS C(J) ¼ C(J) þ W(I)*Y(I)*PJ(I) 30 CONTINUE C(J) ¼ C(J)=GAMJ SC(J) ¼ SQRT(1.=GAMJ) C .. Contribution of this term to the fit DO 40 I ¼ 1, NPTS YFIT(I) ¼ YFIT(I) þ C(J)*PJ(I) 40 CONTINUE C .. Next polynomial term IF (J .LT. NDEGR) THEN

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DO 50 I ¼ 1, NPTS PJPLUS ¼ (X(I) 7 A(J))*PJ(I) 7 B(J)*PJMIN(I) PJMIN(I) ¼ PJ(I) PJ(I) ¼ PJPLUS 50 CONTINUE GAMJMIN ¼ GAMJ ENDIF 100 CONTINUE C -- Weighted sum of squares value SUMSQ ¼ 0. DO 110 I ¼ 1, NPTS SUMSQ ¼ SUMSQ þ W(I)*(Y(I) 7 YFIT(I))**2 110 CONTINUE RETURN END

3. Nonlinear Regression The subroutine MARQFIT performs nonlinear least-squares-fitting according to the Marquardt algorithm (see Sec. IX.C). Input:

Output:

Workspace:

MAXITER X Y W NPTS NTERMS A IERR CHISQR FLAMDA YFIT A SA CRIDIF B, BETA, DERIV ALFA, ARR

Maximum number of iterations Array of independent variable Array of dependent variable Array of weights (wi ¼ 1=s2i ) Number of data points Number of parameters Array of initial values of the parameters Error status, 7 1 indicates failure of fit Reduced chi-square value Marquardt control parameter Array of fitted data points Least-squares estimate of the fitting parameters Standard deviation of A Minimum percent difference in two chi-square values to stop the iteration of size NTERMS of size NTERMS  (NTERMS þ 1)=1

The routine requires two user-supplied subroutines: FITFUNC to evaluate the fitting function y(i) with the current set of parameters a and the DERFUNC to calculate the derivatives of the fitting function with respect to the parameters. SUBROUTINE MARQFIT (IERR, CHISQR, FLAMDA, CRIDIF, MAXITER, > X, Y, W, YFIT, NPTS, A, SA, NTERMS, > B, BETA, DERIV, ALFA, ARR) INTEGER*2 IERR, NPTS, NTERMS REAL*4 CHISQR, FLAMDA, CRIDIF REAL*4 X(NPTS), Y(NPTS), W(NPTS), YFIT(NPTS) REAL*4 A(NTERMS), SA(NTERMS) REAL*4 B(1), BETA(1), DERIV(1), ALFA(1) REAL*8 ARR(1) PARAMETER (FLAMMAX ¼ 1E4, FLAMMIN ¼ 1E 7 6)

Copyright © 2002 Marcel Dekker, Inc.

C -- Evaluate the fitting function YFIT for the current parameters C and save the Chi-square value NITER ¼ 0 CALL FITFUNC(X, YFIT, NPTS, A, NTERMS) CHISQR ¼ CHIFIT(Y, YFIT, W, NPTS, NTERMS) FLAMDA ¼ 0. C -- Set ALFA and BETA to zero, save the current value of the parameters A 100 CONTINUE NITER ¼ NITER þ 1 CHISAV ¼ CHISQR DO 110 J ¼ 1, NTERMS B(J) ¼ A(J) BETA(J) ¼ 0. 110 CONTINUE DO 112 J ¼ 1, NTERMS*(NTERMS þ 1)=2 ALFA(J) ¼ 0. 112 CONTINUE C -- Accumulate Alpha and Beta matrices DO 120 I ¼ 1, NPTS D ¼ Y(I) 7 YFIT(I) C .. Calculate derivatives at point i CALL DERFUNC (X, NPTS, A, NTERMS, DERIV, I) DO 120 J ¼ 1, NTERMS BETA(J) ¼ BETA(J) þ W(I)*D*DERIV(J) JJ ¼ J*(J 7 1)=2 DO 120 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ ALFA(JK) þ W(I)*DERIV(J)*DERIV(K) 120 CONTINUE C -- Test and scale ALFA matrix DO 140 J ¼ 1, NTERMS JJ ¼ J*(J 7 1)=2 JJJ ¼ JJ þ J IF(ALFA(JJJ) .LT. 1.E 7 20) THEN DO 130 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ 0. 130 CONTINUE ALFA (JJJ) ¼ 1. BETA(J) ¼ 0. ENDIF SA(J) ¼ SQRT (ALFA(JJJ)) 140 CONTINUE DO 160 J ¼ 1, NTERMS JJ ¼ J*(J 7 1)=2 DO 150 K ¼ 1, J JK ¼ JJ þ K ALFA(JK) ¼ ALFA(JK)=SA(J)=SA(K) 150 CONTINUE 160 CONTINUE C -- Store ALFA in ARR, modify the diagonal elements with FLAMDA 200 CONTINUE DO 210 J ¼ 1, NTERMS

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JJ ¼ J*(J 7 1)=2 DO 205 K ¼ 1, J JK ¼ JJ þ K ARR (JK) ¼ DBLE(ALFA(JK)) 205 CONTINUE JJJ ¼ JJ þ J ARR(JJJ) ¼ DBLE(1. þ FLAMDA) 210 CONTINUE C -- Invert matrix ARR CALL LMINV (ARR, NTERMS, IERR) IF (IERR .NE. 0) RETURN C -- Calculate new values of parameters A DO 220 J ¼ 1, NTERMS DO 220 K ¼ 1, NTERMS IF(K .GT. J) THEN JK ¼ J þ K*(K 7 1)=2 ELSE JK ¼ K þ J*(J 7 1)=2 ENDIF A(J) ¼ A(J) þ ARR(JK) = SA(J)*BETA(K) = SA(K) 220 CONTINUE C -- Evaluate the fitting function YFIT for the new parameters and Chi-square CALL FITFUNC(X, YFIT, NPTS, A, NTERMS) CHISQR ¼ CHIFIT(Y, YFIT, W, NPTS, NTERMS) IF (NITER .EQ. 1) FLAMDA ¼ 0.001 C -- Test new parameter set IF (CHISQR .GT. CHISAV) THEN C .. Iteration NOT succesful, increase flamda and try again FLAMDA ¼ MIN(FLAMDA * 10., FLAMMAX) DO 300 J ¼ 1, NTERMS A(J) ¼ B(J) 300 CONTINUE GOTO 200 ENDIF C .. Iteration succesful, decrease LAMDA FLAMDA ¼ MAX(FLAMDA=10., FLAMMIN) C .. Get next better estimate if required PERDIF ¼ 100.*(CHISAV 7 CHISQR)=CHISQR IF (NITER .LT. MAXITER .AND. PERDIF .GT. CRIDIF) GOTO 100 C -- Calculate standard deviations and return DO 320 J ¼ 1, NTERMS JJ ¼ J*(J þ 2)=2 SDEV ¼ DSQRT(ARR(JJ)) = SA(J) SA(J) ¼ SDEV 320 CONTINUE RETURN END FUNCTION CHIFIT(Y, YFIT, W, NPTS, NTERMS) REAL*4 Y(NPTS), YFIT(NPTS), W(NPTS) C ** Evaluate chi-square CHI ¼ 0. DO 300 I ¼ 1, NPTS

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300

CHI ¼ CHI þ W(I) * (Y(I) 7 YFIT(I))**2 CONTINUE CHI ¼ CHI = FLOAT(NPTS 7 NTERMS) CHIFIT ¼ CHI RETURN END

4. Matrix Inversion Subroutine LMINV is a general-purpose routine to invert a symmetric matrix. Input:

ARR

Output:

N IERR

Upper triangle and diagonal of real symmetric matrix stored in linear array, size ¼ N(N þ 1)=2. Order of matrix (number of columns) Error status, IERR ¼ 0 inverse obtained, IERR ¼
1 singular matrix

SUBROUTINE LMINV ( ARR, N, IERR ) INTEGER*2 N, IERR REAL*8 ARR(1) REAL*8 DIN, WORK, DSUM, DPIV INTEGER*2 I, IND, IPIV, J, K, KEND, KPIV, L, LANF, > LEND, LHOR, LVER, MIN KPIV ¼ 0 DO 10 K ¼ 1, N KPIV ¼ KPIV þ K IND ¼ KPIV LEND ¼ K 7 1 DO 4 I ¼ K, N DSUM ¼ 0.DO IF (LEND .GT. 0) THEN DO 2 L ¼ 1, LEND DSUM ¼ DSUM þ ARR(KPIV 7 L) * ARR(IND 7 L) 2 CONTINUE ENDIF DSUM ¼ ARR(IND) 7 DSUM IF (I .EQ. K) THEN IF (DSUM .LE. 0.D0) THEN IERR ¼ 7 1 RETURN ENDIF DPIV ¼ DSQRT(DSUM) ARR(KPIV) ¼ DPIV DPIV ¼ 1.D0 = DPIV ELSE ARR(IND) ¼ DSUM * DPIV ENDIF IND ¼ IND þ I 4 CONTINUE 10 CONTINUE IERR ¼ 0 IPIV ¼ N*(N þ 1)=2

Copyright © 2002 Marcel Dekker, Inc.

12

14

20

22

24 30

IND ¼ IPIV DO 20 I ¼ 1, N DIN ¼ 1.DO = ARR(IPIV) ARR(IPIV) ¼ DIN MIN ¼ N KEND ¼ I 7 1 LANF ¼ N 7 KEND IF (KEND .GT. 0) THEN J ¼ IND DO 14 K ¼ 1, KEND WORK ¼ 0.D0 MIN ¼ MIN 7 1 LHOR ¼ IPIV LVER ¼ J DO 12 L ¼ LANF, MIN LVER ¼ LVER þ 1 LHOR ¼ LHOR þ L WORK ¼ WORK þ ARR(LVER) * ARR(LHOR) CONTINUE ARR(J) ¼ 7 WORK * DIN J ¼ J 7 MIN CONTINUE ENDIF IPIV ¼ IPIV 7 MIN IND ¼ IND 7 1 CONTINUE DO 30 I ¼ 1, N IPIV ¼ IPIV þ I J ¼ IPIV DO 24 K ¼ I, N WORK ¼ 0.D0 LHOR ¼ J DO 22 L ¼ K, N LVER ¼ LHOR þ K 7 I WORK ¼ WORK þ ARR(LVER) * ARR(LHOR) LHOR ¼ LHOR þ L CONTINUE ARR(J) ¼ WORK J¼JþK CONTINUE CONTINUE IERR ¼ 0 RETURN END

REFERENCES Aarnio PA, Lauranto H. Nucl Instrum Methods A276:608 (1989). Bertin EP. Principles and Practice of X-ray Spectrometric Analysis. New York: Plenum Press, 1970. Bevington PR, Robinson DK. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1992. Black WW. Nucl Instrum Methods 71:317, 1969.

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Bloomfield DJ, Love G. X-Ray Spectrom 14:8, 1985. Bombelka E, Koenig W, Richter FW. Nucl Instrum Methods B22:21, 1987. Breschinsky R, Krush E, Wehrse R. Diplomarbeit, Fachbereich Physik. Bremen: Universita¨t Bremen, 1979. Brook D, Wynne RJ. Signal Processing Principles and Applications. London: Edward Arnold, 1988. Campbell JL. X-Ray Spectrom 24:307, 1995. Campbell JL. Nucl Instrum Methods B109=110:71, 1996. Campbell JL, Perujo A, Millman BM. X-Ray Spectrom 16:195, 1987. Campbell JL, Maxwell JA, Papp T, White G. X-Ray Spectrom 26:223, 1997. Campbell JL, Millman BM, Maxwell JA, Perujo A, Teesdale WJ. Nucl Instrum Methods B9:71, 1985. Campbell JL, Maenhaut W, Bombelka E, Claytan E, Malmqvist K, Maxwell JA, Pallon J, Vandenhaute J. Nucl Instrum Methods B14:204, 1986. Campbell JL, Cauchon G, Le´py M-C, McDonald L, Plagnard J, Stemmler P, Teesdale WJ, White G. Nucl Instrum Methods A418:394, 1998. Cirone R, Gigante GE, Gualtieri G. X-Ray Spectrom 13:110, 1984. Clayton E, Duerden P, Cohen DD. Nucl Instrum Methods B22:64, 1987. Doster JM, Gardner RP. X-Ray Spectrom 11:173, 1982a. Doster JM, Gardner RP. X-Ray Spectrom 11:181, 1982b. Duffy CJ, Rogers PSZ, Benjamin TM. Nucl Instrum Methods B22:91, 1987. Enke CG, Nieman TA. Anal Chem 48:705A, 1976. Fiori CE, Myklebust RL, Gorlen K. NBS Spec Pub 604, 233, 1981. Gardner RP, Yacout AM, Zhang J, Verghese K. Nucl Instrum Methods A242:299, 1986. Geladi P, Kowalski BR. Anal Chim Acta 185:1, 1986. Gertner I, Heber O, Zajfman J, Zajfman D, Rosner B. Nucl Instrum Methods B36:74, 1989. Gunnink R. Nucl Instrum Methods 143:145, 1977. Heckel J, Scholz W. X-Ray Spectrom 16:181, 1987. Hertogen J, De Donder J, Gijbels R. Nucl Instrum Methods 115:197, 1974. Hoskuldsson A. J Chemometr 2:211, 1988. Jansson PA. Deconvolution with Applications in Spectroscopy. New York: Academic Press, 1984. Janssens K, Van Espen P. Anal Chim Acta 184:117, 1986. Janssens K, Dorrine´ W, Van Espen P. Chemometr Intell Lab Syst 4:147, 1988. Janssens K, Vincze L, Van Espen P, Adams F. X-Ray Spectrom 22:234, 1993. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. New York: Marcel Dekker, 1981. Jensen BB, Pind N. Anal Chim Acta 117:101, 1985. Johansson GI. X-Ray Spectrom 11:194, 1982. Joy DC. Rev Sci Instrum 56:1772, 1985. Kajfosz J, Kwiatek WM. Nucl Instrum Methods B22:78, 1987. Lemberge P, Van Espen PJ. X-Ray Spectrom 28:77, 1999. Lorber A. Wangen LE, Kowalski BK. J Chemometr 1:19, 1987. Lucas-Tooth HJ, Price BJ, Metallurgia 64: 149, 1961. Maenhaut W, Vandenhaute J. Bull Soc Chim Belg 95:407, 1986. Manne R. J Chemometr Intell Lab Syst 2:187, 1987. Marageter E, Wegscheider W, Mu¨ller K. Nucl Instrum Methods B1:137, 1984a. Marageter E, Wegscheider W, Mu¨ller K. X-Ray Spectrom 13:78, 1984b. Marquardt DW. J Soc Ind Appl Math 11:431, 1963. Martens H, Naes T. Multivariate Calibration. Chichester: John Wiley, 1989. Massart DL, Vandeginste BGM, Deming SN, Michotte Y, Kaufman K. Chemometrics: A Textbook. Amsterdam: Elsevier, 1988. McCarthy JJ, Schamber FH. NBS Spec Publ 604:273, 1981. McCrary JH, Singman LV, Ziegler LH, Looney LD, Edmonds CM, Harris CE. Phys Rev A4:1745, 1971.

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McCullagh H. Report EGG-PHYS-5890, Idaho National Engineering Laboratory, Idaho Falls, ID, 1982. McNelles LA, Campbell JL. Nucl Instrum Methods 127:73, 1975. Molt K, Schramm R. Fresenius J Anal Chem 359:61, 1997. Nielson KK. X-Ray Spectrom 7:15, 1978. Nullens H, Van Espen P, Adams F. X-Ray Spectrom 8:104, 1979. Nunez J, Rebollo Neira LE, Plastino A, Bonetto RD, Gue´rin DMA, Alvarez AG. X-Ray Spectrom 17:47, 1988. Op De Beeck JP, Hoste J. Atomic Energy Rev 13:151, 1975. Pella PA Feng L, Small JA. X-Ray Spectrom 14:125, 1985. Pessara W, Debertin K. Nucl Instrum Methods 184:497, 1981. Petersen W, Ketelsen P, Kno¨chel A. Nucl Instrum Methods A245:535, 1986. Phillips GW. Nucl Instrum Methods 153:449, 1978. Philips GW, Marlow KW. Nucl Instrum Methods 137:525, 1976. Pratar A, De Jong S. J Chemometr 11:311, 1997. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical Recipes in C, The Art of Scientific Computing. Cambridge: Cambridge University Press, 1988. Reed SJB, Ware NG. J Phys E5:582, 1972. Robertson A, Prestwich WV, Kennett TJ. Nucl Instrum Methods 100:317, 1972. Routti JT, Prussin SG. Nucl Instrum Methods 72:125, 1969. Ryan CG, Clayton E, Griffin WL, Sie SH, Cousens DR. Nucl Instrum Methods B34:396, 1988. Salem SI, Wimmer RJ. Phys Rev A2:1121, 1970. Salem SI, Saunders BG, Melson C. Phys Rev A1:1563, 1970. Savitzky A, Golay MJE. Anal Chem 36:1627, 1964. Schamber FH. In: Dzubay T, ed. X-Ray Fluorescence Analysis of Environmental Analysis. Ann Arbor, MI: Ann Arbor Science, 1977, p. 241. Schreier F. J Quant Spectros Radiat Transfer 48:743, 1992. Schwalbe LA, Trussell HJ. X-Ray Spectrom 10:187, 1981. Scofield JH. Phys Rev 179:9, 1970. Scofield JH. Phys Rev A9:1041, 1974a. Scofield JH. Phys Rev A10:1507, 1974b. Sherry WM, Vander Sande JB. X-Ray Spectrom 6:154, 1977. Smith DGW, Gold CM, Tomlinson DA. X-Ray Spectrom 4:149, 1975. Statham PJ. X-Ray Spectrom 5:16, 1976a. Statham PJ. X-Ray Spectrom 5:154, 1976b. Statham PJ. X-Ray Spectrom 7:132, 1978. Statham PJ, Nashashibi T. In: Newbury DE, ed. Microbeam Analysis. San Francisco: San Francisco Press, 1988, p. 50. Steenstrup S. J Appl Crystallogr 14:226, 1981. Swerts J, Van Espen P. Anal Chem 65:1181, 1993. Urba´nski P, Kowalska E. X-Ray Spectrom 24:70, 1995. Van Dyck P, Van Grieken R. X-Ray Spectrom 12:111, 1983. Van Espen P, Adams F. X-Ray Spectrom 5:123, 1976. Van Espen P, Nullens H, Adams F. Anal Chem 51:1325, 1979a. Van Espen P, Nullens H, Adams F. Nucl Instrum Methods 142:243, 1977a. Van Espen P, Nullens H, Adams F. Nucl Instrum Methods 145:579, 1977b. Van Espen P, Nullens H, Adams F. X-Ray Spectrom 9:126, 1980. Van Espen P, Janssens K, Nobels J. Chemometr Intell Lab Syst 1:109, 1986. Van Espen P, Nullens H, Maenhaut W. In: Newbury DE, ed. Microbeam Analysis. San Francisco, San Francisco Press, 1979b, p. 265. Van Espen P, Nullens H, Adams F. Anal Chem 51:1580, 1979c. Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. X-Ray Spectrom 23:275, 1994. Vekemans B, Janssens K, Vincze L, Adams F, Van Espen P. Spectrochim Acta 50B:149, 1995.

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Verghese K, Mickael M, He T, Gardner RP. Adv X-Ray Anal 31:461, 1988. Vincze L, Janssens K, Adams F. Spectrochim Acta 48B:553, 1993. Vincze L, Janssens K, Adams F, Jones KW. Spectrochim Acta 50B:1481, 1995a. Vincze L, Janssens K, Vekemans B, Adams F. Spectrochim Acta 54B:1711, 1999. Vincze L, Janssens K, Adams F, Rivers ML, Jones KW. Spectrochim Acta 50B:127, 1995b. Wa¨tjen U. Nucl Instrum Methods B22:29, 1987. Wilkinson DH. Nucl Instrum Methods 95:259, 1971. Wielopolski L, Gardner RP. Nucl Instrum Methods 165:297, 1979. Yacout AM, Dunn WL. Adv X-Ray Anal 30:113, 1987. Yacout AM, Gardner RP, Verghese K. Nucl Instrum Methods A243:121, 1986. Yacout AM, Gardner RP, Verghese K. Adv X-Ray Anal 30:121, 1987. Yule PH. Nucl Instrum Methods 54:61, 1967. Zolnai L, Szabo´, Gy. Nucl Instrum Methods B34:118, 1988.

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5 Quantification of Infinitely Thick Specimens by XRF Analysis Johan L. de Vries* Eindhoven, The Netherlands

Bruno A. R. Vrebos Philips Analytical, Almelo, The Netherlands

I.

INTRODUCTION

Quantitative x-ray fluorescence (XRF) analysis involves the conversion of measured fluorescent intensities to the concentration of the analytes. In most of the literature on this subject, the measured intensities at this stage of the analytical procedure are assumed to be corrected for background and line overlap. The same assumptions will be made here, although some comments will be made on these topics throughout this chapter. Furthermore, energy-dispersive and wavelength-dispersive x-ray spectrometers differ significantly with respect to the counting channel and the resolution or resolving power that can be achieved. This has a direct influence on, for example, the line overlap correction procedures that need to be applied (if any) to obtain net intensities. Wavelength-dispersive x-ray fluorescence spectrometry has been discussed in Chapter 2; energy-dispersive systems are the subject of Chapters 3 (for x-ray tube excitation) and 7 (for radioisotope excitation). The software provided by the vendors of x-ray spectrometry systems for their customers compensates to some extent for the peculiarities of each design and considerable differences in the software packages offered with the instruments are to be expected. It is therefore difficult to give guidelines that are generally applicable. In some cases, most notably those where the intensity of the background is essentially constant (between specimens), the correction for background can be neglected. This is usually the case when the difference in terms of composition of the standard specimens and the unknowns is rather small [i.e., the concentration ranges (not necessarily the concentration levels) of all components are rather small]. This condition of similarity between specimens is sufficient to ensure constant background, but it is not a requirement, as, in certain cases, the intensity of the background is (nearly) constant even when the concentration range of one or more analytes is quite large. Note that the only requirement

*Retired.

