Studies of Light Element X-Ray Fundamental ParametersUsed in PIXE Analysis

by

Christopher Michael Heirwegh

A ThesisPresented to

The University of Guelph

In partial fulfilment of requirementsfor the degree of

Doctor of Philosophyin

Physics

Guelph, Ontario, Canada

© Christopher Michael Heirwegh, April, 2014

ABSTRACT

STUDIES OF LIGHT ELEMENT X-RAY FUNDAMENTAL PARAMETERS

USED IN PIXE ANALYSIS

Christopher Michael Heirwegh Advisor:University of Guelph, 2014 Professor J. L. (Iain) Campbell

New measurements of fundamental parameters (FP), used in reference-free particle in-

duced X-ray emission (PIXE) analysis of materials, may be assisted by incorporating both

state-of-the-art spectral processing methods and methodologies that have minimal reliance

upon other FPs. Under this heading, several light element FPs have been investigated.

Monochromatic spectra [73] have been characterized in terms of the origins and line-

shapes of features arising due solely to photon and electron interactions in the silicon de-

tector crystal. Through the assistance of Monte Carlo simulation results, state-of-the-art fit

treatments were applied to these spectra using a fit routine that combined both non-linear

least squares and manual optimization.

The fitted spectra were used to derive a new estimate of the K X-ray fluorescence yield

(ωK) of Si. This was done using a geometrical expression that relates ωK to the area ratio of

the escape to primary peaks taken from the spectra. The final result (0.0504± 0.0015) was

realized through assistance from new low energy mass-attenuation coefficient (MAC) data.

The accuracy of low-energy (1–2 keV) MACs of light elements was assessed using an

approach incorporating PIXE measurements on pure element (Mg, Al and Si) and oxide

(MgO, Al2O3 and SiO2) targets. Calculated spectrometer efficiency constants, compared

between target pairs (eg, Si vs. SiO2), allow many FPs to cancel. Any non-zero difference

in the comparison indicates errors associated with the remaining FPs. A resultant 4–6 %

discrepancy was attributed to the use of XCOM [10] MACs but this was reduced to 0.5–2.5%

using FFAST [33] MACs.

Additional measurements, performed on silicate micro-probe standards, were analyzed

using the same comparative approach. A light element efficiency-constant discrepancy of 7–

9% was observed and attributed to the use of XCOM MACs. This was reduced to 0.5–3.5%

using FFAST MACs but was reduced further to −0.5–2% using a combination of XCOM

and FFAST MACs. This result suggested that the combination database was superior.

Acknowledgements

The years dedicated toward the completion of my doctorate have been filled with muchsolitude and concentration but they have also involved occasions of helpful discussion, shar-ing of ideas and assistance given by others. Many individuals are deserving of thanks, for ifthey had not been a part of my experience, my own efforts might not have come to fruition.

For their administrative efforts, in keeping the department running smoothly, as wellas managing the affairs of the graduate students, I thank: Linda Stadig, Reggie Vallillee,Janice Hall, Tanya Qureshi, Steve Kempf as well as Department chairs, past and present,Eric Poisson and Leonid Brown.

Experimental work was made possible with the mechanical assistance of Steve Wilsonand accelerator support from Bill Teesdale. I also wish to thank Ralf Geller, Irina Pradlerand John Maxwell each for the stimulating discussions, and John again for software technicalassistance. I also am grateful to Theo Hopman for introducing me to the many fine detailsof this project and for his occasional assistance over the years.

Appreciation is extended to Adam Raegan and John Dutcher of the Polymer Surfaceand Interface Group for atomic force microscopy work. Carbon coating of samples wasmade possible through the assistance of Bob Harris at the Microscope Imaging facility ofthe University of Guelph.

I appreciate the effort and time put forth by the members of my advisory committee,Joanne O’Meara, Paul Garrett and David Chettle. I also thank my advisor, Iain Campbell,for sharing his knowledge and passion for research. I am grateful for the many ways that hehas invested his time for my benefit.

I am thankful to my parents and family for their support and continued encouragementover the years.

I am indebted to my wife for providing editing assistance on this manuscript. Herpatience and encouragement throughout the entire duration of my studies will never beforgotten. Thank you all.

iv

Contents

1 Introduction 11.1 Overview of light element fundamental parameters . . . . . . . . . . . . . . . 21.2 The K X-ray fluorescence yield of silicon . . . . . . . . . . . . . . . . . . . . . 31.3 Assessment of low energy mass-attenuation coefficients used in light element

PIXE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Lineshapes characterization of Si(Li) and silicon drift detectors using 1–8keV X rays 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Data acquisition from spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Fitting the spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Functions fitted to the electron escape shelves . . . . . . . . . . . . . . 142.3.2 Explanation of primary electron escape features . . . . . . . . . . . . . 162.3.3 Fit treatment of the contact contributions . . . . . . . . . . . . . . . . 192.3.4 Fit functions applied to the tail . . . . . . . . . . . . . . . . . . . . . . 202.3.5 Fit treatment of SDD spectra . . . . . . . . . . . . . . . . . . . . . . . 202.3.6 Fitting the escape peak . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4.1 Primary electron escape features . . . . . . . . . . . . . . . . . . . . . 242.4.2 Contact contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2.1 Evidence for a diffusion effect in the contact features . . . . . 302.4.3 Peak tailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Further discussion on the tail feature . . . . . . . . . . . . . . . . . . . . . . . 352.5.1 Comparison of the tailing among different detectors . . . . . . . . . . . 35

2.5.1.1 Influence of the Si-contact work-function . . . . . . . . . . . 362.5.2 Si K X-ray escape peak . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 An accurate determination of the K-shell X-ray fluorescence yield of sili-con 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Previous experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Chemical effects on K X-ray fluorescence yields . . . . . . . . . . . . . 463.3 Present objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Basis of the escape peak measurement . . . . . . . . . . . . . . . . . . . . . . 493.5 Escape peak analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Assessment of Si MAC databases . . . . . . . . . . . . . . . . . . . . . 51

v

3.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.7 Comparison to theoretical predictions . . . . . . . . . . . . . . . . . . . . . . 623.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 An accuracy assessment of photo-ionization cross-section datasets for 1–2keV X rays in light elements using PIXE 664.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 Development of theoretical photo-absorption cross-section formalism . 664.1.2 Current theoretical cross-section datasets . . . . . . . . . . . . . . . . 67

4.2 Previous experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3 Objectives and approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Evaluation of the detector absolute efficiency constant . . . . . . . . . 734.3.2 Consideration for stopping power and X-ray production cross-section . 76

4.3.2.1 Stopping power . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3.2.2 X-ray production cross-section . . . . . . . . . . . . . . . . . 77

4.4 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.2 PIXE measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.3 Magnetic assembly used to deflect scattered protons . . . . . . . . . . 854.4.4 Beam blanking system . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.4.5 Spectra acquisition and fitting . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.6.1 Present data vs. theory . . . . . . . . . . . . . . . . . . . . . . . . . . 934.6.2 Comparison of light element experimental works with theory . . . . . 944.6.3 Attenuation by oxygen in oxide targets . . . . . . . . . . . . . . . . . . 964.6.4 Comparison of Scofield, Cromer, and other theoretical treatments . . . 96

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Quantification of major light elements in silicate glass and mineral stan-dards using PIXE 1005.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.1.1 A brief history on electron-probe micro-analysis . . . . . . . . . . . . . 1015.1.2 Micro-probe use in geochemistry . . . . . . . . . . . . . . . . . . . . . 1025.1.3 PIXE and geological applications . . . . . . . . . . . . . . . . . . . . . 103

5.2 Objectives and approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3.1 Selection and PIXE measurement of silicate standards . . . . . . . . . 1065.3.2 Fitting of silicate standards spectra . . . . . . . . . . . . . . . . . . . . 1095.3.3 Calculation of light element efficiency constant . . . . . . . . . . . . . 1155.3.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.5 Implementation of a GUPIX MAC database of FFAST coefficients . . 117

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5.1 Silicate standard efficiency constant trends . . . . . . . . . . . . . . . . 1255.5.2 Theoretical database assessment and future directions . . . . . . . . . 127

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

vi

6 Concluding Remarks 131

A A contaminant-free polishing procedure for soft metal targets used inPIXE 144

B Procedural details of carbon coating application and layer thickness mea-surement 146B.1 Details on the application of carbon layers . . . . . . . . . . . . . . . . . . . . 146B.2 Techniques in measuring the thickness of carbon layers . . . . . . . . . . . . . 148B.3 Carbon layer removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C Examination of proton deflection by Nd-Fe-B magnet assembly using asurface barrier detector 150C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150C.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.2.1 Detector energy calibration . . . . . . . . . . . . . . . . . . . . . . . . 150C.2.2 PIXE chamber deflection test . . . . . . . . . . . . . . . . . . . . . . . 151

C.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

D Fitting light element spectra with consideration for the radiative Augereffect 155D.1 On the radiative Auger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 155D.2 Fitting light element SDD acquired spectra . . . . . . . . . . . . . . . . . . . 156

E GUPIX determination of PIXE H-value efficiency constants 162E.1 Detector efficiency file and H-value parameters . . . . . . . . . . . . . . . . . 165

F Curriculum vitae of the author 168

vii

List of Tables

3.1 Escape peak intensities (𝜂) from Ti, Cr, Fe and Cu Kα monochromatic spectra. 523.2 Summary of parameters used in analysis and Si ωK’s calculated at the four

X-ray line energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Compilation of theoretical and experimental values of the K-shell fluorescence

yield for Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 Theoretical radiative and non-radiative decay widths (Γ (eV)) for the K-shell

of Si and Ar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1 Summary of chemical-bond corrections to light element X-ray yield. . . . . . . 804.2 Target origins, purity levels and PIXE analyses of trace elements. . . . . . . . 824.3 Average 𝜖a(ℎ𝜈)-values of elements and oxides and the percentage differences

(Δ𝜖a(ℎ𝜈)) calculated using the XCOM and FFAST MACs. . . . . . . . . . . . 924.4 Summary of propagated uncertainties used to derive 𝜎(Δ𝜖a(ℎ𝜈)). . . . . . . . 92

5.1 Information summary of silicate standards measured with PIXE. . . . . . . . 1075.2 Reference compositions of silicate standards. . . . . . . . . . . . . . . . . . . . 1085.3 Average differences in 𝜖a(ℎ𝜈) constants calculated for Mg, Al and Si in silicate

standards using three mass attenuation coefficient databases. . . . . . . . . . 120

D.1 Relative RAE intensities of light elements taken from literature and measuredusing PIXE spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

E.1 H-values derived from an Astimex metal mount containing elements Z =12 . . . 14 and Z = 21 . . . 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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List of Figures

2.1 Non-Gaussian regions (i), (ii) and (iii) in a Fe Kα spectrum. . . . . . . . . . . 82.2 Fitted Fe Kα Si(Li) spectrum showing fit components. . . . . . . . . . . . . . 152.3 Fe Kα Si(Li) spectrum overlaid with Monte Carlo results. . . . . . . . . . . . 162.4 Ag Lα Si(Li) spectrum spectrum overlaid with Monte Carlo results. . . . . . . 172.5 Si K X-ray Si(Li) spectrum overlaid with Monte Carlo results. . . . . . . . . . 182.6 Fitted Ti Kα Si(Li) spectrum showing fit components. . . . . . . . . . . . . . 212.7 Fitted Ti Kα SDD spectrum showing fit components. . . . . . . . . . . . . . . 222.8 Experimental and modelled electron escape shelf intensities plotted against

primary peak energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.9 Experimental and modelled area ratios of the Ni L and M shell DE s relative

to the main peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.10 Simulated Fe Kα spectra showing the Si K Pe edges and escape peaks using

Goto [63] diffusion 𝐶R’s of 1.0, 0.9, 0.8. . . . . . . . . . . . . . . . . . . . . . . 312.11 Simulated Si K Pe edge locations plotted against 𝐶R values. . . . . . . . . . . 322.12 Comparison of Fe Kα Si(Li) spectrum with Monte Carlo predictions using a

Goto 𝐶R of 0.92. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.13 Primary peak relative tail areas plotted against photon energy. . . . . . . . . 34

3.1 Experimental and theoretical [35, 108] values of ωK expressed as ratios to thetheoretical predictions of Walters and Bhalla [164]. . . . . . . . . . . . . . . . 42

3.2 Detector geometry considerations used to derive Si ωK. . . . . . . . . . . . . . 493.3 Percentage difference of Si PTB MACs with respect to the XCOM and FFAST

databases in the 1–5 keV energy range. . . . . . . . . . . . . . . . . . . . . . . 543.4 Percentage difference of Si PTB MACs with respect to the XCOM and FFAST

databases in the 5–30 keV energy range. . . . . . . . . . . . . . . . . . . . . . 553.5 Plot of individual Si ωK values calculated from 28 spectra. . . . . . . . . . . . 59

4.1 Basic schematic diagram of PIXE chamber and electronics setup. . . . . . . . 864.2 PIXE Spectral response in the presence and absence beam deflection. . . . . . 874.3 Plot of yield vs. count rate used to correct residual PIXE system count rate

dependency of observed yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.4 Spectra comparison of Si and SiO2 normalized to the same primary peak height. 904.5 Fit details of a pure Si target PIXE spectrum. . . . . . . . . . . . . . . . . . . 914.6 Experimental [75] Si MAC values plotted with XCOM and FFAST MACs

below the Si K edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.7 Experimental [8] Al MAC values plotted with XCOM and FFAST MACs

below the Al K edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.8 Mg, Al and Si MAC value database comparison to XCOM and FFAST values. 97

ix

5.1 Fitted PIXE spectrum of Silicate BHVO-1 glass standard. . . . . . . . . . . . 1105.2 GUFIT-fitted PIXE spectrum of pure Sc target. . . . . . . . . . . . . . . . . . 1115.3 Summary of tail parameters from light element and 3d transition metal spectra.1135.4 GUPIX-fitted PIXE spectrum of a pure Si target. . . . . . . . . . . . . . . . . 1145.5 Percentage difference of XCOM and FFAST with respect to the Al PTB data. 1195.6 Efficiency constant differences between the pure Mg target and Mg in silicate

targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.7 Efficiency constant differences between the pure Al target and Al in silicate

targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.8 Efficiency constant differences between the pure Si target and Si in silicate

targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.1 SCPD calibration spectrum using a 244Cm 𝛼-emitter. . . . . . . . . . . . . . . 152C.2 Photograph of the arrangement housing the SCPD detector connected to the

PIXE chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.3 SCPD spectrum showing resultant proton deflection by the magnet assembly. 154

D.1 Close-up image of fitted tail and RAE region on low energy side of Si K X-rayprimary peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

x

Chapter 1

Introduction

The task of performing high accuracy X-ray emission analysis of materials has often relied

upon well calibrated standards to extract concentrations from energy dispersed X-ray spec-

tra. While this method can be used to derive meaningful results, calibration standards, if

available, do not always reflect the experimental conditions. A physics fundamental parame-

ter (FP) approach is a useful alternative to this reference-based approach. The FP approach

relies upon tabulated databases of the parameters involved in analysis and is subject to un-

certainty associated with each parameter. The validity of any given database is not always

apparent and as such it is necessary for an analyst to understand the quality, completeness

and accuracy of the parameters extracted for use.

In a recent report, directed to the X-ray FP community, Campbell [19] outlined the

current status of several X-ray FP databases including energies and probabilities of X-ray

transitions as well as yield probabilities of Coster-Kronig and X-ray fluorescence transitions.

In general, reliance upon experimental tabulations may be questionable due to either high

associated levels of uncertainties, use of inferior or obsolete methods to measure a given

value, often in the case of older experiments, or regions in a database in which data are

absent entirely. The latter often results when measurements are exceedingly difficult to

perform. These “information gaps” often necessitate interpolation or extrapolation to fill in

missing values thereby increasing the level of uncertainty in the extracted value. In addition

modern FP studies do not always account for subsidiary effects such as radiative Auger

or multiple vacancy satellite production that may alter observed X-ray yields and impact

the calculation of a new fundamental parameter. Chemical bonding can also influence the

1

X-ray production cross-section for light elements (Z < 20) and must be accounted for in

measurements of low Z materials. As these features may influence final estimates by the

order of several percent, their inclusion becomes increasingly necessary if one is to obtain a

new FP determination of high accuracy.

In partial response to the limitations of experimental datasets, theoretical and semi-

empirical compilations were developed and have served to complement, challenge and even

replace experimental tabulations. While theoretical calculations are generally more exhaus-

tive than experimental values and cover broader ranges in each database, their validity

and accuracy need to be understood. Some databases were created using assumptions and

simplifications to permit calculations to be performed within the limits of available computa-

tional power, though the resultant effect on calculation accuracy was not always understood.

Thus, the challenge for an experimentalist, in performing high accuracy work using funda-

mental parameters, is to make a careful assessment of all databases available: experimental,

semi-empirical or theoretical, so that an informed selection can be made of the best possible

estimate of an FP.

1.1 Overview of light element fundamental parameters

To examine in detail the shortcomings in any database would be an large and exhaustive task.

This work focuses on specific regions of two FP’s, the K X-ray fluorescence yield (ωK) and the

mass attenuation coefficient (MAC). Specific attention is paid to values associated with light

elements and low energy photons. Interest was motivated by the results of a particle-induced

X-ray emission (PIXE) calibration study involving measurements performed by Campbell et

al. [22] on a suite of silicate glass and mineral standards. Their PIXE FP approach revealed

that a 6–8% discrepancy existed between quantification estimates of light elements: Mg, Al

and Si found in the standards relative to pure element targets. This pointed immediately

to issues with one or more of the fundamental parameters used in the calculation. One

parameter considered to be responsible was the K X-ray fluorescence yield corresponding to

each of the light elements. It was thought that differences in the chemical environment of the

light elements in pure form, relative to oxide form, as found in the suite of standards, might

2

be responsible for effects observed on ωK. It was also postulated that the MAC database

values used to represent the self-attenuation of the K X-rays of Mg, Al and Si, through all

major elements in the sample, might be responsible for the discrepancy.

In this work it was hypothesized that if the FPs in question could be better understood

in terms of their accuracy, then the source or sources responsible for the 6–8 % discrepancy

found by Campbell et al. [22] might be determined. It was expected that this could be

accomplished by performing experiments, using “state-of-the-art” techniques to re-evaluate

light element fundamental parameters and to critically examine existing tabulations.

1.2 The K X-ray fluorescence yield of silicon

Two K X-ray fluorescence yield compilations are frequently cited for use in FP X-ray emission

analysis. The first is that of Krause [97] (1979) and the second is from Bambynek [4, 78]

(1984) which was an update of a work published in 1972 [5]. Each author manufactured a

database from a semi-empirical fit to existing experimental tabulations and used their own

prescribed weighting scheme to favour data considered highest in quality. Krause weighted

in favour of newer data while Bambynek selected those considered to be of highest quality.

In the element region Z > 40 agreement between the two databases is very strong but in the

region 10 < Z < 20, the difference is a smooth variation from +7 to −5% leaving the issue of

database accuracy ambiguous. It was expected that resolution of this issue might be assisted

through the creation of new measurements of light element ωK values. Campbell et al. [20]

performed a measurement of the ωK of Si and carried out two separate calculations, using

different MAC databases: XCOM [10, 132, 142] and X-ray form factor, attenuation and

scattering tables (FFAST) [31–33]. This resulted in two distinct estimates of the ωK final

result. An average from both estimates was adopted in the final reported value. This was

a necessary procedure since no reliable MAC data of Si in the low energy photon range was

available that might have allowed for the selection of one MAC database over the other. In

this work, a new measurement of Si was considered to be a good starting point for two main

reasons. First, several high quality measurements of low energy MACs, in Si, now exist [75,

160]. A critical review of these data allow for adoption of one set of MAC values leading

3

to one estimate of Si ωK. Second, the experimental design is relatively convenient, relying

upon monochromatic photon radiation in the presence of a Si semi-conductor detector. A

geometrical calculation of the ratio of escape peak to primary-plus-escape peaks is used in

this derivation [14, 20].

This method relies on accurate peak area extraction of the monochromatic spectra pro-

duced using energy-dispersive semi-conductor detectors and is a technique that may be

considered “state-of-the-art”. A great deal of literature has been dedicated to understanding

the semi-conductor detector physics processes responsible for producing all non-Gaussian

features (tails and shelves) manifesting on the low-energy side of the primary peak. Appli-

cation of empirical fitting functions to individual features has led to better approximation

of the background under the primary and escape peaks. [73, 74] The details of the fit treat-

ments applied to spectra, produced using monochromatic X-ray sources in the range of (1–8

keV), are presented in Chapter 2. Where possible, explanations of the physical processes

responsible for each feature have also been provided.