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here is for the background to be constant; at this point, no restrictions on the magnitude of the background have been imposed. Under these circumstances, correction for background is not required and the gross count rate (or intensity) can be used with equal success as the net intensity for the determination of the specimen’s composition. The situation with line overlap is somewhat more complicated, as it is almost never constant. This is due to the intensity of the overlapping line(s): It is very rarely a constant for different specimens. There are several methods to correct for line overlap, and the preference of the individual analyst as well as the type of spectrometer and its associated software (if any) are some of the factors influencing the final choice. The basic principles of quantitative analysis have not changed much since the early years of x-ray fluorescence analysis, especially for the wavelength-dispersive systems. Energy-dispersive systems have seen more changes due to, for example, development of different solid-state detectors and primary optics, such as capillaries (see Chapter 11), and the increasingly more easily available computing power which allowed for more comprehensive spectral treatment programs at the fingertips of the analysts (see also Chapter 4). In general, the method has always involved some calibration step, during which the intensities of selected elements in a suite of standard specimens are measured according to a scheme or a recipe that has been optimized earlier. These intensities are combined with the composition of the standard specimens, and the calibration curves for the elements of interest are then constructed. These calibration curves might include a correction for matrix effects of some sort. The resulting calibrations could then be used for the analysis of unknown specimens with compositions similar to the standards. This method is rather time-consuming, but it yields, in general, the highest possible analytical accuracy. Also, approximate knowledge of the composition of the specimens prior to the analysis is assumed. Such assumptions can be made very implicitly (e.g., when selecting the calibration curve to be used for analysis). If the method is routinely used and the number of unknown specimens to be analyzed is rather large, the benefits of this instrumental technique become very clear, when compared to ‘‘classical’’ wet chemistry methods, which tend to be much more time-consuming. Since the second half of the 1980s, interest has been growing for a different kind of quantitative analysis on wavelength-dispersive spectrometers, in which the elements present in the specimens and their approximate concentration levels are not known prior to the analysis. Semiquantitative analysis consists of an intimate combination of qualitative and quantitative analyses. During the qualitative analysis, the presence of the elements present in the specimen is established. This can be done in a variety of ways. Qualitative analysis is based on the collection of a spectrum of the specimen, followed by peak search and peak identification. Alternatively, the intensities of a rather large and fixed set of characteristic lines are collected. The lines are selected by the manufacturer and allow for the determination of most of the commonly occurring elements. In both cases, the net intensities are then determined and a quantification is made, involving matrix correction and general calibration lines for each of the elements. These calibration lines have been determined earlier from a set of standard specimens, although some manufacturers insist on calling their method ‘‘standardless.’’ The packages that are available commercially differ on the basis of the qualitative analysis (accumulating a spectrum or measurement on fixed wavelength positions), the number of standard specimens used, the flexibility of the spectrometer configuration that they can cope with, and the method used to correct for matrix effect. The focus of energy-dispersive systems has always been somewhat more qualitative and more oriented toward research, flexibility, and the analysis of small batches of Copyright © 2002 Marcel Dekker, Inc.

unknown specimens. The software allowed for quantitative analysis without the need to have standards similar to the unknowns. These systems are thus more flexible in their use in that respect. This does not mean that such systems are unsuitable for routine analysis in industry; many of them actually are used for that purpose. In this chapter, several methods for the quantification of infinitely thick specimens are discussed. For all methods described here, it is assumed that the specimens are infinitely thick: The intensity of the radiation from the specimen is constant and is not affected by increasing the thickness of the specimen. The analysis of specimens with lessthan-infinite thickness is discussed in Chapter 6. Also, the specimens are assumed to be homogeneous: Their composition is the same throughout the specimen. This assumption is valid for a large range of material types found in many applications. The most important exception are specimens with one or more thin coatings.

II.

CORRELATION BETWEEN COUNT RATE AND SPECIMEN COMPOSITION

A.

Introduction

In quantitative analysis, the measured x-ray fluorescent intensity of a given element is converted into its weight concentration in the specimen. As a first approximation, one would expect a linear relationship. Each atom of the analyte element i has the same probability of being excited by the primary photons and emitting its characteristic photons with wavelength li. Indeed, if separate atoms or ions (e.g., in a gas or in a very diluted solution) are considered, the following relation holds: I i ¼ K i Wi

ð1Þ

where Ii is the measured intensity of the fluorescent radiation of the analyte i and Wi is the weight fraction of the analyte i in the specimen. The proportionality constant Ki consists of many physical and instrumental factors such as the following: The intensity and the distribution of wavelengths of the photons in the primary beam The probability that an atom i emits its characteristic radiation li The probability that these photons li pass through the measuring channel: collimators, diffracting crystal, pulse-height window The probability that these photons are being detected and registered. For a given instrument, the voltage and power on the x-ray tube remain constant for the analyte i and they can be determined by measuring the fluorescent intensity of the pure element i. However, we are generally dealing with compact specimens where the atoms are arranged into chemical compounds. Both the primary x-rays and the fluorescent x-rays will be absorbed by the different atoms in the specimen. B.

General Relationship Between Intensity and Concentration

1. Primary Fluorescence by Monochromatic Radiation First, excitation by monochromatic radiation will be considered. Let the intensity of the incident beam with wavelength l0 at the surface of the specimen be given by I0(l0). This beam strikes the surface of the specimen at an angle c0 (see also Fig. 1). A parallel beam is assumed here and the specimen is considered to be extending to infinity in all three dimensions. The incident radiation is gradually absorbed by the specimen, and at a layer at Copyright © 2002 Marcel Dekker, Inc.

Figure 1 Schematic representation of the geometry involved in the calculation of primary fluorescence emission.

depth t below the surface, the remaining fraction of the intensity It(l0) is given by the Lambert–Beer law: It ðl0 Þ ¼ I0 ðl0 Þ exp½
ms ðl0 Þrs t ðcsc c0 Þ

ð2Þ

ms(l0) is the mass-attenuation coefficient of the specimen for photons with wavelength l0 (the subscript s refers to the specimen) and rs is the density of the specimen. Note that due to the angle of incidence, the path length traveled is given by t csc c0 . The mass-attenuation coefficient ms(l0) in this equation is thus calculated for the specimen; this is done by adding the mass-attenuation coefficients for all elements j present in the specimen, each multiplied with its mass fraction Wj: ms ðl0 Þ ¼

n X

½mj ðl0 ÞWj 

ð3Þ

j¼1

where n is the total number of elements present in the specimen. The fraction of the incident beam absorbed by the analyte i in the layer between t and (t þ dt) is given by Wi mi ðl0 Þrs csc c0 dt

ð4Þ

It is assumed here that the composition of the specimen is uniform throughout. In other words, Wi and Wj are independent of the position of the layer (t, t þ dt) within the specimen. Only a fraction of the photons absorbed creates vacancies in the K shell; this fraction is given by (riK
1)=riK, where riK is the absorption jump ratio of the K shell of element i. The fraction of K vacancies emitting x-rays is given by the fluorescence yield oiK. The fraction of Ka photons in the total of x-rays emitted for the analyte is given by the transition probability fiKa. These factors can be combined in a factor designated Qi(l0, li): Qi ðl0 ; li Þ ¼ oi

riK
1 fiKa riK

ð5Þ

In the above derivation, it is assumed that the characteristic line of interest is a Ka. If another line is considered, the relevant changes need to be made. The characteristic photons thus generated are isotropically emitted in all directions, without a preferential direction. Only a fraction is emitted toward the detector. If O is the solid angle as viewed by the collimator – detector system, expressed in steradians, then the fraction is given by O=4p. The angle O should be small enough so that the beam can be considered to be a parallel beam leaving the specimen at a single well-defined angle c00 with the surface. The fraction of characteristic photons with wavelength li not absorbed between the layer (t, t þ dt) and the surface is given by Copyright © 2002 Marcel Dekker, Inc.

exp½
ms ðli Þrs t csc c00 

ð6Þ

All photons reaching the surface and propagating in the direction indicated by the angle c00 are assumed to be detected. If the detector has absorbing elements (such as windows) or detection efficiencies different from unity, then these can also be taken into account. Also, the attenuation by the medium (e.g., air) between specimen and detector can be calculated using similar expressions. The intensity of element i, as excited by the incident beam with wavelength l0, is labeled Pi(l0) (explicitly denoting the primary fluorescence effect) and is given by Pi ðl0 Þ ¼ I0 ðl0 Þ exp½
ms ðl0 Þrs t csc c0   mi ðl0 ÞWi Qi ðl0 ; li Þrs cscc0 dt

O exp½
ms ðli Þrs t csc c00  4p

ð7Þ

Combining factors leads to O 4p  exp½
ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þrs trs dt

Pi ðl0 Þ ¼ I0 ðl0 Þmi ðl0 ÞWi Qi ðl0 ; li Þ ðcsc c0 Þ

ð8Þ

The contribution of all layers between the surface and the ‘‘bottom’’ of the specimen have to be summed. This can be done by integrating the above expression over dt, from 0 (the surface) to the ‘‘bottom.’’ In practice, for x-rays, the thickness of bulk specimens can be considered to be infinite, so the upper limit is 1 . Note that t is always used in combination with rs, so the integration will be done over rst. Taking all constant factors outside the integral, one obtains Pi ðl0 Þ ¼ I0 ðl0 Þmi ðl0 ÞWi ðcsc c0 ÞQi ðl0 ; li Þ Z1 

O 4p

exp½
ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þrs trs dt

ð9Þ

0

From a textbook on calculus, Z1 exp½
ax dx ¼

1 a

ð10Þ

0

Taking a ¼
(ms(l0) csc c0 þ ms(li) csc c00 ) and noting that rs dt ¼ d(rst), the following expression is obtained for the primary fluorescence of the analyte i in the specimen s: Pi ðl0 Þ ¼

I0 ðl0 Þmi ðl0 ÞWi ðcsc c0 ÞQi ðl0 ; li ÞðO=4pÞ ms ðl0 Þ csc c0 þ ms ðli Þ csc c00

ð11Þ

Often, the element specific factors, given by Eq. (5), are combined with the instrument specific factor O=4p. This leads to a simpler expression for the primary fluorescence: mi ðl0 Þ Wi Pi ðl0 Þ ¼ Ki I0 ðl0 Þ ð12Þ ms ðl0 Þ þ Gms ðli Þ where Ki ¼ ðO=4pÞQi ðl0 ; li Þ and G ¼ csc c00 =csc c0 ¼ sin c0 = sin c00 . In the above derivation, the following assumptions have been made: (a) The specimen is completely homogeneous. (b) The specimen extends to infinity in three dimensions. Copyright © 2002 Marcel Dekker, Inc.

(c) The primary rays are not scattered on their way to the layer dt. (d) No enhancement effects occur. (e) The characteristic radiation is not scattered on its way to the specimen surface. The simplest case is excitation of Ka (or Kb) photons. For the characteristic photons associated with the L lines, other effects, such as Coster–Kronig transitions and so forth have to be taken into account when describing the fraction of absorbed primary photons that give rise to characteristic photons. For a more detailed discussion on these effects, refer to Chapter 1. 2. Secondary Fluorescence Excited by Monochromatic Radiation Under certain conditions, the characteristic photons of an element j can excite atoms of the analyte i. This will lead to additional characteristic photons of element i. The term ‘‘additional’’ is used here in the sense that these photons are not considered in the above derivation for the primary fluorescence and, consequently, are not predicted by Eq. (12). It is beyond the scope of this chapter to derive the mathematical expressions following a rigorous approach, but certain aspects of secondary fluorescence emission are readily seen from the above derivation and from Figure 2. In Figure 2, two layers are now considered. The layer on the left, indicated by j, at a depth of t1 with a thickness of dt1, is where the primary fluorescence of element j is excited. This is described by Eqs. (2)–(5). Obviously, the element considered at this stage is the enhancing element j, so the fundamental parameters for j need to be used instead of these for i in Eqs. (2)–(5). Once the fluorescent radiation is created, it will be absorbed on its way through the specimen. The precise direction is of no concern, as we are now no longer considering the fraction that travels toward the detector. The attenuation along its path is described by equations such as Eq. (2) or Eq. (6). Keep in mind that the angles and the distances involved are now different and that the wavelength l for which the mass attenuation needs to be calculated [Eq. (3)] is now lj instead of l0. At the second layer, indicated by i, at a depth of t2 with a thickness of dt2, part of this (primary fluorescence) radiation will be absorbed. Again, this effect is described by an equation such as Eq. (4), with the angle c0 replaced by the angle of this path. The factors to be applied in order to lead to characteristic radiation of the analyte i are given by Qi(lj, li); Eq. (6) describes the attenuation (wavelength is now li) on the way toward the detector. These factors can easily be recalculated. The complication is in the description of the geometry, especially in the description of the distance betwee the two layers: The position of the two layers relative to one another is not restricted, the path of the photon lj does not have to be in the plane of the figure, nor is the second layer

Figure 2 Schematic representation of the geometry involved in the calculation of secondary fluorescence emission.

Copyright © 2002 Marcel Dekker, Inc.

always at a larger depth than the first layer. The angle X 0 in Figure 2 can take any value between 0 and 2p, and the three paths shown are not necessarily in one plane. This aspect has been discussed by Li-Xing (1984). Anyway, after integration, the final result Sij ðl0 ; lj Þ is given by Sij ðl0 ; lj Þ ¼

I0 ðl0 Þðcsc c0 Þmj ðl0 ÞWj Qj ðl0 ; lj Þmi ðlj ÞWi Qi ðlj ; li ÞðO=4pÞ 2ðms ðl0 Þ csc c0 þ ms ðli Þ csc c00 Þ
    sin c0 ms ðl0 Þ sin c00 ms ðlj Þ ln 1 þ ln 1 þ  þ ms ðl0 Þ ms ðlj Þ ms ðlj Þ sin c0 ms ðlj Þ sin c00

ð13Þ

where Sij ðl0 ; lj Þ describes the secondary fluorescence (enhancement) of the analyte i by characteristic photons with wavelength lj; these photons have been excited by primary photons with wavelength l0. The photons to be considered for the enhancement of the analyte are not limited to Ka only, as the Kb lines and other characteristic lines that have sufficient energy to excite the shell of interest of the analyte i must also be taken into account. For this reason, all individual contributions need to be added in order to calculate the total enhancement of the analyte i by element j: X Sij ðl0 Þ ¼ Sij ðl0 ; lj Þ ð14Þ i

There are two criteria to be satisfied for a characteristic photon lj to be able to cause secondary fluorescence: 1. It must be excited by the incident photon l0 (this means that the energy of the incident photon must be higher than the energy of the absorption edge associated with lj). 2. The energy of the photon lj must be higher than the energy of the absorption edge for the analyte. The derivation of the intensities of characteristic x-rays as a function of specimen composition has first been done by Sherman (1955). The resulting equations, however, were rather unwieldy. Equations (9) and (13) have been published later by Shiraiwa and Fujino (1966) and Sparks (1976). In practice, many computer programs only consider a few lines for enhancement, and lines that have a low transition probability f are usually neglected. Data regarding these transition probabilities for a line within a series can be found, for example, in Appendix II of Chapter 1. The enhancement phenomenon will be more pronounced if the x-rays of the enhancing elements are only slightly more energetic than the energy of the absorption edge of the element i. The enhancement may contribute up to 40–50% of the total fluorescent radiation Ii , especially where the concentration of the enhancing elements is much greater than the concentration of the analyte. This applies even more for the light elements, where the primary spectrum may not be very effective as the most intense wavelengths are far away from the absorption edges of these light elements. The effect of scattered radiation is generally ignored, although its contribution can also be calculated (Pollai et al., 1971). If the characteristic line of the analyte considered is one of its L lines, then it is possible that other lines of the analyte itself enhance the characteristic line considered. The energy of the K edge of La (atomic number Z ¼ 57) is 38.9 keV (see Chapter 1), so the K lines of all elements with Z < 57 are excited if, for example, an x-ray tube is used at Copyright © 2002 Marcel Dekker, Inc.

Table 1

Data for Selected Absorption Edges and Characteristic Line of Pb

Absorption edge K L1 L2 L3

Energy (keV)

Characteristic line

Energy (keV)

88.04 15.86 15.20 13.04

La1(L3–M5) Lb1(L2–M4) Lg1(L2–N4)

10.55 12.61 14.76

Source: Appendix I and Appendix II of Chapter 1 (this volume).

voltages above 40 keV, as is commonly the case with wavelength-dispersive spectrometers. Under these conditions, the L lines of these elements are enhanced by the K lines. The situation for elements with high atomic numbers is more complex. Let us consider Pb (Z ¼ 82). Lead is very often determined through its La1 or Lb1 line because the K lines of Pb are too energetic to be measured with reasonable efficiency by scintillation or Si(Li) detectors. Also, excitation voltages higher than 88 keV are required in order to excite the K lines. Thus, one would not readily expect enhancement by Pb on its own L lines. From the data in Table 1, however, it follows that the PbLa1 line (L3 –M5 ) is enhanced by PbLg1 (L2 –N4 ) because its energy (14.76 keV) is higher than the energy of the L3 edge (13.04 keV). The total excited characteristic intensity is then given by adding the primary and the secondary fluorescence contributions: X Sij ðl0 ; lj Þ ð15Þ Ii ðl0 Þ ¼ Pi ðl0 Þ þ j

Also, tertiary fluorescence is possible, where the incident photon excites element k (primary fluorescence), whose radiation excites element j (secondary fluorescence), whose radiation, in turn, excites element i, causing tertiary fluorescence. This contribution is generally lower than 3% of the total fluorescence and is commonly ignored, as shown by Shiraiwa and Fujino (1967, 1974). 3. Excitation by Continuous Spectra If the incident beam is polychromatic rather than monochromatic, Eq. (15) needs to be calculated for each wavelength. Wavelengths longer than the wavelength of the absorption edge ledge,i of the analyte cannot excite fluorescence, so these need not to be considered. If JðlÞ is the function representing the tube spectrum, then Eqs. (11) and (13) can still be used, provided that I0 ðl0 Þ is replaced by JðlÞ, representing the intensity of the incident spectrum at wavelength l: lZedge; i

Ii ¼

lZedge; i

Ii ðJðlÞÞ dl ¼ lmin

" Pi ðlÞ þ

lmin

X

# Sij ðl; lj Þ dl

ð16Þ

j

This equation allows the calculation of the intensity of characteristic radiation of a given analyte in an infinitely thick specimen. Equations such as Eqs. (11), (14), and (16) are often referred to as fundamental parameter equations because they allow the calculation of the intensity of fluorescent radiation as a function of the composition of the specimen (weight fractions Wi ), the incident spectrum [JðlÞ], and the configuration of the spectrometer used (c0 ; c00 , and O). All other variables used are fundamental constants, such as the mass-attenuation coefficients for a given element at a given wavelength or its fluorescence yield and so forth. Copyright © 2002 Marcel Dekker, Inc.

C.

Some Observations

The integration over t in Eq. (9) is taken from zero to infinity. It is obvious that the first layers contribute more to the intensity of li than the more inward layers. Theoretically, even at large values of t, a very minor contribution to the intensity is still to be expected. Often, the (minimum) infinite depth is defined arbitrarily as that thickness t where the contribution of the layer ðt; t þ dtÞ is 0.01% of that of the surface layer. In this case, it is defined relative to the surface layer. Alternatively, it is defined as the thickness where the contribution to the total intensity is less than 1%. In this case, it is relative to the total intensity from a ‘‘truly infinitely’’ thick specimen. The value of the infinite thickness depends on the value of the absorption coefficients and the density of the specimen. In practice, it may vary from a few micrometers for heavy matrices and long wavelengths to centimeters for short wavelengths and light matrices, as in solutions. For a given element and a fixed geometry, an efficiency factor Cðl0 ; li Þ can be introduced: mi ðl0 Þ Pn ð17Þ j¼1 Wj mj ðl0 Þ þ G j¼1 Wj mj ðli Þ P It is obvious that the terms Wj mj are the origin of nonlinear calibration lines, as variations in Wj influence the value of the denominator. For a pure metal, this reduces to Cðl0 ; li Þ ¼ Pn

Cðl0 ; li Þ ¼

mi ðl0 Þ mi ðl0 Þ þ Gmi ðli Þ

ð18Þ

In the general case, the wavelengths in the primary spectrum close to the absorption edge are the most effective in exciting the analyte i. The efficiency factor Cðl0 ; li Þ is thus a combination of the absorption curve of analyte i as a function of l and the spectral distribution. In a first approximation, an ‘‘effective wavelength’’ le can be introduced, which has the same effect of excitation of element i as the total primary spectrum. The exact value of this le will be influenced by the characteristic tube lines if they are active in exciting i. Otherwise, le can, in general, be assumed to have a value of approximately two-thirds of the absorption edge ledge. Its actual value is, however, dependent on the chemical composition of the specimen. For instance, for Fe, the wavelength of the K edge ledge is 0.174 nm (7.11 keV) and an effective wavelength le of 0.116 nm is obtained using this rule of thumb. In the ZnO–Fe2O3 system, le was found to vary from 0.130 nm for 100% Fe2O3 to 0.119 nm for 10% Fe2O3 in ZnO. The estimated value of 0.116 nm is in good agreement with the experimental one for Fe2O3. Another interesting case is the analysis of a heavy element i in a light matrix. In the summation in the denominator of Eq. (12), the terms Wi mi ðl0 Þ and Wi mi ðli Þ are then the most important. The other terms can, in first approximation, be neglected if Wi is not small. However, that means that the terms Wi in the numerator and the denominator cancel and the measured intensity becomes independent of Wi ; thus, the analysis becomes impossible in this extreme case. A solution to this problem is found by making the influence of the terms Wi mi ðlÞ in the denominator less dominating by adding a large term Wa ma ðlÞ. This can be done be making Wa large (e.g., diluting) or ma ðlÞ large, by adding heavy absorber. Equation (17) enables one to calculate beforehand how large this term should be to eliminate fluctuations in the concentration of the other elements. In the derivation of the fluorescence of the analyte i in the preceding paragraphs, the following simplifications were made: Copyright © 2002 Marcel Dekker, Inc.

1.

First, it was assumed that the primary rays follow a linear path to the layer dt at depth t. However, the primary rays may also be scattered. In general, the loss in intensity of the primary beam of photons due to scattering may be neglected. These scattering effects become more important when the primary x-rays are more energetic and the average atomic number of the matrix decreases. This scatter may give a higher background in the secondary spectrum, thus leading to a poorer precision of the analysis. On the other hand, the excitation efficiency may be enhanced, as the primary rays ‘‘dwell longer’’ in the active layers, thus having a higher probability to encounter atoms of element i. This effect may overrule the increase in intensity of the background radiation. A case in hand is the determination of Sn in oils which gives better results using the SnK lines at a high x-ray tube voltage, than using the SnL lines at moderate voltages. Incidentally, this scattering of the primary radiation makes it possible to check the voltage over the x-ray tube. According to Bragg’s law, the intensity of the primary spectrum is zero at an angle y0, given by
 nlmin
1 y0 ¼ sin ð19Þ 2dcrystal where 2dcrystal is the 2d spacing of the crystal used and n is an integer number. lmin (in nm) is given by

1:24 ð20Þ V where V is the voltage on the x-ray tube, expressed in kilovolts. In practice, a lower value for V will be found when Compton scattering dominates over Rayleigh scattering and thus lmin found has a value too low by the Compton shift, which is about 0.024 nm for most spectrometers; the actual value depends on the incidence and exit angle (see also Sec. V.C.1.) 2. The integral in Eq. (9) was taken from zero to infinity; further, it was assumed that the specimen is completely homogeneous. This, of course, is never realized in practice, as we are dealing with discrete atoms in chemical compounds. In powders, the different compounds may have a tendency to cluster. The particles will, in general, have different sizes and shapes. Putting the sample into solution, either aqueous or solid (melt) may overcome this problem. lmin ¼

It was stated formerly that infinite thickness may vary from 20 mm to a few centimeters. However, the most effective layers are much thinner. Thus, the number of discrete particles actually contributing to the fluorescent radiation may be rather small.

III.

FACTORS INFLUENCING THE ACCURACY OF THE INTENSITY MEASUREMENT

A.

Introduction

The total uncertainty of the analysis consist of many errors whose source may be the following: The measurement of the intensity The reproducibility of the specimen preparation The conversion of intensity into concentration Copyright © 2002 Marcel Dekker, Inc.

The uncertainty of a single determination may be found from n determinations of the same analysis, given a mean result xmean, of the all individual results xi. If n is sufficiently large, the standard deviation s may be found from sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ðxi
xmean Þ ð21Þ s¼ n
1 In this total standard deviation s, random and systematic uncertainties are combined. Random uncertainties give an indication of the precision of an analysis (the scatter of results around the mean value), whereas systematic errors are the reason for deviations of the mean value from the ‘‘true’’ value. An analysis may thus be precise, but not very accurate, if systematic errors are present, whereas accurate values could be found from the mean of widely scattered measurements if only large random uncertainties were present. The total error of a measurement is composed of all the separate errors. If only random errors s1, s2, and so forth are considered, the resulting standard deviation is given by s2 ¼ s21 þ s22 þ s23 þ    þ s2n

ð22Þ

where s1, s2, and so on are the errors associated with, for example, intensity measurements, specimen preparation, instrumental settings, and so on. In practice, it is often found that s is dependent on the concentration of the analyte Wi (Johnson, 1967; Hughes and Hurley, 1987): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ K Wi þ Wb ð23Þ where Wb is a small concentration offset (typically 0.001 in weight fraction). Thus, K, rather than s, becomes an indication of the accuracy of the determination. B.

Random Errors

1. Counting Statistics If an x-ray measurement consisting of the determination of a number of counts N is repeated n times the results N1,N2, N3, . . . ,Nn would spread about the true value N0. If n is large, the distribution of the measurements would follow a Gaussian distribution, " # 1 ðN
N0 Þ2 WðNÞ ¼ pffiffiffiffiffiffiffiffiffi exp
ð24Þ 2N 2pN pffiffiffiffiffiffiffiffiffiffiffiffi provided N is also large. The standard deviation s of the distribution is equal to Nmean , again if n and N are large, where Nmean is the mean of n determinations. From the properties of the Gaussian distribution, the following hold: 68.3% of all values N will be between N0
s and N0 þ s. 95.4% of all values N will be between N0
2s and N0 þ 2s. 99.7% of all values N will be between N0
3s and N0 þ 3s.

pffiffiffiffi Similarly,pthere ffiffiffiffi is a certain probability that the true result N0 will lie between N
N and N þ N, assuming the same distribution for N and N0. Measurement results are commonly expressed as a count rate (intensity per unit time) instead of an intensity, which gives the number of counts collected in the counting interval. This allows an easier comparison of measurements made with different counting times, but the measuring time needs to be specified in order to be able to assess the counting statistical error. Copyright © 2002 Marcel Dekker, Inc.

The determined concentration is dependent on the net count rate, which is the peak count rate Rp minus the background count rate Rb. The total measuring time T equals tp þ tb , where tp and tb are the measurement times for peak and background, respectively. In modern equipment, there is no significant statistical error in the measurement of t. We can thus assume that R follows the same Gaussian distribution as N with the same relative standard deviation eN. eN is defined as sN ð25Þ eN ¼ N Hence,

and

pffiffiffiffi N 1 1 ¼ pffiffiffiffi ¼ pffiffiffiffipffiffi ¼ eR eN ¼ N N R t

ð26Þ

pffiffiffiffi R sR ¼ eR R ¼ pffiffi t

ð27Þ

It is obvious that the relative counting error decreases as t increases. When a net count rate has to be determined the peak, Rp and the background, Rb, have to be measured; there are thus two independent variables. The standard deviation of the net intensity, sd, is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rp Rb sd ¼ s2p þ s2b ¼ ð28Þ þ tp tb and the relative standard deviation ed by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rp =tp þ Rb =tb ed ¼ Rp
Rb

ð29Þ

When using a sequential wavelength-dispersive x-ray fluorescence spectrometer and when the count rates are rather low and time is limited, it is of interest to divide the total counting time available over tp and tb in the best possible way. In principle, there are three methods to split up the total counting time T: 1.

2.

Fixed time. Peak and background are measured for the same time; tp ¼ tb ¼ T=2. The resulting standard deviation in the difference between the peak and the background count rate is then given by rffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðRp þ Rb Þ ð30Þ sd ¼ T Fixed count. The same number of counts, N, is collected on the peak and on the background: Np ¼ Nb or tp Rb ¼ tb Rp The resulting standard deviation for the difference is given by rffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rp Rb 1 ðRp þ Rb Þ þ sd ¼ T Rb Rp

Copyright © 2002 Marcel Dekker, Inc.

ð31Þ

ð32Þ

3. Fixed time optimal. The optimum division of the total measurement time T over tp and tb can be found by differentiating Eq. (29) with respect to tp, where tb ¼ T
tp. It is found that for optimal results, the ratio of the measuring times is sffiffiffiffiffiffi tp Rb ¼ ð33Þ tb Rp The resulting standard deviation is given by rffiffiffiffi 1 pffiffiffiffiffiffi pffiffiffiffiffiffi sd ¼ ð Rp þ Rb Þ T and the relative standard deviation by 1 1 ed ¼ pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi T Rp
Rb

ð34Þ

ð35Þ

It can easily be demonstrated that sFTO  sFT  sFC

ð36Þ

where sFTO refers to the method of fixed time optimized, sFT is for fixed time, and sFC is for fixed counts. When Rp is very large compared with Rb, sFTO is close to sFT. Thus, the fixed time method is often used in practice because Rp and Rb are not known beforehand. If there is a great difference between peak and background count rate, then usually all the available time is spend counting the peak. Thus, the background is not measured. A constant value for the background can be assumed or it can be ignored altogether. Compared with method of fixed time, the following observation can be made: If, Rb is not measured, the standard deviation in the net count rate (calculated as the difference between Rp and Rb) is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rp þ s2b ð37Þ sd ¼ T pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi This approximation pffiffiffiffiffiffiffiffiffiffiffi is allowed if this value is smaller than ð2=TÞ ðRp þ Rb Þ. This is the case if sb 5 Rb =T. This applies if the background is more or less constant between specimens and a fixed value for the background can be deducted. If the background pffiffiffiffiffiffiffiffiffiffiffiis altogether ignored, sb ¼ Rb . This is only allowed when Rb is much smaller than Rp =T. follows that when one aims for optimum instrumental conditions, the expression pffiffiffiffiffiffi It p ffiffiffiffiffiffi This is often used as a figure of merit Rp
Rb is a good quality function. p ffiffiffiffiffiffi parameter pffiffiffiffiffiffi (FOM). Obviously, the highest value of R gives the best result. When it can be
R p b pffiffiffiffiffiffi pffiffiffiffiffiffi assumed that Rp
Rb approximates Rp, for low intensities and high background, optimizing this term, equals optimizing M pffiffiffiffiffiffi Rb

ð38Þ

where M (the slope of the calibration line in counts per second per percent) is proportional to Rp
Rb. If the ratio of two count rates R1 and R2 has to be determined, the methods of fixed time and fixed count give the same result, whereas the method of optimum division, where rffiffiffiffiffiffi t1 R2 ¼ ð39Þ t2 R1 always gives the best result. Copyright © 2002 Marcel Dekker, Inc.

The total counting uncertainty is the combination of instrumental error and counting statistics. When the count rate is very high, the counting uncertainty is small and it may be worthwhile to apply a ratio method to reduce possible instrumental uncertainties. However, if the count rate is low, it is better to spend all the available time in analyzing the specimen to reduce the counting error. 2. Instrumental Errors If the instrumental and counting uncertainties are random and independent variables, then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi etot ¼ e2instr þ e2count ð40Þ einstr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2tot
e2count

ð41Þ

Although the counting error is influenced by the instrumental error, Eq. (41) is still a good approximation. stot can be found from a series of repeated results of one measurement; thus, scount is known. To check the instrumental instability, all the functions should be measured separately. A radioactive source 55Fe, for example, can be used to check the detector and electronic circuitry. The x-ray tube can be checked by repeating the measurements with specimen and goniometer in a fixed position, eliminating errors stemming from the mechanics of the spectrometer. In another series of experiments, recycling between different angles checks the reproducibility of the goniometer, whereas repositioning the specimen between measurements checks the specimen holder, the reproducibility of the specimen loading mechanics, and so forth. A comprehensive series of tests for wavelength-dispersive x-ray fluorescence spectrometers is described in the Australian Standard 2563-1982 (1982). Sometimes the error stot found is smaller than expected, or even smaller than scount . This may indicate that an unexpected systematic error is involved, or there may be an uncorrected deadtime in the equipments, (i.e., the true counting rate is higher than measured, which means that the relative error is smaller). 3. Detection Limit A characteristic line intensity decreases with decreasing concentration of the analyte and finally disappears in the background noise. The true background intensity may be constant, but the results of the measurements fluctuate around a mean value Rbmean . To be significantly different from the background, a signal Rp must, although it is larger than Rbmean , be distinguished from the spread in Rb . In other words, if we measure a signal Rp larger than Rb and we assume the analyte is present, what is the probability that our assumption is correct? If the results of the measurements are random and follow a Gaussian distribution, then this probability is determined by sRb . If the measurement Rp is higher than Rb þ 2sRb , then the probability that our assumption is correct is approximately 95% if a higher certainty is required (e.g., 99.7%), then Rp should be larger than Rb þ 3sRb . Thus, the net intensity is 3sRb and the detection limit, DL, would be DL ¼

3sRb M

ð42Þ

where M is the sensitivity in counts per second per percent. So, the detection limit in the above equation is the concentration corresponding to a net peak intensity of 3sRb . However, in x-ray spectrometry, the background signal is specimen dependent and cannot Copyright © 2002 Marcel Dekker, Inc.

be measured independently, as in radioactivity measurements. Hence, Rb has to be measured in an off-peak location in the spectrum. The resultffi Rb found in measuring time t s is pffiffiffiffiffiffiffiffiffiffiffiffiffi assumed to be Rbmean and sRb is assumed to be ðRb =tÞ. Thus, two measurements have to be made: Rp and Rb , each in time t s. The detection limit thus becomes pffiffiffi rffiffiffiffiffiffi 3 2 Rb ð43Þ DL ¼ M T where T ¼ 2t. If we are satisfied with a 95% probability that our assumption is correct, then pffiffiffi rffiffiffiffiffiffi 2 2 Rb ð44Þ DL ¼ M T which is roughly equal to rffiffiffiffiffiffi 3 Rb DL ¼ M T

ð45Þ

It is obvious that the detection limit decreases if the counting time increases. However, the total error in Rb contains the instrumental error as well. Thus, there is no sense to increase the counting time when the instrumental error dominates. Ingham and Vrebos (1994) have shown that the detection limit can be improved by carefully selecting a primary filter. If the application of such a primary beam filter reduces the intensity of the (scattered) continuum from the tube more than it affects the sensitivity, the detection limit is improved. The loss in sensitivity M needs to be more than compensated for by the reduction in background intensity; as from Eq. (45), the detection limit is proportional to the square root of Rb and inversely proportional to the sensitivity M. 4. Variation in X-ray Spectrum The fluorescent intensity of the analyte Ii is, in first approximation, dependent on the primary spectrum according to Ii ¼ KiðV0
Vc Þp

ð46Þ

where K is a constant, i is the current of the x-ray tube, V0 is the working voltage, Vc the excitation voltage, and p varies between 1 and 2, depending on the ratio of excitation by characteristic tube rays and white continuum. In modern instruments, the tube voltage is not dependent on the mains cycle, but they run on constant potential, which still may fluctuate. If the working voltage V0 , or the region or line of highest excitation probability is rather close to Vc , then small fluctuations in V0 will introduce a considerable error in Ii . For instance, if V0 ¼ 1:5 Vc , then a 1% error in V0 gives an error in Ii of 6% when p ¼ 2. It is therefore better to run the tube at three to five times the excitation voltage of the analyte. A too high voltage, however, might introduce an unproportionally high background. 5. Other Instrument Errors Other possible random instrumental errors include positioning the specimen and setting the goniometer in wavelength-dispersive x-ray spectrometers. These errors have to be checked by repeated measurements of one specimen in a systematic way: Copyright © 2002 Marcel Dekker, Inc.

Repeated Repeated Repeated Repeated

counting counting counting counting

with with with with

stationary stationary stationary stationary

specimen and goniometer specimen and repositioning goniometer goniometer and repositioning specimen goniometer and reloading specimen holder.

Some diffracting crystals have a rather high coefficient of thermal expansion; therefore, their d value may fluctuate with fluctuations in temperature. This results in a setting of the goniometer, slightly off-peak, which introduces a change in measured intensity. Therefore, most modern equipment are operating at a stabilized temperature. 6. Particle Statistics Only a limited volume of the specimen can actually contribute to the fluorescent radiation. As long as this active volume is the same in standards and actual specimens and the atomic distribution is completely homogeneous, this poses no problem. However, the atoms are bound into chemical compounds, forming finite particles with different chemical compositions. The analyte may only occur in particles with a certain chemical composition and not in other particles. Then, only these specific particles can contribute to the fluorescent radiation of the analyte i. Therefore, the count rate Ri measured depends on the number of those particles present in the active volume, where, evidently, the first layers contribute most of the fluorescent radiation. Table 2 gives an indication of the penetration depth of radiation of various wavelengths into matrices with varying absorption power. It is evident that for most solid specimens, the fluorescent radiation originates within 20 mm or less from the surface. To get an idea of how many particles can actually contribute to the fluorescent intensity of analyte i, let us assume that the irradiated area of 10 cm2 is covered with cubic particles of 100 mm dimension in a random fashion. Assuming a filling factor of 0.8 and assuming that the analyte i is only present in 10% of the particles, then 10  108  10
4  10
1  0:8 ¼ 8000 particles could be actually contributing. Assuming pffiffiffiffiffiffiffiffiffiffi a Gaussian distribution, then this number would have a standard deviation of 8000 ¼ 90 particles, or a relative standard deviation of approximately 1.1%. If the concentration of analyte is only 1%, then this relative standard deviation would be roughly 3.3%. In practice, these errors might even be larger, as the radiation of the specimen is not homogeneous because the primary spectrum originates in a rather small anode and passes through a large window and is, thus, conically shaped. Spinning the specimen in its own plane during the analysis will reduce this error. Furthermore, the first layers are the most effective, having

Table 2

‘‘Infinite’’ Thickness (in mm for Certain Analytical Lines as a Function of the Matrix)

Analytical line SnKa MoKa NiKa CrKa AlKa NaKa CKa

Fe base

Mg base

H2O solution

Borate

Borate La2O3 10%

300 100 12 33 1.5 0.7 0.3

10,000 3,700 340 120 4 20 0.3

100,000 30,000 2,400 900 10 9 4

70,000 30,000 2,000 800 15 6 1

10,000 2,600 300 250 5 5 0.4

Note: Both the incidence and exit angles are 45 and excitation is by a Rh tube at 60 kV. The influence of elementspecific absorption can be seen from, for example, the values for NiKa and CrKa in the Fe-base matrix.

Copyright © 2002 Marcel Dekker, Inc.

only a small number of particles containing the analyte with corresponding larger relative errors. In extreme cases, it is evident that the effective layer is very thin, less than 1 mm. Care should thus be taken that the specimen surface is as smooth as possible; as surface irregularities (e.g., grooves, ridges) will introduce a considerable error. Spinning the specimen will, again, reduce this error. It is therefore vital that the specimen be completely homogeneous. If this is not possible and powders have to be analyzed, care should be taken that the particles are very small, less than a few micrometers in diameter. Specimen preparation will be discussed in full detail in Chapter 14. C.

Systematic Errors

1. Dead T|me After an x-ray photon is detected in the counter and accompanying electronics, it takes a certain time before the counting circuit is ready to accept the next photon. Any photon entering the counter within this period, called the dead time of the counter circuit, is simply not registered and is thus lost. This dead time is of the order of a few microseconds. The counting losses are thus dependent on the actual count rate. The measured count rate Rm is always lower than the true count rate RT . Their relation can be approximated by the expression RT ¼

Rm 1
td Rm

ð47Þ

where td is the dead time. For instance, if td ¼ 1 ms and Rm ¼ 105 counts per second, the dead-time loss is approximately 10%. With modern equipment, very high count rates can be handled, to reach sufficient precision in a short time. It is therefore necessary to reduce these losses. In most wavelength-dispersive instruments, an automatic dead-time correction circuit is included. Energy-dispersive instruments, on the other hand, tend to collect counts for the specified time; thus, the measurement takes longer because the total time required in this case consists of the specified measuring time (‘‘lifetime’’) and the dead time. 2. Matrix Effects The fluorescent intensity of the analyte i is, as discussed earlier, not only dependent on its concentration but can also be strongly dependent on the composition of the specimen itself. The primary rays will be absorbed and scattered and secondary fluorescence may occur. All of these effects depend on the chemical composition of the specimen. The importance of these effects depends on the concentration of these matrix elements and their influence (e.g., their absorption of primary and secondary x-rays). These matrix effects may introduce large systematic errors when they are not properly accounted for, as discussed in Sec. V. D.

Choice of Optimal Conditions

1. Selecting the Analytical Line Some considerations in choosing the analytical line are the following: 1. High sensitivity, thus preferably the strongest line in the emission spectrum is used; this is commonly the Ka Copyright © 2002 Marcel Dekker, Inc.

2. 3. 4.

Low intensity of background radiation Constant angle of Bragg diffraction (wavelength-dispersive instruments) No coincidence (line overlap) with lines of other elements

Each of these items will be discussed in more detail below. 1.

2.

3.

4.

For the light- and medium-Z elements, the Ka line is by far the strongest line in their spectrum and is therefore often chosen as the analytical line. For the elements with K-edge energies exceeding 40 keV, the L lines are preferred because the K lines, in general, cannot be used, the maximum voltage for most spectrometers being limited to 100 kV or less. Typically, a voltage of three times Vc or more is needed to get a high characteristic intensity. Furthermore, the resulting K lines are very energetic and are only detected with mediocre efficiency by the Si(Li) detectors or NaI scintillation detectors. Ge detectors have a higher efficiency for that energy range and these are the preferred detectors. Another reason the L lines are preferred for the heavier elements is that in direct tube-excited XRF, the background due to scattering of the primary x-rays is much lower in the L region of the spectrum. The wavelength of some analytical lines may shift slightly with the valence state of the elements, especially for the light elements and the L lines of the transition metals (Wood and Urch, 1978); thus, the standard used in setting the goniometer to the analytical line should correspond to the specimen in this respect. Another reason for an apparent shift in angle may be the change in d value of analyzing crystal with temperature. The analytical line should ideally be completely free of any disturbing lines.

However, there are many sources of distributing influences; some of these are described next. The following sections deal with line overlap in the case of wavelength dispersive spectrometers. For energy dispersive spectrometers, please refer to Chapter 4. 2. Spectral Overlap Two or more characteristic lines may not be completely separated from the analytical line. This separation may be improved by using a crystal with better dispersion (e.g., a lower d value). However, the choice must often be made between high intensity and high dispersion. If the disturbing line is due to a high-order crystal reflection, its influence may be strongly reduced by the proper setting of the pulse-height selector. However, in some cases, the escape peak of the interfering line may be within the pulse-height selector window. For instance, the third-order reflection using a penta erythritol (PE) crystal, of the characteristic tube lines of a Sc anode, scattered by the specimen will slightly interfere with the analysis for Al in a light matrix, as the energy of the ScKa escape peak in an Arfilled gas detector is very close to the energy of the AlKa line. Often, the overlap is due to a diagram line of an element of which another diagram line is free of overlap. As, in general, two diagram lines of one element have a constant intensity ratio, the measured intensity of the nonoverlapped line of the disturbing element multiplied by a constant factor (experimentally determined) may be subtracted from the measured intensity of the analytical line to give the characteristic intensity. Absorption effects can, however, strongly influence the ratio. This is most clearly the case when there is an absorption edge of a major element between the two diagram lines considered. Copyright © 2002 Marcel Dekker, Inc.

3. Primary Radiation Scattered by the Specimen Photons of all wavelengths present in the white spectrum of the incident beam are scattered by the specimen, including the characteristic lines of the tube anode, giving rise to a continuous background. If, however, the specimen consists of rather coarse grains, it may happen that a crystallite is in a favorable position for Bragg diffraction for a wavelength of the continuum; thus, a sharp peak will be found in the spectral analysis. The influence of primary tube lines, coherently or incoherently scattered, may be eliminated by the proper choice of the anode. 4. Spurious Reflections by the Analyzing Crystal LiF(110) is a common analyzing crystal. The second order of reflection is used because the first order is crystallographically forbidden; therefore, the actual planes used for the diffraction are the 220 planes. Hence, some manufacturers refer to it as the LiF(220) crystal; the same applies to the LiF(100) and LiF(200) denominations. However, when a very, very high intensity is observed, a first-order reflection will still be found, due to asymmetry of the electronic cloud. Similarly, a second order may be observed using a Ge crystal. 5. Satellite Lines The common wavelength tables give only characteristic K, L, and M lines of most elements. The M lines of heavy elements may interfere with the K lines of light elements. Nondiagram satellite lines may also occur, often giving rise to an unexpected background level. One of the tables that includes such lines is provided by NIH (Bethesda, MD) (Garbauskas and Goehner, 1983).* The lines that are most obvious to observe are some of the satellites lines of Al, Si, and P with a wavelength-dispersive spectrometer. In Figure 3, a spectrum over aluminum is shown, on which some of these lines have been annotated. IV.

CALIBRATION AND STANDARD SPECIMENS

A.

Introduction

As shown earlier, standard specimens must cover the concentration range of interest, be stable over time, and have a certified composition. They are, however, not the only specimens required to set up a calibration for routine use and to maintain it over extended periods of time. Quality control specimens (also called quality assurance specimens) are used to assess the quality of the analysis obtained over time, whereas drift-correction monitors are used to correct for long-term drift of the equipment. Finally, recalibration standards can be used if the calibration graph must be reconstructed. B.

Quality Control Specimens

The use of at least one sample with known composition to assess the accuracy at the time of the calibration is highly recommended. The sample(s) used for this purpose should be typical for the unknowns and should not be used for calibration to avoid biases and overly optimistic estimates or accuracy. Furthermore, it is a sound practice to select at least one specimen, with a composition similar to the unknowns, as a quality control specimen to verify the repeatability over extended periods of time. Should the results on the quality *Database made available through C. Fiori, National Institutes of Health, Bethesda, Maryland.

Copyright © 2002 Marcel Dekker, Inc.

Figure 3 Spectrum of aluminum-containing specimen. The peaks are the following emission lines: (1) AlKb1, Kb3 doublet; (2) AlSKb0 ; (3) AlSKa7; (4) AlSKa5; (5) AlSKa4; (6) AlSKa3; (7) AlKa1, Ka2 doublet. The satellite lines are explicitly labeled with S. Note the logarithmic scale on the intensity axis. Conditions: wavelength-dispersive spectrometer with PE crystal, Rh tube.

control specimen fall outside a predetermined range, then adequate measures must be taken. The first step is usually to perform a correction for drift. DeGroot (1990) has described how the use of statistical process control (SPC) can be beneficial in this respect. The SPC charts that can be maintained in this way allow one to check the performance of the spectrometer system. C.

Drift-Correction Monitors

Correction for drift can be made by measuring selected specimens for each of the analytes and calculate the ratio between the observed intensities and those obtained when the calibration was performed. The ratios can then be applied as a correction to the slope of the calibration graphs or, as is done more often, the measured intensities of the unknowns are corrected for drift, prior to the conversion to concentration. If drift correction is performed regularly (e.g., once a day), the drift between subsequent measurements is very small and, thus, measurements with high precision are required, otherwise the counting statistical error of the measurement would become dominant. Drift correction should not even be applied unless the correction is significant. Because the precision of an intensity measurement is determined by the number of counts collected, drift-correction monitors are, ideally, specimens on which high count rates can be obtained. Drift-correction monitors do not have to be specimens with a composition similar to the unknowns. As different components of the spectrometer (such as the x-ray tube) age, not only will the sensitivity be affected (this is generally a downward trend), but in some cases, the background will vary also. To correct for this change in background intensity, measurements of the background must be performed. This poses a problem, inasmuch as the count Copyright © 2002 Marcel Dekker, Inc.

rate on the background is usually low and thus measurements are not vary precise or they require a long measuring time. However, the considerations on the counting statistical error made earlier (Sec. III.B.1) offer some suggestions. First, the intensity of the background is irrelevant when the intensity of the analyte peak is much higher. Obviously, this has to apply for all specimens to be measured. If that is the case, then variations in the background intensity are also negligible. When the background matters, it should already be measured anyway and then drift correction on the background intensity is not required, as the net count rate is corrected for drift. The only case that is not covered here is the contribution of spectral contamination from the x-ray tube (e.g., Cu, W, Ag, Fe, etc.). In that case, the background, including the contamination, must be measured on peak. This implies that measurements must be done for specimens with zero analyte concentration. Again, two cases can be distinguished: If the background including the contamination is not important compared to the count rate observed, then no corrections are required. On the other hand, if one of the analytes is present at low concentration, then the contribution to the background due to the contaminant must be checked periodically and taken into account. After the drift correction is performed, a quality control specimen should be measured to verify the procedure. D.