In Chapter 3 the author demonstrates how the monochromatic spectra of the previous

chapter, combined with a detailed spectrum fitting treatment and greater understanding of

the silicon MAC databases, has been used to derive a new, high-accuracy estimate of the

K-shell X-ray fluorescence yield (ωK) of silicon [75]. The new Si ωK result may be considered

the most accurate determination of this parameter to date.

1.3 Assessment of low energy mass-attenuation coefficients

used in light element PIXE analysis

The electron micro-probe is a device that predates PIXE as a method of X-ray emission

analysis of materials, though the physics fundamentals of the technique are much the same.

Since its inception by Castaing in 1951 [30], there has been a focus on perfecting its use

in materials analysis. J. V. Smith [149] outlined the details of a calibration approach to

quantify measurements performed on various silicate targets. His method involved creating

an efficiency curve relating X-ray generation to the mean atomic number of a sample. This

relationship was used to quantify all targets. The resultant empirical absorption curves

4

derived for light elements in silicate targets were found to differ significantly from those

derived for pure element targets. No solid explanation was provided though the author

suggested that if the MACs are incorrect, then they may be responsible for this discrepancy.

In PIXE analysis of silicate standards, Campbell et al. [22] encountered a similar prob-

lem in an attempt to quantify light elements. Their approach reinforced the issue that light

element analysis, using any micro-probe in a fundamental parameters approach, is prob-

lematic. Both techniques rely upon correction of attenuation of the outgoing X rays as a

necessary step in analysis. The results are thus affected by the accuracy of the chosen MAC

values. Of the many MAC databases available for use in X-ray emission analysis, the XCOM

database [10, 142], supported by the National Institute of Standards and Technology (NIST)

is perhaps the most widely used. The convenience of its web-based interactive program, al-

lowing for fast interpolation of user-inputted photon energies, makes the XCOM program

an appealing choice for users. The FFAST database [31–33], also hosted by NIST, merits

attention given that its more recent development has incorporated changes to the theoretical

treatments used to create earlier existing databases, including XCOM. Both databases are

large, spanning broad energy ranges for almost all known elements, and are useful tools to

analysts, though the issue of database superiority is again questioned. To probe for inter-

database discrepancies across the entire range would be a formidable task. The comparisons

presented in this work are therefore limited to specific regions. Specifically, it is shown that

for low-energy (1–8 keV) photons in selected light elements (8 6 Z 6 30), inter-database dis-

crepancies range from about 1–12%. The MAC discrepancies are highest in the low energy

(∼ 1 keV) region, below the respective K absorption edge of elements (11 6 Z 6 14).

The author presents, in Chapter 4, a PIXE-examination into the effects of using each

database to obtain relative quantification of light element targets in pure (Mg, Al and Si) and

oxide (MgO, Al2O3 and SiO2) forms. The conditions of this experiment were kept similiar to

those of the Campbell et al. [22] study, to assist in probing the discrepancies they found. The

design of this experiment has permitted the uncertainties associated with many of the PIXE

FPs not to influence the resultant calculation so that an assessment of high precision could

be obtained. These results show that a discrepancy similar to Campbell suggest that the

coefficients from FFAST are more accurate than XCOM for light elements at low energies.

5

Within this work, care was taken to account for subsidiary effects of radiative Auger and

multiple vacancy satellite influence on the total observed X-ray yield.

Most of the targets analyzed by Campbell et al. [22] have been well characterized [43,

85, 86] for use as micro-probe calibration standards. Their compositions were determined

primarily using chemical analysis for the purpose of avoiding possible errors present in mi-

croprobe measurements. In Chapter 5, the experimental design presented in Chapter 4 was

expanded to include the silicate suite. PIXE analysis was performed on some of these stan-

dards to assess if substitution of the FFAST database over XCOM might resolve the light

element quantification discrepancy seen previously. The results have shown that discrepan-

cies similar to Campbell et al. [22] may be reproduced using the XCOM database but that

superior agreement may be found using the FFAST values of light elements below their K

absorption edges.

6

Chapter 2

Lineshapes characterization of Si(Li)and silicon drift detectors using 1–8keV X rays

2.1 Introduction

Since the beginning of their use in practical science, semi-conductor detectors have been

studied to understand the photon and electron physics of their operation as well as the

resultant spectral lineshapes they produce. A rather extensive collection of literature has

been developed toward this goal and serves two purposes. The first is that the lineshape

information can be used to assess performance and detector design, including engineering

of electrical contacts. Additionally, knowledge of lineshapes components may assist in the

formation of mathematical or semi-empirical algorithms used in computer codes designed

for fitting peak and background features to obtain accurate measure of peak area.

The discussion of spectral lineshapes in this work focuses on those produced from rela-

tively low energy monochromatic radiation (1–8 keV) incident upon lithium-drifted Si semi-

conductor detectors (Si(Li)). At low energy, several non-Gaussian features appear on the

low energy side of the primary peak that extend to almost zero energy. In earlier works, this

entire region has been classified generally as the low-energy peak tail. However, the recent

work of Scholze and Procop [135] has provided an effective review of the many processes

responsible for the creation of the tail region and has delineated the numerous shelf, mound

and tail features that comprise this entire region. These have been categorized as (i), (ii)

and (iii) and their regions are indicated in Figure 2.1 which shows a monochromatic Si(Li)

7

0 1 2 3 4 5 6 71 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

S i ( L i ) D a t a

r e g i o n o f( i ) a n d ( i i )

( i i i )

p r i m a r yp e a k

S i K X - r a ye s c a p e p e a k

Coun

ts

P h o t o n e n e r g y ( k e V )Figure 2.1: Monochromatic spectrum produced using Fe Kα keV X-rays displaying regionsof non-Gaussian features (i), (ii) and (iii) and principal Gaussian features including theprimary peak and the Si K X-ray escape peak.

spectrum produced by 6.4 keV X rays.

(i) Flat or gentle sloping shelf-like continua extend from the primary peak downwards

in energy and originate from the escape of energetic primary (photo- and Auger)

electrons from Si. They have well-defined low-energy cut-offs corresponding to the

maximum energy with which the electron can escape. Their production might be

considered analogous to the well-known Si K X-ray escape peak. A shelf arises because

the electron may lose an undefined amount of energy in the detector crystal prior to

ejection. These features are hereafter identified as the primary electron escape shelves.

(ii) Additional X-ray peaks and shelf-like continua can arise from photon interactions

occurring in the front contact material or, if present, any external deadlayer (DL).

Contacts in Si(Li) detectors are usually metallic and in silicon drift detectors (SDD)s

may be either metalic, implant doped or deposited layers [93]. Unlike the escape

of particles as described in (i), these events correspond to the injection of electrons

8

into the bulk Si. The energy of the injected contact electrons is less than that of

the incoming photon as some energy is used in the work-function to promote bound

electrons to the continuum. Like (i), the features appear on the lower energy side of

the primary peak. Their distribution appears as a continuum due to the collisional

energy losses of the contact electron in the metal contact before entry into the active

crystal. These are referred to as contact features in this chapter.

(iii) The low energy side tailing on the primary peak is the most intense non-Gaussian

feature in the low energy region and does not typically extend far from the primary

peak. It has been attributed principally to out-diffusion or recombination of secondary

electrons from the detector bulk as well as issues encountered with signal transmission

and pulse processing electronics. This feature is hereafter referred to as the peak tail.

The processes responsible for (i) and (ii) have been covered extensively in the literature

but have not always been distinguished from those responsible for (iii). Of the goals set

forth in this chapter, one is to review and distinguish in detail the individual features (i, ii

and iii) observed in spectra acquired with various incident energy radiation (1–8 keV). The

second is to demonstrate the detailed fitting treatment used to provide high-accuracy fits of

spectra at varying energies. The last goal is to extract hitherto unrecognized information

from the fine detail of contributions (i) and (ii).

Emphasis is placed on producing high-accuracy fits to extract information about line-

shapes. One added benefit is the use of the fitted spectra in deriving a new estimate of the

K shell fluorescence yield of Si (Chapter 3). The energy-dependent trends in the intensities

of the individual fit features are also monitored for their possible future use in construct-

ing energy-dependent, fixed parameter empirical descriptions of each feature for use in fast,

high-accuracy fitting of multi-peak spectra. Monte Carlo evidence is also given in support

of a diffusion effect manifesting in a downward energy shift of several shelves and contact

features.

The processes responsible for the non-Gaussian features have been predicted by both

Monte Carlo simulations [25, 63] and analytical modelling [51, 104, 135]. Generally however,

the simplification of the electron interaction treatments in these approaches is such that

9

precise agreement with experimental data is unlikely but rather, serves in assisting with the

understanding of each process. Earlier experimental observations by Campbell et al. [21]

suggested that electron escape shelves in Si(Li) spectra were approximately flat though the

current simulation code [25] suggests that the electron escape shelves have finite slope. This

slope becomes apparent when looking at individual feature outputs on a linear intensity

scale. The analytical approach of Lowe [104] predicted that these features would slope

downward in intensity toward increasing energy, and his experimental work incorporating

spectra produced using both Si(Li) and high-purity germanium (HPGe) detectors confirmed

this prediction.

The sloping behaviour was further supported by results from an electron spectroscopy

study. In a novel approach, Papp et al. [120] and Papp [118] measured the kinetic energies of

the photo- and Auger electrons ejected from a slab of Si, following electron excitation. They

observed for each electron, shelf-like continua with a sloped intensity distribution. Unlike

the electron escape shelves seen in energy-dispersive spectra, their shelves sloped upwards

towards increasing energy. The high energy edges of the shelves were produced by escaping

electrons having maximum kinetic energy, which is a mirror image of the shelves found in

the energy-dispersive escape shelves. Allowing for this difference, the agreement between

the two situations is quite strong.

Scholze and Procop [135] modelled the electron escape shelves as a series of flat shelves,

which at first glance appears contrary to the literary evidence presented thus far. However,

their situation involved a Au contact which has been shown [25] to produce rather complex,

L- and M-shell contributions that may result in a flattening effect on the total shelf region.

Also, a flat shelf approach may have been a compromise to allow for a simplification of their

fit procedure to make implementation feasible. It is interesting to note that the features

of (i) were not studied until quite recently given that gold contacts were widely used in

early Si(Li) detectors. Additionally, the shelves only become distinguishable in detail, on a

logarithmic scale, when very large full energy peak counts (> 106) are collected.

For many years, the source of the most intense tailing feature (iii) has been investigated.

Most studies utilized monochromatic radiation incident on HPGe and Si(Li) detectors. The

challenge in interpretting some of the earlier works is that the individual sources responsible

10

for features (i, ii and iii), contributing to the peak tail, were not well distinguished from

each other. Still, various mechanisms have been proposed to explain the many non-Gaussian

features in this region, which has helped to provide insight to possible origins of (iii). The

individual sources have been well summarized elsewhere [25] and are briefly mentioned here.

In each proposed mechanism the tail is produced as a result of photon absorption occurring

very near to the contact boundary, where charge or energy is lost by a specific process.

Several studies suggest that if photon absorption occurs in the vicinity of an external DL,

or incomplete charge collection layer (ICCL), both of which may be present at the crystal

boundary, then there is a high probability that thermalized electron-hole (e–h) carriers may

become trapped or recombine prior to collection at the contacts. One theory [81] describes

how the size of the apparent DL is directly linked to the choice of the contact material

used. In this case, the thickness of the DL is directly proportional to the tail intensity.

Others have suggested that surface defects, including oxide contamination as a result of the

manufacturing technique used, might explain increased recombination.

The concept of out-diffusion of hot (secondary) electrons from the Si wafer boundary is

an idea that has attracted repeated interest as a means to explain (iii). Llacer et al. [103]

suggested that the tail in Ge detectors arises from the escape of hot electrons at the Ge

boundary. In this situation the hot electron is more energetic than a thermalized electron,

from an e–h pair, but does not possess the minimum energy needed to thermalize a new

pair. As the hot electron cools through interactions with the crystal environment, its mean

interaction path length increases to as much as several hundred nanometers, which increases

its chances for escape. In Llacer et al.’s model, all thermalized e–h pairs are collected at the

contacts but the tail is explained by energy loss due to the escape of hot electrons. Several

authors [63, 136] have since created models built on the concept of diffusion but mitigated by

the possibility of electron reflection at the boundary. This model treats secondary electron

loss in a more general way. Here, low energy photon interactions in the crystal create an

approximately spherical (radius = 0.1𝜇m) distribution of thermalized electron-hole pairs.

If the interaction occurs close to the surface, up to half of the pairs may be lost, resulting

in a tail extending down to half of 𝐸P, the primary peak energy. The introduction of the

possibility of reflection at the boundary has the effect of reducing the loss and shortening the

11

model tail. Goto’s solution [63] to the continuity equation differs in that his model allows

the possibility of complete charge loss, which would correspond to zero reflection. Eggert et

al. [51], on the other hand, argued that at most half the secondary electron cloud could be

lost into the contact.

These authors have found that implementation of simulation models built on this concept

provides good agreement with experimental data, which suggest that it may be valid in at

least partially describing the source of (iii). As of yet, a full characterization of its source

has not been completed. However, a greater understanding of the features in (i) and (ii)

now exist. Through assistance of Monte Carlo code such as that found in ref. [25], their

contributions may now be well distinguished from the effects responsible for (iii).

2.2 Data acquisition from spectra

The spectra analysed in this work were recorded by Dr. T. L. Hopman, under the supervision

of Professor J. L. Campbell of the University of Guelph. Acquisition of spectra has been

described elsewhere [73, 74] though some of the pertinent details are mentioned in this

section. The dataset comprises primarily monoenergetic X-ray sets collected using two Si(Li)

detectors, each having a Ni contact, and one Ketek Vitus SDD (part number AV450H7-

ZR1BE-144). The two Si(Li) detectors were Oxford Instruments devices and referred to as

Detector 1 and Detector 2.

Various curved crystals: Si111, LiF and Quartz were used to select principal X-ray lines

from the X-ray fluorescence of replaceable anodes. This was done to avoid introducing multi-

line complexities in the spectra of radiative Auger and shake satellite effects. Monochromatic

spectra were acquired using all three detectors and monochromatic beams of Kα X rays

from Ti, Cr, Fe and Cu anodes. Using Detector 1, additional spectra were acquired using

monochromatic Lα X-ray beams of Mo and Ag and polychromatic beams of Al and Si K X

rays. The detectors were oriented such that the monochromatized beams had perpendicular

incidence upon the detector crystal. The polychromatic set was acquired using Ti K X rays

as a means to excite the X rays from Al and Si foils placed in front of the window of the

detector. The foils were placed in a closed sleeve filled with He gas to minimize attenuation

12

losses. Using this method, it was not possible to obtain separation of Kβ, radiative and

satellite lines from the Kα lines of Si and Al. However, the low intensity and closeness of

separation of these lines from the the principal Kα line was not considered to hinder this

examination substantially. Generally, high total counts were acquired in each spectrum. For

the Ti to Cu spectra, these were in the range of 40–100 million.

Si(Li) data were acquired using an Oxford Instruments XP3 analogue signal processor

(ASP) and a Cambridge Scientific CSX4 digital signal processor (DSP) device. Processing

time was optimized to ensure maximum quality of data aquisition. An Ortec 672 ASP am-

plifier was used to process pulses from the SDD. Some electronics issues were reported to be

present during the collection of the SDD data which may have had an impact on their overall

quality. For this reason, only the Si(Li) data are discussed in detail throughout this chapter.

As a future project, SDD characterization will be redone using the tunable monochromatic

source of low energy X-rays (Solex) at the Laboratoire National Henri Becquerel - CEA

Saclay, 91190 Gif-Sur-Yvette Cedex, France.

2.3 Fitting the spectra

To assist in describing the fitting details, one of the acquired Si(Li) spectra, produced using

incident Fe Kα X rays, is shown in Figure 2.2. The empirical fit prescription used in this

work is provided for the Fe Kα spectrum seen in this figure. Its features arise exclusively from

photon interactions in the Si bulk and front contact, and behaviour of the electronic system

(Gaussian broadening). There are two prominent peaks and several non-Gaussian features

that make up the low-energy region of this figure. All spectra were fitted with an in-house

non-linear least-squares code (GUFIT). In the example shown, the fitted region extends

from zero energy to the primary first-order diffraction peak. Results from the simulation

program [25] were used to interpret the locations and approximate shapes and intensities of

the features, to assist in fitting. The energy resolution parameters derived from fitting the

measured spectra were used in the production of the modelled spectra.

13

2.3.1 Functions fitted to the electron escape shelves

Fitting of the electron escape shelves requires a function that can account for the sloping

behaviour of their shelves when observed on a logarithmic scale. Using the energy distribu-

tions of his electron spectroscopy results, Papp [118] has suggested that the electron features

should be described as a difference of two exponentials (DE ), convolved to a Lorentzian func-

tion at its edge. A similar functional form (without the Lorentzian component at the edge)

is used in the fitting routine which has the basic formula:

𝐷𝐸(𝑖) =1

2𝐻𝐸

[︀𝐸′(𝑖, 𝑆1)− 𝐸′(𝑖, 𝑆2)

]︀(2.1)

where S denotes reciprocal slope, i is channel number and E ′ is the exponential expres-

sion. [73, 74] The reciprocal slope of the first exponential determines the slow downward

trend towards higher energy while that of the second exponential determines the roundness

of the low-energy edge. The exponential expression is given as:

𝐸′(𝑖, 𝑆) = exp

(︂𝑖− 𝑖𝑝𝜎𝑝𝑆

+1

2𝑆2

)︂{︂erfc

[︂1√2

(︂𝑖− 𝑖𝑝𝜎𝑝

+1

𝑆

)︂]︂− erfc

[︂1√2

(︂𝑖− 𝑖𝑐𝜎𝑐

+1

𝑆

)︂]︂}︂,

(2.2)

where 𝑖𝑝 and 𝜎𝑝 are the channel number and Gaussian width of the peak respectively,

and 𝑖𝑐 and 𝜎𝑐 are identically described for the tail cutoff. Three separate DE functions were

fitted to the electron escape features. These have been denoted as A, B and C in Figure 2.2

and are described in detail in Section 2.3.2. Each of these slope downward toward higher

energy (Figure 2.2).

The shapes and positions of the non-Gaussian features depend on the energy of the

incident radiation. There are marked differences between spectra collected using different

incident energies in the 1–8 keV range. The eight different energy sets have been sub-

categorized into three groups based on similar appearence of the shelf region. The Ti, Cr,

Fe and Cu spectra are in one group, the Mo and Ag in the second and the Al and Si are in

the third. Their differences are discussed in detail below and a sample spectrum is shown

14

0 1 2 3 4 5 6 71 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

( 2 )

( 1 )

i . A ) S i K L V A u g e r i . B ) S i K P e i . C ) S i L P e & c o i n c i d e n t ( S i K P e , K L V A u g e r ) i i ) N i L P e i i ) N i M P e i i i ) t a i l - e x p o n e n t i a l ( 1 ) a n d s h o r t - s t e p ( 2 ) t o t a l f i t d a t a

Co

unts

P h o t o n E n e r g y ( k e V )

- 404

Resid

uals

Figure 2.2: Si(Li) Detector 1 Fe Kα spectrum acquired using Si(111) diffraction and XP3analogue processor. The total fit (𝜒2 = 2.97) to the data is shown with individual fitcomponents to non-Gaussian features, each corresponding to primary electron escape (i),contact contributions (ii) and tailing (iii); all were fitted using the modified GUFIT program.’Goodness of fit’ is indicated by the fit residuals in the top panel, on a linear scale, and haveunits of the standard deviation (𝜎).

from each sub-group: Fe Kα, Ag Lα and Si K X rays have been chosen to help illustrate

the unique features. The simulation results for Si(Li) spectra of these elements have been

overlaid onto the corresponding data, seen in Figures 2.3, 2.4 and 2.5 respectively. The

focus in this work is to separate the effect(s) responsible for (iii) from those of (i) and (ii).

To this end, charge loss models based on diffusion effects were not included in the presented

simulations. Rather, the simulations demonstrate the contributions from (i) and (ii), only.