Recalibration Standards

Sometimes (e.g., after a major maintenance on a spectrometer), drift correction does not bring the quality control specimens back in line with the expectations. In those cases, the calibration curve must be reconstructed. This can be done by measuring all the standard specimens again and repeating the complete calibration procedure. Because calibrations often use many standards and validating each calibration is required, this can be a timeconsuming process, even if the validation is limited to a quick visual inspection of the calibration graphs. In such cases, a recalibration can be performed based on only a few standards. The idea of recalibration is to reconstruct the calibration graph, without having to measure all the standards again. This is done by selecting a few standard specimens for each analyte and measuring these (the top and bottom point in Fig. 4). Subsequently, when determining the parameters (such as slope and intercept) of the regression, for the concentrations of these specimens in the calibration the certified values are no longer used, but the values as found on the calibration line at the time of the original calibration (the x-ray values) are. These x-ray values have been found based on all standard specimens used, and the idea is to ‘‘fix’’ the calibration line again through these points. As a result, the statistical data are now skewed, but the values for the slope and the intercept are very close to the original ones; the small differences between old and new values are due to the counting statistical errors in the measurements and these are also present when unknowns are measured. The specimens used for recalibration have also been used for the calibration. The only requirement is that the recalibration specimens have count rates that are different enough so that the determination of the slope is accurate enough. For each of the calibration lines, the number of selected specimens must be at least the same as the number of parameters to be determined. If the slope and intercept are determined, at least two recalibration standards are required. However, if in this case three or four standards are used, it is possible to detect gross counting artifacts (e.g., caused by mislabeled standards, incorrect loading, etc.). The root mean square error or the correlation coefficient on the calibration line (which can only be calculated if more specimens are used than parameters determined) has no relationship with the accuracy of the analysis. In fact, a near-perfect correlation should be obtained. As in the case of drift correction, it is recommended to Copyright © 2002 Marcel Dekker, Inc.

Figure 4 The original calibration line, based on seven data points (h) can be reconstructed using only two data points (in this case, top and bottom), with concentrations (x-ray values) modified to those obtained on the calibration (j).

measure, after recalibration, the quality control specimen(s), as gross errors might thus be identified before unknown specimen are analyzed. E.

Conclusion

Setting up a calibration to be used over extended periods of time requires considerable amounts of work and preparation. It involves not only the selection and procurement of standard specimens but also drift correction monitors and recalibration standards. Also, the specimen preparation method is an essential part of the whole procedure. The result, however, is the ability to produce quantitative results to a previously assessed degree of accuracy and precision over extended periods of time with minimal work, once the specimen preparation procedure is set up and the initial calibration is performed.

V.

CONVERTING INTENSITIES TO CONCENTRATION

A.

Introduction

The simplest equation relating intensity to concentration is Ii ¼ Ki0 Wi Ki0

ð48Þ

is assumed to be a constant. The equation holds in general for cases where the where total effect of the matrix on the analyte i is constant (e.g., elements at minor and trace concentration levels in low-alloy steels or thin-film specimens). The intensity Ii in Eq. (48) is a net intensity: measured intensity corrected for background, line overlap, and so forth. Copyright © 2002 Marcel Dekker, Inc.

In practice, the measured intensity is often used directly without the subtraction of background, leading to a more general equation: Ii ¼ Bi0 þ Ki0 Wi

ð49Þ

where Bi0 is the measured intensity when Wi ¼ 0. If there is no uncorrected line overlap, Bi0 is the background. This equation can be rearranged to Wi ¼ Bi þ Ki Ii

ð50Þ

The constant Ki is called the sensitivity and is expressed in counts per second per unit concentration (e.g., percent, mg=L, etc.). The most common method of determining the constants Bi and Ki (or Bi0 and Ki0 ) is linear regression on a number of standard specimens. Linear regression can be done by minimizing the sum of squared residuals of W, or I; see Figure 5. In theory, the method of least squares assumes that the errors of the dependent variable [W in Eq. (50)] are normally distributed. The two lines obtained (one by minimizing DW, the other by minimizing DI ) are not the same. Because, for the analysis, the intensity Ii is measured, it is recommended to minimize for Wi . Also, in general, the relative error of the measured intensities is smaller than the relative error of the concentrations in the standard specimens. This is especially true for the determination of trace elements. The following formulas can be used for the determination of the values of the parameters Ki and Bi : Wi ¼ Bi þ Ki Ii Pn Pn Iij Þ=n j¼1 Wij Iij
ð j¼1 Wij Pn 2 Pn Pn j¼1 j¼1 Iij
ð j¼1 Iij j¼1 Iij Þ=n

ð50Þ

Pn Ki ¼

ð51Þ

Figure 5 Straight lines through the data points can be determined by minimizing either the sum of the squares of the residuals DI or DW.

Copyright © 2002 Marcel Dekker, Inc.

and

Pn Bi ¼

j¼1

Wij
Ki

Pn

j¼1 Iij

n

ð52Þ

where the sums are over the standard specimens, j ¼ 1; 2; . . . ; n, with n the number of standard specimens used for analyte i. Equally important, however, are the variances on the parameters determined. Formulas to calculate the variances can be found in the literature [Draper and Smith (1966)], and many commonly used computer programs such as spread sheets and statistical packages include the relevant calculations as well. Some important conclusions are as follows: 1. 2. 3.

The concentrations of the standard specimens must cover the expected range of concentrations. The calculated concentration is more accurate at the center of the line than at the extremities; the estimated variance for Wx increases with (Wx
Waverage). Because a calibration line is derived using data from several standards, the analysis of the unknown can sometimes be more accurate than the accuracy of the individual standard specimens; this is due to the effect of averaging.

When the background is properly subtracted from the gross count rate, the background Bi is equal to zero and Eq. (50) reduces to Wi ¼ K i I i In this case, the value for the slope Ki is found from Pn Pn Iij j¼1 Wij Pn 2j¼1 Ki ¼ j¼1 Iij

B.

ð53Þ

ð54Þ

Matrix Effect

Applying Eq. (50) or (53) requires that all standards must be similar to the unknown in all aspects considered: matrix effect, homogeneity, and so on. This would lead to the use of standard specimens with a very limited concentration range. Such a requirement is in disagreement with the observation that the variance on the slope factor is smaller with increasing range: The use of standard specimens covering only a small concentration range will lead to a calibration graph with large uncertainty on the slope and intercept. On the one hand, this advocates the use of a set of standards with a wide range of concentrations; on the other hand, the requirement of similarity in matrix effects tends to limit the range. Obviously, a compromise must be made. Equation (50) is a simplification of the more general equation describing the relationship among analyte concentration Wi , specimen homogeneity Si , measured intensity Ii , and matrix effect Mi : Wi ¼ Ki Ii Mi Si

ð55Þ

The term ‘‘specimen homogeneity’’ also includes the grain size effect and the mineralogical effect. These are notoriously difficult to treat mathematically; in fact, most methods describing the grain size effect rigorously assume, for example, the dispersed phase to be perfect spheres of a given diameter or an arrangement of cubes (Bonetto and Riveros, 1985). Other methods allow more variability, but these also require a priori more information about the specimen, such as the composition of the individual granular phases, Copyright © 2002 Marcel Dekker, Inc.

the average shape and size of the phases, and so forth (Hunter and Rhodes, 1972; Lubecki et al., 1968; Holynska and Markowicz, 1981). The fact that the specimen homogeneity is as yet not described by a single successful method is one of the reasons that Eq. (55) is commonly reduced to Wi ¼ K i I i Mi

ð56Þ

Fortunately, by using adequate specimen preparation methods the effect of Si between specimens (standards as well as unknowns) can be rendered constant. This constant factor is then absorbed by the sensitivity Ki . As a first approximation, the degree of variation in matrix effect between two specimens for a given analyte i can be estimated by calculating, for both compositions, the following parameter: Ii ¼ Pi

mi ms ðl0 Þ þ Gms ðli Þ

ð57Þ

where ms ðl0 Þ and ms ðli Þ are the mass-attenuation coefficients of the specimen considered for wavelengths l0 and li , respectively, and G is the geometrical factor for both compositions. The relative difference between these expressions should not exceed a few percent, otherwise the matrix effects become too important to ignore. If the specimens include flux or a binding agent, then these must also be considered in Eq. (57). With increasing range of concentrations, deviations from linearity will be observed due to variations in matrix effects between specimens and standards. The analyst must then resort to other methods to obtain accurate results. Matrix effects are studied most easily by considering binary systems (i.e., specimens with only two elements or compounds). In the case of absorption (both primary and secondary absorption must be considered), three cases can be distinguished (see Fig. 6): 1. A simple, linear relationship between relative intensity R and weight fraction W (Curve 1 in Fig. 6). In this case, there is no matrix effect: The analyte and the matrix have very similar (in principle, the same) attenuation coefficients for the incident and the characteristic radiations. 2. Curve 2 in Figure 6 is obtained when the matrix has a higher attenuation coefficient for the analyte’s characteristic radiation than the analyte itself: The characteristic radiation is primarily absorbed by the matrix element. This is usually called positive absorption. 3. Curve 3 in Figure 6 is obtained when the matrix absorbs less than the analyte itself: The matrix has a smaller value for the attenuation coefficient for the characteristic radiation than the analyte itself. This can be the case, for example, if the matrix has a much lower atomic number than the analyte, such as Mo in Al. This effect is called negative absorption. Enhancement will generally lead to a calibration graph like curve 4 in Figure 6. The effect of enhancement is usually smaller than that of positive or negative absorption, as indicated by the position, relative to curve 1, of curves 2 and 3 and curve 4. It can be shown that the behavior of the calibration curves can be explained in terms of attenuation coefficients only if absorption is the only matrix effect. Furthermore, if monochromatic excitation is used, a single constant (calculated from attenuation Copyright © 2002 Marcel Dekker, Inc.

Figure 6 Calibration curves for binaries. Curve 1: no net matrix-effect; curve 2: net absorption of the analyte’s radiation by the matrix (positive absorption); curve 3: net absorption of the analyte’s radiation by the analyte (negative absorption); curve 4: enhancement of the analyte’s radiation by the matrix.

coefficients) suffices to express the effect of one element on the intensity of another. In the following section, various methods to deal with matrix effects will be discussed. C.

Elimination or Evaluation of the Total Matrix Effect: Compensation Methods

1. Scattered Radiation: Compton Scatter (in cooperation with Mark N. Ingham, British Geological Survey, Keyworth, UK) If the variation in matrix effects is mainly due to absorption, scattered x-rays can be used to obtain an estimate of the absorption coefficient of the specimen at a certain wavelength ls . The intensity of the scattered radiation can be shown to be inversely proportional to the mass-attenuation coefficient ms of the specimen: Is ðls Þ ffi

1 ms ðls Þ

ð58Þ

where ls is the wavelength of the scattered radiation and ms ðls Þ is the mass-attenuation coefficient of the specimen for wavelength ls . This is illustrated in Figure 7. The intensity of the fluorescent radiation, Ii ; is also inversely proportional to the mass-attenuation coefficient [Eq. (57)], but at a different wavelength: Ii ffi

Wi Wi ¼  ms ðl0 Þ þ Gms ðli Þ ms

ð59Þ

where ms ¼ ms ðl0 Þ þ Gms ðli Þ Copyright © 2002 Marcel Dekker, Inc.

ð60Þ

Figure 7 The intensity of Compton-scattered radiation is inversely proportional to the massattenuation coefficient (mac) of the specimen. Conditions: Rh tube at 60 kV.

Mass-attenuation coefficients at two different wavelengths are virtually proportional, independent of matrix composition, provided there are no significant absorption edges between the two wavelengths considered (Hower, 1959). Hence, the ratio Ii =Is is proportional to the concentration of the analyte. Both coherently and incoherently scattered primary radiation, such as tube lines for tube excited x-ray fluorescence (XRF), as well as the scattered continuous radiation can be used. The method corrects also, to some degree, for surface finish, grain size effects, and variations in tube voltage and current, but it does not correct for enhancement, thus limiting its use to analytes that are influences by absorption only. Furthermore, no absorption edges of major elements may be situated between the two wavelengths considered. If that is the case, the ratio between the mass-attenuation coefficients for the wavelengths considered depends to a large degree on the concentration of the major element(s) and the ratio is no longer constant between specimens. This reduces the range of analytes that can be covered using the scattered tube lines. The use of scattered continuum radiation close to the analyte peak may be disadvantageous due to limited intensity, leading to either long measurement times or poor counting statistics. The intensity of the scattered Compton radiation is higher for specimens mainly consisting of low-atomic-number elements than for specimens with a higher average atomic number. This is illustrated in Figure 8, where the scattered radiation is plotted as a function of wavelength for two specimens: iron and magnesium. In this experiment, the spectrometer was equipped with a Rh anode x-ray tube. The two sharp peaks that can be observed are the RhKb (at 0.0546 nm) and the RhKa (at 0.0613 nm), respectively. The other two, much broader peaks are the Compton-scattered RhKb and the RhKa. The Compton-scattered peaks are much broader than Rayleigh-scattered Copyright © 2002 Marcel Dekker, Inc.

Figure 8 Intensity of scattered radiation as a function of wavelength. Two different specimens have been used: iron and magnesium. A Rh anode tube was used in this experiment. Note the large difference in intensity between the specimens. For the specimen consisting of the element with the higher atomic number (Fe), the intensity of the Compton (incoherent)-scattered radiation is lower than the intensity of the Rayleigh (coherent)-scattered radiation, compared to the specimen with the lower-atomic-number element (Mg).

characteristic lines. They are also shifted toward longer wavelengths by an amount Dl, which is given by Dl ¼ 0:00243ð1
cos cÞ

ð61Þ 0

00

where c is the angle through which the radiation is scattered: c ¼ c þ c and Dl is expressed in nanometers. In the spectrometer used for the recording of the spectra in Figure 8, c is 100 . The Compton shift, Dl, is thus 0.0029 nm for this configuration. The maxima of the Compton peaks (at 0.0576 nm and 0.0644 nm) in Figure 8 are in good agreement with the theoretical values: 0.0546 nm þ 0.0029 nm ¼ 0.0575 nm and 0.0613 nm þ 0.0029 nm ¼ 0.0642 nm, respectively. The intensity of the Compton-scattered radiation is higher for shorter wavelengths and for specimens consisting of elements with low atomic numbers. For a given wavelength (e.g., the characteristic radiation of a tube line), the intensity of Compton scatter decreases as the specimen consists of more and more elements with higher atomic numbers. For specimens made up of oxides, the scattered intensity is usually so intense that it can be measured with sufficient precision in a relatively short time. On the other hand, for specimens made up predominantly of heavier elements, such as steel and even more so for brasses and solders, the intensity of the scattered radiation is very low (see Fig. 8) and counting the statistical error can preclude precise analysis in a reasonable amount of time. The most common application where this approach (or a variant thereof) is used is in the determination of trace elements in specimens of geological origin. This is illustrated in Figure 9 for the determination of Sr in specimens of widely Copyright © 2002 Marcel Dekker, Inc.

varying geological origin. In Figure 9a, the net count rate for SrKa is plotted against the concentration for a large number of specimens. There is a considerable spread around the calibration line established. The scatter is greatly reduced when the net count rate of

Figure 9 (a) Net count rate of SrKa as a function of Sr concentration for a large number of specimens of varying geological origin. There is considerable spread around the calibration line. (b) The ratio of the count rates of the SrKa radiation and the Compton-scattered tube line is plotted against the concentration of Sr. The spread of the data points around the calibration line is now much reduced compared to (a); this is especially the case for the point labeled A.

Copyright © 2002 Marcel Dekker, Inc.

SrKa is divided by the count rate of the RhKa Compton-scattered tube radiation, as indicated in Figure 9b. Feather and Willis (1976) have shown that the intensity of the Compton peak can also be used as an estimate for the background under characteristic lines; this eliminates measuring the intensity of the background near the peak. With this method, not only is the measurement time per specimen reduced (using WDS) but the difficult task of finding interference-free background positions is no longer required. 2. Internal Standard In this method, an element is added to each specimen in a fixed proportion to the original sample. This addition has to be made to the standard samples as well as to the unknowns. The characteristic radiation of the element added should be similar to the characteristic radiation of the analyte in terms of absorption and enhancement properties in the matrix considered. Such an element is called an ‘‘added internal standard’’, or ‘‘internal standard’’ for short. In practice, the method works equally well if a pure element or a pure compound is added, or if a solution with the internal standard element is used. If a solution is used, care must be taken that the solution itself does not contain any elements that are to be analyzed. The composition of the solution used as the additive must be constant, otherwise it might affect the matrix effect. The intensity of the internal standard is affected by matrix effects in much the same way as the intensity of the analyte, provided there are no absorption edges (leading to difference in absorption) or characteristic lines including scattered tube lines (leading to difference due to enhancement) between the two wavelengths considered. Because Wi ¼ K i I i Mi

ð62Þ

for the analyte i and Ws ¼ Ks Is Ms

ð63Þ

for the internal standard s, the following ratio can be obtained by dividing Equation (62) by Equation (63): Ii ¼ Kis Wi Is

ð64Þ

where Kis ¼

K s Ms K i Mi W s

ð65Þ

Because the same amount of internal standard is added to all specimens, Ws is essentially a constant and can be included in the constant Kis. It should be noted that Mi (and Ms) is not a constant over the concentration range of interest (otherwise linear calibration would suffice) but depends on the matrix elements. However, if both Mi and Ms vary in a similar manner with the matrix elements, the ratio Mi=Ms is less sensitive to variation in the matrix effect and, in practice, can be considered a constant. In practice, the constant Kis is determined using linear regression. The main advantage of the internal standard method over the scattered-radiation method is its ability to correct effectively for enhancement as well as for absorption. It also corrects—at least partially—for variations in density of pressed specimens. The requirement that the intensity of the characteristic radiation of both the analyte and the internal standard element vary in the same manner with the Copyright © 2002 Marcel Dekker, Inc.

matrix effects imposes that there should be no absorption edges and no characteristic radiation from other elements between the measured line of the analyte and that of the internal standard. Furthermore, ideally, the analyte should not enhance the internal standard or vice versa. If Ka radiation is measured and if the atomic number of the analyte is Z (with Z > 23), then, very often, the elements with atomic number Z
1 or Z þ 1 are very good candidates. This assures that there are no K absorption edges and no K emission lines of other elements between the two elements considered. The element with atomic number Z is not enhanced by Ka radiation from an element with Z þ 1, but only by the much weaker Kb radiation, whereas for elements with atomic number Z þ 2 and higher, both Ka and Kb contribute to enhancement. In practice, some enhancement between the internal standard element and the analyte or vice versa is allowed, as the concentration of the internal standard is constant and the ratio is based on intensities. The absence of major elements in the specimens with L absorption edges and emission lines, however, must be checked for. The situations that must be avoided are (1) the case where a major line of a major matrix element is between the absorption edges of the analyte and the internal standard and (2) the case where a major absorption edge of a major matrix element is situated between the measured characteristic lines of the analyte and the internal standard. In the first case, the matrix element would enhance either the analyte or the internal standard element, but not both; in the second case, the matrix element absorbs strongly either the radiation from the analyte or the internal standard, but not both. In both of these cases, varying concentrations of the matrix element will lead to variable and different effects on the intensities of the analyte and of the internal standard, and the ratio used in Eq. (64) will not compensate for such events. The method, however, has some important limitations: The specimen preparation is made more complicated and is more susceptible to errors. The addition of reagents and the requirement of homogeneity of the specimen tends to limit the practical application of the method to the analysis of liquids and fused specimens, although it sometimes finds application in the analysis of pressed powders. Although the rule Z þ 1 or Z
1 can serve as a rule of thumb, it is quite clear that for samples where many elements are to be quantified, a suitable internal standard cannot be found for every analyte element. Sometimes, more elements are used in one internal standard solution to provide suitable internal standards for more analytes. Also, the fact that the internal standard method is easier to apply to liquids can generate some problems. Heavier elements (e.g., Mo) are more difficult to determine using this method, because liquid specimens are generally not of infinite thickness for the K wavelengths of these heavier elements. In such cases, the L line can be used, with an appropriate internal standard. The method will, however, also provide some compensation for the effects of noninfinite thickness, especially if the wavelength of the internal standard selected is very similar to that of the analyte line. Theoretically, L lines of a given element can be used as internal standards for K lines of other elements and vice versa if these wavelengths are reasonably close to each other and neither interfering lines nor edges occur between them. In principle, the method allows the determination of one or two elements in a specimen without requiring analysis (or knowledge) of the complete matrix. Copyright © 2002 Marcel Dekker, Inc.

The range of concentration over which this method is suitable can be quite large, up to 10–20 % in favorable situations, but the internal standard technique is most effective at low concentrations (or high dilutions). The method finds, for instance, application in the determination of Ni and V in petroleum products, where MnKa is used as an internal standard (ISO, 1995). The internal standard method allows an accurate determination of these elements in a much wider variety of petroleum products than the method based on linear calibration. 3. Standard Addition Methods Another method of analysis involves the addition of known quantities of the analyte to the specimen and is referred to as the standard addition method. If the analyte element is present at low levels, and no suitable standards are available (e.g., the matrix is unknown), standard addition and=or dilution may prove to be an alternative, especially if the analyst is interested in only one analyte element. The principle is the following: Adding a known amount of the analyte i ðDWi Þ to the unknown specimen will give an increased intensity ðIi þ DIi Þ. Assuming a linear calibration, the following equations apply: Wi ¼ K i I i

ð62Þ

for the original specimen and Wi þ DWi ¼ Ki ðIi þ DIi Þ

ð66Þ

for the specimen with the addition. Thus, the method assumes that linear calibration is adequate throughout the range of addition, because it assumes that an increase in the concentration of the analyte by an amount DWi will increase the intensity by Ki DIi .

Figure 10 Standard addition method. The net intensity is plotted versus weight fraction of the element added to the sample and a ‘‘best-fit’’ line is determined. The intercept of that line with the concentration axis is
Wi.

Copyright © 2002 Marcel Dekker, Inc.

These equations can be solved for Wi. To check the linearity of the calibration, the process can be repeated by adding different amounts of the analyte to the specimen and plotting the intensity measured versus the concentrations added (Fig. 10). The intercept of the line on the concentration axis equals
Wi. Note that the concentration in the unknown is actually found through extrapolation of the linear calibration toward zero intensity. The intensities used for calibration must be corrected for background and line overlap. If this correction is not performed (or not performed accurately), the value of the concentration determined will be overestimated by an amount proportional to the intensity of the background (or line overlap) and inversely proportional to the sensitivity. The method is suitable mainly for determination of trace and minor concentration levels, as the amount DWi added to the sample must be in proportion to the amount Wi in the sample itself. The extrapolation error can be quite large if the slope of the line is not known accurately. Adding significant amounts of additives to the sample might, however, lead to nonlinearity, as it will alter the matrix effect. Compounds and solutions can be used for the standard addition. If the analyte i in the original sample is in a different phase than in the additive, care must be taken in the calculation of the concentration sought. The relevant stoichiometric or gravimetric factors must be included. This also applies if the analyte is present under elemental or ionic form in, for example, the original sample and in a compound phase in the additive, or vice versa. Another way to alter the concentration of the analyte is by diluting the liquid or solid solution of the sample. By diluting several times by known amounts, a line can be established. By repeating this procedure with a standard solution containing a known amount of i, the unknown concentration can be found. 4. Dilution Methods Dilution methods can also eliminate or reduce the variation of the matrix effect, rather than compensating for such variation. The dilution method can be explained using Eq. (17): mi ðl0 Þ Pn j¼1 Wj mj ðl0 Þ þ G j¼1 Wj mj ðli Þ

Cðl0 ; li Þ ¼ Pn

ð17Þ

which can be rewritten as Cðl0 ; li Þ ¼

mi ðl0 Þ ms ðl0 Þ þ Gms ðli Þ

ð67Þ

where ms ðl0 Þ and ms ðli Þ are the mass-attenuation coefficients of the specimen for the primary wavelength l0 and analyte wavelength li, respectively. Apparently, deviations from linearity are due to variations in ms ðl0 Þ and=or ms ðli Þ. Enhancement is ignored at this stage. If one adds, to the sample, D grams of a diluent (d ) for each gram of sample, the denominator or Eq. (67) becomes 1 D ½m ðl0 Þ þ Gms ðli Þ þ ½m ðl0 Þ þ Gmd ðli Þ 1þD s 1þD d

ð68Þ

If the term D=ð1 þ DÞ½md ðl0 Þ þ Gmd ðli Þ is much larger than 1=ð1 þ DÞ½ms ðl0 Þ þ Gms ðli Þ, the factor Cðl0 ; li Þ becomes essentially a constant and variations due to varying matrix effects between samples become negligible. This can be done in two ways. (a)

Making D=ð1 þ DÞ large by diluting each sample by adding a large, known amount of a diluent.