The primary escape shelves (i) present in Figures 2.3, 2.4 and 2.5 are identified with grey

corresponding to the K and L Pe’s, and blue as the Auger electrons. Contact contributions

(ii) are denoted in green and the total model spectra are provided in red.

15

0 1 2 3 4 5 6 71 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

N i M P eN i L P e

N i L V V

S i L P e S i K P e

S i K L V S i L V V

S i P e & c o i n c i d e n c e S i A u g e r N i c o n t a c t S i m u l a t i o n s u m D a t a

Coun

ts

P h o t o n e n e r g y ( k e V )Figure 2.3: Experimental Fe Kα spectrum showing experimental data overlaid with MonteCarlo simulation predictions of Fe Kα radiation incident on Si bulk, with a 10 nm Ni contact,all normalized to the same primary peak intensity.

The region beyond the high side of the primary peak was not of interest as it represents

regions of only very low intensity pile-up. All spectra were fitted first using the manual fit

routine that allows for single parameter, single step adjustments by the user. In this way, all

functions could be placed appropriately according to the approximate locations given by the

Monte Carlo predictions. As a final step, an automatic data-fit minimization step was run

to optimize the fit. In certain cases, some features did not always remain in their prescribed

locations and certain parameters had to be fixed in place to maintain their stability.

2.3.2 Explanation of primary electron escape features

The primary electron escape features (i) arise foremost from the escape of Si KLV Auger

electrons and Si K or L photo-electrons. In the former, the “V” in KLV denotes an L or

M electron from a Si atom. Three distinct regions appear in spectra arising from various

16

0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

S i L P e

N i L V V S i K L V N i M P eN i L P e

S i K P e S iL V V

S i P e & c o i n c i d e n c e S i A u g e r N i c o n t a c t S i m u l a t i o n s u m D a t a

Coun

ts

P h o t o n e n e r g y ( k e V )Figure 2.4: Experimental Ag Lα spectrum showing experimental data overlaid with MonteCarlo simulation predictions of Ag Lα radiation incident on Si bulk, with a 10 nm Ni contact,all normalized to the same primary peak intensity.

combinations of individual particle escape events. These have been sub-categorized (A– C)

as follows.

(A) The Si KLV Auger escape feature (grey line in Figure 2.2) appears as a sloped shelf ex-

tending from the primary peak downward in energy to a maximum cutoff of 𝐸P−1.62

keV. This cutoff corresponds to the maximum kinetic energy of an Si KLV elec-

tron [143]. Its low energy edge is obscured by the high side of the escape peak. Si LVV

Auger electrons also contribute an escape feature to the spectrum that is analogous

to that of KLV Auger electrons [120]. Based on the model predictions (Figures 2.3

– 2.5), the intensity of this contribution appears to be quite small and, due to its low

energy (∼ 90 eV), extends as a very short step down from the primary peak. The

electron spectroscopy results of Papp [118] suggests a greater intensity of this feature.

This discrepancy is likely due to the difference in the electron vs. photon bombard-

ment of the detector materials. As well, the simulation code used relies on the electron

17

0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 1 . 2 5 1 . 5 0 1 . 7 5 2 . 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

N i M P eN i L V V & L P e

S i L P e E s c a p e f r o m A r K a

S i L V V

S i L P e S i A u g e r N i c o n t a c t S i m u l a t i o n s u m D a t a

Coun

ts

P h o t o n e n e r g y ( k e V )Figure 2.5: Experimental Si K X-ray spectrum showing experimental data overlaid withMonte Carlo simulation predictions of Si K X rays incident on Si bulk, with a 10 nm Nicontact, all normalized to the same Kα peak intensity. Contributions from the Si KLV ra-diative Auger lines and the Si Kβ lines were included in the simulation and are distinguishedon the left and right side of the Kα peak, respectively.

transport calculations of Iskef et al. [83] and, due to the simplified treatment of elec-

tron transport, may not be completely reliable at very low energies. The combined

KLV and LVV feature has been fitted as one shelf unit.

(B) The K (Pe) escape shelf (red line in Figure 2.2) extends from the primary peak down

to a maximum fixed energy-independent average cutoff of 1.738 keV (range: 1.715–

1.753 keV). The physical source of this cutoff is the Si K binding energy edge of the

Si atom (1.84 keV). The escaping K Pe will have, at most, the incident photon energy

less the binding energy required to promote it to the continuum. The mismatch of the

shelf cut-off energy with the value of the binding energy is a topic of discussion that

appears later in this chapter.

(C) The longest shelf (orange line in Figure 2.2) corresponds to a combination feature that

has a cutoff value of 0.2 to 0.4 keV. This feature represents two distinct processes: the

18

escape of a Si L Pe and the coincident escape of both a Si K Pe and Si KLV Auger

electron. Both effects contribute to the production of a shelf that extends from the

primary peak down to almost zero energy. In most cases, the shelf is terminated by

the detector’s own low-energy discrimination setting. This shelf also includes the effect

of the escape of Si M Pe’s which is expected to produce a shelf very similar to L Pe

escape, having a cut-off effectively at zero energy. Since its intensity is much lower

than the L Pe shelf, its inclusion has been approximated by the L Pe fit alone and it

is not mentioned further.

In these features a slight downward slope towards higher energies is apparent. This

observation agrees with the predictions from Lowe’s analytical model [104] and Papp’s photo-

electron spectroscopy work [118].

In the Al and Si K X-ray spectra, the features associated with the Si K-shell were not

present. The incident X-ray energies did not possess the energy needed to promote a Si K

electron to the continuum. Features not present are the Si KLV Auger and Si K Pe escape

shelves as well as the Si K X-ray escape peak. Only one of the three DE s mentioned above

was needed to fit the Si L Pe shelf (Figure 2.5). The slight asymmetries on the immediate

low and high energy sides of the primary peak, shown in red in Figure 2.5, are due to model

inclusions of the Si RAE and Kβ features, respectively.

2.3.3 Fit treatment of the contact contributions

The approximate shapes and locations of spectral contributions from the Si(Li) detector

contacts (feature ii) are presented in Figures 2.2 – 2.5. The principal contributions arise

from the excitation of the Ni L and M shells from which Ni L and M Pe’s and Ni LVV Auger

electrons may be emitted incident into the detector. Each feature can deposit its energy in a

continuum fashion, similar to the escape of primary electrons (i) and are thus appropriately

fitted with DE functions. The Ni LVV Auger feature has a fixed energy upper limit less than

1 keV. The Ni L and M features are each fitted with a DE that slopes upward with increasing

energy and has an upper energy limit set as the incident X-ray energy less the approximate

Ni shell binding energy. For the Pe edge of the Ni L-shell, this is 𝐸P − 0.855 keV (dark blue

19

line in Figure 2.2) and for the M-shell, 𝐸P − 0.068 keV (cyan line in Figure 2.2). Two DE

functions, one fitted to each of the Ni L and M Pe locations were used. The convolution

of the DE s to a Gaussian resolution function appropriately models the high energy sides

of each feature. The rounded shape of the top of the feature is apparent in the fit and

supported by the model prediction. The Ni LVV feature had greater intensity in the lower

energy (1–3 keV) spectra and was fitted in these data with an additional, almost flat, DE.

2.3.4 Fit functions applied to the tail

There are well-defined tails (green lines in Figure 2.2) on the low-energy side of the escape

peak (#1 in Figure 2.2) and the primary peak (#2 in Figure 2.2); these were referred

to earlier as feature (iii). The tails on the primary peaks of the Si(Li) spectra were best

described by a short rounded step function having three parameters which were varied in

the fitting procedure: the height, the low-energy cut-off and the Gaussian width of the low-

energy edge. An exponential function sufficed in the fit of (iii) on the escape peak. For

some DSP spectra, an exponential tail was necessary as an empirical solution to fit a small

tail-like distortion on the high-energy side of the primary peak. This feature may be present

due to pile-up with noise that resulted from a slight mis-setting of the pile-up discriminator.

2.3.5 Fit treatment of SDD spectra

While the SDD differs in construction from the traditional Si(Li) device, it is still fundamen-

tally a slab of Si that exhibits most of the bulk effects described thus far. For every primary

peak, an escape peak corresponding to the escape of Si K X-rays appears, as do a series of

shelves and tails on the two peaks that are similar to the Si(Li) spectra. The similarities

and differences between the two detector types have been illustrated in Figures 2.6 and 2.7

for incident Ti Kα rays.

The various primary electrons described as escaping in A, B and C are all present in

the SDD spectra with shelves terminating at the same approximate energy positions. There

are, however, subtle differences to the total low-energy region that deserve mention. The

higher intensity region on the immediate high side of the escape peak is of approximately

20

0 1 2 3 4 51 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

( 2 )

( 1 )

i . A ) S i K L V A u g e r i . B ) S i K P e i . C ) S i L P e & c o i n c i d e n t ( S i K P e , K L V A u g e r ) i i ) N i L P e i i ) N i M P e i i i ) t a i l - e x p o n e n t i a l ( 1 ) a n d

s h o r t - s t e p ( 2 ) t o t a l f i t d a t a

Co

unts

P h o t o n E n e r g y ( k e V )

- 1 50

1 5

Resid

uals

Figure 2.6: Fitted spectrum of Ti Kα X rays collected from Si111 diffraction into Si(Li)Detector 1. Data points and the total fit (𝜒2 = 20.4) are shown in black. Individual fitsto non-Gaussian features have been identified. Gaussian functions that were fitted to theprimary and escape peaks were excluded from this figure to avoid cluttering. A small peakappearing at 1.5 keV corresponds to K X rays of Al emitted by detector collimators viascattering from the primary radiation. Fit residuals (𝜎) are indicated in the top panel.

equal height to the low energy side while a two-fold difference exists in Si(Li) spectra due

to the presence of the Ni contact features. This difference is attributed to the different

construction of the SDD device. In an SDD the traditional metal contact has been replaced

with a B-doped (3× 1019cm−3) Si window having an approximate 10–20 nm thickness [93].

This construction was implemented, in part, to diminish the presence of contact features in

spectra background. Given the appearance of the background in Figure 2.7, this goal appears

to have been largely successful. Due to the apparent absence of contact contributions, only

three DE s have been fitted to the SDD spectra, corresponding to the primary electron escape

features.

Another key difference of the fit treatment applied to the SDD spectra is the functional

form of the primary tail. While a short step function served adequately in the Si(Li) fits, the

21

1 2 3 4 51 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

i . A ) S i K L V A u g e r i . B ) S i K P e i . C ) S i L P e & c o i n c i d e n t ( S i K P e , K L V A u g e r ) i i i ) t a i l - e x p o n e n t i a l t o t a l f i t d a t a

Coun

ts

P h o t o n E n e r g y ( k e V )

- 1 0

0

1 0

Resid

uals

Figure 2.7: Fitted spectrum of Ti Kα X rays collected from Si111 diffraction into the SDD(𝜒2 = 7.95). Data points and the total fit are shown in black. Individual fits to non-Gaussianfeatures present have been identified. Gaussian functions that were fitted to the primaryand escape peaks were excluded to avoid cluttering. Fit residuals (𝜎) are indicated in thetop panel.

tails of the SDD spectra were more appropriately fitted using an exponential function. This

is the same function used to fit the tail on the escape peak. For some spectra possessing

large tails, like those presented in Figures 2.6 and 2.7, the residuals show strong variation

of the fit at the location of the tails. This coincides with rather large reduced-𝜒2 values

(5 < 𝜒2 < 20) presented for the overall fit. The more typically observed reduced-𝜒2 values

range from about 1.5 to 4. At first glance, the higher values suggest that a poor fitting

treatment of the spectra has occurred. However, it is worth noting that the very high

counts acquired in each spectrum are partly responsible for this inflation. The proportional

dependence of 𝜒2 on acquired counts has been demonstrated previously in very high count

spectra [18, 144]. Also, the tails have been fitted with a single empirical function that has

been found to describe the shape satisfactorily. It is quite likely that the physical processes

22

contributing to the tail are complex such that one function may not always be sufficient

when the tail’s intensity is high. This seems to be the case for the two Ti spectra presented

here as the smooth variations of the tail data about the fits may indicate finer structure.

2.3.6 Fitting the escape peak

The well-known escape peak is located at a position corresponding to an energy separation

of 1.74 keV down from the primary peak. Its fit included a Gaussian function containing

six components [21]. Of the six Gaussian components, one function each was fitted to the

Kα lines (1.74 keV), Kβ lines (1.836 keV), KLV RAE (1.61 keV) feature and 3 multivacancy

satellite lines [64]: Kα3 (1.751 keV), Kα4 (1.753 keV) and Kα′ (1.746 keV). These satellites are

produced by an atom possessing a spectator L shell vacancy in the presence of K vacancy

leading to K X-ray emission. Greater detail on the source of this feature is explained in

Section 4.3.2.2.

The intensities of each of the six components were fixed relative to the parent Si Kα peak.

The calculated relative intensities of the Kα3, Kα4 and Kα′ satellite lines are approximately

9 : 5 : 1, respectively, with the Kα3 intensity being 6% of the Si Kα line. [6] The Kβ intensity

was set as 4% and the KLV RAE as 0.9% [21]. The overall intensity of the escape feature

was not linked to that of the primary peak. This was done in part for the reason that

since the peak areas are so high for these spectra, any error associated with a pre-defined

escape-to-parent peak ratio would manifest as a worsened fit on the escape peak. Since one

of the tasks of this work is to acquire high quality fits of the full low-energy region, the

escape peak intensity was made to vary to help achieve this goal.

For some fits of higher energy spectra, the residuals suggested that a slight misfit of the

escape peak had occurred. These peak fits appeared to be offset slightly from the location

of the data; a smooth variation of the fit residuals from +6𝜎 to −6𝜎 indicated this presence.

The misfit occurred despite running the automatic final optimization step of GUFIT. To get

a superior fit of the escape peak, the energy calibration parameters were adjusted manually

prior to running the final optimization step. This indicates that the routine is not sensitive

enough on its own to fit finely the smaller intensity escape peak, in the presence of the high

intensity primary peak. The existence of this issue for higher energies is not surprising given

23

that the escape peak is much smaller due to the increased penetration depth of the incident

radiation.

The result of this manual treatment is best illustrated by the residuals in Figure 2.2

in which a variation not greater than about ±3𝜎 about the escape peak is apparent. The

GUFIT program library for elemental X-ray lines does not contain any description of the

most intense shake satellites (Si Kα3, Kα4 and Kα′) [64] that may appear in the spectra.

The 10–15 eV separation of the satellite lines from the primary Si Kα1,2 lines is enough

to produce a slight asymmetric effect in the peak data. Campbell et al. [26] demonstrated

that Si satellite inclusion in the escape peak fit description results in a better fit of the

escape peak. Therefore, a six-component peak description was added to the spectrum using

a temporary element that included the main Si lines, the shake satellites and the KLV

radiative Auger line.

2.4 Results and discussion

2.4.1 Primary electron escape features

The Si(Li) spectra acquired using the Ti, Cr, Fe and Cu Kα radiation were grouped together

under the criterion that their shelf features are well pronounced in appearance in the spectra.

This allowed for a manageable fit of each shelf. For example, the Si K Pe shelf appears on

the left side of the escape peak and is easily distinguished from each of the other two primary

escape shelves in all four elemental spectra. The task of fitting the Mo and Ag Lα spectra

was more difficult for several reasons. First, at lower incident energies, the K Pe shelf

becomes narrower such that its placement becomes obscured by the numerous features in

the vicinity of the escape to primary peak region. The K Pe shelf in the case of the Mo Lα

spectra is so short that it is obscured by the presence of the short-step (iii) on the primary

peak. Secondly, the K Auger and the coincident escape features have an increased overlap

that makes meaningful separation nearly impossible. Finally, the intensity of the contact

contributions is greater at lower energies whereby superposition of (ii) with (i) results in

further obscuring of the shelf edges. The Ag Lα simulation result (Figure 2.4) suggest that

the intensity of the contact contributions is roughly equivalent to the primary escape shelves

24

for 3 keV photons. This might partly explain why the K edge feature is obscured in the Ag

Lα spectrum. For spectra produced with incident energy less than that of Ag Lα X rays, the

intensity of the contact contributions exceeds that of the primary electron escape features.

The ability to discern the various shelves and apply appropriate fits is therefore exceed-

ingly difficult. A common fitting issue is that one or two fitted DE s may overtake some

of the others during the automatic fit mode, such that the applied background under the

escape peaks may become compromised. Therefore, only the primary escape (i) data in the

energy range of 4–8 keV are reported. For this reason, only the spectra acquired in this

energy range are used in the following chapter, in the determination of the Si ωK parameter.

The energy dependent trends of the intensities of the primary escape shelves, as measured

by the fitting routine, are plotted in Figure 2.8. Also plotted are the intensities of the same

features derived from the Monte Carlo simulation results.

It is important to note that the intensities of all features (i, ii and iii) are independent of

the type of processor. It was expected, in general, that the intensities of all escape features

would decrease with increasing incident photon energy. As energy increases, average inter-

action depth increases, which must reduce the probability of electron escape. In Figure 2.8

the data and Monte Carlo predictions of the Si KLV Auger and combined escape features

(A and C) show this trend. The same trend was expected in the Si K Pe escape (B) plot but

this was found to be nearly flat. This trend might be explained as follows. As the incident

photon energy is increased, the maximum energy that can be imparted to the photo-electron

also increases. Both particles therefore experience an increase to their mean interaction path

distance which likely has a cancelling effect when incident energy is varied.

One further interesting observation was made regarding the Si K Pe features [73]. In each

of the Ti, Cr, Fe and Cu spectra, the K Pe shelf extends downward from the primary peak

to the fixed (average) energy location of 1.738 keV. When compared to the edge location

expected based on the Si K shell binding energy (1.84 keV), the observed location was

substantially less though this has been reported by several other authors. A low energy

cutoff of 1.74–1.75 keV was observed in spectra obtained using both radionuclides [26] and

synchrotron radiation [21]. Lowe [104] also observed a cut-off at 1.74 keV in his spectra. For

25

0 . 0 0 0 5

0 . 0 0 1 0

A ) S i A u g e r

S i ( L i ) + A S P S i ( L i ) + D S P S i L i M C

0 . 0 0 1 0

0 . 0 0 2 0

B ) S i K P e

0 . 0 0 1 0

0 . 0 0 2 0

C ) C o i n c i d e n t

Escap

e feat

ure / p

rimary

peak

area r

atio

5 6 7 8

0 . 0 0 2 0

0 . 0 0 4 0

D ) S u m ( A + B + C )

P h o t o n E n e r g y ( k e V )Figure 2.8: Intensity plots of experimental and modelled Si(Li) primary electron escapeshelves vs incident photon energy where area of the feature is normalized to the area of theprimary peak. The intensity trends for the Si KLV Auger (A), K Pe (B), combined escapefeature (C) and the sum of all three (D) are shown in the four panels.

26

the statistically superior spectra obtained in the present work the edge location agrees with

the earlier work mentioned. Hopman [73] and Hopman et al. [74] demonstrated through

simulations that the most likely cause of this shift is the same effect responsible for the

production of the tail feature (iii). Hopman also suggested that this effect is manifested as a

slight downward shift of the location of the Ni Pe contact features. This effect has also been

observed in these spectra by this author. Additional simulations were performed to assist

in the prediction of the absolute shift of the contact features. These are explained further

in Section 2.4.2.

The coincident escape feature (C) in Figure 2.8 was found to have a slightly higher in-

tensity, when compared against the Monte Carlo predictions. Again, the low energy electron

simplifications of the model may be responsible in part for this discrepancy. As well, the

fitted DE includes the effect of the Ni LVV Auger contribution which may be over-estimated

due to again, model simplification. Papp [118] also observed this effect and attributed it to

a difference in pulse processors. In comparing his own results, he suggested that their digital

processor possessed more refined processing capabilities than their analogue unit and was

therefore able to reject degraded events more effectively. This suggests that the combined

shelf feature from the present data might be displaying effects related to signal processor

noise discrimination, in addition to the physical escape of electrons.

The sum of the first three escape events is given in the final panel (D) in Figure 2.8. An

overall downward trend of the sum of the shelf features is observed which is consistent with

total shelf intensities reported elsewhere [21, 136].

In summary, the empirically chosen DE functions, one fitted to each of the Si KLV, K Pe

and combined Pe regions, provided a satisfactory description of the lineshapes suggested by

Monte Carlo modelling. Lowe [104] predicted the shape of the contact contribution on the

basis of his analytical approximation but his plots did not include convolution of this shape

with a Gaussian resolution function. Allowing for this difference, the agreement between

the description presented here and his is quite good.