Copyright © 2002 Marcel Dekker, Inc.

(b)

Adding a smaller quantity of diluent than in the previous case, but with a much larger value for md ðl0 Þ þ Gmd ðli Þ. This is called the technique of the heavy absorber.

Both of these procedures, however, require the addition of reagents to the sample. This can easily be done for dissolved samples, either in liquids or fused samples, but it is more difficult for powdered samples (homogeneity!). These methods do not eliminate the matrix effects completely, but reduce their influence. On the other hand, they also reduce the line intensity of the analyte; thus, a compromise must be sought. Dilution methods also have the advantage of reducing the enhancement effect if one uses a nonfluorescing diluent (e.g., H2O or Li2B4O7). In this case, the effect is reduced by the fact that the concentrations of both the enhancing element and the analyte are reduced. If the diluent contains elements whose characteristic x-rays can excite the analyte, as well as some other matrix elements, then the contribution of those unknown quantities of matrix elements to the total enhancement is reduced: The enhancement of the analyte by the diluent would then be determining and can be considered to be constant. This method allows the determination of all measurable elements in the sample, as opposed to the standard addition method, where an addition must be made for each element of interest. D.

Mathematical Methods

1. General The term ‘‘mathematical methods’’ refers to those methods that calculate rather than eliminate or measure the matrix effect. Mathematical methods are independent of the specimen preparation in the sense that specimen preparation is taken into account if the composition of the specimen presented to the spectrometer has been changed (e.g., by fusion), but mathematical methods do not prescribe the specimen preparation method as is done, for example, by the standard addition method. The actual calculation method used to convert intensities to concentrations does not affect the choice of the specimen preparation method. The aim of the specimen preparation is limited to the presentation to the spectrometer of a specimen that is homogeneous (with respect to the XRF technique) and that has a well-defined, flat surface representative for the bulk of the specimen. Mathematical methods usually require knowledge of all elements in the standard specimens and allow determination of all measurable elements in the unknowns. In practice, trace elements can be neglected in the calculations for the analytes present at higher concentrations, as these trace compounds are neither subject to an important (and variable) matrix effect nor do they contribute significantly to the matrix effect of other elements. Their concentrations are often found by straightforward linear regression. The mathematical methods are divided in two main categories: the fundamental parameter method and the methods using influence coefficients. 2. The Fundamental Parameter Method a. Introduction The fundamental parameter method is based on the theory that enables one to calculate the intensity of fluorescent radiation, originating from a specimen of known composition. The equations used usually consider both primary and secondary flourescence (enhancement). Copyright © 2002 Marcel Dekker, Inc.

Higher-order effects and effects due to scattered radiation are usually neglected. Formulas to calculate intensities of fluorescent radiation were proposed very shortly after the introduction of the commercial XRF spectrometers, in the early 1950s (Gillam and Heal, 1952). The equation describing the intensity of the fluorescent radiation as a function of specimen composition, spectrometer configuration, and incident spectrum was derived earlier (see Sec. II.B) and is repeated here: lZedge;i "

Ii ðJðlÞÞ ¼

Pi ðlÞ þ lmin

X

# Sij ðl; lj Þ dl

ð69Þ

j

where Pi ðlÞ is the contribution of the primary fluorescence caused by incident photons with wavelength l, and Sij ðl; lj Þ is the contribution of the secondary fluorescence (enhancement) by characteristics photons lj which have been excited by primary photons l. The summation in Eq. (16) or (69) is over all elements j that have characteristic lines that can excite the analyte i. For each of these elements j, all characteristic lines must be considered. This is quite simple if none of the L lines or M lines of element j can excite the analyte. In that case, only the Ka and Kb lines are to be considered. If the L lines of an element j are energetic enough for enhancement of the analyte, the sheer number of L lines (e.g., W has more than 20 characteristic L lines that can be considered) would make the calculation very time-consuming. Therefore, most programs consider only three to five L lines for each element. A similar reasoning holds for the M lines. Application of these formulas was originally limited to the prediction of intensities for specimens with given composition. The application of those formulas for analysis was not pursued until the 1960s. The method of analyzing specimens by fundamental parameter equations has been developed independently around the same time by Criss and Birks (1968) and by Shiraiwa and Fujino (1966). Due to the large amount of calculations involved (especially the integration of the incident spectrum and the calculation of the contribution of enhancement), these programs initially ran on mainframes and mini computers. A PC version of the same program was proposed by Criss in 1980 (Criss, 1980a). The application of a fundamental parameter method for analyzing specimens consists of two steps: calibration and analysis. Both steps will be discussed in more detail in the following subsections. b. Calibration The fundamental parameter equation is used to predict the intensity of characteristic lines for a composition identical to that of the standard used. If more than one standard specimen is used, the calculations are repeated for each of the standards. The calculations are performed using the appropriate geometry (i.e., incidence and take-off angles are taken in agreement with those of the spectrometer used) and the parameters determining the tube spectrum (such as the anode material, voltage, thickness of beryllium window, and so on) correspond to the ones used in the spectrometer for the measurements. The intensities predicted are (almost always) net intensities, void of background, line overlap, crystal fluorescence, and so on. Hence, the measured intensities must be corrected for such spectral artifacts. These theoretically predicted intensities are then linked to the actually measured ones. If only one standard specimen is used, the ratio between the measured intensity and the calculated intensity is calculated. If more than one standard is used, the net intensities obtained from the measurements are plotted versus the calculated intensities and a straight line can be determined for each characteristic line measured. The slope of Copyright © 2002 Marcel Dekker, Inc.

such a line is the proportionality factor between predicted (calculated) and measured intensities. In general, this relationship will be determined more accurately if more standards are used. A special case of calibration ensues when one uses pure elements as standards. Dividing each of the measured (net) intensities of the corresponding pure element gives the relative intensity. This relative intensity can be calculated directly by some fundamental parameter programs. In fact, some programs use equations that express the intensity of characteristic radiation directly in terms of relative intensity. The relative intensity in this respect is thus defined as the intensity of the sample, divided by the intensity of the corresponding pure element (or compound, if the concentration of the analyte is defined in compound concentration), under identical conditions for excitation and detection. Basically, the calibration function, which is determined using the measured intensities of the standards and the calculated intensities, accounts for instrument-related factors only; matrix effects are accounted for by using the physical theory as described in the fundamental parameter equation. The instrument-related parameters for a wavelength-dispersive spectrometer are as follows: Collimation Crystal reflectivity Efficiency of detector(s) Fraction of emergent beam, allowed into the detectors, after Bragg reflection (is also dependent on Bragg angle) For energy-dispersive spectrometers, the instrument-related parameters are collimation and detector efficiency. The effect of the windows of the detector, the dead layer, and so forth can also be taken into account. c.

Analysis Step 1. For every unknown specimen, a first estimate of the composition is made. There are several ways to obtain such a first estimate. They vary from using a simple, fixed composition (e.g., equal to 100%, divided by the number of elements considered; in the first estimate, all the concentrations of each of the elements are thus taken equal to one another) to the composition derived from the measured intensities in combination with the calibration curves. Using the calibration data, it is possible to estimate for each element the intensity that would be obtained if the pure elements were measured. These numbers are then used to divide the intensity of the corresponding element, measured in the unknown specimen. The resulting fractions are scaled to 100% and used as the first estimate. Step 2. For this estimate of composition, the theoretical intensities are calculated. These are converted to measured intensities, using the calibration data, so that these two sets of intensities for the same specimen can be compared. Step 3. The next estimate of composition is obtained based on the difference between the measured and calculated intensities. Again, there are different methods available: 1.

2.

The simplest method is based on linear interpolation. If, for a given element, the measured intensity is 10% higher than the calculated intensity, the concentration of that element is increased by 10%. Rather than a linear relationship, some authors (Criss and Birks, 1968) use an interpolation based on three points. This is done because the relationship between concentration and intensity is usually non linear over a wider range. If the specimen is a pseudobinary, hyperbolic relationships have proven to be

Copyright © 2002 Marcel Dekker, Inc.

better approximations. For more complex specimens, it still works out quite well, because the concentrations of the other elements are considered fixed at this stage. The hyperbolic equation requires a minimum of three points for its parameters to be determined. The points requiring the least additional calculation time are the following: The origin (net intensity zero, at concentration zero). The pure element, W ¼ 100%; the corresponding intensity has already been calculated. The point W ¼ current estimate, the intensity has already been calculated. These three points allow an hyperbolic relation to be established around the current estimated composition. From this curve and the measured intensity, the new concentration estimate is derived. This approach is repeated for every analyte element. This method usually provides a faster convergence than using the simple straight line. 3. It is also possible to use gradient methods to determine the next composition estimate. The formula for the first derivative, with respect to concentration, of the fundamental parameter equation have been published (Shiraiwa and Fujino, 1968), but the calculation is cumbersome and time-consuming. Also, the derivatives can be obtained by a finite-difference method, where the effect of a small change in composition on the intensity is observed. Step 4. The process, starting at Step 2, is now repeated until convergence is obtained. Different convergence criteria exist. The calculation can be terminated if one of the following criteria is satisfied for all the elements (compounds) concerned: 1. The intensities, calculated in Step 2, do not change from one step to another, by more than a present level. 2. The intensities, calculated in Step 2 agree, to within a preset level, with the measured intensities 3. The compositions, calculated in Step 3, do not change from one step to another by more than a present level (e.g., 0.0005 or 0.0001 by weight fraction). One or more of these criteria might be incorporated in the program. These criteria, however, are no guarantee that the final result is accurate to within the level, specified in the convergence criteria. Furthermore, especially with convergence criteria based on concentrations (such as criteria 3), it must be realized that a convergence criterion of 0.0005 is unacceptable when determining elements at levels below 0.0005. d. Extensions to the Method More complex scenarios are possible. The most common one includes the calculation of a set of influence coefficients (based on theoretical calculations) to obtain a composition quickly, close to the final result. Next, the fundamental parameter method is applied (Rousseau, 1984a). This should yield faster convergence in terms of computation time, because the calculation by influence factors of the preliminary composition is very fast. This method reduces the number of evaluations of the fundamental parameter equation. Also, the different programs available differ quite markedly in their treatment of the intensities of the standards measured. Some programs use a weighting of the standards, stressing the standard(s) closest (in terms of intensity) to the unknown (Criss, Copyright © 2002 Marcel Dekker, Inc.

1980a). Such programs use a different calibration for each unknown specimen. The unknown specimen dictates which standards will be given a high weighting factor and which standards will be used with less weighting. Other programs use all standards with equal weighting. e. Typical Results Early results of the fundamental parameter method on stainless steels are given by Criss and Birks (1968). The average relative differences between x-ray results and certified values were about 3–4%. Later, Criss et al. (1978) reported accuracies of about 1.5% relative for stainless steels, using a more accurate fundamental parameter program. Typical results for tool steel alloys are given in Table 3. Often, the fundamental parameter method is considered to be less accurate than an influence coefficient algorithm. This is caused primarily by the fact that the fundamental parameter method has extensively been used and described as a method that allows quantitative analysis with only a few standards. This is obviously an advantage, but it does not imply that fundamental parameter methods cannot be used in combination with many standards similar to the unknown. As a matter of fact, on several occasions and with a variety of matrices, the authors have obtained results of analysis with an accuracy similar to that of influence coefficient algorithms when using the same standards in both cases. f. Factors Affecting Accuracy The accuracy of the final results is determined by the following: The The The The

measurement specimen preparation physical constants used in the fundamental parameter equation limited description of the physical processes that are considered in the fundamental parameter equations The standard and the calibration In the following discussion, the effect of measurements and specimen preparation will not be considered.

Table 3 Analysis of Seven Tool Steels with a Fundamental Parameter Program [XRF11, from CRISS SOFTWARE, Largo, MD (Criss, 1980)] Element W Co Mn Cr Mo S P Si C

Minimum conc. (%)

Maximum conc. (%)

Standard deviation (%)

1.8 0.0 0.21 2.9 0.2 0.015 0.022 0.14 0.65

20.4 10.0 0.41 5.0 9.4 0.029 0.029 0.27 1.02

0.52 0.20 0.01 0.013 0.04 0.003 0.003 0.03 0.16

Note: One standard has been used. The minimum and maximum concentrations refer to the minimum and maximum concentrations in the set of analyzed specimens, respectively. The standard deviation is calculated from the difference between concentration values found and certified. Source: Data courtesy of Philips Analytical, Almelo, The Netherlands.

Copyright © 2002 Marcel Dekker, Inc.

g.

Physical Constants

The physical constants used in the fundamental parameter equations are as follows: Incidence and exit angles Spectrum of incident beam Mass-attenuation coefficients Flourescence yields Absorption jump ratios Ratios of intensity of different lines within a given series (e.g., Ka=Kb ratio) Wavelengths (or energies) of absorption edges and emission lines Incidence and exit angles. The incidence angle in most wavelength-dispersive (WD) and energy-dispersive (ED) spectrometers is, in fact, defined by a relatively wide cone with a different intensity at the boundaries compared to the center. This incident cone is neglected and the incident radiation is considered parallel, along a single, fixed direction. A similar observation holds for the exit angle. The effect is far less pronounced if diffraction from a plane crystal surface is used for dispersion, as is done, for example, in most sequential WD spectrometers. This has been studied to some extent by Mu¨ller (1972). To our knowledge, none of the fundamental parameter programs available takes this effect into account. Its influence, however, is, to some extent, compensated for by calibration with standard specimens. Spectrum of incident beam. The spectrum of the incident beam from an x-ray tube spectrum requires more attention. Parts of the primary spectrum might excite an element B that, in turn, excites element A very efficiently. In such cases, this enhancement may make the intensity of element A sensitive to small errors in the tube spectrum representation, which would not be compensated for if the pure A was used for calibration. This can arise, for example, in the analysis of silica–zirconia specimens with a Rh tube (Criss, 1980b). Pure silica is relatively insensitive to the intensity of the characteristic K lines of Rh. In combination with Zr, however, the situation is different. Indeed, the RhK lines are strongly absorbed by Zr. Zr then emits K and L lines that enhance Si. As a result, the Si intensity is more sensitive to the RhK lines in SiO2–ZrO2 mixtures than it is in pure SiO2. Tube spectra have been calculated using, for example, the algorithm of Pella et al. (1985). Mass-attenuation coefficients. There are several compilation of mass-attenuation coefficients, published in the literature. A continuing effort to compile the most comprehensive table has been undertaken by the National Institute of Standards and Technology (formerly National Bureau of Standards), Gaitherburg, MD. When selecting a table of mass-attenuation coefficients for use in a fundamental parameter program, the following question must be addressed: Does the table cover all the analytical needs? (In practice, does it cover the complete range of interest from the longest wavelength considered, down to the excitation potential of the tube?) The analyst should be aware that the use of formulas to generate mass-attenuation coefficients can lead to values that can be significantly different from the corresponding table values. Presently, for applications in XRF, the complications of McMaster et al. (1969), Heinrich (1966), Leroux and Thinh (1977), or Veigele (1974) are most often used. A short discussion on the agreement between some of these compilations has been presented by Vrebos and Pella (1988). A more recent compilation has been published by de Boer (1989). Fluorescence yields. A comprehensive reference to fluorescence yields, including Coster–Kronig transitions, can be found in the work of Bambynek et al. (1972). (see also Chapter 1, and Appendix VI). Copyright © 2002 Marcel Dekker, Inc.

Absorption jump ratios. These can be derived from the tables of attenuation coefficients. Ratios of different fluorescent lines within a family. Data for the K spectra can be found in the work by Venugopalo Rao et al. (1972) (see also Chapter 1). Wavelengths of absorption edges and emission lines. A comprehensive table was published by Bearden (1967) and is also presented in the appendices to Chapter 1. Because attenuation coefficients are wavelength dependent, an error in a wavelength of any characteristic line will automatically lead to a bias in the corresponding attenuation coefficients. h.

Limited Physical Processes Considered

The fundamental parameter equation [Eq. (16)] does not consider all physical processes in the specimens. Three of the most obvious that are missing are described here. Tertiary fluorescence. Although the formula for tertiary fluorescence has been derived by, for example, Shiraiwa and Fujino (1966) and Pollai and Ebel (1971), it is not included in most fundamental parameter programs. Usually, the tertiary fluorescence effect is considered small enough to be negligible. Shiraiwa and Fujino (1967, 1974) have presented data showing a maximum contribution of tertiary fluorescence of about 3% relative to the total intensity of Cr in Fe–Cr–Ni specimens. Therefore, even in Fe–Cr–Ni specimens whose characteristic lines and absorption edges are ideally positioned relative to one another to favor enhancement, the effect of tertiary fluorescence is quite limited. Higher-order enhancement is also possible, but it is even less pronounced than tertiary fluorescence. Scatter. Other processes not considered in most of the fundamental parameter methods are coherent and incoherent scatter of both the primary spectrum and the fluorescent lines. This is usually justified by pointing out that the photoelectric effect is, by far, the major contribution to the total absorption. It is believed that the contribution by scattered photons to the excitation of characteristic photons is negligible. However, in some cases the scattered primary spectrum may have a considerable influence, as illustrated earlier in this chapter. The equations describing the contribution of scatter to fluorescent intensity have been derived by Pollai et al. (1971). These equations have obviously many similarities to those for secondary fluorescence. Photoelectrons. The processes that are probably the must unknown in the fundamental parameter method are related to the contributions of the photoelectrons and of the Auger electrons that are produced as a result of absorption of the primary and fluorescent x-ray photons. These electrons have sufficient energy to excite other atoms and thus create additional fluorescence. This is especially important in the case of low-atomic-number elements, as has been described by Mantler (1993) and has been illustrated, for example, by Kaufmann et al. (1994). i.

Standards and Calibration

The use of good standards (similar to the unknown) will almost always lead to more accurate results, compared to a situation where the standards used have a widely different composition from that of the unknown. This is because most of the uncertainties, caused by inaccuracies in the physical constants, cancel. The degree of similarity between standards and unknown has an important effect on the accuracy of the analysis. 3. Influence Coefficient Algorithms Another class of mathematical methods calculates the matrix effect by means of coefficients, rather than by evaluating the fundamental parameter equation for each unknown. It will be shown that these coefficients can also be calculated from theory, using fundaCopyright © 2002 Marcel Dekker, Inc.

mental parameters. Many of such influence coefficient algorithms have been proposed and they have been divided and subdivided in different ways (Lachance, 1979). It is not the intention to discuss all of the algorithms here; only a few selected ones will be discussed. This selection is based on the popularity of the methods and=or on some interesting characteristics of their underlying theory. Some of these algorithms use only one single coefficient per interfering element; others use more than one. The distinction used here, however, depends on whether the influence coefficients is considered to be a constant for a given application or whether the value of the coefficient varies with composition. The latter methods will be discussed in Sec. V.5. Only two algorithms that use constant influence coefficients will be discussed here: the Lachance–Traill and the de Jongh algorithms. The practical application of the resulting equations (i.e., calibration and analysis) will be treated separately in Sec. V.7. One has to emphasize that the approach based on constant influence coefficients has many common aspects with all the influence coefficient algorithms discussed in Secs. V.4 and V.5. All of the influence coefficient models express the total matrix effect Mi for a binary mixture ij as follows: Mi ¼ ½1 þ mij Wj 

ð70Þ

where mij indicates the true binary influence coefficient describing the matrix effect of j on the analyte i in binaries ij. More generally, 3 2 n X 7 6 Mi ¼ 6 mij Wj 7 5 41 þ

ð71Þ

j¼1 j6¼e

for a multielement specimen, with n being the total number of elements or compounds. In most of the influence coefficient algorithms, one element is eliminated from the summation {i.e., one influence coefficient is used when dealing with binaries, as in Eq. (70), and n
1 coefficients deal with a specimen consisting of n compounds [Eq. (71)]}. In Eq. (71) this is explicitly indicated by the j 6¼ e under the summation sign. The eliminated compound e can be any of the ones present in the specimens; however, most authors eliminate the analyte. The expression for Mi is then used as follows: 2 3 n X 6 7 Wi ¼ R i 6 1 þ mij Wj 7 4 5

ð72Þ

j¼1 j6¼e

which links the relative intensity and the influence coefficients to the composition of the specimen. The relative intensity Ri for a given analyte is defined as the ratio of the net measured intensity Ii in the specimen and the intensity that would have been measured on the pure analyte IðiÞ under identical conditions: Ri ¼

Ii IðiÞ

ð73Þ

In practice, the relative intensity is often derived indirectly from measurements on standards, and the pure element (or compound) is not required. Equation (56) can be rewritten in terms of Ri : Wi ¼ R i Mi Copyright © 2002 Marcel Dekker, Inc.

ð74Þ

By definition, Mi ¼

Wi Ri

ð75Þ

The ratio of the weight fraction of the analyte Wi and its relative intensity Ri is the matrix effect Mi . When the analyte radiation is absorbed (or when the absorption effects are dominating over enhancement), Mi is larger than 1. On the other hand, when enhancement is dominant, Mi is smaller than 1. Also, Mi is smaller than 1 in the absence of enhancement, but when the absorption by the analyte is significantly higher than that of the matrix elements, as indicated by curve 3 in Figure 6. One of the consequences of Eq. (74) is that for the pure analyte Wi ¼ 1, Ri ¼ 1 and, thus, Mi is also equal to 1. This implies that the matrix effect as introduced here should be viewed as relative to the pure element and not in absolute terms. Even in the pure element specimen, x-ray photons are subject to matrix effect. It is possible, even in the pure analyte, to be subjected to enhancement effects. This is, for example, the case when L lines are analyzed if the K lines of the same analyte are also excited. Furthermore, the L lines of elements with large atomic numbers can be fluoresced by other L lines, as indicated in Table 1. However, it is customary to refer to the situation as a situation without matrix effect. Also, if Mi ¼ 1, Eq. (74) reduces to Wi ¼ R i

ð76Þ

In other words, in the absence of matrix effect, the concentration of the analyte is equal to the relative intensity. For a specimen containing, for example, 25% (by weight) of the analyte, an intensity will then be measured that is 25% of that of the pure analyte. Comparing Eqs. (56) and (74) and considering the definition of the relative intensity given by Eq. (73) yields Ki ¼

1 IðiÞ

in other words, the sensitivity is the reciprocal of the intensity of the pure analyte. So, the intensity of the pure analyte can also be obtained without making measurements for a specimen of the pure element. It can be obtained from the slope of the calibration line or even from the measurement on a single specimen: IðiÞ ¼

Ii M i Wi

ð77Þ

where Mi is calculated and Ii is measured. Mi can be calculated using Eq. (75), where Ri is calculated from theory for the standard specimen of known composition. Therefore, although many influence coefficient algorithms are presented using the format of Eq. (72), involving the relative intensity, there is no real need to perform measurements of the pure analyte, as Eq. (72) can be written as 2 3 n X 6 7 mij Wij 7 Wi ¼ K i I i 6 41 þ 5

ð78Þ

j¼1 j6¼e

where Ki is then determined during the calibration phase. Furthermore, if the background is not subtracted, Eq. (78) can be written as Copyright © 2002 Marcel Dekker, Inc.

3

2 6 Wi ¼ Bi þ Ki Ii 6 41 þ

n X j¼1 j6¼e

7 mij Wij 7 5

ð79Þ

where Bi is the background expressed as a concentration equivalent. The constants Ki and Bi are then determined during regression analysis. 4. Algorithms with Constant Coefficients a.

The Lachance–Traill Algorithm Formulation. In 1966, Lachance and Traill proposed a correction algorithm based on influence coefficients (Lachance and Traill, 1966). The equations are, for a ternary consisting of the elements (or compounds) A, B, and C: WA ¼ RA ½1 þ aAB WB þ aAC WC 

ð80aÞ

WB ¼ RB ½1 þ aBA WA þ aBC WC 

ð80bÞ

WC ¼ RC ½1 þ aCA WA þ aCB WB 

ð80cÞ

RA , RB , RC are the relative intensities of A, B, and C, respectively. The coefficients aAB , aAC , aCA , and so forth are called influence coefficients. A more general notation of the Lachance–Traill algorithm is, for analyte i, 2 3 n X 6 7 Wi ¼ R i 6 1 þ aij Wj 7 4 5

ð81Þ

j¼1 j6¼i

where the summation covers all n elements (or compounds) in the specimen, except the analyte itself. Hence, there are n
1 terms in the summation. This is common to all currently used algorithms. Most of the algorithms, developed earlier [such as Sherman’s (1953) and Beattie and Brissey’s (1954)], used to have n terms, rather than n
1, for specimens with n elements. Equations (80a), (80b), and (80c) are linear equations in the concentrations of the elements WA , WB , and WC , respectively. Note that there are only two coefficients for each analyte element. Consider, for example, the first equation of the set 80, namely Eq. (80a): Element A is the analyte and its concentration is equal to the relative intensity RA , multiplied by the matrix correction factor (1 þ aAB WB þ aAC WC ). This matrix correction factor has only two coefficients: one (aAB ) to describe the effect of element B on the intensity of A and, similarly, one to describe the effect of element C on the intensity of A. The value of the coefficient aAA , which would correct for the effect of A on its own intensity (sometimes—but incorrectly—referred to as self-absorption) is zero. Similarly, aBB and aCC are also zero. The effect of A on A, however, is taken into account, as will be shown in the next subsection. Calculation of the coefficients. Lachance and Traill also showed that the influence coefficients, ij can be calculated for monochromatic excitation by photons with wavelength 0 (assuming absorption only) from the expression aij ¼

mj ðl0 Þ cscðc0 Þ þ mj ðli Þ cscðc00 Þ
1 mi ðl0 Þ cscðc0 Þ þ mi ðli Þ cscðc00 Þ

Copyright © 2002 Marcel Dekker, Inc.

ð82Þ

When secondary fluorescence (enhancement) is involved, the coefficients are calculated in the same way. Thus, enhancement is being treated as negative absorption. This assumption is not valid when enhancement is quite severe. Differences in primary absorption may easily be confused with enhancement (see Fig. 6). From Eq. (82), it follows clearly that aii is always zero. Therefore, it is not included in the summation of Eq. (81). Also, from Eq. (82), it follows that the value of the coefficients for the Lachance–Traill algorithm cannot be less than
1. It must be stressed that Eq. (82) is only valid strictly for monochromatic excitation and for those analytes that are only subject to absorption (no enhancement). In this case, the influence coefficient is concentration independent: It is a constant, even for the complete concentration range from 0% to 100%. It does, however, depend on parameters, such as the wavelength of the primary photons and the incidence and exit angles. In all other cases (polychromatic excitation and=or enhancement), Eq. (82), strictu sensu, cannot be used. A polychromatic beam (from, e.g., an x-ray tube) can be ‘‘replaced’’ by a monochromatic one, by resorting to the effective wavelength. The effective wavelength, however, is composition dependent (see Sec. II.C). The value of the coefficients, calculated using Eq. (82), is also dependent on composition, although Wi nor Wj figure explicitly in Eq. (82). If enhancement is dominant, another method must be applied to calculate the coefficients. Influence coefficients for the algorithm of, for example, Lachance–Traill can also be calculated based on actual measurements. Rewriting Wi ¼ Ri ½1 þ aij Wj 

ð83Þ

to aij ¼

ðWi =Ri Þ
1 Wj

ð84Þ

yields an expression that can be used to obtain aij , based on the composition of the binary and the relative intensity Ri . The drawbacks associated with this method are as follows: 1.

2. 3.

4.

The calculation of Ri requires the measurements of the intensity on the pure i (element or compound). This could lead to large errors if the intensity of i in the binary is much lower than that of the pure, due to, for example, nonlinearity of the detectors. The pure elements (or compounds) are not always easily available or could be unsuitable to present to the spectrometer (e.g., pure Na or Tl). Equation (84) is very prone to error propagation when Wi is close to 1. The numerator is then a difference between two quantities of similar magnitude, and the denominator is then close to zero, magnifying the errors. Also, the availability of suitable binary specimens can present problems: Some alloys tend to segregate and homogeneous specimens are then difficult to obtain.

The coefficients aij can also be calculated from theory: Calculate Ri for the binary with composition ðWi ; Wj Þ rather than obtain it from measurements and substitute in Eq. (84). This method eliminates drawbacks 1, 2, and 4. However a better method—without the problem associated to error propagation—is to use Eq. (75) directly with the values for Ri and Wi . Lachance (1988) has also presented methods to calculate the values of the coefficients from theory. Copyright © 2002 Marcel Dekker, Inc.

The Lachance–Traill algorithm assumes the following: 1.

b.

The influence coefficients can be treated as constants, independent of concentration; this limits the concentration range in cases where the matrix effects change considerably with composition. 2. The influence coefficients are invariant to the presence and nature of other matrix elements. So aFeCr , determined for use in Fe–Cr–Ni ternary specimens, is the same as aFeCr in Fe–Cr–Mo–W–Ta or Fe–Cr specimens. The de Jongh Algorithm

Formulation. In 1973, de Jongh proposed an influence coefficient algorithm (de Jongh, 1973), based on fundamental parameter calculations. The general formulation of his equation is " # n X aij Wj Wi ¼ E i R i 1 þ ð85Þ j¼1 j6¼e

where Ei is a proportionality constant (and is usually determined during the calibration). The summation covers n
1 elements (as is the case with the algorithm of Lachance and Traill), but the eliminated element, e, is the same for all equations. If, for a ternary specimen, element C is eliminated, the following equations are obtained: WA ¼ EA RA ½1 þ aAA WA þ aAB WB 

ð86aÞ

WB ¼ EB RB ½1 þ aBA WA þ aBB WB 

ð86bÞ

WC ¼ EC RC ½1 þ aCA WA þ aCB WB 

ð86cÞ

Note that in order to obtain the concentration of elements A and B, the concentration of C, WC , is not required. This is different from Lachance and Traill’s algorithm: In order to calculate the concentration of A, using Eq. (80a), the concentrations of both B and C are required. If the user is not really interested in element C (e.g., element C is iron in stainless steels), Eq. (86c) need not be considered and the analysis of the ternary specimen can be done by measuring RA and RB and solving Eqs. (86a) and (86b). Calculation of the coefficients. de Jongh also presented a method to calculate the coefficients from theory. The basis is an approximation of Wi =Ri by a Taylor series around an ‘‘average composition’’: Wi ¼ Ei þ di1 DW1 þ di2 DW2 þ    þ din DWn Ri where Ei is a constant given by
 Wi Ei ¼ Ri average

ð87Þ

ð88Þ

and DWi ¼ Wi
Wi;average

ð89Þ

dij are the partial derivatives of Wi =Ri with respect to concentration: dij ¼

@ðWi =Ri Þ @Wj

Copyright © 2002 Marcel Dekker, Inc.

ð90Þ

In practice, these derivatives are calculated as finite differences. Wi =Ri is calculated for a specimen with the average composition W1;average ,W2;average , . . . ,Wn;average (symbol ½Wi =Ri average Þ. Then, the concentration of each element j in turn is increased by a small amount [e.g., 0.1% (0.001 in weight fraction)] and Wi =Ri is calculated for that composition (symbol [Wi =Ri Wj þ0:001 ). Substituting in Eq. (90) yields dij ¼

@ðWi =Ri Þ ½Wi =Ri Wj þ0:001
½Wi =Ri average ¼ 0:001 @Wj

ð91Þ

This process is repeated for each of the elements j to calculate all the coefficients for analyte i. This is also repeated for the other analyte elements. The coefficients, calculated from Eq. (91), can be used in Eq. (87) for analysis. Equation (87), however, has n factors rather than n
1 and uses DWj rather than Wj . Using the fact that n X

DWj ¼ 0

ð92Þ

j¼1

or DWe ¼
DW1
DW2
  
DWn

ð93Þ

one element e can be eliminated. The resulting equation is similar to Eq. (87), but has only n
1 terms: Wi ¼ Ei þ bi1 DW1 þ bi2 DW2 þ    þ bin DWn Ri

ð94Þ

bi1 ¼ di1
die

ð95Þ

with

for all bij except bie , which is equal to zero. Equation (94) has n
1 terms, but they are still in DW, rather than W. Transformation of DW to W is done by substituting Eq. (89) in Eq. (94): n n X X Wi ¼ Ei
bij Wj;average þ bij Wj Ri j6¼e j6¼e

ð96Þ

which can be rearranged to Eq. (85) with aij ¼

Ei


bij j6¼e bij Wj;average

Pn

ð97Þ

Combining Eq. (85) with Eqs. (75) and (88) yields 3 2 n X 7 6 aij Wj 7 Wi ¼ Mi;average Ri 6 5 41 þ

ð98Þ

j¼1 j6¼e

indicating that the weight fraction of the analyte is calculated from its relative intensity, a matrix correction term, and the matrix correction term for the average composition (which for a given composition is a constant). The value for Mi,average can be calculated using the influence coefficients calculated and taking the composition (Wi, Wj) equal to the average Copyright © 2002 Marcel Dekker, Inc.

Table 4

Analysis of Stainless Steels with Theoretical Influence Coefficients (de Jongh)

Element Mn Cr Ni

Min.a (%)

Max.a (%)

Std. dev.b (%)

0.64 12.40 6.16

1.47 25.83 20.70

0.015 0.06 0.06

a

Min., Max.: minimum and maximum concentration in the set of analyzed specimens, respectively. Std. dev.: standard deviation, calculated from the difference between concentration values found and certified. Source: Data courtesy of Philips Analytical, Almelo, The Netherlands. b

composition. This equation is very similar at first sight to the Lachance–Traill equation [Eq. (81)], except for the term Mi,average. Typical results. Tables with de Jongh’s coefficients have been used for a wide variety of materials, including high-temperature alloys, brass, solders, cements, glasses, and so forth. An example for stainless steels is given in Table 4. Results are shown for Mn, Ni, and Cr only. For these analytes, the matrix effects are the most important. Other elements, such as Si, P, S, and C, are also present at trace level. The coefficients are calculated at a given composition [see Eq. (91)]. The practical range of concentration over which these coefficients yield accurate results varies from 5% to 15% in alloys to the whole range from 0% to 100% in fused oxide specimens. Comparison between the algorithms of de Jongh and Lachance–Traill. The following points can be noted: 1. The basis of the de Jongh algorithm is a Taylor series expansion, around an average (or reference) composition. The values of the coefficients calculated depend on this composition. 2. De Jongh can eliminate any element; Lachance and Traill eliminate the analyte itself: aij is zero. De Jongh eliminates (i.e., fixes the coefficient to zero) the same element for all analytes. Eliminating the base material (e.g., iron in steels) or the loss on ignition (for beads) generally leads to smaller numerical values for the coefficients and avoids the necessity to determine all elements. 3. De Jongh’s coefficients are calculated at a given reference composition. They are composition dependent and take into account all elements present. A coefficient aij represents the effect of element j on the element i in the presence of all other elements: They are multielement coefficients rather than binary coefficients. This is seen in Table 5, where the values of the coefficients aCrCr and aCrNi are shown for several different specimens, but with identical concentrations for the analyte

Table 5 Values for Influence Coefficients aCrCr and aCrNi, Calculated According to the Algorithm of de Jongh, for Selected Specimens WCr

WFe

WNi

WTi

WW

aCrCr

aCrNi

0.18 0.18 0.18

Bal. Bal. Bal.

0.08 0.08 0.08

0 0 0.08

0 0.08 0

2.06 2.30 2.49

0.750 0.817 0.868

Note the variation of the values of the coefficients, depending on the presence of Ti or W.

Copyright © 2002 Marcel Dekker, Inc.

and the interferent. In all cases, Fe has been eliminated and W and Ti are either present at 0.08 or not. 4. The calculation of the coefficients is based on theory, treating both absorption and enhancement effects. Hence, the coefficients are susceptible to the errors, described earlier (Sec. V.D.2). However, the calculation of the coefficients involves a division of the matrix correction terms for the ‘‘slightly affected’’ composition by the corresponding term of the reference composition [Eq. (97)]. This compensates to a large degree for some of the biases introduced by the fundamental parameters. 5. For a specimen, containing n elements, there are n equations (one for each of the elements) if one uses the Lachance–Traill algorithm. Each of these equations has n
1 influence coefficients; the coefficient of the analyte in each of the equations has been set to 0. For the same specimen, de Jongh only requires n
1 equations, using n
1 coefficients per equation. One element has been eliminated throughout. This element (or compound) is usually one that is of no or little interest to the analyst (e.g., iron in steels or the loss on ignition in fused beads) (de Jongh, 1979). The nth equation (for the eliminated element) can also be written: Its form is identical to the others, it has also n
1 terms, and the coefficients can be calculated following exactly the same procedure as for the other coefficients. 6. The coefficients in the Lachance–Traill equation have been calculated empirically, using measured data from many more standards than used in the de Jongh’s algorithm, which always used theoretical coefficients. These coefficients could be obtained from Philips, eliminating the need for a large computer at each users’ site. With increased computer capabilities at the disposition of every analyst, the calculation of the coefficients from theory has now been feasible for some years. It is possible to calculate the coefficients for the Lachance–Traill algorithm from theory as well (Lachance, 1988). 5. Algorithms withVariable Coefficients a. Introduction Both algorithms discussed so far use a single, constant coefficient for each (except one) interfering element: The expression from Lachance and Traill uses coefficients, expressing the effect of one element on the characteristic intensity of the analyte, relative to the analyte, ignoring all other elements. Such coefficients are therefore referred to as binary coefficients. The algorithm of de Jongh calculates multielement influence coefficients effectively. Such coefficients predict the effect of one element on the intensity of another in a given matrix. This distinction between binary and multielement coefficients can clearly be seen in, for example, a ternary specimen. Algorithms based on binary coefficients add interelement effects from each of the constituent elements. They calculate the matrix effect using influence coefficients that have been calculated for binaries. Assume a Ni–Fe–Cr specimen. The total matrix effect of Cr is accounted for using a coefficient expressing the influence of Ni on Cr (aCrNi) and a similar coefficient for Fe on Cr (aCrFe). Both of these coefficients are calculated for the corresponding binaries (Ni–Cr and Fe–Cr, respectively). In a ternary specimen (e.g., Ni–Fe–Cr), however, the effect of, for example, Ni on Cr is affected by the presence of Fe (and Ni similarly affects the effect of Fe on Cr). This effect is called the crossed effect and will be discussed in a following section. Copyright © 2002 Marcel Dekker, Inc.

The original expression presented by Lachance and Traill (1966) to calculate the influence coefficients required the use of monochromatic excitation. An equivalent wavelength was used to calculate the coefficients when the polychromatic excitation is applied. The equivalent wavelength, however, has been shown to vary with composition, as treated earlier in this chapter. The theoretical coefficients, calculated according to de Jongh, are also composition dependent, as the reference composition is used explicitly in the calculations. This variation is due to the fact that the composition of the matrix varies considerably if analysis is required over a wider range of concentrations. This has been recognized early in the development of influence coefficient algorithms, and many different algorithms with variable coefficients have been proposed. A variable influence coefficient in this respect is an influence coefficient that varies explicitly with concentration of one or more components in the specimen. Some of these will be discussed in the subsequent subsections. b. The Claisse–Quintin Algorithm Formulation. Claisse and Quintin (1967) extended Lachance and Traill’s algorithm by considering a polychromatic primary beam. The resulting equation for WA can be expressed as " # n n n n X X X X 2 aAj Wj þ aAjj Wj þ aA j k W j W k ð99Þ WA ¼ RA 1 þ j6¼A

j6¼A

j6¼A k6¼A;k>j

where the summation over j has n
1 terms (all n elements, except i), and the summation over k has (n
2)=2 terms (all n elements, except the analyte i and element j; furthermore, if aAj k is used, then aAkj is not). For a binary specimen, Eq. (99) reduces to WA ¼ RA ½1 þ a0AB WB 

ð100Þ

a0AB ¼ aAB þ aABB WB

ð101Þ

with clearly showing that the influence coefficient a0AB varies linearly with composition (i.e., WB ). For binaries, WA ¼ 1
WB ; hence, Eq. (101) can also be rearranged to a0AB ¼ aAB þ aABA WA

ð102Þ

Equations (101) and (102) are, at least theoretically, identical. It has been shown, however, that Eq. (102) is preferable to Eq. (101) if specimens with more than two elements (or compounds) are analyzed (Lachance and Claisse, 1980). This will be discussed in more detail in Sec. V.D.6. Note that the value of aAB in Eq. (101) is different from its value in Eq. (102). Cross-product coefficients. For a ternary specimen, the Claisse–Quintin algorithm can be written WA ¼ RA ½1 þ aAB WB þ aABB W2B þ aAC WC þ aACC W2C þ aABC WB WC  The terms aAB WB þ aABB W2B and aAC WC þ aACC W2C Copyright © 2002 Marcel Dekker, Inc.

ð103Þ

are the matrix corrections, due to B and C, respectively. The term aABC WB WC corrects for the simultaneous presence of both B and C and is referred to as a cross-product coefficient. Calculation of the coefficients. Claisse and Quintin (1967) also published methods to calculate the coefficients from measurements on binary and ternary mixtures or from theory. These methods, however, are now generally superseded by theoretical calculations, such as discussed in Sec. V.C.4.c and V.C.4.d. Rousseau (1984b) has presented a calculation method for the coefficients in the Claisse–Quintin algorithm, and Wadleigh (1987) has commented upon this approach. c. The Rasberry–Heinrich Algorithm Following a systematic study of the Fe–Cr–Ni ternary system, Rasberry and Heinrich (1974) concluded that the two phenomena—absorption and enhancement—are to be described by two different equations. They introduced the following algorithm: " # n n X X Bik Aij Wj þ Wk ð104Þ Wi ¼ R i 1 þ 1 þ Wi j6¼i k6¼i where only one coefficient is used for each interfering element. The coefficients Aij are used for cases where absorption is the dominant effect. In this case, the coefficient Bik is taken equal to zero. If, for a given analyte, all Bik coefficients are zero, Eq. (104) reduces to the Lachance–Traill expression. When enhancement by element k dominates, a Bik coefficient is used. The corresponding Aij coefficient is then taken equal to zero. Hence, the total number of terms in both summations is n
1. The correction factor for enhancement by element k can be rewritten as aik ¼

Bik 1 þ Wi

ð105Þ

showing that aik varies with concentration in a nonlinear fashion. The algorithm is very popular when analyzing stainless steels and steels in general. Among the disadvantages of the Rasberry–Heinrich algorithm are the following: 1.

2.

3.

It is not always clear which interfering elements should be assigned a B coefficient and which one an A. In Pb–Sn alloys, the SnLa line is fluoresced (enhanced) by both SnK and PbL lines. Yet, the calibration curve for SnLa clearly shows that absorption is dominant (Fig. 11). Furthermore, Eq. (105) suggests that the value of Bik at Wi ¼ 0 is twice the value at the ‘‘other’’ end of the calibration range when Wi ¼ 1. This is not generally valid. Mainardi and co-workers (1982) have therefore suggested replacing the 1 in the denominator by an additional coefficient. Rasberry and Heinrich did not publish a method for calculating the coefficients from theory.

Some of the disadvantages of calculating empirical coefficients have been discussed in Sec. V.D.4.a. For these reasons, the Rasberry–Heinrich algorithm is not generally applicable. However, the concept of a hyperbolically varying influence coefficient has been incorporated in the three-coefficient algorithm of Lachance. d.

The Three-Coefficient Algorithm of Lachance

Formulation. In 1981, Lachance (1981) proposed a new approximation to the binary influence coefficient B ij given by Copyright © 2002 Marcel Dekker, Inc.

Figure 11 Calibration curves for PbLa (solid line), SnKa (dashed line), and SnLa (dotted line) in Pb–Sn binaries. The SnLa is apparently dominated by absorption, rather than by enhancement of SnK and PbL lines.

mij ffi aBij ¼ aij1 þ

aij2 Wm 1 þ aij3 ð1
Wm Þ

ð106Þ

with Wm ¼ 1
Wi

ð107Þ

Wm is the concentration of all matrix elements. It has been shown by Lachance and Claisse (1980), as well as by Tertian (1976), that variable binary coefficients must be expressed in terms of Wm (or 1
Wi). For binary specimens, Eq. (106) can be rewritten using Wj for Wm and Wi for (1
Wm): mij ffi aBij ¼ aij1 þ

aij2 Wj 1 þ aij3 Wi

ð108Þ

For specimens with more than two compounds, however, the difference between Eqs. (106) and (108) becomes clear. The value for the influence coefficient mij is approximated over the complete concentration range for the binary by the function in Eq. (106), which relies on three coefficients only. The excellent agreement between the true influence coefficient mij and the approximation of Eq. (106) is shown in Figure 12 for Fe in FeNi (severe enhancement) and for Fe in FeCr (pronounced absorption). For multielement specimens, cross-product coeffcients aijk are used to correct for the crossed effect, similar to Eq. (99). The general equation for a multielement specimen is " #  n
n n X X X aij2 Wm aij1 þ aijk Wj Wk Wi ¼ R i 1 þ ð109Þ Wj þ 1 þ aij3 ð1
Wm Þ j6¼i j6¼i k6¼i;k>j Copyright © 2002 Marcel Dekker, Inc.

Figure 12 The binary influence coefficient mFeNi in FeNi binary systems (j, enhancement, top) and mFeCr in FeCr (j, absorption, bottom) and the approximation by the hyperbolic threecoefficient algorithm for Lachance (COLA). Note the excellent agreement in both cases. Conditions: W tube at 45 kV, in a spectrometer with an incidence angle of 63 and 33 take-off angle.

where the summation over j has n
1 terms (all n elements, except i) and the summation over k has (n
2)=2 terms (all n elements, except the analyte i and element j; furthermore if aijk is used, then aikj is not). Vrebos and Helsen (1986) have published some data on this algorithm, clearly showing the accuracy of the algorithm, using theoretically calculated intensities. The use of Copyright © 2002 Marcel Dekker, Inc.