27

2.4.2 Contact contributions

Hopman [73] concluded that both of the detectors possessed a Ni contact by observing the

presence of Ni K X rays when these Si(Li) detectors were fluoresced with photons having

energy above the Ni K-shell binding energy. The Ni L3 absorption edge energy is 0.855 keV

and that of the M2,3 edge is 0.068 keV. In the spectra, the contact contributions are expected

to begin at these energy offsets, from the main peak, and extend downward in energy. This

is consistent with the observed spectra given the presence (Figure 2.2) of the mound-like

shape on the immediate high side of the escape peak. The DE functions fitted to the Ni

features provide a reasonable match to the model predictions. The fits shown in Figures 2.2

and 2.6 reveal that their inclusion with the other primary escape shelves conspire to produce

a good fit to the region surrounding the escape peak.

Although the contribution from the Ni M Pe is not very well distinguished in the data,

its presence is in part responsible for the intensity of the region just left of the primary

peak. Its intensity is partially obscured by the primary peak tail and the Si KLV Auger

shelf. The Auger shelf is itself obscured by the presence of the escape peak and contact

contributions. It is thus a formidable challenge to fit these Si(Li) spectra incorporating the

two Ni DE ’s and the Si K Auger DE in such close proximity. The fit treatment by this

author has required close monitoring of the fitting results created in the final automatic

run of the GUFIT program. It was not uncommon for one of the three DE s in this region

to overwhelm the others in intensity, yet still produce an overall decent fit. The resultant

effect in this case is an unphysical fit representation. Any fit results found to exhibit this

behaviour were refitted to acquire appropriate intensities of each DE.

The results of fits to the Ni L and M features are presented in Figure 2.9 as the ratios

of total area to the primary peak. The corresponding Monte Carlo derived intensities are

included for comparison. The general trend of decreasing intensity with increasing incident

photon energy is apparent experimentally and is supported by the model. However, the

intensity of the fitted M DE, in some spectra, is very similar to the intensity of the L DE

while model predictions suggest the former to be less intense by a factor of 2–3 fold. This

difference may be attributed to the previously mentioned difficulty in distinguishing the

28

4 5 6 7 80 . 0

2 . 0

4 . 0

6 . 0

8 . 0

A S P N i L P E D S P N i L P E M C N i L P E A S P N i M P E D S P N i M P E M C N i M P E

Relat

ive In

tensity

( •10

-4 )

P h o t o n e n e r g y ( k e V )Figure 2.9: Experimental and modelled area ratios of the Ni L and M shell DE s relative tothe main peak.

three features in this region during fitting.

The fitting routine also produces a visually satisfactory fit of the Ni M region often by

inflating the intensity of its DE feature to a value similar to that of the Ni L DE. While the

resultant total fit is quite good, caution is warranted given that this fit may be unphysical

based on the modelled predictions. Due to the close proximity of the Ni M feature to the

parent peak, the functions fitted to both the tail and the Ni M feature have a substantial

overlap and competition between the two functions being fitted to this region may have

resulted in the Ni M DE function compensating for some of the tailing. This may be

responsible for the larger-than-predicted fit estimates of the area under the Ni M DE.

The simulations, incorporating a 10 nm Ni contact, show rather good agreement with

the higher energy (4–8 keV) data. This suggests that the contact thickness [25] has been

modelled appropriately and that it was reasonable to not include any DL or ICCL that

may exist. These layers are expected to be very small if present. For these higher energy

photons, this might be expected. However, agreement of the lower energy spectral photon

data with the simulated spectra is somewhat less satisfactory. This is most apparent near

29

the lower energy cutoff region where it is likely that the MC program oversimplifies electron

transport.

2.4.2.1 Evidence for a diffusion effect in the contact features

The initial attempts to fit the DE functions to the two Ni Pe features resulted in the fitted

mounds appearing at energies higher than the location of the experimental data. Close

inspection of Figure 2.3 will reveal the slight horizontal mismatch of the Ni L Pe data

with the Monte Carlo results, an observation that is analogous to the mismatch seen in the

initial fits. All Ni Pe transport must occur close to the front surface, and hence the contact

contributions may be shifted downwards in energy by the processes responsible for (iii).

To estimate the absolute amount of the energy shift, a semi-quantitative method was

implemented on the Detector 1 Fe Kα spectrum. Spectra were simulated by Hopman [73],

using a 10 nm Ni contact, with a series of different reflection coefficient (𝐶R) values; these

were generated using the Goto model [63]. The effect on the K region of the simulated

spectra are shown in Figure 2.10. There is a clear trend in the downward shift of the Pe

lower edge energy with the decreasing reflection coefficient. Incidentally, the effect also

manifests as a tail on the escape peak seen in the figure. This suggests that the edge feature

can be used to assess the combined diffusion and reflection process. The regions around the

simulated Si K Pe edge were fitted by Hopman with an error function of the form

𝑓 (𝐸) = 𝐴 · 𝑒𝑟𝑓 [(𝐸 − 𝐸0) /𝑠] + 𝑦0 (2.3)

Edge energy values, interpolated from the center height of the error function, were plotted

against the 𝐶R’s used in the simulation; the resulting linear relationship was plotted in

Figure 2.11. The Pe edge location corresponding to the analysed Fe Kα spectrum (1.73

keV) was used in this study to interpolate a 𝐶R value of 0.92, corresponding to the detector

used (Detector 1 in this case). This coefficient was used as an input to a new Goto diffusion

model simulation for Fe Kα X rays. The simulation results were overlaid in Figure 2.12 onto

the same Fe Kα data displayed in Figures 2.2 and 2.3. The modelled locations of the Si K Pe

edge and Ni L Pe mound in Figure 2.12 show better visible agreement with the data using

30

1 2 3 4 51 0 2

1 0 3

1 0 4

Coun

ts

P h o t o n e n e r g y ( k e V )

N o d i f f u s i o nC R = 0 . 9 0C R = 0 . 8 0

Figure 2.10: Simulated Fe Kα spectra showing the Si K Pe edges and escape peaks usingGoto [63] diffusion 𝐶R’s of 1.0, 0.9, 0.8.

this model. The new modelled cut-off location of the Ni M Pe high energy edge was shifted

by about 150 eV, down from its original location. The empirically chosen shift values of

100±50 eV, for both Ni L and M DE functions, were less than predicted but still permitted

satisfactory fits to this region. The GUFIT program permits the degree of curvature of the

high energy side of the DE functions to vary (slope parameter S2) as needed. If the chosen

fit energy shift is off by several tens of eV then this parameter may be adjusted to smooth

the function into a satisfactory fit.

The connection of the diffusion process with the downward shifting of the escape and

contact features does not appear to have been reported in the literature. Given the linear

relationship apparent in Figure 2.11, it would be worthwhile to monitor how different detec-

tors, having different reflection coefficients, respond in the production of this downward shift

effect. These spectra show us that the processes responsible for (iii) have the effect of both

31

0 . 7 5 0 . 8 0 0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 01 . 4 51 . 5 01 . 5 51 . 6 01 . 6 51 . 7 01 . 7 51 . 8 01 . 8 51 . 9 0

Pe ed

ge loc

ation

(keV

)

R e f l e c t i o n c o e f f i c i e n t ( C R )Figure 2.11: Si K Pe edge locations from simulated Fe Kα spectra plotted against 𝐶R valuesin the range of 0.7 to 1.0. Linear fit: 𝐸𝑑𝑔𝑒(keV) = (1.476 ± 0.022)𝐶R +(0.36502 ± 0.019),𝑅2 = 0.9996.

producing a significant part of the tail on the primary and escape peaks and, causing the

downard shifts of the contact features and shelf edges. The immediate shelf edge represents

those electrons that have escaped without undergoing any collisional energy losses ie. those

that have been emitted from the immediate detector boundary. If the downward extension

is due to diffusional losses then these would be expected to maximize at the boundary.

Since the contact is inherently at the boundary, the argument supports the effect of diffu-

sion losses in the contact features resulting in this shifting effect. There is some degree of

uncertainty associated with the semi-empirical process used given that the DE function is

only an approximate description of the true contact contribution. As well, the Goto model

is an approximation that may fail to account for the full details of the effects contributing

to the tail.

For these reasons, the conclusions made from this procedure are largely qualitative. The

diffusion effect appears to be responsible for the downward energy shift of both the observed

32

0 1 2 3 4 5 6 71 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

1 0 6

1 0 7

N i M P e

N i L P e

N i L V V

S i L P e S i K P e

S i K L V S i L V V

S i P e & c o i n c i d e n c e S i A u g e r N i c o n t a c t S i m u l a t i o n s u m D a t a

Coun

ts

P h o t o n e n e r g y ( k e V )Figure 2.12: Fe Kα spectrum from Figure 2.3 overlaid with simulated spectrum using a Gotoreflection coefficient of 0.92.

primary escape shelves and contact contributions. Additionally, the predicted downward

shift of the Ni Pe’s appear to agree quite well with the empirically chosen locations of the

shelves. Furthermore, these effects are expected to manifest in these features given that each

represents interactions occurring at the contact boundary, where the process(es) responsible

for the production of (iii) will have maximum effect.

2.4.3 Peak tailing

Si(Li) spectra in the 1–8 keV energy range have been fitted with the assistance of model

predictions. For all spectra, the tail is not predicted by the contributions from (i) and (ii)

mentioned in Section 2.1. The intensity ratio of the tail, relative to its parent peak, is

shown as a function of the incident X-ray energy in Figure 2.13. The data show a sharp

rise at the location of the Si K binding energy which is expected given the sudden increase

in attenuation of photons (>1.84 keV) by the Si K shell. The data on both sides of the

33

1 2 3 4 5 6 7 8 90 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

Area

intens

ity ra

tio

P h o t o n e n e r g y ( k e V )

A S P D S P

Figure 2.13: Ratio of area of the short step diffusion feature to area of the first order peakusing both processors and Detector 1 spectra.

edge show a decrease in intensity, with increasing photon energy. This is consistent with

the decrease in escape probability due to increased depth of average interactions found to

occur using higher energy photons. This trend has been reported extensively by others. The

relative intensities seen in Figure 2.13 match closely with the earlier results of Campbell et

al. [21]. As indicated above, this feature is frequently attributed to diffusion of secondary

electrons from the bulk into the contact. These data show that there is no apparent effect by

the choice of processor. The possibility of using the plotted relative intensities in a fixed fit

prescription of spectra, acquired using other detectors, is unlikely given that the tail shape

and size are features that are usually unique to the specific detector used. However, the

similarity of this plot, with previous plots of the same feature, indicate that the spectral

processing settings of the electronics were optimized for the Si(Li) data.

In previous lineshapes studies, excessive amounts of tailing on the primary peak have

been linked to improper settings used with the pulse processing electronics, effects that are

entirely separate from physics interactions occurring in the detector crystal. If spectra are

34

acquired with shaping times that are set too short to collect all electron-hole charge carriers,

then excessive tailing on both the low and high side can manifest on the primary peak [118,

121, 122]. The processing time was optimized in this experiment to eliminate it as a possible

source of the tailing. The remaining tail may be attributed principally to physical processes

in the detector. The simulations presented in Figures 2.3 – 2.5 incorporated primary electron

escape and contact electron injection into the crystal but does not include the diffusion and

reflection effect. This was done to emphasize that the source of the tail feature is independent

of the two other processes.

2.5 Further discussion on the tail feature

2.5.1 Comparison of the tailing among different detectors

Spectra measured with different detector types have shown wide variation in the tail fea-

ture. Various works have attempted to assign a source to the immediate tailing on the main

peak. However, it is important not to confuse those tail effects that may not be distin-

guished from the effects of primary electron escape and contact injection. Papp et al. [120]

identified numerous early works that report the tailing feature as extending from 80% to

below 50% of the energy of the primary peaks. However, these early works do not clearly

identify contributions from the various primary electron escape and contact electron injec-

tion features discussed in this work. For example, in their recent work, Eggert et al. [51]

successfully analytically modelled the tailing seen to extend down to half the energy of the

primary peak, in the case of an Al contact SDD. Their approach assumed the presence of a

layer of incomplete charge collection and focused on very low energy incident photons (<2

keV). In their approach, they did not separate the events corresponding to features (i) and

(iii). However, the strong agreement of their method with their data does support that a

rather complex approach is required to effectively model the overall tailing. The Guelph

MC treatment does not allow for ICCL effects and likely lacks the sensitivity to effectively

model the full picture produced at very low energies, for all three features (i, ii and iii).

The slightly worsened agreement of the Si K X-ray Monte Carlo results with the data in

Figure 2.5 demonstrate that a full representation of the processes responsible for (iii) have

35

not yet been realized. This figure demonstrates that the modelled effects from sources of (i)

and (ii) break down slightly in the production of lower energy spectra.

Returning to the higher energy spectra seen in this work, the tail feature manifests

as the short step that extends down to about 90–95% of the primary peak energy. It is

interesting that HPGe detectors with Ni contacts appear to display little to no tailing. The

work of Campbell et al. [21] compared Si(Li) and HPGe detector spectra at 2, 5 and 8.3

keV. They used lower count spectra than that used in this work but their comparison clearly

demonstrated that the tailing is much smaller for HPGe detectors.

2.5.1.1 Influence of the Si-contact work-function

Consideration of the contact used in each detector’s construction helps, in part, to explain

this detector-specific difference [73]. A typical semiconductor detector design incorporates

a reverse-biased diode with a contact material that produces either an Ohmic or Schottky

juction with the Si bulk. An Ohmic junction has an energy potential at its interface that

is low such that it permits most electrons from the Si conduction band to flow from the Si

into the contact. A Schottky junction possesses a higher potential boundary that restricts

this flow to only higher energy electrons. The potential boundary is proportional to the

work-function (Φ𝑝+) that exists between the boundary materials. Calculation of Φ𝑝+ for a

multitude of materials is possible [53] and has assisted in the discussion below.

The Si(Li) and HPGe detectors studied by Campbell et al. [21] possessed Ni contacts.

The Φ𝑝+’s of Ni with Si and Ni with Ge were found to be 0.45 and 1.32 eV, respectively [53].

The higher potential barrier in the case of the HPGe detector implies that fewer secondary

electrons are able to escape and further suggests that they must be reflected at this boundary.

The reduction of secondary electron losses, explained by this argument, is consistent with

the reduced tailing seen in HPGe spectra. Other detectors have been constructed with

materials having very low Φ𝑝+ values. Eggert et al. [51] describe the interface between the

Al contact and the bulk Si in their SDD as Ohmic; here Φ𝑝+ = 0.20 eV. For very low energy

photons, the tailing in their spectra was quite large, as might be expected given the low

inherent value of the work-function. The commonly produced Au contact Si(Li) detector

also possesses a low value of 0.25 eV while the newer SDD devices, having a boron-doped

36

window of Si, possess an approximate 1.13 eV work-function. On the latter, the potential

to see a reduction in tailing when compared against the Ni contact Si(Li) devices is evident.

It appears that appropriate choice of contact can greatly reduce the diffusion loss fea-

ture. A relatively small increase in work-function results in a large reduction in diffusion

losses. The tail produced by the simulation using a Goto reflection coefficient of 0.92, seen

in Figure 2.12, is still smaller in intensity than seen in the data. This is not surprising given

the simplicity of the model. The model does not account for discrete losses and reflection

of individual electrons at the contacts, as would occur in a physical situation. This model

uses a mathematical formalism in which charge loss is calculated using an equation incorpo-

rating a decaying exponential that is a function of depth.In principle, this model would also

approximate an effect of e–h recombination at the boundary rather than that of secondary

electron out-diffusion as it does not distinguish the mechanism of (iii). A more complex

means of modelling the diffusion effect would be required to improve our understanding of

the tail. Still, our Si K Pe edge shift results show some support in favour of the Goto model

of charge loss. This, coupled with the potential barrier explanation presented above, seems

to support the concept of out-diffusion and reflection as the most likely source of (iii).

2.5.2 Si K X-ray escape peak

The primary focus of this work has been to review and identify the sources of the numerous

non-Gaussian features found in the low energy side of the primary peak. Little mention has

been made so far of the well-known Si K X-ray escape peak that appears 1.74 keV below the

primary. In Chapter 3 the derivation of a new and highly accurate value for the Si K-shell

fluorescence yield based upon the escape peak intensity is reported. Here we merely note that

previous measurements of this quantity, which ignored the presence of contact contributions

and the Si KLV feature underlying the escape peak, are subject to error. The variation

among detectors of the contact contributions and the electron escape shelves, together with

the earlier neglect of the escape peak satellites, are likely sources of significant variation in

escape peak relative intensities that have been reported in the literature and demonstrated

in ref. [119].

37

2.6 Conclusion

The work described here shows that most of the non-Gaussian features of X-ray lineshapes

in Si(Li) detectors can be attributed to specific physical processes. Monte Carlo modelling

and recent literature [104, 118, 120, 135] have provided assistance in delineating specific

features resulting from primary electron escape (i), metal contact electron injection (ii) and

the intense tailing (iii). The latter is not explained by an influence from either (i) or (ii).

The use of the Goto approximation of charge loss and reflection in the model has provided

some further agreement with the data. The Goto model was of assistance in presenting

evidence that a downward shift effect of the Si K Pe edge and contact contributions arises

from the same mechanism responsible for the tail (iii). While it alone does not allow further

distinguishing of the exact processes responsible for the tail, the work-function discussion

regarding detector contact selection lends credence to the process of secondary electron

out-diffusion and reflection as the most likely source behind its production.

Trends in the total relative fitted areas of the various features discussed (i, ii and iii)

have been presented for their potential future use in deriving an automated fit function that

may be applied to the entire low-energy region of peaks in a multipeak spectrum. While

agreement of data with Monte Carlo predictions is rather good in the higher energy range

(4–8 keV), caution is warranted at lower energies due to the reduced agreement likely caused

by the oversimplification of the model. The model predictions indicate that the Monte Carlo

method is not ready for creating reproducible models of the spectra for fast, automatic fits

applied to more complex spectra. The Monte Carlo routine lacks the sensitivity at lower

energies and cannot be used to appropriately model the tail.

The fits produced in this work were accomplished by making use of all knowledge of

lineshape specifics available in the literature. The final product has undergone one of the

most sophisticated treatments to date. The fits applied to data in the 4–8 keV range are

considered to be sufficiently accurate to warrant their use in Chapter 3 in deriving a new

estimate of the K X-ray fluorescence yield of Si.

38

Chapter 3

An accurate determination of theK-shell X-ray fluorescence yield ofsilicon

3.1 Introduction

The increasing demand of a FP approach to materials analysis, when calibration standards

are not available, drives the development of FP tabulations. Material analysis performed us-

ing characteristic X-ray emission requires a number of atomic FPs to convert the X-ray peak

area to an estimate of composition. ωK is one of these parameters. This chapter provides

an overview of the current status of available ωK databases with particular focus in the light

element region. A new ωK measurement of one of the light elements is presented. Details of

this analysis and the results are discussed by comparing them to earlier measurements of the

same parameter. Finally, some of the shortcomings in the available theoretical predictions

of light elements ωK’s are reviewed.

The experimental ωK databases of Krause [97] and Bambynek [4] are most frequently

cited for use in X-ray emission analysis. Both authors implement a unique selection criterion

for data that they consider most accurate, as mentioned in Chapter 1. Using selected data,

they applied empirical fits to interpolate values for each element. Since then, Hubbell et

al. [78, 79] have published compilations adopting the tabulations of Bambynek, over Krause,

along with additional measurements obtained since Bambynek’s most recent (1984) work.

This conclusion was also reached by Schönfeld and Janβen in their atomic FP compila-

tion [137, 138].

39

More recently, an exhaustive dataset [90] was published presenting average ωK values for

elements in the 6 6 Z 6 99 range. For their ωK interpolations, the authors incorporated

numerous estimates of the ωK measurements found throughout the literature. Their fit did

not involve any pre-selection to data that might be considered higher in quality. They

compared their data to the Bambynek and Krause tabulations and found that for light

elements, their interpolated ωK estimates tended to be much lower. This result was not

surprising given that older light element data were prone to high levels of error. It is likely

that the incorporation of the older data in their compilation has compromised the accuracy

of their approach. For this reason, this author believes it is appropriate to revert to making

a choice to adopt one of the older databases, those of Krause or Bambynek.