Table 6 Composition of the Specimens Used for the Calculations of the Coefficients for Lachance’s Three-Coefficient Algorithm, in Weight Fraction Specimen No. 1 2 3 4 5 6 7

Wi

Wj

Wk

0.999 0.001 0.5 0.999 0.001 0.5 0.30

0.001 0.999 0.5 0.0 0.0 0.0 0.35

0.0 0.0 0.0 0.001 0.999 0.5 0.35

theoretically calculated intensities has the advantage that it avoids errors due to specimen preparation and measurement errors associated with actual measured data. Pella and coworkers (1986) have presented a comparison of the algorithm with several others and with a fundamental parameter method using experimental data. Calculation of the coefficients. The coefficients aij1 , aij2 , and aij3 are calculated using fundamental parameters at three binaries i
j. The cross-product coefficients are calculated from a ternary. The compositions of the specimens concerned are listed in Table 6. The ‘‘specimens’’ referred to in Table 6 are hypothetical specimens. The intensities are calculated from fundamental parameters and require no actual measurements on real specimens. Step 1. Calculate the relative intensity Ri for the first composition in Table 6. If the analysis of interest has more than three elements, then the system is divided in combinations of three elements i, j, k at a time. The analyte is element i, and j and k are two interfering elements. If the system considered is with compound phases, such as oxides, then the compositions in Table 6 are assumed to be for the oxides. Step 2. Using Eq. (84), the corresponding influence coefficient aBij can be calculated. Step 3. For this composition, Wm ¼ 1
Wi ¼ Wj ¼ 0:001, which is small enough to be considered zero. Hence, Eq. (106) reduces to aBij ¼ aij1

ð110Þ

aBij has been calculated in Step 2, so aij1 can be computed. Step 4. Calculate the intensity for the second composition of Table 6 and use Eq. (84) to calculate aBij . In most cases, this value will be different from the one found in Step 2 because the compositions involved are different. Step 5. 1
Wm ¼ Wi ¼ 0:001 is small enough to be considered zero; hence, Eq. (106) reduces to aBij ¼ aij1 þ aij2

ð111Þ

aij1 and aBij are known so aij2 can be calculated. Step 6. Calculate the intensity for the third composition of Table 6 and use Eq. (84) to calculate aBij . In most cases, this value will be different from the one found in Step 2 or 4 because the compositions involved are different. Step 7. Using Wm ¼ 1
Wi ¼ 0:5 ¼ Wi , Eq. (106) reduces to aBij ¼ aij1 þ

aij2 ð0:5Þ 1 þ aij3 ð0:5Þ

Copyright © 2002 Marcel Dekker, Inc.

ð112Þ

which can be rearranged to aij2 aij3 ¼ B
2 aij
aij1

ð113Þ

all coefficients on the right-hand side are known, so aij3 can be calculated. Step 8. Repeat Steps 1–7 for Wi and Wk , to compute the coefficients aik1 , aik2 , and aik3 . Step 9. Calculate the intensity Ri for the ternary (composition 7 in Table 6). Calculate aBij and aBik , using Eq. (106) and the coefficients determined earlier. Step 10. Eq. (109) combined with Eq. (106) for a ternary specimen i–j–k reduces to Wi ¼ Ri ½1 þ aBij Wj þ aBik Wk þ aijk Wj Wk 

ð114Þ

which can be rearranged to solve for aijk: aijk ¼

ðWi =Ri Þ
1
aBij Wj
aBik Wk Wj Wk

ð115Þ

All variables on the right-hand side of Equation (115) are known, so aijk can be calculated. Step 11. Repeat for other interfering elements ( j and k) and repeat for other analytes i. Tao et al. (1985) published a complete computer program illustrating the method and allowing the calculation of the coefficients and analysis of unknowns. This program suffers from an oversimplification in that only the measured analytical lines are considered for enhancement. This would generate erroneous values for the coefficients in cases such as Cu–Zn alloys, where the ZnKa line cannot fluoresce the K shell of Cu, but the ZnKb can do so. If the ZnKa line is used for analysis, the effect of the ZnKb line (enhancement of Cu) is not taken into account by the program. In practice, however, the only lines that are considered for enhancement are the characteristic lines used for the analysis of the other elements. This can be seen, for example, by calculating the coefficients twice for Fe–Si with identical conditions; once indicating FeKa and SiKa lines are to be used and once indicating that the FeLa be used (Table 7). The value of the coefficients for Fe will change; this is quite obvious because the magnitude and the sort of the matrix effects on the FeKa and the FeLa characteristic lines are quite different. The value for Si, however, should not change: in both cases, the same elements are present, and using the same excitation conditions, there is no reason why the coefficients should be different as the matrix effects are the same. e.

The Algorithm of Rousseau

Formulation. Rousseau and Claisse (1974) used a linear relationship to approximate the binary coefficients and cross-product coefficients: " # n n n X X X ðaij1 þ aij2 Wm ÞWj þ aijk Wj Wk Wi ¼ R i 1 þ ð116Þ j6¼i

j6¼i k6¼i;k>j

The binary influence coefficients are thus approximated by mij ffi aBij ¼ aij1 þ aij2 Wm

ð117Þ

This model can be used as a stand-alone influence coefficient algorithm, but it has also been proposed as the starting point for a fundamental parameter algorithm (Rousseau, 1984a). The degree of agreement between the influence coefficient mij and the approxiCopyright © 2002 Marcel Dekker, Inc.

Table 7

Values for the Coefficients for Eq. (109) for Si in Fe–Si Binaries

aij1 aij2 aij3

SiKa(FeKa)

SiKa(FeLa)

5.396
1.890
0.409

6.284
0.015
0.846

Note: In the second column, Fe is measured using the Ka line and in the last column, the La line is used. Conditions: W tube at 45 kV, in a spectrometer with an incidence angle of 63 and a 33 take-off angle.

mation is shown in Figure 13 for the FeNi and the FeCr binaries. The agreement for the straight line is obviously not as good as with the COLA algorithm, especially in those cases where the value of the true influence coefficient varies markedly, as is the case for Fe in FeCr (absorption). Equation (117) has been compared to the three-coefficient algorithm of Lachance by Vrebos and Helsen (1986). They show that the accuracy is somewhat less than for Lachance’s method, but for most practical purposes, the Rousseau algorithm should give acceptable results. Calculation of the coefficients. Rousseau has shown that the fundamental parameter equation can be rearranged to " # n X Wi ¼ R i 1 þ aij Wj ð118Þ j6¼1

and he also proposed a method to calculate the a coefficients directly from fundamental parameters, without calculating the intensity first (Rousseau, 1984a). As a matter of fact, Rousseau first calculates the coefficients for a given composition and then calculates the intensity, using Eq. (118). The coefficients in Eq. (116) are calculated in a way very similar to the method described in Sec. V.D.5.d. The compositions involved are given in Table 8. The ‘‘specimens’’ referred to in Table 8 are hypothetical specimens. The intensity is calculated from fundamental parameters and requires no actual measurements on real specimens. For the first two binaries of Table 8, the influence coefficient is calculated [symbol aij(0.20,0.80) and aij(0.80,0.20), respectively]. Then the corresponding values are substituted in Eq. (117): aij ð0:20; 0:80Þ ¼ aij1 þ aij2 ð0:80Þ

ð119aÞ

aij ð0:80; 0:20Þ ¼ aij1 þ aij2 ð0:20Þ

ð119bÞ

These equations can be solved for aij1 and aij2 . Similarly, using compositions 3 and 4 from Table 8, the corresponding coefficients for i–k can be calculated. The cross-product coefficients aijk are calculated using Eq. (115). 6. Specimens with More thanTwo Compounds The methods described by Lachance [Eq. (106)] and Rousseau [Eq. (117)] explicitly describe algorithms to calculate the value of the binary influence coefficient by a rather simple, hyperbolic or linear, relationship. Combining these binary coefficients to describe the matrix effect for specimens with more than two elements (or compounds) is described in this subsection. The ternary system FeNiCr is taken here as an example. Figure 14 gives Copyright © 2002 Marcel Dekker, Inc.

Figure 13 The binary influence coefficient mFeNi in FeNi binary systems (j, enhancement, top) and mFeCr in FeCr (j, absorption, bottom) and the approximation by the straight line as suggested by Rousseau and Claisse. Note the rather large deviations, especially at the high concentration ranges in FeCr. Conditions: W tube 45 kV, in a spectrometer with an incidence angle of 63 and 33 take-off angle.

the relative intensity of FeKa as a function of the weight fraction of Fe in FeNiCr specimens. There is considerable spread of the intensity of FeKa, even for a constant weight fraction of Fe. For specimens with a weight fraction of 0.10 Fe, the relative intensity of FeKa varies between 0.036 and 0.16 (points marked 1 and 2 in Fig. 14). This is due to the Copyright © 2002 Marcel Dekker, Inc.

Table 8 Composition of the Specimens Used for the Calculations of the Coefficients for the Linear Approximation According to Rousseau’s Algorithm, in Weight Fraction Specimen No.

Wi

Wj

Wk

1 2 3 4 5

0.20 0.80 0.20 0.80 0.30

0.80 0.20 0.0 0.0 0.35

0.0 0.0 0.80 0.20 0.35

rather different effect that Ni and Cr have on Fe:Cr is an absorber for FeKa radiation, whereas the NiK radiation can enhance FeK radiation through the process of secondary fluorescence (enhancement). For these specimens, the matrix effect MFe can be calculated from Eq. (75). The total matrix effect on Fe, MFe(FeNiCr), in these specimens, at a fixed Fe concentration of 0.1, for example, varies from 0.63 (for 0.1 Fe in FeNi, point 2) to 2.8 (for 0.1 Fe in FeCr, point 1). Now, the problem is how to calculate the matrix effect in this case, based on influence coefficients. Assume a specimen with the following composition: WFe ¼ 0.1,

Figure 14 The relative intensity of FeKa as a function of the concentration of Fe, in the presence of Ni and Cr. For every given weight fraction of Fe, the highest value of the intensity is obtained for the binary system FeNi (enhancement), whereas the lowest intensity is for the binary FeCr (absorption). The intermediate values are for ternary specimens, where the concentrations of Ni and Cr vary in steps of 0.1 weight fraction. At a weight fraction of Fe ¼ 0.7, the four data points labeled a, b, c, and d represent the following specimens (WFe, WNi, WCr): a ¼ (0.7, 0.0, 0.3), b ¼ (0.7, 0.1, 0.2), c ¼ (0.7, 0.2, 0.1), and d ¼ (0.7, 0.3, 0.0). Points labeled 1 and 2: see text. Experimental conditions: W tube at 45 kV, 1-mm Be window, incidence and take-off angles 63 and 33 , respectively.

Copyright © 2002 Marcel Dekker, Inc.

WCr ¼ 0.3, and WNi ¼ 0.6 (again, all concentrations are expressed as weight fractions). The total matrix effect on Fe will be caused by both Ni and Cr and its magnitude will be between MFe(FeNi; WFe ¼ 0.1) ¼ 0.63 (for Fe in FeNi) and MFe(FeCr; WFe ¼ 0.1) ¼ 2.8 (for Fe in FeCr). It is assumed that the total effect MFe(FeNiCr; WFe ¼ 0.1) is proportional to the concentrations of Ni (0.6) and Cr (0.3) in this example. Applying the law of weighted averages, the total matrix effect is given by WNi WNi þ WCr WCr ¼ 0:1Þ WNi þ WCr

MFe ðFeNiCr; WFe ¼ 0:1Þ ffi MFe ðFeNi; WFe ¼ 0:1Þ þ MFe ðFeCr; WFe

ð120Þ

There is not a strict derivation indicating the validity of Eq. (120) in the general case. For cases involving absorption only, the derivation is rather straightforward and based on the additivity law for absorption. For now, let it suffice to indicate that the matrix effect will change gradually when adding an element or when changing the composition of the specimen slightly; this is described by Eq. (120). By substituting the numerical values, MFe ðFeNiCr; WFe ¼ 0:1Þ ffi 0:63

0:6 0:3 þ 2:8 ¼ 1:35 0:6 þ 0:3 0:6 þ 0:3

ð121Þ

a value of 1.35 is obtained. This is in good agreement with the theoretical value of 1.31. Equation (120) is based on the availability of binary influence coefficients calculated at specimen compositions given by WFe ¼ 0.10; WNi ¼ 0.90 and WFe ¼ 0.10; WCr ¼ 0.90. Using the more general expressions for matrix effects, MFe ðFeNiÞ ¼ ½1 þ mFeNi;bin WNi;bin 

ð122aÞ

MFe ðFeCrÞ ¼ ½1 þ mFeCr;bin WCr;bin 

ð122bÞ

and

the following is obtained: WNi WNi þ WCr WCr þ ½1 þ mFeCr;bin WCr;bin  WNi þ WCr

MFe ðFeNiCrÞ ffi ½1 þ mFeNi;bin WNi;bin 

ð123Þ

where WNi,bin is the concentration of Ni in the binary FeNi (for example, CNi,bin ¼ 0.90). Because WNi;bin ¼ WCr;bin ¼ 1
WFe ¼ WNi þ WCr

ð124Þ

Equation (123) can be rearranged to MFe ðFeNiCrÞ ffi ½1 þ mFeNi;bin WNi þ mFeCr;bin WCr 

ð125Þ

stressing the point again that the influence coefficients are to be calculated for binaries ij with the composition Wi ; Wj ¼ 1
Wi . This is the reason why Eqs. (106) and (117) use Wm instead of Wj. However, applying binary coefficients to multielement specimens leads to an incomplete matrix correction because we are trying to describe the effect of Cr and Ni on Fe in FeNiCr based on the matrix effects in the corresponding binaries only. This effect Copyright © 2002 Marcel Dekker, Inc.

is referred as the crossed effect and has been described by Tertian (1987), who also proposed a method to correct for this. The proposed method (Tertian, 1987) involves the use of weighting factors based on the reciprocals of the relative intensities of the binaries involved. It is a rather cumbersome method, but theoretically valid, and it does not imply any approximation whatsoever; a discussion is outside the scope of this work. An easier method is to use the cross-product coefficients as used in Eq. (99). The derivation from Tertian and Vie le Sage (1977) offers some insight in this matter. Tertian and Vie le Sage (1977) assume that a multielement influence coefficient aM ij can be approximated as the sum of the binary coefficient aBij and a linear variation with the other elements: B aM ij ¼ aij þ tijk Wk

ð126Þ

where tijk is a coefficient expressing the effect of element k on the influence coefficient aM ij . Similarly, B aM ik ¼ aik þ tikj Wj

ð127Þ

Substituting Eqs. (126) and (127) in M ½1 þ aM ij Wj þ aik Wk 

ð128Þ

(the superscript M is used to explicitly indicate the use of multi element influence coefficients) yields ½1 þ aBij Wj þ aBik Wk þ aijk Wj Wk 

ð129Þ

aijk ¼ tijk þ tikj

ð130Þ

with

It is to be realized that crossed effect is introduced by the use of binary coefficients; use of multi element coefficients would not lead to crossed effect. Thus, the equation expressing the matrix effect using binary influence coefficients for specimens with more than two compounds is " # n n
1 X n X X mij;bin Wj þ aijk Wj Wk Mi ¼ 1 þ ð131Þ j¼1 j6¼i

j¼1 k¼jþ1 j6¼i k6¼i

and is based on cross-product coefficients to correct for crossed effect introduced by the use of binary influence coefficients. The use of the cross-product coefficients is not mandated by the concentration range to be covered (the binary coefficients as calculated by, for example, the algorithm of Lachance are more than adequate) but is a consequence of the use of binary coefficients. 7. Application In Secs.V.D.4 and V.D.5, several influence coefficient algorithms have been discussed. Application of the resulting equations for calibration and analysis will be discussed here and is equally valid for any of the influence coefficient algorithms. a.

Calibration

Step 1. It is assumed that the coefficients have been calculated from theory, for example, using Eq. (84) or (97). Copyright © 2002 Marcel Dekker, Inc.

Step 2. Calculate the matrix correction term [the square brackets in Eq. (81), (85), (99), (104), and (109)] for all standard specimens and for a given analyte. The coefficients are known (Step 1), and for standard specimens, all weight fractions Wi and Wj are known. Step 3. Plot the measured intensity of the analyte, multiplied by the corresponding matrix correction term against analyte weight fraction. Then, determine the ‘‘best’’ line, Wi ¼ Bi þ Ki Ii ½1 þ   

ð132Þ

by minimizing DWi (see Sec. V.A). Note that Eq. (132) is more general than Eq. (50), which does not correct for matrix effects. This process is repeated for all analytes. Other methods are also feasible. The most common variant is the one where W ¼ Bi þ Ki Ii ½1 þ   

ð133Þ

is used. This is nearly equivalent to Eq. (132) but with brackets: Wi ¼ ðBi þ Ki Ii Þ½1 þ   

ð134Þ

The term ðBi þ Ki Ii Þ is related directly to the relative intensity Ri. Corrections for line overlap should only affect this term. b. Analysis For each of the analytes, a set of equations has to be solved for the unknown Wi, Wj and so forth. If the matrix correction term used is the one according to Lachance and Trail [Eq. (81)] or de Jongh [Eq. (85)], then the set of equations can be solved algebraically (n linear equations with n unknowns for Lachance and Traill and n
1 equations with n
1 unknowns for de Jongh). Mostly, however, an iterative method is used. As a first estimate, one can simply take the matrix correction term equal to 1. This yields a first estimate of the composition Wi, Wj, and so on. This first estimate is used to calculate the matrix correction terms for all analytes. Subsequently, a new composition estimate can be obtained. This process is repeated until none of the concentrations changed between subsequent iterations by more than a preset quantity. If the matrix correction is done using algorithms which use more than one coefficient {e.g., Claisse and Quintin [Eq. (99)] or Rasberry and Heinrich [Eq. (104)]}, then the equations are not linear in the unknown concentrations and an algebraic solution is not possible. An iterative method, such as described earlier can be used. 8. Algorithms with Empirical Coefficients Empirical coefficients are coefficients that are not calculated from theory but from actually measured specimens using regression analysis (Anderson et al., 1974). They were the basis of the earliest correction methods, but now they are largely superseded by more theoretical ones. Such empirically determined coefficients tend to mix the matrix correction with the sensitivity of the spectrometer. On the one hand, the matrix effect is determined by the composition of the sample and ‘‘physical’’ parameters such as take-off and incidence angles and tube anode and voltage. These are the same for spectrometers of similar design. The sensitivity of the spectrometer, on the other hand, depends on the reflectivity of the crystals, the efficiency of the detectors, and so on. These parameters are unique for each spectrometer. Also, if one of the analyte lines is overlapped by another x-ray line, some of this effect can also affect the value of the influence coefficients. The Copyright © 2002 Marcel Dekker, Inc.

coefficients thus determined are instrument-specific and are not transferable to other instruments. Stephenson (1971) has noted that the regression equations involved in the determination of the coefficients in such an empirical way become unstable as the degree of correlation between the independent variables increases. This mandates careful planning of the experiment, including the composition of the synthetic standards. Klimasara (1994, 1995) has illustrated the use of standard spreadsheet programs for the calculation of the values of empirical influence coefficients and composition. a.

The Sherman Algorithm

Sherman (1953) was among the first to propose an algorithm for correction of matrix effects. For a ternary system, the algorithm can be represented by the following set of equations: ðaAA
tA ÞWA þ aAB WB þ aAC WC ¼ 0 aBA WA þ ðaBB
tB ÞWB þ aBC WC ¼ 0 aCA WA þ aCB WB þ ðaCC
tC ÞWC ¼ 0

ð135Þ

where aij represents the influence coefficient of element j on the analyte i and ti is the time (in s) required to accumulate a preset number of counts. The constants aij are determined from measurements on specimens with known composition. Determination of the composition of an unknown involves the solving of the above set of linear equations [Eq. (135)]. This set, however, is homogeneous: Its constant terms are all equal to zero. So, only ratios among the unknown Wi can be obtained. In order to obtain the weight fractions Wi, an extra equation is required. Sherman proposed using the sum of all the weight fractions of all the elements (or components) in the specimen, which ideally, should be equal to unity. For a ternary specimen, WA þ WB þ WC ¼ 1

ð136Þ

Using Eq. (136), one of the equations in the set of Eqs. (135) can be eliminated. The solution obtained, however, is not unique: For a ternary, any one of the three equations can be eliminated. This yields three different combinations. Furthermore, any of the three elements can be eliminated in each of the combinations. Hence, a total of 363 ¼ 9 different sets can be derived from Eqs. (135) and (136), and each of these sets will generate different results. In general, the algorithm yields n2 different results for a system with n elements or compounds. This is clearly undesirable, because it is hard to determine which set will give the most accurate results. Another disadvantage is the fact that the sum of the elements determined always equals unity, even if the most abundant element has been neglected. Furthermore, the numerical values of the coefficients depend, among other parameters such as geometry and excitation conditions, also on the number of counts accumulated. Nonquantifiable parameters, such as reflectivity of the diffracting crystal used in wavelength-dispersive spectrometers, or tube contamination will also affect the value of the coefficients. The coefficients determined on a given spectrometer cannot be used with another instrument; they are not transferable. The other algorithms discussed use some form of a ratio method: The Lachance and Traill algorithm, for example, uses relative intensities. The measurements are then done, relative to a monitor; this reduces, or eliminates, the effect of such nonquantifiable parameters. Copyright © 2002 Marcel Dekker, Inc.

b.

The Algorithm of Lucas-Tooth and Price

Lucas-Tooth and Price (1961) developed a correction algorithm, where the matrix effect was corrected for, using intensity (rather than concentration) of the interfering elements. The equation can be written as " # n X Wi ¼ Bi þ Ii k0 þ kij Ij ð137Þ j6¼i

where Bi is a background term and k0 and kij are the correction coefficients. A total of n þ 1 coefficients have to be determined, requiring at least n þ 1 standards. Usually, however, a much larger number of standards is used. The coefficients are then determined by, for example, a least-squares method. The corrections for the effect of the matrix on the analyte are done via the intensities of the interfering elements; their concentrations are not required. The method assumes that the calibration curves of the interfering elements themselves are all linear; the correction is done using intensities rather than concentrations. The algorithm will, therefore, have a limited range. Its use will be limited to applications where only one or two elements are to be analyzed (it still involves measurements of all interfering element intensities) and where a computer of limited capabilities is used (although calculation of the coefficients involves much more compute capabilities than the subsequent routine analysis of unknowns). The advantages of the method are as follows: The method is very fast, because the calculation of composition of the unknowns requires no iteration. Analysis of only one element is possible; this requires, however, the determination of all relevant correction factors. Very simple algorithm, requiring very little calculation. c.

Algorithms Based on Concentrations

Algorithms similar to Eq. (137) have been proposed, using corrections based on concentrations rather than intensities. The values of the coefficients were then to be derived from multiple-regression analysis on a large suite of standards. The main aim was to obtain correction factors that could be determined on one spectrometer and used, without alteration, on another instrument. In practice, the coefficients still have to be adjusted because of the intimate and inseparable entanglement of spectrometer-dependent factors with matrix effects. Furthermore, compared to the algorithms based on intensities, some of the advantages of the latter are not retained: A calibration for all elements present is now required, calculation of the composition of unknowns requires iteration, and so forth. In principle, methods based on theoretically calculated influence coefficients are recommended.

VI.