The publication of Chen et al. [35] is perhaps the most reliable theoretical compilation

of ωK for reasons that will be elaborated upon later in this work. The values of Bambynek

show rather strong agreement with those of Chen et al. computed using Dirac-Hartree-Slater

(DHS) formalism. In the 20 < 𝑍 < 30 region, this agreement is within 1%, and at higher Z,

the agreement improves to within 0.25%. At first glance, this appears to support adopting

the Bambynek tabulation. However, two issues remain in validating database accuracy.

The first issue is the smoothly varying (−7% to +5%) discrepancy between the Krause and

Bambynek tabulations for light elements, mentioned in Chapter 1. The recent, carefully

performed study of Ménesguen et al. [110], in the atomic number range 22 6 𝑍 6 30, has

demonstrated an average difference of their ωK measurements to Krause and Bambynek of

3.5 and 0.6 %, respectively. This study shows strong support in favour of Bambynek and

appears to resolve the matter of choice between the two tabulations for Z > 20.

The second issue is the scarcity of modern experimental data for light elements, partic-

ularly in the region Z 6 20. This issue is further complicated by lack of accuracy in the

existing data. Most of these data were derived decades ago using now antiquated techniques

and should not be considered absolutely reliable. Unfortunately, Chen et al. did not provide

DHS predictions for Z < 18. However, two other DHS theoretical compilations provide an

extension of the ωK data into the light element region. The first is that of McGuire [108,

109] whose method incorporates non-relativistic wavefunctions and the Herman and Skill-

man potential. The second is that of Walters and Bhalla [164], which uses the Kohn-Sham

40

and Gaspar potential approximation. The Chen and McGuire theoretical values of ωK for

elements 10 6 Z 6 20 are presented in Figure 3.1 as ratios with respect to the corresponding

values of Walters and Bhalla. This figure also shows a selection of experimental ωK values,

the details of which will be covered more extensively in Section 3.2.

In Figure 3.1, the Chen et al. [35] data are seen to overlap with the Walters and Bhalla

data only for Z > 18. In this small region, the Chen data agree with Walters and Bhalla to

within 0.5% but are 3–4% less than those of McGuire. Based on this observation, it seems

probable that a downward atomic number extension of the Chen ωK calculations would find

further agreement with Walters and Bhalla numbers.

All three of the Chen, McGuire and Walters and Bhalla theoretical predictions were

formulated using the model of an isolated atom. A practical representation of this model

would be an atom in a neutral gas such as an atom of unbonded Ar. Krause stressed

that the values in condensed matter may differ from those in the isolated case as chemical

bonding of an element to its environment may impact the experimentally derived values of

ωK. However, the magnitude of this effect is not known. The issue of chemical bonding will

be discussed further in Section 3.2.1.

3.2 Previous experimental work

Generally, experimental techniques that have minimal reliance upon other FPs in the calcu-

lation of ωK are ideal. The use of fewer FPs helps to minimize the total uncertainty on the

final result. Many existing measurements do not satisfy or only partially satisfy this crite-

rion. Additionally, many measurements have been calculated long ago using now-obsolete

techniques and are of questionable validity. For this reason, only those ωK measurements

produced since the late 1960’s have been reviewed here.

These reviewed ωK measurements are presented in Figure 3.1 [3, 8, 14, 20, 46, 54, 116,

117, 153] and have been categorized according to the experimental approach used in their

calculation. Each value is presented as a ratio relative to the theoretical ωK predictions of

Walters and Bhalla [164]. Measurements lower than Z = 10 have not been included since only

a few values exist and their associated uncertainties are very high. The ωK measurements in

41

1 0 1 2 1 4 1 6 1 8 2 0

0 . 6

0 . 8

1 . 0

1 . 2

Ratio

of ω K

E l e m e n t ( Z )Figure 3.1: Experimental and theoretical [35, 108] values of ωK expressed as ratios ofthe theoretical predictions of Walters and Bhalla [164]. Circles represent comparisonsbetween theories. Upward pointing triangles indicate thick and thin target fluorescencedata (Approaches 1 and 3), downward pointing triangles indicate escape data (Approach2) and squares represent electron capture data (Approach 4). Open symbols indicate caseswhere the target elements were incorporated within a molecule while solid colour symbolsreflect pure element forms. References are identified in the legend below.

∙ McGuire [108] (1970)∙ Chen et al. [35] (1980)� Espenschied and Hoffman [54] (1978)N Bailey and Swedlund [3] (1967)N Solé et al. [153] (1993)N Dick and Lucas [46] (1970)N Beckhoff [8] (2008)M Bailey and Swedlund [3] (1967)H Brunner [14] (1987)H Campbell et al. [20] (1998)H Pahor et al. [117] (1972)O Pahor et al. [116] (1968)O Pahor et al. [117] (1972)

42

Figure 3.1 were each derived from one of the four principal methods described below. In the

figure, the experimental values are further distinguished according to whether the element

was found in a pure-form target or incorporated in a chemical compound.

The first and perhaps most straightforward approach involves measuring the character-

istic X-ray yield from a target containing the element of interest. Excitation of the charac-

teristic lines is most commonly achieved from exposure to an X-ray excitation source. This

method relies upon the theoretical photo-ionization cross-sections of the incident energies,

among many other parameters, to deduce a value for ωK. The reliance on this parameter

is a weakness of this method. Furthermore, calibration of the X-ray detection efficiency re-

mains a significant challenge for the low energy photons emitted by light atoms. A common

practice is to create a calibration curve using multiple radionuclide standards. However,

most radionuclide standards emit photons having energy higher than those associated with

light element K X-ray emission. This necessitates extrapolation of the efficiency curve to

the lower energy region which may introduce substantial error. Considering the complex

nature of spectra produced by low energy photons, as reviewed in Chapter 1, this approach

may not be justified.

Also within this approach, several ωK studies were performed using charged particles

instead of photons as a means of ionization. This method requires the use of theoretical

tabulations to extract values of the ionization cross-sections of these particles. In some ωK

works, older unrefined values of charged particle cross-sections were used. Inclusion of these

works in the initial recommended dataset by Hubbell [78] was found to introduce significant

divergence from the earlier Bambynek database which was based only on photo-ionization

methods. The charged particle experimental ωK values were later withdrawn by Hubbell

for this reason [79]. The dependence upon charged particle ionization cross-section FPs is a

clear weakness in this approach.

The second approach involves photo-ionization of an element where the ratio of the

element escape peak to primary peak, measured from an acquired spectrum, is used in the

analysis. In this setup, the element under study is incorporated as a component of the

detector’s sensitive region. This element may be found in the detector gas, as used in a

multiwire proportional counter or the primary constituent of the sensitive bulk, as found in

43

a semi-conductor detector. Radiation incident upon the detector interacts via the photo-

electric effect such that full absorption corresponds to the formation of a spectral primary

peak. Some characteristic X rays are created by the medium and have a probability of

escaping the detector. The occurrence of these events corresponds to a loss of energy in

the spectrum, which produces an escape peak at lower energy relative to the primary peak.

Through geometrical considerations, the ratio of the escape to the primary peak can be used

to derive ωK, as outlined in Chapter 2. A necessary step in this process is to assign a value

to the ratio of the MACs of the incident photons to the escaping X-rays within the detector

medium. One benefit to the use of a ratio of coefficients is that systematic uncertainties in

the MAC database might cancel out. The uncertainty in the ratio is therefore less than that

associated with the individual coefficient. A second benefit to this approach is that detector

efficiency is not a factor in analysis. This approach is therefore slightly superior to the first

approach.

The third approach involves fluorescence measurements of thin foil targets. The basic

approach is similar to the first group with a few exceptions. Here the intensities of the

incident and transmitted beams are measured. As well, the emitted characteristic X rays

are measured, usually from a position that is at an angle with respect to the path of the

beam. The thin foil method allows for calculation of the number of interactions that occur

in the foil. This calculation eliminates the reliance upon some FPs. The ωK parameter is de-

duced using the ratio of the characteristic X-ray intensity to the number of foil interactions.

With this method, detector efficiency corrections are necessary. Some earlier studies imple-

mented quasi-monochromatic beams created using filters in front of polychromatic radiation.

However, this method has become obsolete with the advent of synchrotron radiation [8, 9].

Again, reliance upon detector efficiency corrrection is a clear drawback to this approach.

The fourth approach involves the measurement of characteristic X rays emitted by the

electron-capture decay of a radionuclide. The element of interest is introduced, as a gas,

into a multiwire ‘wall-less’ proportional counter, a device that prevents escaping X-rays and

electrons from being registered [47]. In this approach, two independent measurements are

made of the gas radionuclide under both low and high pressure. Under low pressure, only the

K Auger spectrum is registered as almost all X-rays escape the active volume. This measures

44

the K Auger emission count rate of the gas. The second measurement, under high pressure,

measures the total (K Auger plus K X-ray) count rate. Subtracting the first rate from the

second yields the total X-ray intensity from which ωK is derived. With this device, using

the ratio of the two measurements, it is possible to eliminate reliance on several FPs [54,

58] including: the detector volume and the radionuclide total decay and K-capture rates.

One limitation with this method is the large uncertainty that arises from incomplete charge

collection. There is also a small amount of X-ray contamination seen in the Auger-only

spectra.

In a comparison of measurement approach accuracy, it is useful to compare the uncer-

tainties associated with each measurement reported in Figure 3.1. The mean uncertainty

reported for the escape peak measurements (approach 2) was found to be ±3.6%, whereas

the corresponding value for the fluorescence methods (approaches 1 and 3) is ±7%. The

increase in the latter estimate may be attributed to the reliance on detection efficiency cor-

rections. The error is smaller in the escape approach since efficiency corrections are not

required.

It appears that some of the uncertainties assigned to earlier measurements may under-

estimate the true level of possible error. The recent work of Beckhoff [8] presented a new

ωK measurement of Al using a thin foil method. In his sophisticated analysis, he quoted an

uncertainty of ±10% which exceeds most of the earlier reported uncertainties. This suggests

that the earlier uncertainties should be larger. The ωK measurement of Mg in Figure 3.1,

reported by Dick and Lucas, provides an uncertainty range that is perhaps a more accurate

depiction of the experimental error associated with light element experiments. However,

their Mg ωK value is substantially less than the theoretical predictions of McGuire and

Walters and Bhalla, suggesting that this measurement may be inaccurate.

Overall, the experimental values support the predictions of Walters and Bhalla. The

error-weighted mean value of the ωK ratios, in the range 13 6 𝑍 6 20, is 0.980 ± 0.018

(standard error of the mean). As mentioned earlier, the Walters and Bhalla work used

the Kohn-Sham and Gaspar exchange potential. This is a model that has seen repeated

use since the 1970’s, and was utilized in the development of the FFAST MAC database by

Chantler [31–33]. A more in-depth discussion on this model will take place in Chapter 4.

45

3.2.1 Chemical effects on K X-ray fluorescence yields

A relatively large amount of literature has been dedicated to the investigation of chemical

effects on the ωK parameter. These include effects produced by multiple vacancy production

during ionization and dependence on chemical bond environment. Statistical consideration

of the multiple vacancy production was calculated for the Ar atom by Larkins [100]. His

work demonstrated that removal of additional L-shell electrons prior to the de-excitation

of the atom can lead to a reduction of the Auger probability and therefore an increase

in the X-ray emission probability. This effect has been observed in experiments involving

light elements bombarded with heavy, charged ions. Multiple vacancy production, in these

atoms, was found to increase the emission probability of satellite X rays, which represents

an increase in ωK [48].

The issue of chemical bonding of elements in pure form, relative to elemental compounds,

has been examined for its influence on ωK. Hartmann [67] demonstrated, through theoretical

considerations, that differences in the local electron density distribution of a light atom

participating in several bond configurations can affect ωK. His detailed theoretical treatment

gives a ωK range of 0.00215–0.00297 for a set of six carbon compounds; the Walters and

Bhalla elemental value is 0.0024. While this one calculation suggested that large differences

may exist due to bonding, it is not easy to extrapolate this effect in slightly heavier elements

such as those in the range 10 6 𝑍 6 20. For one, the carbon atom valence bonding involves

L-shell electrons while for the elements discussed in this chapter, the M-shell is involved.

Therefore, the effects may not be quite as prominent. Still, this bonding effect was one

suggestion presented to explain the Campbell et al. [22] PIXE discrepancies between light

elements bonded in pure form vs oxide form.

Given the possibility of chemical effects on ωK, it is reasonable to infer that the most

accurate test of the isolated atom theories will be done using measurements performed on

pure elements. In Figure 3.1, the experimental data from pure elements do not show any

significant difference from data derived using elemental compounds. It is most likely that the

data presented do not possess the necessary accuracy to indicate this effect. More recently,

large chemical effects on ωK have been attributed to element bonding in various compounds.

46

In one study, measurements performed on 13 K compounds and 6 Ca compounds revealed

ωK values, normalized to the Walters and Bhalla predictions, that span ranges of 0.78–1.18

and 0.97–1.28, respectively [99]. Another study revealed a similar large discrepancy of 0.08-

–0.33 and 0.09–0.44 for absolute values of ωK for K and Ca compounds [158]. In light of the

earlier discussion of problems associated with fluorescence measurements, caution should be

exercised in interpreting these results. These studies utilized a fluorescence method that is

likely not sensitive enough to identify chemical effects on the order of a few percent. Still,

the discrepancies found suggest that there is merit to the reproduction of their experiment,

using more sensitive techniques.

3.3 Present objectives

The experimental ωK values, presented in Figure 3.1, do not suggest the presence of chemi-

cal effects, even on the order of a few percent. The error estimates attached to some of the

values are large and yet, many have likely been underestimated. Overall, the experimental

data appear to agree quite well with the Walters and Bhalla theoretical predictions and less

well with the McGuire values. If the Walters and Bhalla database were to be validated with

a higher degree of confidence, then it may be incorporated into an X-ray fundamental pa-

rameters database. This would be highly useful in providing users with highly accurate ωK

values for the complete set of elements in the low energy region. A stringent test of the Wal-

ters and Bhalla theory might be performed using a new, carefully conducted measurement

of an element in this region.

The escape peak approach has the greatest potential for high accuracy measurements.

This approach has a low dependence on FPs and does not require correction for detector

efficiency effects. The two measurements of Si ωK [14, 20] referenced in Figure 3.1, utilized

the Si(Li) detectors in the escape approach and produced results that differed by 7%. The

large discrepancy suggests that one or more systematic errors may be responsible. The

earlier of these works, that of Brunner [14], was performed using an 55Fe radionuclide source

that emits the full spectrum of Mn K X-ray lines (Kα, Kβ, shake satellites and radiative

Auger photons). This gave rise to rather complex spectra and background underlying the

47

escape peak. The escape peak in this work had a 4 : 1 peak-to-background ratio and the

background treatment involved fitting only a simple polynomial function.

In the later work of Campbell et al. [20] the experimental approach of Brunner was sim-

plified by implementing monochromatic synchrotron radiation instead of radionuclides. This

was done to eliminate all subsidiary emission features present in the spectra of the earlier

source-based method. Also, Campbell et al.’s peak-to-background ratio was greater than

20:1 which implied that the spectral quality had been improved. These factors suggested

greater accuracy in results but their work still did not account for the various physical pro-

cesses that occur in the detector to culminate in the production of the background under the

escape peak. The various electron escape, contact and tail features, that were the subject of

Chapter 2, were treated in only a rudimentary fashion; most of the extended features were

fitted with only flat step functions.

Further concerns with the second approach were expressed in a work [119] that compared

escape peak intensities found using different Si(Li) detectors under the same incident energy.

A spread of peak to primary ratios of about ±15% was observed. This would translate

directly into a ±15% spread in measured ωK values which represents a very large amount

of uncertainty. Additionally, there is a need to assume a ratio of attenuation coefficients in

the calculation of ωK using this method. With insufficient reliable data on Si in the lower

energy region (1–10 keV), the selection of any one theory could not be critically determined.

The last study utilized the two prominent theoretical databases: XCOM and FFAST and

arrived at two estimates of Si ωK that differed by 4%. This effect also doubled their final

uncertainty value.

A new measurement of Si ωK is warranted given that new techniques and complementary

data now exist that may be used to address these concerns directly. In particular, the

lineshapes information presented in the previous chapter might be used to fit the background

underlying the escape peak with a higher level of accuracy. It is hoped that this might

prevent the high level of peak area variation, observed in earlier studies [119], from being

repeated here. As well, new MAC data for Si in the pertinent region now exist and can be

used to put the two databases (XCOM and FFAST) to test. The goal here is ultimately to

find an accurate test of the Walters and Bhalla predictions in the Z < 20 region. This will

48

dθ

xIncident Photon

dx

Si K X ray

Detector

dφ

φθ

Figure 3.2: Detector geometry considerations used to derive an expression that relates SiωK to the ratio of escaping Si K X rays to incoming photon beam.

be accomplished by utilizing the fits produced in Chapter 2 as well as new Si MAC data [75,

160].

3.4 Basis of the escape peak measurement

Previous discussions of escape peak intensities tend to assume that the detector is very thick,

with all incident interactions absorbed in the detector and that escape of Si K X-rays is only

possible from the front face [127, 154]. The physical geometry for perpendicular incidence

on an infinitely thick detector, leading to the production and escape of a Si K X-ray, is

illustrated in Figure 3.2. In calculating the probability of the escape of a Si K X ray, the

sides of the detector wafer are assumed to extend to infinity in all directions. The limits of

integration are 𝑥 = 0 . . .∞, θ = −𝜋2 . . .

𝜋2 and φ = 0 . . . 2𝜋.

For the Si ωK experiments mentioned previously, infinite thickness is a safe assumption

as most Si(Li) detectors have wafers of thickness that may be approximated as infinite

below 10 keV. However, the more recent development of SDDs has put this assumption into

question. A typical SDD incorporates a much thinner wafer than that found in most Si(Li)

49

detectors which implies that backface photon transmission and Si K X-ray escape need to be

considered. Dyson [49] published a formula that was derived to account for the possibility

of both front and back face escape of the detector’s characteristic X-rays. His expression

was used to determine if such considerations are needed for corrections in this experiment.

For the most energetic photons (Cu Kα: 8.041 keV) in the thinnest detector (SDD: 0.45

mm thick), the contribution to the escape ratio of the backface transmission was found to

be about 0.01%. This was considered to be small enough to justify reverting to the simple

geometric analysis that assumes infinite thickness.

The escape peak intensity ratio (𝜂) is defined as the ratio of the escape peak intensity

(𝐼esc) to the combined intensities of the primary (𝐼inc) and escape peaks. Under the as-

sumption that the detector is an infinitely thick slab of Si, a geometrical calculation for

perpendicular incidence produces an expression (Equation 3.1) that relates 𝜂 to the Si ωK

parameter. This expression is:

𝜂 =𝐼esc

𝐼inc + 𝐼esc=

1

2ωK

(︂𝜎KPE

𝜎PE

)︂[︂1−𝐴 ln

(︂1 +

1

𝐴

)︂]︂(3.1)

where:

ωK is the K X-ray fluorescence yield

𝜎KPE is the Si K-shell photo-electric cross-section for the incident radiation

𝜎PE is the total (𝐾 +𝐿+𝑀 + ...) photo-electric cross-section for the incident radiation

𝐴 is the MAC ratio of Si K X rays in Si ((𝜇/𝜌)Si Kα) to that of the incident anode(Ti, Cr, Fe and Cu Kα) X rays in Si ((𝜇/𝜌)anode Kα

).

This equation relies on only the physics interactions occurring in the detector crystal

upon entry of an incident photon. It is therefore not necessary to correct for detector

efficiency or to consider the systematics of the X-ray device responsible of production of

the incident X rays. Equation 3.1 demonstrates that only a few FPs are relevant in the

derivation of ωK, ie. the ratio of photo-ionization cross-sections and the MACs of Si at

several energies.

50

3.5 Escape peak analysis

Incident monochromatic Kα X-ray spectra from anodes of Ti, Cr, Fe and Cu have been used

in this work to derive Si ωK. The details of the acquisition and fitting of these spectra were

discussed in Chapter 2. Details of the analysis and information processing of these spectra

are described in this section. A total of twenty-eight spectra were processed, six each from

the Ti and Cu anodes and eight each from the Cr and Fe anodes. Spectra acquired from

each anode were collected using various combinations of the monochromator crystals, the

three (two Si(Li) and one SDD) detectors and the two signal processors (one ASP and one

DSP).

𝐼inc was obtained by adding together the individual intensities of the primary peak,

the primary peak tail and the three DE features attributed to electron escape (feature (i)

in Chapter 2). These non-Gaussian features are a part of this summation as each count

represents the entry of one photon into the detector active crystal. The contact electron

counts are not included since they represent photons absorbed prior to entry into the active

crystal.