CONCLUSION

Among the advantages of XRF analysis are the facts that the method is nondestructive and allows direct analysis involving little or no specimen preparation. Analysis of major and minor constituents requires correction for matrix effects of variable (from one specimen to another) magnitude. If the matrix varies appreciably from one specimen to the next, then even the intensity of elements present at a trace level can be subject to matrix effects and a correction is required. Several methods for matrix correction have been Copyright © 2002 Marcel Dekker, Inc.

described. Each of these methods have their own advantages and disadvantages. These, by themselves, do not generally lead to the selection of ‘‘best’’ method. The choice of the method to use is also determined by the particular application. From the previous sections, it may appear that the mathematical methods are more powerful than the compensation methods. Yet, if only one or two elements at a trace level in liquids have to be determined, compensation methods (either standard addition or the use of an internal standard) can turn out to be better suited than, for examples, rigorous fundamental parameter calculations. Compensation methods will correct for the effect of an unknown, but constant, matrix. Also, they do not require the analysis of all constituents in the specimen. The mathematical methods (fundamental parameters as well as methods based on theoretical influence coefficients), on the other hand can handle cases in which the matrix effect is more variable from one specimen to another. In this respect, they appear to be more flexible than the compensation methods, but they do require more knowledge of the complete matrix. All elements contributing significantly to the matrix effect must be quantified (either by x-ray measurement or by another technique) even if the determination of their concentrations is not required by the person who submits the sample to the analyst. Once a particular algorithm is selected, it is customary to use for all analytes. However, it must be stressed that this is not a requirement. There is only one requirement for adequate matrix correction: Each analyte should be corrected adequately, by whatever method. If complete analysis (covering all major elements) is required, the analyst has the choice between the fundamental parameter method and algorithms, based on influence coefficients. Commonly, fundamental parameter methods are (or were) used in research environments rather than for routine analysis in industry. This choice is more often made on considerations such as the availability of the programs and computers than on differences in analytical capabilities. Influence coefficient algorithms tend to be used in combination with more standards compared to fundamental parameter methods, because their structure and simple mathematical representation facilitates interpretation of the data (establishing a relationship between concentration and intensity, corrected for matrix effect). The final choice, however, has to be made by the analyst.

REFERENCES Anderson CH, Mander JE, Leitner JW. Adv X-Ray Anal 17:214, 1974. Australian Standard 2563-1982, Wavelength Dispersive X-ray Fluorescence Spectrometers— Methods of Test for Determination of Precision. North Sydney, NSW: Standards Association of Australia, 1982. Bambynek W, Crasemann B, Fink RW, Freund HU, Mark H, Swift CD, Price RE, Venugopala Rao P. Rev Mod Phys 44:716, 1972. Bearden JA. Rev Mod Phys 39:78, 1967. Beattie HJ, Brissey RM. Anal Chem 26:980, 1954. Bonetto RD, Riveros JA. X-Ray Spectrom 14:2, 1985. Claisse F, Quintin M. Can Spectrosc 12:129, 1967. Criss JW. Adv X-Ray Anal 23:93, 1980a. Criss JW. Adv X-Ray Anal 23:111, 1980b. Criss JW, Birks LS. Anal Chem 40:1080, 1968. Criss JW, Birks LS, Gilfrich JV. Anal Chem 50:33, 1978. de Boer DKG. Spectrochim Acta 44B:1171, 1989. de Jongh WK. X-Ray Spectrom 2:151, 1973.

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de Jongh WK. X-Ray Spectrom 8:52, 1979. DeGroot PB. Adv X-Ray Anal 33:53, 1990. Draper NR, Smith H. Applied Regression Analysis. New York: Wiley, 1966. Feather CE, Willis JP. X-Ray Spectrom 5:41, 1976. Garbauskas MF, Goehner RP. Adv X-Ray Anal 26:345, 1983. Gillam E, Heal HT. Br J Appl Phys 3:353, 1952. Heinrich KFJ. In: McKinley TD, Heinrich KFJ, Wittry DB, eds. The Electron Microprobe. New York: Wiley, 1966, p 296. Holynska B, Markowicz A. X-Ray Spectrom 10:61, 1981. Hower J. Am Mineral 44:19, 1959. Hughes H, Hurley P. Analyst 112:1445, 1987. Hunter CB, Rhodes JR. X-Ray Spectrom 1:107, 1972. Ingham MN, Vrebos BAR. Adv X-Ray Anal 37:717, 1994. ISO, Determination of Nickel and Vanadium in Liquid Fuels—Wavelength-Dispersive X-Ray Fluorescence Method, ISO 14597. Geneva: ISO, 1995. Johnson W. International Report BISRA MG=D=Conf Proc=610=67, 1967. Kaufmann M, Mantler M, Weber F. Adv X-Ray Anal 37:205, 1994. Klimasara AJ. Adv X-Ray Anal 37:647, 1994. Klimasara AJ. Workshop on the Use of Spread Sheets in XRF Analysis, 44th Annual Denver X-Ray Conference, Colorado Springs, CO, 1995. Lachance GR. X-Ray Spectrom 8:190, 1979. Lachance GR. International Conference on Industrial Inorganic Elemental Analysis, Metz, France, 1981. Lachance GR. Adv X-Ray Anal 31:471, 1988. Lachance GR, Claisse F. Adv X-Ray Anal 23:87, 1980. Lachance GR, Traill RJ. Can Spectrosc 11:43, 1966. Leroux J, Thinh TP. Revised Tables of Mass Attenuation Coefficients. Quebec: Corporation Scientifique Claisse, 1977. Li-Xing Z. X-Ray Spectrom 13:52, 1984. Lubecki A, Holynska B, Wasilewska M. Spectrochim Acta 23B:465, 1968. Lucas-Tooth HJ, Price BJ. Metallurgia 64:149, 1961. Mainardi RT, Fernandez JE, Nores M. X-Ray Spectrom 11:70, 1982. Mantler M. Adv X-Ray Anal 36:27, 1993. McMaster WH, Delgrande NK, Mallet JH, Hubbel JH. Compilation of X-Ray Cross Sections, UCRL 50174, Sec II, Rev 1, 1969. Mu¨ller RO. Spectrochemical Analysis by X-Ray Fluorescence. New York: Plenum Press, 1972, chap 9. Pella PA, Feng LY, Small JA. X-Ray Spectrom 14:125, 1985. Pella PA, Tao GY, Lachance GR. X-Ray Spectrom 15:251, 1986. Pollai G, Ebel H. Spectrochim Acta 26B:761, 1971. Pollai G, Mantler M, Ebel H. Spectrochim Acta 26B:733, 1971. Rasberry SD, Heinrich KFJ. Anal Chem 46:81, 1974. Rousseau RM. X-Ray Spectrom 13:121, 1984a. Rousseau RM. X-Ray Spectrom 13:3, 1984b. Rousseau RM, Claisse F. X-Ray Spectrom 3:31, 1974. Sherman J. The Correlation Between Fluorescent X-Ray Intensity and Chemical Composition. ASTM Special Publication No 157. Philadelphia: ASTM, 1953, p 27. Sherman J. Spectrochim Acta 7:283, 1955. Shiraiwa T, Fujino N. Jpn J Appl Phys 5:886, 1966. Shiraiwa T, Fujino N. Bull Chem Soc Japan 40:2289, 1967. Shiraiwa T, Fujino N. Adv X-Ray Anal 11:63, 1968. Shiraiwa T, Fujino N. X-Ray Spectrom 3:64, 1974. Sparks CJ. Adv X-Ray Anal 19:19, 1976.

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Stephenson DA. Anal Chem 43:310, 1971. Tao GY, Pella PA, Rousseau RM. NBSGSC, a FORTRAN program for quantitative X-Ray Fluorescence Analysis. NBS Technical Note 1213. Gaithersburg, MD: National Bureau of Standards, 1985. Tertian R. Adv X-Ray Anal 19:85, 1976. Tertian R. X-Ray Spectrom 16:261, 1987. Tertian R, Vie le Sage R. X-Ray Spectrom 6:123, 1977. Vrebos BAR, Helsen JA. X-Ray Spectrom 15:167, 1986. Vrebos BAR, Pella PA. X-Ray Spectrom 17:3, 1988. Veigele WJ. In: Robinson JW, ed. Handbook of Spectroscopy. Cleveland, OH: CRC, 1974, p 28. Venugopala Rao P, Chen MH, Crasemann B. Phys Rev A 5:997, 1972. Wadleigh KR. X-Ray Spectrom 16:41, 1987. Wood PR, Urch DS. J Phys F: Metal Phys 8:543, 1978.

SUGGESTIONS FOR FURTHER READING Bertin EP. Principles and Practice of X-Ray Spectrometric Analysis. 2nd ed. New York: Plenum Press, 1975. Jenkins R, Gould RW, Gedcke D. Quantitative X-Ray Spectrometry. 2nd ed. New York: Marcel Dekker, 1995. Tertian R, Claisse F. Principles of Quantitative X-Ray Fluorescence Analysis. New York: Wiley, 1982.

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6 Quantification in XRF Analysis of Intermediate-Thickness Samples Andrzej A. Markowicz Vienna, Austria

Rene´ E. Van Grieken University of Antwerp, Antwerp, Belgium

I.

INTRODUCTION

A number of approaches have been developed for quantitation in x-ray fluorescence (XRF) analysis of intermediate-thickness samples whose mass per unit area m fulfills the relation mthin < m < mthick

ð1Þ

where mthin and mthick are the values of mass per unit area for thin and thick samples [for a definition, see Chapter 1, Eqs. (93) and (95)]. Intermediate samples can be preferable to thick specimens because remaining uncertainties about mass-attenuation coefficients have a smaller effect on the analysis results, less material is required, the sensitivity is more favorable for low-Z elements, and secondary enhancement effects are less important. Historically, the oldest correction method applied in quantitative XRF analysis of intermediate-thickness samples is the emission–transmission (E–T) method in which the specific x-ray intensities from a sample are measured successively with and without a target positioned adjacent to the back of the sample in a fixed geometry. Recently, the E–T method has thoroughly been evaluated, and both the advantages and limitations of the technique are well identified. A number of modifications of the E–T method, developed in recent years, allowed an essential extension of its applicability range. To avoid additional measurements that are inevitable in the emission–transmission method, some alternative correction procedures based on the use of scattered primary radiation were also developed. In recent years, however, only a few papers have been published in this field. The underlying principles as well as the ranges of applicability and the limitations of the correction procedures applied to XRF analysis of both homogeneous and heterogeneous intermediate-thickness samples are outlined here.

Copyright © 2002 Marcel Dekker, Inc.

II.

EMISSION---TRANSMISSION METHOD

In the absence of enhancement effects and assuming monochromatic excitation, the mass per unit area of the element i; mi , for homogeneous intermediate-thickness samples, can be calculated from [see Eq. (90) in Chapter 1]: mi ¼

Ii ðEi Þ Abcorr Bi

ð2Þ

where GI0 ðE0 Þ"ðEi Þ
i0 ðE0 Þ!i pi ð1
1=ji Þ sin 1 and Abcorr is the absorption correction factor given by Bi ¼

Abcorr ¼

½ðE0 Þ csc 1 þ ðEi Þ csc 2 m 1
expf
½ðE0 Þ csc 1 þ ðEi Þ csc 2 mg

ð3Þ

ð4Þ

[The symbols used in Eqs. (2)–(4) are explained in Chapter 1, Eqs. (88) and (89).] The value of the constant Bi (sometimes called the sensitivity factor) can be determined either experimentally as the slope of the straight calibration line for the ith element obtained with thin homogeneous samples or semiempirically based on both the experimentally determined GI0 ðE0 Þ value and the relevant fundamental parameters (Yap et al., 1987; Markowicz et al., 1992a). The absorption correction factor Abcorr represents the combined attenuation of the primary and fluorescent radiations in the whole specimen and can be determined individually for each sample by transmission experiments (Leroux and Mahmud, 1996; Giauque et al., 1973). These are done by measuring the x-ray intensities with and without the specimen from a thick multielement target located at a position adjacent to the back of the specimen, as shown in Figure 1. If ðIi ÞS ; ðIi ÞT ; and ðIi Þ0 are the intensities after background correction from the sample alone, from the sample plus target, and from the target alone, respectively, then the combined fraction of the exciting and fluorescent radiations transmitted through the total sample thickness is expressed by expf
½ðE0 Þ csc

1

þ ðEi Þ csc

2 mg

¼

ðIi ÞT
ðIi ÞS H ðIi Þ0

ð5Þ

After a simple transformation, Eq. (2) can be rewritten as

Figure 1 Experimental procedure used in the emission–transmission method for the correction of matrix absorption effects.

Copyright © 2002 Marcel Dekker, Inc.

Ii ðEi Þ ð
ln HÞ ð6Þ Bi 1
H The emission–transmission method can only be applied in the quantitative XRF analysis of homogeneous samples of which the mass per unit area is smaller than the critical value, mcrit ; defined by ln Hcrit ð7Þ mcrit ¼
ðE0 Þ csc 1 þ ðEi Þ csc 2 where Hcrit is the critical value of the transmission factor defined by Eq. (5); in practice, Hcrit ¼ 0:1 ðor 0:05Þ: To minimize possible absorption correction errors resulting from enhancement of the specimen radiation by scattered target radiation, targets that yield a high ratio of scattered to fluorescent radiation should not be used. Giauque et al. (1979) developed a modified version of the emission–transmission method. Using data from the attenuation measurements and Eq. (5), the values of ½ðE0 Þ csc 1 þ ðEi Þ csc 2 m ð¼ T mÞ are calculated for the energies of characteristic x-rays of all elements present in a thick multielement target. If these values are plotted versus the fluorescence x-ray energy on a log–log scale (Fig. 2), an approximate value for mi ¼

Figure 2 Curves of [m(E0) csc 1 þ (Ei) csc 2]m and mm(Ei) csc 2 values versus fluorescence x-ray energy for an NBS SRM 1632 coal specimen. (From Giauque et al., 1979. Reprinted with permission from Analytical Chemistry. Copyright American Chemical Society.)

Copyright © 2002 Marcel Dekker, Inc.

ðE0 Þ csc 1 can be obtained by extrapolation of the curve to the energy of the excitation radiation. In turn, values for mðEi Þ csc 2 can be calculated, a curve for these values drawn, and a new value for mðE0 Þ csc 1 established. This last step is iterated several times. Using data from the latter curve, the absorption correction factors for all radiations of interest can be calculated from Eq. (4). If some elements to be determined are major or minor constituents, a few separate curves for mðEi Þ csc 2 values should be plotted between the preselected x-ray energies corresponding to the relevant absorption edges. In the emission–transmission method of Giauque et al. (1979), the incoherent scattered radiation corrected for matrix absorption is used as the internal standard to compensate for variations in sample mass, x-ray tube output, and sample geometry. In practical applications of the emission–transmission method, the values of the transmission factor H [Eq. (5)] determined for a few energies of the characteristic x-rays of some elements present in a multielement target are used to construct a curve ln lnðH
1 Þ ¼ lnðT mÞ versus ln E; which enables one to calculate the absorption correction factor Abcorr for any energy. In some cases, however, the relationship of ln lnðH
1 Þ versus ln E exhibits some discontinuities which correspond to the absorption edges of minor (major) elements present in the unknown samples. In such cases the approach proposed by Giauque et al. (1979), based on the construction of separate curves for the predefined energy regions, can only be used if at least two experimental points are available for each energy region. When only one or no experimental point is available, a modified version of the emission–transmission method can be used (Markowicz and Haselberger, 1992). To explain the modified procedure, let us assume that a multielement target consists of Ca, Ti, Fe, Zn, Sr, Zr, and Pb, and the material to be analyzed contains some minor elements, such as Fe and Ca. From the E–T measurements, the values of A ðA ¼ H
1 Þ are easily obtained for higher energies (e.g., for the characteristics x-rays of Fe, Zn, Sr, Zr, and Pb in this case) (see Fig. 3). First, a straight line is fitted to the points corresponding to Zn, Pb, Sr, and Zr (region I). Second, a straight line of the same slope and passing through a point for Fe is constructed (region II). If no experimental point is available for Ca, the discontinuity for the K absorption edge of Ca has to be taken into account by applying a computation routine. In the first step, the product 1 ¼ WCa ðabove
below Þ is calculated, where WCa is the weight fraction of Ca obtained by using a direct extrapolation of curve II and above and below are the mass absorption coefficients just above and below the K absorption edge of Ca, respectively. Next, the corrected value of the total mass absorption coefficient cor T ¼ ðln AÞ=m
1 csc 2 is calculated for the energy of the CaKabs edge, where A is taken from curve II and 2 is the emerging angle for the characteristic x-rays. Based on the cor T value, a straight line passing through point C and having the same slope as that one in region II is constructed (the coordinates of the point C are x ¼ ln CaKabs and y ¼ ln½ln A
m1 csc 2 ). In the next step, the values of ln T m and the absorption correction factor Abcorr are calculated for the energy of Ca characteristic x-rays based on the straight line for region III. Because the true value of the weight fraction for Ca is unknown, all calculations for the absorption edge of Ca (region III) are iterated until the following convergence is obtained: ðWCa Þn
ðWCa Þn
1  0:001 ðWCa Þn

ð8Þ

where ðWCa Þn and ðWCa Þn
1 are the values of the weight fractions for Ca obtained in the subsequent iterations. Copyright © 2002 Marcel Dekker, Inc.

Figure 3 Graphical explanation of the idea for the modified version of the E–T method: see text for details. (From Markowicz and Haselberger, 1992. Reprinted with permission of Pergamon Press Ltd.)

A calculation procedure applied for the Ca absorption edge can easily be repeated for any other absorption discontinuity at a lower energy and, finally, a total absorption curve ln ln A versus ln E is obtained. The importance of the additional correction for the discontinuities in the absorption properties of the samples to be analyzed was demonstrated, among others, for the analysis of coal samples. Neglecting the absorption edges correction for Fe and Ca at the concentration level of around 1% resulted in considerable errors of the analysis, up to 60% for the determination of Ca (Markowicz and Haselberger, 1992). Van Dyck et al. (1980) developed a correction method that allows calculations of the absorption coefficients (and absorption correction factors as well) at any energy for intermediate-thickness samples, without additional measurements, by using the ratio of the x-ray signals from a Zr wire positioned in front of the sample and from a Pd foil placed behind the sample, both in a fixed geometry, as shown in Figure 4. The Zr wire provides an external reference signal ðZrKÞ; which is applied for normalization of all measured fluorescent intensities to reduce considerably the effect of variations in exciting x-ray tube intensity and of dead-time losses. The coefficients for higher energies are calculated with an iterative program from the experimentally measured absorption coefficient at the PdL energy (2.9 keV), ðEPd Þ. In the first step, the total attenuation coefficient at the PdL energy, caused exclusively by the lowZ elements (e.g., Z < 17) in the sample that show no characteristic peak above 3.0 keV in the spectrum, is calculated from the normalized measured intensities, ðIPd ÞT and ðIPd Þ0 and Copyright © 2002 Marcel Dekker, Inc.

Figure 4 Measurement geometry applied in an automatic absorption correction method. (From Van Dyck et al., 1980. Reprinted by permission of John Wiley & Sons, Ltd.)

the different characteristic peaks recorded in the spectrum. This attenuation coefficient due to low-Z elements, low-Z ðEPd Þ, is given by (Van Dyck et al., 1980) low-Z ðEPd Þ ¼

ln½ðIPd Þ0 =ðIPd ÞT =m csc

2


ðE0 Þ csc 1 =csc P0 1
nj¼1 Wj

2




Pn 0 j¼1

Wj j ðEPd Þ

ð9Þ

where n0 is the number of characteristic peak in the XRF spectrum and j ðEPd Þ is the mass attenuation coefficient for the PdL energy in the element j (McMaster et al., 1969), giving rise to a characteristic peak in the XRF spectrum. The weight fraction of the element j in the sample, Wj , is correlated with the recorded characteristic peak through the sensitivity factor, routinely obtained by measuring thin single or multielement standards. As a first approximation, the characteristic peak intensities are not corrected for absorption. In a second step, based on the low-Z ðEPd Þ value, the absorption coefficients of the low-Z matrix for other energies, low-Z ðEÞ, are calculated quantitatively by assuming nearly parallel properties of the logarithmic absorption curves ln ðEÞ versus ln E. The total mass attenuation coefficient ðEÞ for the characteristic x-rays of an element in the whole sample can now be calculated, taking into account the contributions j ðEÞ of the high-Z elements: ! n0 n0 X X ðEÞ ¼ 1
Wj low-Z ðEÞ þ Wj j ðEÞ ð10Þ j¼1

j¼1

The overall procedure is summarized and schematically represented in Figure 5. Better ðEÞ values [Eq. (10)] are obtained in the second and following loops by carrying out appropriate absorption corrections to the characteristic intensities from which the Wj values are derived using ðEÞ and ðE0 Þ values from the previous loop and by including in Eq. (9) the ðE0 Þ values, taken as zero in the first loop. The iteration is stopped when the difference in ðEÞ between two loops is negligible. Subroutine ENHANC is applied to evaluate the enhancement effect of the PdL x-rays caused by all elements in the sample of which the x-ray energy is higher than the L1 ; L2 and L3 absorption edges of Pd. The intensity caused by this enhancement effect can be considered to result in an apparent increase in ðIPd Þ0 . The enhancement contributions, IenhPd , are added to the ðIPd Þ0 in Eq. (9). This total ðIPd ÞOeff value and the detected ðIPd ÞT signal allow us to calculate the correct  values via Eqs. (9) and (10). A comprehensive discussion of the influence of secondary enhancement of the PdL x-rays by the samples as well as of the influence of two other complicating factors, grain Copyright © 2002 Marcel Dekker, Inc.

Figure 5 The calculation steps. From the experimentally measured total absorption coefficient at the PdL energy, m(EPd), the calculated contribution from high-Z elements (giving a characteristic peak in the spectrum) is substracted; through this low-Z matrix contribution at the PdL energy, the total low-Z absorption curve (dashed line) is calculated; the contribution from high-Z elements (e.g., Ca and Fe) is then added, to yield the total absorption curve (thick line). (From Van Dyck et al., 1980. Reprinted by permission of John Wiley & Sons, Ltd.)

size effects, and a heterogeneous sample load is presented in the work of Van Dyck et al. (1980). The grain size and sample heterogeneity effects induce inaccuracies on the absorption coefficient determinations that may well reach 20% for particulate samples, such as intermediate-thickness deposits of geological materials. Thus, this approach has the same limitations as all emission–transmission methods applied to heterogeneous samples. A.

Accuracy and Limitations of the Emission--Transmission Method

The overall (total) absolute uncertainty of the mass per unit area of the ith element, mi , can be calculated from the law of error propagation applied to Eq. (2) (Markowicz et al., 1992b): "
#1=2 

 Abcorr 2 Ii ðEi Þ 2 Ii ðEi ÞAbcorr 2 2 2 2 mi ¼ ½Ii ðEi Þ þ ðAbcorr Þ þ ðBi Þ ð11Þ Bi Bi B2i where Ii ðEi Þ is the absolute uncertainty of the characteristic x-ray intensity Ii ðEi Þ (usually provided by a computer routine applied for spectrum evaluation), Abcorr is the absolute uncertainty of the absorption correction factor, and Bi is the absolute uncertainty of the sensitivity (or calibration) factor Bi (usually provided by a computer calibration routine, as a mean percent difference between the experimental and fitted values for the calibration factors). The total relative uncertainty of the mass per unit area mi =mi is calculated from "

 #1=2 
mi Ii ðEi Þ 2 Abcorr 2 Bi 2 ¼ þ þ ð12Þ Ii ðEi Þ mi Abcorr Bi Copyright © 2002 Marcel Dekker, Inc.

In order to calculate Abcorr , one has to apply the law of error propagation to Eqs. (4) and (5): Abcorr ¼

A
1
ln A ðA
1Þ2

A

ð12aÞ

with A ¼ H
1 [Eq. (5)] and " A ¼

½ðIi Þ0 2

½ðIi ÞT
ðIi ÞS 2

þ

½ðIi ÞT 2 ½ðIi Þ0 2 ½ðIi ÞT
ðIi ÞS 4

þ

½ðIi ÞS 2 ½ðIi Þ0 2 ½ðIi ÞT
ðIi ÞS 4

#1=2 ð12bÞ

where ½ðIi Þ0  and ½ðIi ÞT  are the absolute uncertainities of the characteristic x-ray intensities obtained from the target alone and from the sample plus target, respectively. After a simple transformation, mi =mi can be given by "

 #1=2  mi Ii ðEi Þ 2 ðA
1
ln AÞ2 Bi 2 2 ¼ þ ðAÞ þ ð13Þ mi Bi Ii ðEi Þ ðA
1Þ2 A2 ðln AÞ2 It is obvious that