The escape intensity (𝐼esc) was calculated as the sum of the escape peak, which included

its six Gaussian components, and an exponential tail fitted to the low energy side of the

escape peak. The tabulated values of 𝜂 are given for each fitted spectrum in Table 3.1.

The GUFIT program provides a means of extraction of the peak and shelf areas through

a manual-mode output option. This option exports the channel-by-channel intensity values

of each fitted feature. The total intensity of each feature is therefore calculated by summing

the values in each channel across the relative energy range.

3.5.1 Assessment of Si MAC databases

The previous measurement [20] of Si ωK was limited in that a single Si MAC database could

not be adopted. No reliable data in the low energy range for Si was available to lend sup-

port to one theoretical treatment over the other. Since that time, two independent sets of

low energy MAC measurements for Si have been performed. The first is that of Tran et

al. [160] who reported very high accuracy measurements, with well defined uncertainties, in

51

Z Detector Crystal Processor 𝜂 𝜂

22 Si(Li)1 Si111 ASP 0.00716 0.00708± 0.00012Si(Li)2 Quartz ASP 0.00693Si(Li)1 Si111 DSP 0.00717Si(Li)2 Quartz DSP 0.00693SDD Si111 ASP 0.00709SDD Quartz ASP 0.00720

24 Si(Li)1 Si111 ASP 0.00500 0.00503± 0.00005Si(Li)2 Si111 ASP 0.00500Si(Li)2 LiF ASP 0.00500Si(Li)1 Si111 DSP 0.00512Si(Li)2 Si111 DSP 0.00498Si(Li)2 LiF DSP 0.00500SDD Si111 ASP 0.00509SDD LiF ASP 0.00503

26 Si(Li)1 Si111 ASP 0.00346 0.00349± 0.00004Si(Li)2 Si111 ASP 0.00342Si(Li)2 LiF ASP 0.00349Si(Li)1 Si111 DSP 0.00354Si(Li)2 Si111 DSP 0.00354Si(Li)2 LiF DSP 0.00350SDD Si111 ASP 0.00348SDD LiF ASP 0.00347

29 Si(Li)1 Si111 ASP 0.00205 0.00203± 0.00003Si(Li)2 LiF ASP 0.00199Si(Li)1 Si111 DSP 0.00207Si(Li)2 LiF DSP 0.00202SDD Si111 ASP 0.00205SDD LiF ASP 0.00200

Table 3.1: Measured escape peak intensities (𝜂) calculated for four element anode Kα ener-gies. Uncertainties in the average are given as a one standard deviation spread of the six toeight measurements taken using each incident energy.

52

the energy range of 5-20 keV. Data lower than 5 keV were not present given the challenge of

finding both a sufficiently thin wafer of Si and a reliable photon source. Most synchrotron

monochromators are equiped with sets of monochromators that measure from 3 keV and up

and diffraction gratings that measure energies lower than about 1.8 keV. The energy gap

in this case made coverage around the Si K-edge difficult. Recent measurements from the

second study were performed by Scholze and Krumrey at Physikalisch-Technische Bunde-

sanstalt (PTB) [75]. Their measurements in the 1–10 keV range were made possible using

monochromators of Si(111) at higher energies and InSb(111) at low energies. They imple-

mented a very thin (6.5 𝜇m) Si photodiode and measured transmittance using a GaAsP

photodiode as a radiation detector. In both works, care was taken to accurately determine

the sample thickness and to ensure that high-order Bragg diffraction of upper energy pho-

tons did not contaminate their results at a significant level. They also took measurements at

higher photon energies, from 10–30 keV, but did not publish these due to the questionable

accuracy of using high energy photons in so thin a sample. These two works provide the

means for assessment of the XCOM and FFAST databases in the pertinent energy range,

though the overlap region between the two datasets (5–10 keV) should be compared for

consistency.

One means of comparison is to assess energy regions where the selected sample thickness

is considered optimal. The Nordfors criterion (2 < 𝜇𝑡 < 4) [114], and more recently, the

extended range criterion (0.5 < 𝜇𝑡 < 6) [34] are conditions generally accepted as necessary

to optimize measurement accuracy. Here 𝜇𝑡 represents the product of the attenuation coef-

ficient with the thickness of the sample. The extended range condition was strictly followed

in the work of Tran et al. [160] across their measured energy range. They used several dif-

ferent thicknesses of wafers to make this possible. The recent (PTB) study was focused on

making a highly reliable measurement of the lower energy region, specifically in the vicinity

of the Si K edge. The authors selected a 6.5 𝜇m Si photo-diode wafer which corresponds to

the thickness necessary to satisfy the extended range criterion in the low energy region up

to about 4.5 keV. For photons higher than 4.5 keV in the thin wafer, the extended range

criterion is exceeded and therefore it is not entirely clear how much reliance may be placed

in this region. However, it is fortunate that the Tran data begin at 5 keV, providing a

53

1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0- 4

- 2

0

2

4

6

8

1 0

Perce

ntage

differ

ence in

µ/ρ (

%)

P h o t o n e n e r g y ( k e V )

X C O M F F A S T P T B

Figure 3.3: Percentage difference of Si PTB MACs with respect to the XCOM and FFASTdatabases for the 1–5 keV energy range. The dotted line through zero represents the PTBdata where dots represent individual measurements (𝑁 = 534) of the MAC.

means to assess the two theories with high confidence in accuracy. A comparison of the

XCOM and FFAST theories with the PTB Si data is plotted for the energy region of 1–5

keV in Figure 3.3 and for 5–30 keV in Figure 3.4. Both figures are expressed as percentage

differences of (𝜇THEORY − 𝜇PTB)/𝜇PTB.

The raw data of Scholze and Krumrey were provided for use in this work. To compare

the theories directly to the data, the Si XCOM and FFAST tables were downloaded from

NIST and fitted piecewise to interpolate MAC values at the energies that the PTB data

were measured. Piecewise fitting was accomplished using power functions or polynomial

functions of degree (𝑛 = 3...6). An R2 value, corresponding to the fit, was required to be at

least 0.9998 to ensure high quality fits. The fitting was done piecewise across the range as

it was found that fitting the full range with one function produced unacceptably high levels

of error. As well, the fit was made to incorporate data that extended by several data points

54

5 1 0 1 5 2 0 2 5 3 0

0

5

1 0

1 5

2 0

2 5

Perce

ntage

differ

ence in

µ/ρ (

%)

P h o t o n e n e r g y ( k e V )

X C O M F F A S T P T B

Figure 3.4: Percentage difference of Si PTB MACs with respect to the XCOM and FFASTdatabases for the 5–30 keV energy range. The dotted line through zero represents the PTBdata where dots represent individual measurements (𝑁 = 112) of the MAC.

beyond the region of interpolation for that piecewise function. This was done to minimize

discontinuities in connecting the interpolated values across the range.

The PTB data in Figure 3.3 show much closer agreement with FFAST (within 2%) than

with XCOM (within 8%) below the Si K edge. From this assessment, a MAC value, for Si K

X-rays in Si, was taken as the average of the PTB and FFAST values. Just above the K edge

(within 0.4 keV), the presence of fine structure makes direct comparison difficult. Beyond

this region it is apparent that there are regions where either XCOM or FFAST have superior

agreement. At the energy location corresponding to Ti Kα X rays (4.509 keV) the data are

still considered reliable enough to assess the two theories. It is apparent that agreement falls

between both theories, though much closer to FFAST. Beyond 5 keV, the data comparison

in Figure 3.4 show a diverging trend of the PTB data with both theories. This increases

with increasing energy to the maximum measured data point at 30 keV. The divergence may

be attributed directly to use of too thin a sample for the given incident energy. Since the

55

extended range criterion cutoff for the PTB data are surpassed beyond about 4.5 keV, the

data above this energy may not be reliable. The data of the Tran et al. [160] study show a

very strong agreement with XCOM in the 5–10 keV range which supports that the XCOM

MAC values should be used for incident Cr, Fe and Cu Kα lines.

For the Ti Kα MAC, the corresponding energy of 4.509 keV marks the limit of confidence

that has been placed on the PTB data. Still, at this energy, the PTB value of the Si MAC

data is considered to be reasonably accurate. The PTB 4.509 keV MAC shows a slight trend

toward FFAST (Figure 3.3). However, it is also shown through the work of Tran et al. that

by 5 keV, there is full agreement with XCOM. The combination of the two experimental

works suggest that the percentage difference in the data seen in Figure 3.3 must transition

from near agreement with FFAST at 4.509 keV to agreement with XCOM by 5 keV, which

is a rather large jump. This jump seems improbable and it is more likely that perhaps the

PTB value at 4.5 keV may not be entirely accurate. Therefore, a compromise was made to

use the average of the two theoretical works, XCOM and FFAST, as this resultant value is

not all that different from the PTB value itself.

Another parameter in Equation 3.1 that requires input is the ratio of the photo-electric

cross-sections (𝜎KPE/𝜎PE). The values in this ratio has been derived from two theoretical

sources. The FFAST tables provide tabulation of the photoelectric cross-sections of the K

shell alone and also the total atom (𝐾+𝐿+𝑀+ ...). The XCOM program does not provide

the K-shell Pe cross-sections directly. These numbers were taken directly from Scofield [142],

the same report that provides the numbers used by XCOM. The 𝜎KPE/𝜎PE ratio has been

plotted for both databases in the pertinent energy range. An overall ratio difference not

exceeding 0.5% was found across the 4–10 keV range. It is not apparent which of the

databases may be superior in this regard so the final value of the ratio for each energy was

chosen as the average ratio of the two databases.

3.6 Results and discussion

The measured escape ratios (𝜂) in Table 3.1 show a decrease with increasing photon energies.

This is consistent with the increased penetration depth leading to a decreased probability

56

of Si K X-ray escape. While this is nothing new, a comparison of the observed ratios

with earlier similar studies might provide insight into the effect of improving the fit to the

background under the escape peak. One benefit to comparing the raw 𝜂 values is that they

are not influenced by the other FP’s used to calculate ωK. 𝜂-values from the previous Si ωK

measurement [20] were reported for energies: 3, 4, 5, 6, and 8.3 keV. A plot of these data

was fitted with a 3rd degree polynomial. 𝜂-values corresponding to the energies used in this

work were interpolated and compared directly to these data. The difference at the energy

corresponding to Ti Kα was not signicant, though a difference was observed for the other

three elements. The 𝜂 values in this work were larger than in the previous study by 2% for

Cr, 3.6% for Fe and 2.3% for Cu. It is most likely that the differences have manifested in

the higher energy elements, given that the escape peak areas are generally much smaller.

Therefore, any changes made to fitting the background, such as the improvements made in

this work in the fitting of the primary electron escape shelves and Ni contact features, will

manifest as a stronger effect on the measured peak area. While 2–4% may seem small, this

relative difference appears to be the effect of improving the fit description of the features

below the escape peak.

The spread in the values of 𝜂 at each energy provide some measure of the level of uncer-

tainty associated with the fitting technique. The peak areas of the Ti data are approximately

half a million counts. The associated Poisson statistics suggest a ±0.14% spread. However,

the standard deviation spread for the Ti 𝜂 values is 1.7% which is an order of magnitude

greater than the Poisson spread. This magnitude is not surprising given the complexity of

the background that was fitted under the peak. Still, as a measure of spread, this is not

a very large amount and suggests that the method has reasonably high precision. From

Table 3.1 the relative spread is actually smaller for the higher energy values where the peak

areas are relatively less, though still large on an absolute scale (100,000–500,000). These

data suggest that there is no overall dependence on physical experimental components at

each energy value, including the type of detector, pulse processor and monochromator, as

was expected.

The MACs in Equation 3.1 are presented in each instance as a ratio (A) of the Si K x-rays

to the primary anode X-rays both in Si. The use of a ratio may reduce the overall uncertainty

57

Energy (keV) 𝜎KPE/𝜎PE (𝜇/𝜌)Si ωK

Database Value (cm2/g) 𝜇Si/𝜇

1.740 PTB+FFAST 328.7 1.004.509 0.9250 Sco+FFAST 321.6 1.02 0.050575.412 0.9265 Sco 196.4 1.67 0.050236.401 0.9275 Sco 122.4 2.69 0.050178.041 0.9280 Sco 63.74 5.17 0.05073

Table 3.2: Summary of parameters used in analysis for each X-ray energy dataset. Shownare the photoelectric cross-section ratios, the MAC database(s) used and their correspondingvalues, the MAC ratio of the Si K X rays to the incident primary X rays and the average SiωK values calculated for each energy.

when compared to the use of absolute values of the coefficients. In some situations use of the

same database for the numerator and denominator, as in A, could allow certain systematic

uncertainties, present across the whole range, to cancel out. It is unfortunate that no

single experimental database can be relied upon for the attenuation coefficients in the 1–8

keV range. Rather, two experimental tabulations (PTB [75] and Tran et al. [160]) have

been used to select from the two frontline theoretical databases (XCOM and FFAST) at

two distinct energy regions. Any systematic errors present in either database are therefore

unlikely to cancel out. Therefore, it is uncertain that the benefits of a ratio approach will

apply to this measurement.

Values of ωK have been calculated individually for each spectrum. The average ωK

values from each of the four energy sets have been calculated. These are shown alongside

the parameters used in their calculation in Table 3.2. The average calculated ωK, from all

28 spectra, is 0.0504± 0.0015. A plot of the individual ωK values is provided in Figure 3.5.

The relative spread in the data points is 2.8% (±2𝜎, 95% confidence). This represents a

reasonably conservative measure of the uncertainty associated with the fitting process. The

results across the energies of the four elements are encouraging as there does not appear to

be an energy dependence in the results.

Sources of uncertainty, beyond statistical scatter, also need to be addressed. When

considering that the XCOM and FFAST theories differ by as much as 11% below the Si K

edge, the total uncertainty associated with the use of either tabulation alone becomes rather

58

2 2 2 4 2 6 2 8 3 0

0 . 0 4 9

0 . 0 5 0

0 . 0 5 1

0 . 0 5 2

ωK of

Si

ZFigure 3.5: The full dataset of 28 individual calculated Si ωK values, plotted against theatomic number (Z) of the anode used in their production. This dataset represents all spectracollected using a total of 3 detectors and 3 monochromators at four different incident photonenergies. The average ωK value calculated from all measurements is denoted by the horizontalline in the centre of the plot.

large. Comparison with the newer data therefore goes a long way to eliminating the bulk

of this uncertainty. In their work, Tran et al. quoted very small (6 0.5%) uncertainties on

each of their MAC data points. This suggests that the uncertainties associated with the

MACs of the incident Kα X-ray radiation used in this work are small. Uncertainties were

not quoted for the PTB measurements [75] but the authors took precautionary measures to

minimize those factors known to contribute major sources of error [37]. They ensured that

harmonic contributions to the measured photon flux were less than 0.1% for all energies.

They also verified their source thickness, with high accuracy, using two complementary

methods. These considerations justifiy the confidence placed in this work on the PTB data

in the range of 1–4.5 keV.

The uncertainty associated with ratio A gets its greatest contribution from the value

corresponding to the Si K X-rays (1.74 keV). The average MAC (PTB + FFAST) adopted

at this energy differs from the PTB and FFAST by 1%, allowing this value to be considered

59

a reasonable measure of uncertainty in the ratio A. Since the MAC value at 1.74 keV is used

for all four energy sets, this 1% difference is the governing uncertainty introduced by ratio A

to all four measurement sets. Two calculations of the curly bracketed term in Equation 3.1

were performed to assess how this uncertainty might manifest in the final report on ωK.

This was done by using the Si K X-ray value of FFAST in the first calculation and the value

of PTB in the second calculation, keeping the incident X-ray MAC’s constant. This was

found to introduce a corresponding ±1% variation as might be expected. Overall, the use

of the new experimental MAC data has helped to reduce the uncertainty associated with

the selection between the two theoretical treatments.

For the ratio 𝜎KPE/𝜎PE, the mean of the values from the two theoretical sources (XCOM

and FFAST) was used. These differ by only approximately 0.5%, which has been considered

to be a reasonable measure of this uncertainty. The final uncertainty reported in this work

is therefore the quadrature sum of the standard deviation spread (2.8%), the 1% factor cor-

responding to MAC ratio A and the 0.5% factor from the ratio of the cross-sections. This

value totals 3.0%. When compared to the uncertainties reported by earlier determinations

of this parameter, this does not appear to be much of an improvement. However, several

advancements have been made since the last measurement. The first is the improved high

accuracy fit treatment of spectra outlined in Chapter 2. The full characterization of all

lineshape features appearing as background under the escape peak was a problem in ear-

lier works and likely a significant source of error. In particular, the fit description of the

Ni contact Pe features was not treated in a meaningful way in the earlier work that was

representative of the true shape. Now that a more complete characterization of all the es-

cape shelves and contact features has been carried out, a greater reliance might be placed

on this new ωK estimate. Another improvement includes the adoption of one set of MAC

values from the choice of the two different theoretical treatments. Selection was assisted by

comparisons between the new high quality experimental measurements of low energy MACs

for Si and the two theoretical databases. Ultimately, the results in this work suggest that

the uncertainties associated with the older tabulations of Si ωK, and those of other light

elements (Figure 3.1), were underestimated. Given the 3% uncertainty of this result and the

considerations made in its determination, it is difficult to see how this 3% might be further

60

Source type Source Value

Experimental Brunner [14] (Mn Kα) 0.0476 ± 0.0006Brunner [14] (Mn Kβ) 0.0511 ± 0.0037Campbell et al. [20] (synchr. & monochr., FFAST) 0.050 ± 0.001Campbell et al. [20] (synchr. & monochr., XCOM) 0.052 ± 0.001This work (X-ray set and monochr., mixed 𝜇/𝜌) 0.0504 ± 0.0015

Theoretical McGuire [108] 0.0592Walters and Bhalla [164] 0.0514

Semi-empirical Bambynek [4, 78] 0.0504Krause [97] 0.050

Table 3.3: Compilation of theoretical and experimental values of the K-shell fluorescenceyield for Si.

reduced.

The final value obtained in this work is compared with other estimates of this param-

eter in Table 3.3. The calculation from this work agrees with the semi-empirical estimates

of Bambynek and Krause, within uncertainty. The new result is about 16% lower than

McGuire’s prediction and 2.0% lower than the Walters and Bhalla value. The Walters and

Bhalla estimate falls inside the uncertainty range of this measurement.

The bulk of the work summarized in this chapter has been published by Hopman et

al. [75]. A minor improvement to that work has since been made and is described below.

Fundamentally, in this work, the method used and conclusion drawn previously have not

changed. The Hopman result for Si (ωK = 0.0495 ± 0.0015) was derived using only the

SDD data. This result represents the median value with the absolute spread of these mea-

surements used to define the uncertainty. The Hopman uncertainty was 3%, as is reported

in this chapter. The data were found to exhibit a slight Z dependence manifesting as a

linear increase in calculated ωK’s with increasing Z. An explanation of this behaviour could

not initially be offered though the discovery of some minor technical issues with the former

analysis have now made this possible.

The Z dependence issue was investigated by beginning with a careful review of the

method used to fit the PTB MAC data. The original method involved fitting a log-linear

plot of the MAC data vs. energy with a ln function of the form:

61

ln(︂𝜇

𝜌

)︂= 𝑎+ 𝑏 · ln (𝑐+ 𝐸) (3.2)

where a, b, c are fit coefficients and E is the photon energy (keV). This one function

was applied across the full 1–10 keV dataset and appeared to produce a satisfactory fit.

Interpolation of the PTB data using this function indicated a strong agreement with FFAST

across the entire energy range, hence its adoption in the earlier analysis. However, in a closer

examination, this function was found to neglect the fine differences between the data and

fit at higher energies. In fitting the data using the piecewise description described in this

work, a more refined assessment of the PTB comparison with XCOM and FFAST was made

possible (Figures 3.3 and 3.4). This led to the realization that the PTB data is discrepant

from either theory at higher energies. However, this was not surprising given that the 6.5

𝜇m sample thickness places the measurement outside of the extended range criterion. At

this point deferral was made to the Tran et al. data for > 5 keV where their results showed

support in favour of XCOM. In making the switch to this database for the > 5 keV elements,

the linearity was diminished entirely, yielding the data spread seen in Figure 3.5. The final

estimate of Si ωK presented here (0.0504) is about 2% greater than that in the former work

but is still within the ±3% uncertainty range of the previous measurement.

The comparison of the earlier estimate with this one has demonstrated the strong effect of

uncertainties associated with FPs and how they can influence final results. In this case, the

method of information extraction from the PTB data was shown to introduce its own level

of error. The correction of this effect has reinforced the care one must place in maintaining

precision in all aspects of performing a new measurement of a FP, with high accuracy.

3.7 Comparison to theoretical predictions

The apparent support of the new Si ωK result for the Walters and Bhalla prediction war-

rants careful examination. Theoretical considerations for probabilities of radiative and non-

radiative transitions may be calculated using theoretically obtained energy widths (Γ). This

is done using the simple expression:

62

ωK =ΓR

ΓTOTAL=

ΓR

ΓR + ΓNR(3.3)

where ΓR and ΓNR are the radiative and non-radiative widths (eV) of the K-shell tran-

sitions, respectively. The radiative and non-radiative widths used in the three theoretical

treatments (Walters and Bhalla, McGuire and Chen), presented in Figure 3.1, have been

summarized for Si and Ar in Table 3.4.

The Walters and Bhalla and McGuire theories both used DHS formalism as a basis to

the calculation of their radiative and non-radiative widths. The Chen approach used their

own DHS calculations, but only for the non-radiative widths. For the radiative widths, they

used the well-regarded Dirac-Fock (DF) radiative widths generated by Scofield [139], shown

in Table 3.4. The Chen approach is perhaps the most sophisticated because of the adoption

of the DF formalism over DHS in the calculation of radiative widths. The differences in the

DF vs. DHS approaches will be covered in greater detail in Chapter 4 but to summarize in

brief, the rigorous treatment of the DF approach should be expected to produce superior

results in calculation of ωK. The superiority of the DF approach has been demonstrated

when used to accurately predict measured (Kβ/Kα) X-ray intensity ratios [138] and the

intensities of KMM radiative Auger emissions [29].

Chen et al.’s use of the Scofield DF widths is further supported by the strong agreement

of their Ar ωK value with the experimental data, shown in Figure 3.1. The pre-cursor DHS

radiative and non-radiative widths of Scofield [140, 141] are also given in Table 3.4. The

differences in the radiative widths of Si and Ar between Scofield’s DF and DHS approaches

are 18% and 11%, respectively. With this difference, it becomes clear that if Chen et

al. had adopted Scofield’s earlier DHS calculations then their Ar estimate would have

disagreed substantially with the experimental data. It calls to question then why the Walters

and Bhalla DHS approach finds such strong agreement with the Chen calculations. The

agreement may be merely coincidental or it could demonstrate the superiority of some of

the particulars of the Walters and Bhalla approach, such as their use of the Kohn-Sham

potential.

63

Treatment 𝑍 = 14 𝑍 = 18

Radiative width McGuire [108] 0.0242 0.0811Walters and Bhalla [164] 0.0226 0.0782Scofield DHS [141] 0.0202 0.0717Scofield DF [139] 0.0238 0.0799

Non-radiative width McGuire [108] 0.358 0.544Walters and Bhalla [164] 0.417 0.566Chen [35] 0.576

Table 3.4: Theoretical radiative and non-radiative decay widths (Γ (eV)) for the K-shell ofSi and Ar.

3.8 Conclusion

The new measurement of Si ωK reported in this chapter (0.0504 ± 0.0015) is only slightly

lower than the Walters and Bhalla prediction (0.0514). It agrees with the Bambynek and

Krause semi-empirical values and with the earlier Campbell et al. [20] FFAST result. The

3% relative uncertainty reported with this measurement is a rather conservative estimate

which represents primarily the observed spread of the 28 individual spectral estimates. At

first glance, one might presume that little has been gained in the painstaking effort presented

here. However, it has been shown that the earlier works may have had significant systematic

errors resulting from various technique and database limitations. In this work, efforts have

been made to resolve these errors. When considering the problems that plagued many of

the earlier ωK measurements, as mentioned in Section 3.1, it is likely that the present result

is the most rigorously performed measurement, in this atomic number region, to date.

This work has demonstrated that high-accuracy extraction of escape-peak areas from

monochromatic spectra is a complex process. A great deal of dependence is placed upon

understanding the detector fabrication and how different contact contributions will effect the

background under the escape peak. The complexity of the present fit treatment strongly

suggests that the simplified fit treatments of earlier measurements may have been a signifi-

cant source of systematic error. The present study has also demonstrated that a great deal of

uncertainty may arise from the incorrect use of attenuation coefficients in analysis. However,

this uncertainty has been largely remediated with the advent of new Si MAC measurements.

Although the agreement of the new result with Walters and Bhalla is quite good, this

64

should not be misinterpreted to suggest that this particular theoretical approach will be

accurate for the entire light element range (10 < 𝑍 < 20) in question. Still, the Walters and

Bhalla approach might be further validated if the Chen et al. calculations, incorporating

Scofield DF radiative widths, were to be extended below Z = 18.

65

Chapter 4

An accuracy assessment ofphoto-ionization cross-sectiondatasets for 1–2 keV X rays in lightelements using PIXE

4.1 Introduction

The attenuation of photons of energy less than about 20 keV is determined primarily by

photo-electric absorption. Coherent (Rayleigh) and incoherent (Compton) scattering con-

tribute to attenuation, but at low energy, play only a subsidiary role. In Si for example, at 1

keV, coherent and incoherent scattering contribute 0.16% and 0.0008%, respectively. Even

at 10 keV, coherent and incoherent scattering are 1.8% and 0.32%. The photo-electric cross-

section (𝜎PE) parameter is therefore highly relevant to application areas involving X-ray

transmission analysis, medical physics and X-ray emission analysis of a material’s chemical

composition. Tabulations of these parameters and the related MACs are necessary as they

have an immediate practical use. For applications involving low energy photons (1–2 keV),

reliable experimental data are sparse or non-existent. Reliance upon theoretical calculation

databases is therefore necessary.

4.1.1 Development of theoretical photo-absorption cross-section formalism

The development of photo-absorption theory was initiated shortly after the advent of the

quantum mechanical wave-equation formalism by Schrödinger. Using Schrödinger’s equation,

66

D. R. Hartree [68, 69] illustrated that the iterative self-consistent field (SCF) approach could

be used to solve for the distribution of electrons in an atom and to find solutions to the corre-

sponding wavefunctions. This approach was aptly named the Hartree approach. Noteworthy

advancements to this method were supplied soon after by Dirac, Fock and Slater. At the

time of Hartree’s discovery, Dirac had introduced the famous Dirac equation which provides

a means to treat relativistically the wavefunctions used in the Hartree approach. In 1930,

Fock [57] and Slater [148] independently identified that Hartree’s formalism neglected the

antisymmetry properties of fermion wavefunctions. The Hartree method was henceforth

known as the Hartree-Fock (HF) method.

Early on, the application of the HF method to calculate 𝜎PE for photon attenuation was

realized. With unlimited computational power, this method could be used to construct com-

plete databases of 𝜎PE for any and all elements. However, the iterative based HF approach,

for a multi-electron system, was computationally expensive. The available computer power

of the mid-twentieth century could not meet the demands of this approach when building a

complete database. As a result, scattering factors for only a few specific elements and ener-

gies were computed. A solution came with the next major theoretical advancement involving

an approximation to the HF method by Slater [147]. His 1951 publication demonstrated

that all electrons in the system may be approximated as experiencing the same exchange

potential. Before, the HF solution required each electron’s potential to be calculated indi-

vidually. This milestone has made possible, in the decades to follow, the computation of

extensive datasets of scattering factors for multiple elements across broad energy ranges.

4.1.2 Current theoretical cross-section datasets

By the late 1960’s a number of researchers were directing their efforts toward the construction

of attenuation coefficient databases. Most of these databases used some form of the Slater

approximation. In this work, discussion is limited to the works responsible for providing the

𝜎PE calculations which are used to build the current XCOM [10] and FFAST [33] datasets.

The XCOM database derives its 𝜎PE values from the 1973 tabulation of Scofield [142].

His DHS calculations incorporated relativistic wavefunctions using the Hartree-Slater po-

tential [147]. Scofield provided 𝜎PE values for each element in the range 1 6 Z 6 101,

67

at numerous energies from 1 to 1500 keV. This dataset was the most complete extensive

cross-section compilation at the time. Scofield further recognized that a full HF treatment

would be superior to any approach incorporating the Slater approximation. He presented

renormalization factors for the range 2 6 Z 6 54 to convert his HS results to approximate

HF values. These were not calculated using a rigorous HF treatment but rather by using the

basis of potential model calculations performed by Pratt et al. [124, 125, 132, 134]. These

renormalization factors introduce a 5–15% difference in 𝜎PEvalues corresponding to lower en-

ergy photons [31]. Upon first impression, a reader may be inclined to adopt the renormalized

factors as they might approximate the HF treatment more closely than the un-renormalized

DHS values. However, Saloman and Hubbell [131, 132] reviewed the extensive experimental

compilation of Henke et al. [72] and concluded that Scofield D+HS calculations find better

overall agreement with the data. For this reason, the current XCOM program was made to

incorporate the original un-renormalized values.

The FFAST database was constructed from the more recent calculations of Chantler [31,

32] who used the DHS formalism introduced by Cromer et al. in the late 1960’s [39–

42]. Cromer’s approach was similar to Scofield’s, and many others at the time, with a

few exceptions. The first was that Cromer et al. adopted the Kohn-Sham and Gaspar

exchange potential [96] which, in turn, was a modification to the original Slater potential.

The other major difference was in the use of experimental binding energies for the iterative

process in calculating wave-functions. Scofield used the self-consistent iterative approach

of calculating both binding energies and wavefunctions. In this manner, both parameters

are calculated using each other until reasonable convergence is found. Cromer, rather than

calculate energy, used the the experimental binding energies of the extensive compilation

by Bearden and Burr [7]. Cromer’s calculations were performed for only 5 elements in a

broad energy range and was therefore not as exhaustive as Scofield’s dataset. In adopting

Cromer’s approach, Chantler produced many data points for each element in the range

1 6 Z 6 92, for energies betweeen 1–10 eV to 400–1000 keV. Additionally, he provided a

large amount of detail in his calculated data by producing many closely-spaced values near

absorption edges. The earlier Chantler treatment [32] had a problem just above absorption

edges, causing large peaks and oscillatory behaviour which is not related to extended fine

68

structure that results from solid state effects. Chantler’s second paper [31] highlighted

improvements to his treatment near edges and consequent solution of earlier convergence

issues. This improvement resulted in the elimination of these unwanted features and the

desired smoothing of near-edge behaviour.

Some clarification on the acronym usage of these theoretical treatments is required.

A large amount of literature exists in which authors have performed calculations using

Hartree-Fock and related methods as a basis and have used a wide variation of acronyms

like HF, DHF, HS, DHS, DF, DS and HFS to describe their work. In some cases, the

letter combinations clearly indicate the specific treatment adopted while in others, it is

not so apparent. Clarification is necessary to explain how the lettering has been used in

various circumstances. To begin, the use of the letter “D” implies that Dirac relativisitic

effects have been considered. Its absence may be interpreted as an indication that non-

relativistic wavefunctions were used. On the other hand, if the author states “relativistic

HS”, this is equivalent to DHS. In general the meaning behind the D is quite consistent

in the literature. The use of the letters “H” and “F” implies Hartree-Fock formalism has

been used as a basis. Some authors of DS, DHS and DF calculations have abandoned the

use of one or both of these letters, perhaps in making the assumption that readers would

understand that Hartree-Fock formalism is inherent to the formalism’s foundation. The “S”

is used to define that the Slater approximation to the electron exchange potential is used,

rather than a full HF approach. Its presence in the literature is also generally consistent

with this implementation. The acronyms DHF or DF suggest the use of a full relativistic

HF treatment, without simplification using some form of approximation, such as the Slater

exchange potential. It seems that very few full HF calculations have been performed. The

appearance in the literature of acronyms DHF and DF should therefore be interpreted with

caution. In these situations it is necessary to review the work carefully to determine what,

if any, deviation from the full HF treatment, has been applied.

NIST has produced two databases of MAC values using the Scofield and Chantler 𝜎PE

calculations. In both of these databases, the sources for coherent [77] and incoherent [80]

scattering components are the same. Therefore any differences existing in the X-ray energy

range may be attributed to the photo-electric cross-sections alone. Both the XCOM and

69

FFAST databases are tabulated for almost all known stable elements, across a broad photon

energy range, and each is a useful tool in application. The question of database superiority

is therefore somewhat open-ended. Regions of disagreement between the two databases have

been identified [31], as illustrated in Chapter 3, but generally, have have not been shown in

great detail. Some modern experimental MAC data corresponding to low energy photons

will be discussed in Section 4.2 as they may provide support of one theoretical database over

the other.

This chapter is focused on very low X-ray energies, specifically in the region below the

K absorption edge of light elements. The primary motivation for this work comes from the

results of the 1997 PIXE calibration study by Campbell et al. [22]. Their work involved

PIXE measurements performed on pure element targets (Mg, Al and Si) and multi-element

silicate mineral and glass standard targets containing various compositions of MgO, Al2O3,

SiO2 and other elemental oxides. In a unique FP approach, utilizing known concentrations

of samples, they solved for the solid-angle efficiency term (H-value) for each element in each

target. They found that the Mg, Al and Si H-values in the silicate targets were consistently

less than those of the pure targets by 6–8%. This was found despite taking into account

all composition and target FP differences. These results pointed to issues present in one or

more of the FPs, used in analysis. The MAC values used are one of these FPs that have

been questioned and are the subject of analysis in this chapter.

For characteristic X-ray emission by Mg, Al and Si, the energy region of interest falls

between 1–2 keV. In this range, the photo-electric effect is confined to the L and higher shells.

Here, the XCOM and FFAST database values differ by about 10%. In this chapter the results

of thick target PIXE measurements performed on pure Mg, Al and Si elemental and oxide

targets are presented. Unlike the earlier calibration study, the silicate standards have been

substituted by pure oxide targets: MgO, Al2O3 and SiO2 as this might allow simplification

of the analysis and reduce uncertainties associated with complex target compositions. The

X-ray yields acquired from each target have been compared against yields predicted from a

FP calculation. This has been done to assess which of the two theoretical databases provide

superior results for this particular energy and element region. Given the small differences

expected, efforts have been made to reduce all possible systematic errors to a negligible level.

70

In general, this work strives to improve the understanding of 𝜎PE’s for energies below the K

absorption edge of elements 11 6 Z 6 26. This is highly relevant to improving the accuracy

of elemental analysis in geological samples by X-ray emission methods.

4.2 Previous experimental work

Measurements of MACs can be corrected for the contributions of Rayleigh and Compton

scattering to provide 𝜎PE’s. For very low energy photons (a few keV) in light elements, the

scattering contributions are negligible. The conventional approach to measuring a MAC is

rather simple. A chemically pure film of the material of interest is inserted into a geomet-

rically well-characterized X-ray beam and the decrease in intensity is measured. While this

appears straightfoward, many challenges involved with this type of measurement have been

identified by Creagh [36] and by Creagh and Hubbell [37, 38] through the IUCr X-ray

Attenuation Project of 1979–1987. In general, MAC measurements become more suscep-

tible to error as the photon energy decreases, primarily due to the challenge of obtaining

a sufficiently thin, well characterized sample. Measurement accuracy depends strongly on

the choices of experimental configuration, detection technology and characterization of the

absorber material. For measurements on thin foils, small errors in the thickness estimate or

even slight foil profile contour deviations can substantially impact the final result. Another

issue is whether the material is entirely homogenous or if it has pockets of empty space that

cannot be seen or measured. This may cause erroneous assumptions regarding the mate-

rial’s density and could contribute significant small angle scattering. The neglect of various

factors including dead-time corrections, higher energy beam harmonic contamination and

excessive beam divergence has been identified in many early MAC measurements [36–38].

Still, the large amount of existing experimental data may be used as evidence of database

accuracy. Saloman et al. [131, 132] used an extensive experimental dataset to demonstrate

the overall superiority of Scofield’s un-renormalized cross-sections over his HF renormalized

values. While this appears to resolve the issue of the choice of Scofield’s database, it is

important to point out that the Saloman compilation contains little reliable data for low

X-ray energies and for light elements. Therefore this general statement cannot be taken to

71

imply that in the very low energy range the Scofield HS un-renormalized predictions are

necessarily superior.

In light of the previously discussed problems found to be present in older MAC data,

it is worthwhile to make database comparisons to newer, presumably more accurate exper-

imental data. A recent and extensive set of synchroton radiation MAC measurements has

been reported by one group [34, 60, 61, 84, 88, 89, 126, 159, 160] in which the authors

have demonstrated painstaking efforts to eliminate systematic errors identified in earlier

measurements [36–38]. They examined attenuation coefficients for elements Si, Cu, Zn, Ag,

Mo, Sn and Au with an overall five-fold improvement in measurement accuracy (typically to

within 0.1–0.2%) with respect to earlier work. This has allowed, for the first time, a critical

analysis of the available theories, generally extending from the K edge of each element and

upward in energy. MACs immediately below the K edge for Ag [159], Mo [89], Cu [61] and

Zn [126] were also included. Care was taken to incorporate measurement energy spacings at

close enough increments near the K edges to show fine structure detail. Just above the K

edge, both theories fail to reproduce the fine structure seen in the data. This is not surpris-

ing given that both theories involve independent particle approximations for isolated atoms

and do not consider physical environment. It also became apparent that neither theoretical

approach is superior on the whole or across the full energy ranges examined. Rather, the

two theories vary smoothly about the data such that agreement of one or the other is found

in select energy regions. Tran et al. [160] reported Si measurements in the range of 5–20

keV. These data showed stronger agreement with Scofield 𝜎PE’s, as discussed in Chapter 3

though this agreement worsens when approaching 20 keV. They illustrated further that ear-

lier measurements performed above 25 keV favoured the theoretical values of Chantler over

Scofield. However, coherent and incoherent scatter at this energy contribute about 15% of

the total attenuation in Si and this contribution continues to rise with increasing energy.

Examination of the accuracy of the theories is therefore more complex at higher energies

since the total attenuation is no longer principally determined by 𝜎PE.

Another new set of MAC measurements has been produced by Ménesguen and Lépy [110]

for 3d transition metals Ti, V, Fe, Co, Ni, Cu and Zn, using monochromatic radiation. The

authors presented MACs in the energy range of about 4–11 keV, which incorporate data on

72

both sides of the K edges of elements. In general, their results indicated an overall stronger

agreement with Scofield’s predictions, as Chantler’s MAC values were consistently lower.

Although they did not report on their treatment of systematic uncertainties, their work for

Cu can be compared directly to the new synchrotron Cu MAC measurements of Glover et

al. [61]. In the energy range of 5–7 keV both works show superior agreement with XCOM.

In the energy region (8–9 keV) where the Cu Kα (8.041 keV)) and Cu Kβ (8.905 keV) lines

are found, the data of Glover et al. favoured FFAST and those of Ménesguen and Lépy [110]

still favoured XCOM. New high accuracy experimental work is needed to examine specific

regions to improve a somewhat confused situation. As well, the works mentioned here still do

not extend as far down in energy as would be relevant to this study, with the one exception

being the PTB measurements of Si MACs mentioned in Chapter 3 [75]. Therefore, it is

difficult to extrapolate how agreement with either XCOM or FFAST might manifest in the

1–2 keV range.

4.3 Objectives and approach

4.3.1 Evaluation of the detector absolute efficiency constant

In the 1–2 keV range, Compton and Rayleigh scattering account, at most, for 0.8% of

all photon interactions occuring in elements Mg, Al and Si. Measurements of MACs at

this energy provide a direct test of photo-absorption theory. However, self-absorption of

K X-rays occurs by interaction with the L shell and higher energy shells only. Therefore,

analysis of MACs in this energy region will provide a direct test of photo-absorption theories

corresponding to the L and higher energy shells. This is an energy range and region of data

(L shell and higher) that have been largely neglected by experimenters.

The conventional approach to measuring attenuation coefficients is to measure directly

the relative photon absorption in a material or foil of well known thickness, as has been

detailed in Section 4.2. The advent of synchrotron radiation has revolutionized the task

of performing low-energy MAC measurements and has minimized some of the technolog-

ical challenges present in older measurements [36]. Modern synchrotron facilities provide

well focused monochromatic beams corrected for higher-order beam harmonics while still

73

providing a reasonable photon intensity. Beam divergence is also less of an issue. However,

in the soft X-ray range, (<5 keV) the primary technological challenge is the determination

of sample thickness. For 1–2 keV photons, foils of several microns thick, at most, are re-

quired to perform a measurement. Precise methods of thickness determination that work for

thicker foils are less reliable here and attenuation measurements on thinner foils are more

sensitive to lateral variations in foil thickness. Furthermore, if self-supporting foils cannot

be obtained, the additional complication of correcting for absorption in the plastic or mylar

support layer is presented. Direct measurements of this sort are non-trivial and are sensitive

to systematic error.

The approach adopted here employs PIXE as an alternative to the conventional method.

The K X-ray yields generated by a 3 MeV proton beam incident on thick targets (0.5–3

mm) of the three elements: Mg, Al and Si and their oxides: MgO, Al2O3 and SiO2 are

compared. The use of thick targets, which can be treated as infinitely thick relative to the

proton range, circumvents the need for accurate measurement of sample thickness. The

proton beam penetrates, at most, to a depth of about 120 𝜇m. Most of the X-ray signal

emerges from within the first 20 𝜇m or less. The equation [87] of the K X-ray yield 𝑌l(Z)

for a given line (𝑙) of element Z, having concentration 𝐶Z is:

𝑌𝑙(Z) =𝑁a𝑁PωK𝑏

𝑙Z𝑡Z𝐶Z𝜖a(ℎ𝜈)

𝐴Z

∫︁ 0

𝐸0

𝜎Z(𝐸)𝑇Z(𝐸)

𝑆(𝐸)d𝐸 (4.1)

where:

𝐴Z is the atomic mass

𝑁a is Avogadro’s number

𝑁P is the number of protons incident on the target, i.e. integrated beam charge

ωK is the K-shell fluorescence yield

𝑏𝑙Z is the fraction of total K X rays appearing in the K𝑙 line where 𝑙 = 𝛼, 𝛽

𝑡Z is the transmission factor through any absorbing materials between target anddetector (= 1 in this case since the experiment is conducted in vacuum)

𝜖a(ℎ𝜈) is the detector’s absolute efficiency (including solid angle factor and intrinsic effi-ciency) at X-ray energy ℎ𝜈

𝜎Z(𝐸) is the energy-dependent proton ionization cross-section

𝑆(𝐸) is the energy-dependent proton stopping power in the target

74

𝑇Z(𝐸) describes the transmission of outgoing X rays defined as:

𝑇Z(𝐸) = exp{︂−(︂𝜇

𝜌

)︂Z

sin 𝜃

sin𝜑

∫︁ 𝐸

𝐸0

d𝐸

𝑆(𝐸)

}︂(4.2)

where:

𝜃 is the angle of the incident proton beam relative to the surface plane

𝜑 is the angle of emitted X rays relative to the surface plane

𝜇/𝜌 is the MAC of the X ray of interest within the target.

𝐸 is the proton energy ranging from the incident maximum energy (𝐸0) to 0.

The yield expression (Equation 4.1) can be rearranged to solve for concentration (𝐶Z), as

is done in application. In this experiment, concentration is already known from the chemical

formula and is used as one of the inputs to analysis. Using 𝐶Z, the efficiency term 𝜖a(ℎ𝜈) can

be determined by substituting known values into each parameter of Equation 4.1. Parameter

inputs include the measured X-ray yield, the well-defined incident and outgoing angles, the

integrated beam charge, and the Guelph PIXE (GUPIX) Software Package database values

of the remaining FPs. In this manner, the yield equation is solved explicitly for the efficiency

term. This has the benefit of circumventing the need to calculate detection efficiency that

is otherwise necessary for the calculation of 𝐶Z. Solving for 𝜖a(ℎ𝜈) provides a meaningful

measure of the accuracy of using the yield equation to compare calculation results from

specific X-ray lines, or more generally, specific photon energies. For example, if 𝜖a(ℎ𝜈) is

calculated for Si K X-ray yields from Si and SiO2 targets, and full FP calculations are

performed in which FP systematic errors are negligible, then the results are expected to be

identical, within the statistical uncertainty. This rationale is justified since the efficiency of

collecting a Si K X-ray photon should be the same regardless of its target source, ie. Si in

a pure element or in an oxide.

This method of comparing pure element efficiency to element-oxide efficiency, has the

added benefit that parameters shared in common, in Equation 4.1, may cancel out. Also,

any systematic errors in detector efficiency and charge measurement will cancel. Quantities

which are not common to the two targets are the target stopping power and the target

attenuation coefficient, both of which appear within the integrand of Equation 4.1. There

75

is also the possibility that chemical bonding may alter the X-ray production cross-section

(𝜎Z(𝐸)ω𝐾); this effect would likely manifest itself through the presence of L spectator va-

cancies created in double ionization interactions. Each of these will be discussed in greater

detail in the following section.

4.3.2 Consideration for stopping power and X-ray production cross-section

4.3.2.1 Stopping power

When 3 MeV protons are incident on thick targets of Mg, Al and Si, they will produce

ionizations along their travel path, losing small increments of energy along the way, until

maximum depth is reached and proton energy is zero. Most of the light element K X rays

emerge from the first 20 𝜇m of depth in the target which corresponds to the initial loss of

∼ 0.3 MeV of the proton’s kinetic energy. The remaining ∼2.7 MeV of energy is transferred

to the sample at a greater depth in the target in which most X rays are self-attenuated by

the sample. The stopping power effects at 2.7 MeV are quite similar to 3.0 MeV. Therefore,

the considerations on stopping power in the following discussion have been simplified by

restricting attention to the effects produced at 3 MeV only.

Ishiwara et al. [82] tabulated the results of four independent studies of proton stopping

power of Al. The claimed accuracy ranges from ±0.3 to ±0.5% and the four results at 3.0

MeV have a 1% overall range. When the 1–3 MeV experimental data are compared to the

theoretical Bethe-stopping calculations of Ziegler and Biersack [165], used in GUPIX, both

find agreement to within ±1%. This indicates that the level of uncertainty associated with

the pertinent energy region is very small and that the theory used in our data analysis is

highly accurate. While Al has been extensively measured in this energy range, there are

fewer data for proton stopping in Mg and Si. However, given the demonstrated accuracy of

the stopping theory for Al, we can infer that the theoretical stopping power values for these

elements are valid at the same 1% level. Additionally, stopping power data for oxygen show

a similar agreement trend to the theory, as found for Al [165].

The question still remains as to how an assumed 1% stopping power error would alter

the theoretical PIXE yields. This has been assessed by solving the transmission expression

76

(Equation 4.2) for the Mg and Si Kα X-ray yields from respective Mg and Si targets. A

numerical integration was performed, in which a ±1% variation in 𝑆(𝐸) was introduced.

The resultant effect on 𝑇Z(𝐸) was ±0.18% for Si and ±0.14% for Mg. A similar result is

expected for Al. These two values demonstrate that the introduced uncertainty is effectively

diminished by the transmission equation. Therefore the uncertainties associated with stop-

ping powers are not expected to compromise significantly the accuracy of the conclusions

made in this work regarding 𝜎PE.

4.3.2.2 X-ray production cross-section

This section addresses how differences in the target chemical environment may alter 𝜎Z(𝐸)ωK.

Specifically, it is shown an element bonded to other like elements, as in a pure target, will

experience a different 𝜎Z(𝐸)ωK value than the same element bonded to oxygen, such as in

the oxide targets. For chemical effects, it is necessary to consider the finite probability of the

simultaneous production of one or more spectator L vacancies with the K shell ionization in

light atoms. It is possible to have multiple L vacancies such that the total initial vacancy

may be described as KL𝑛 where 𝑛 = 0, 1, 2.... It is also possible to have multiple initial K

vacancies. However, the probability of producing more than one K vacancy and (𝑛 > 2)

L vacancies is negligibly small under the conditions of this experiment. Therefore in this

study, consideration is given only to the KL1 initial vacancy state.

In Mg, Al and Si, the presence of the KL1 initial vacancy state is responsible for the

production of the Kα3, Kα4 and Kα′ shake-satellite lines, the same lines introduced as

appearing in the Si escape peaks in Chapter 2. The intensities of these satellite lines have

been studied using wavelength dispersive X-ray spectroscopy (WDXRS). Energy-dispersive

X-ray spectroscopy (EDXRS) is unable to resolve the satellite lines from the principal lines.

As such, any differences in their intensity can be registered only as an overall change in

the principal Kα line intensity. For this reason and since EDXRS is used in this study, the

chemical effects modifications to the KL1 intensity that are discussed in this section have to

be calculated using the WDXRS data available in the literature. A number of studies have

reported measurements of the KL1 intensity for light elements Mg, Al and Si [6, 45, 55, 56,

64, 98, 112, 133, 155, 162]; most of these studies incorporated photo- or electron-ionization

77

as an excitation source. In general, the intensity of the KL1 satellite lines in light elements,

relative to the principal Kα line, is on the order of 10%. The chemical environment (eg. pure

element vs. oxide) has been shown, in these works, to affect the total observed intensity of

the KL1 satellites [133, 155]. In particular, the KL1 intensity of atoms bonded to oxygen

is about 10–15% greater than the intensity of atoms bonded to like atoms. Therefore, the

impact on the total measured K X-ray yield, due to chemical environment, is the product

of the KL1/KL0 satellite intensity (∼ 10%) and the KL1 intensity increase (10–15%) due to

environmental differences. This results in an effect that is on the order of one percent.

This approximate 1% effect can be calculated in this straightforward approach, when

photo-ionization is involved. Under photo-ionization, the production of the L vacancy in the

KL1 vacancy state only occurs via the shake process. This is a process that is independent

of photon energy and is attributed to the sudden change in atomic potential resulting from

the immediate production of a K vacancy [106]. The situation involving charged particle

ionization is somewhat more complex. In addition to the shake process probability, the

charged particle has the probability of directly ionizing the L shell via Coulomb interaction

to produce the KL vacancy state. The total probability (𝑝L) of producing the L spectator

vacancy via charged-particle ionization is therefore:

𝑝L = 𝑝DIL + 𝑝shake

L (4.3)

where 𝑝shakeL is the probability due to shake and (𝑝DI

L ) is the probability of direct ionization

(DI) due to Coulomb interaction. Introduction of the 𝑝DIL term adds complexity to the

situation since it depends on both particle energy and mass. The proton induced KL1

intensity can be calculated for proton ionization using a method outlined by Kavčič et

al. [92]. In their work, Equation 4.3 is written in the equivalent form:

𝑝expL = 𝑝SCA

L + 𝑝shakeL (4.4)

where 𝑝expL is extracted from experimental tabulations and has the same meaning as 𝑝L

and 𝑝SCAL is the Coulomb ionization probability calculated using a semi-classical approxima-

tion (SCA) approach which is equivalent to 𝑝DIL .

78

The Kavčič method makes use of an extensive WDXRS dataset of KL1/KL0 X-ray ratio

measurements performed using proton ionization at varying energies. Using these data

with a series of calculations, their method was used to predict the X-ray intensity ratio

(rX = X(KL1)/X(KL0)) of the satellite lines to primary peak that would arise in 3 MeV

proton ionization. The numerical steps of this process are many but the procedure is well

documented by Kavčič et al. [92]. Rather than present the full details of the calculation,

only the key steps in the calculation and slight differences in values adopted to cater to light

elements (Mg, Al and Si) are outlined below.

The first step of the process is to consider that observed spectral X-ray satellite intensity

ratios rX, found in literature, are the end result of the X-ray emission process. This ratio does

not directly reflect the initial KL1 to KL0 vacancy probability ratio (rV) = 𝜎(KL1)/𝜎(KL0))

of the atom as several processes can affect how this ratio translates into rX. The first is

that a finite probability exists that the electron shell may undergo a fast Coster-Kronig

re-arrangement of the L spectator vacancy, prior to primary or satellite X-ray emission.

In this process the L vacancy can be promoted to a higher unoccupied orbital or to the

continuum. This returns the atom to a configuration corresponding to diagram line emission

and ultimately reduces the intensity of the effects introduced by the KL1 multi-vacancy state

de-excitation. The second effect is that the presence of the L spectator vacancy can increase

the fluorescence yield parameter. In this situation, ωKL1 > ωK where ωKL1 is the K X-ray

fluorescence yield corresponding to an atom with an initial KL1 multi-vacancy. For light

elements, ωKL1 is approximately 10–14% larger than ωK. This increases the relative intensity

of KL1 satellites with respect to K diagram lines, which would increase the overall X-ray

yield. This counteracts the first effect to some degree. The task at hand is to determine the

extent of these two effects and how rX relates to rV. Only then can a correction be applied

to this work. The equation relating rX to rV is given as:

𝑟X =𝑟V(1−𝑅)

1 + 𝑟V𝑅

ωKL1

ωK(4.5)

where R is the factor used to correct for the rearrangement process. This factor has the

expression:

79

Z 𝑟V 𝑟X 𝑟XT Δ𝑟X 𝑟X

T ·Δ𝑟X

12 14.2± 1.4 14.8± 1.5 12.9± 1.3 11.7± 2.0 1.50± 0.3013 11.2± 1.1 11.7± 1.2 10.5± 1.0 12.5± 2.4 1.31± 0.2814 8.2± 0.8 8.5± 0.8 7.8± 0.8 12.2± 2.2 0.96± 0.19

Table 4.1: Columns from left to right correspond to, element atomic number, calculatedinitial vacancy ratios (𝑟V), calculated X-ray intensity ratios (𝑟X), the KL1 X-ray intensitiesrelative to the total X-ray intensities (𝑟X

T=X𝐾𝐿1 :X𝐾𝐿0+𝐾𝐿1), average differences in X𝐾𝐿1

intensity between oxides and pure elements extracted from literature (Δ𝑟X), and the producteffect on the total X-ray yield (𝑟X

T ·Δ𝑟X). All values in this table are given as percentages.

𝑅 =3∑︁

𝑖=1

𝑤𝑖(ΓL𝑖(𝑅) + ΓL𝑖

(𝐴))

ΓL𝑖+ ΓKL1

(4.6)

where 𝑤𝑖 are weighting factors, ΓL𝑖(𝑅) are radiative L-shell widths, ΓL𝑖

(𝐴) are the L-

shell Auger widths, ΓL𝑖are the total subshell widths and ΓKL1 is the width of the K shell in

the KL1 multi-vacancy configuration.

Values recommended by Kavčič et al. [92], for each of the variables in Equations 4.5

and 4.6, were substituted for this calculation. However, some exceptions exist due to lack of

data available in the literature for the light elements of interest. The ratios of the ωKL1/ωK

values for each of the Mg, Al and Si elements were derived exclusively from the interpo-

lation of the calculations of Tunnell and Bhalla [161]. Weighting factors (𝑤𝑖) were taken

from the work of Mauron et al. [106]. The initial vacancy ratio (𝑟V) was calculated using

Kavčič’s method of extracting values from their Figure 4. Their figure was a plot of proba-

bility (𝑝expL /𝑝SCA

L ) vs. proton reduced velocity (v/vL), constructed using the data of previous

satellite measurements. Most of these measurements were made with heavier elements and

lower photon energies than that corresponding to this study. As such, an extrapolation of

data in their figure was necessary.

The 𝑟V and 𝑟X ratios calculated using the Kavčič method are presented in Table 4.1. As

seen in the table, 𝑟V and 𝑟X are approximately equal with the latter being only about 4–5%

larger. The calculated 𝑟X value for Si at 3 MeV is in agreement with the measured Si value of

Kavčič (8.5± 0.8) [91]. The same 10% relative uncertainty was applied by Kavčič [91] to his

value. This uncertainty was applied to the ratios calculated in this work. The magnitude of

80

this uncertainty was believed to provide a meaningful measure of probable error introduced

by certain aspects of the calculation, such as the extrapolation procedure used to derive 𝑟V.

The 𝑟X ratios are also calculated in terms of the total relative satellite intensity (𝑟XT), ie.

X𝐾𝐿1 as a ratio to X𝐾𝐿1 plus X𝐾𝐿0 . This value represents the satellite contribution present

in the PIXE spectra collected in this work.

To calculate chemical effects on 𝑟XT , the WDXRS KL1 satellite works mentioned earlier [6,

45, 55, 56, 64, 98, 112, 133, 155, 162] were used. Each compilation provides experimental

satellite relative intensities for light elements in pure and oxide forms. Their results for Mg,

Al and Si have been used to determine the relative differences in 𝑟XT that exist due to bonding

environment, ie. elements bonded to other like elements vs. elements bonded to oxygen.

The average increase in ωK calculated for each element is presented in Table 4.1 under the

heading Δ𝑟X. In general oxide satellite yields are about 12% higher. Uncertainties here

are given as the standard deviation spread making use of the number (N ) of data points

for Al (𝑁 = 6) and Si (𝑁 = 5). For Mg, only two reliable works were available and so

an uncertainty similar to the other elements was applied. The product result in the final

column of Table 4.1 represents the increases that must be applied to the predicted oxide

yields to correct for differences in 𝜎Z(𝐸)ω𝐾 . These values are on the order of a percent and

are expected to provide a satisfactory correction of chemical effects on yield.

Two of three parameters of Equation 4.1, that are not shared in common, are the stopping

power and the K X-ray production cross-section. The uncertainties and effects introduced

by these two parameters have been carefully estimated so that they do not detract from the

accuracy of this work. The approach adopted here appears to provide a means of comparing

the last remaining FP not shared in common, ie. the MAC. At 1–2 keV energy, in the

light elements Mg, Al and Si, the XCOM and FFAST MACs differ by about 10%. This

FP therefore represents the largest source of uncertainty and is the object of analysis. The

objective is to determine which MAC database, XCOM or FFAST, provides the smallest

difference between the element and oxide 𝜖a(ℎ𝜈)-values.

81

Purity (%)

Target Manufacturer Quoted GUPIX Contaminants

Mg Kamis Inc. 99.95 verifiedAl Kamis Inc. 99.999 verified surface SiO2

Si Kamis Inc. 99.999 verifiedMgO SPI Supplies 99.99 99.97 0.03 Ca & FeAl2O3 recycled sapphire window 99.95 0.05 SiSiO2 GE 214 fused silicate 99.998 verified

Table 4.2: Target origins, purity levels and PIXE analyses of trace elements.

4.4 Materials and methods

4.4.1 Sample preparation

Thick targets of Mg, Al and Si pure elements, and the corresponding oxides MgO, Al2O3 and

SiO2, were acquired from various sources. The reported purities (by wt.), where available,

are given in Table 4.2. The results of purity tests performed using PIXE measurements,

interpreted through the GUPIX [107] program are also presented in the table. Attention

was primarily focused upon the possible Ca, Ti, Fe and Cu contaminants, each of which

may influence the results due to their higher Z values. The PIXE detection limts (by wt)

were typically: Ca (60–230 ppm), Ti (50–280 ppm), Fe (40–110 ppm) and Cu (30–130

ppm). For targets that were verified to match the reported purities, no detectable levels

of additional trace element contaminants were observed. In the MgO target, traces of Ca

and Fe were apparent; inclusion of these when evaluating Equation 4.1 had less than a

0.1% effect on predicted yields. The presence of Si peaks in the spectra from the Al target

was also observed. This has been attributed to trace amounts of SiO2 deposited on the

surface from the SiO2-based polishing method used in the finishing process. SiO2 deposits

appear primarily in small surface troughs etched by the SiO2 grains. Raster scanning over

the sample surface minimizes the apparent SiO2 contribution. It is likely that most of the

emergent Al K X-ray intensity comes from the flat regions where little to no SiO2 is found

because in the troughs, the Si concentration is high but the surface density of the troughs

is very small relative to the smooth and clean Al regions.

82

Preliminary experiments on elemental targets having ±5 𝜇m of surface height variation

were found to give inconsistent X-ray yields per unit of beam charge. This may be expected

when using very low energy X-rays that are collected at an angle with respect to the surface

plane [24]. When the surface is rough, it may be thought of as a plane of alternating

mountains and valleys. For X rays exiting at 45∘ relative to this rough surface, some will

have to traverse greater distances in the material before emerging from the metal. Those

having to pass through surface mountains will likely experience greater overall attenuation

which would reduce X-ray yield. This produces variations in observed X-ray yields which

is unfavourable to accuracy and reproducibility of the experiment. The metal targets were

therefore polished to <1 𝜇m to eliminate this problem. Polishing involved the use of fine

grit sandpaper of consecutively decreasing grit coarseness. The final step implemented a

mechanical buffing to near mirror finish. The crystalline oxide targets did not require further

surface preparation. The issue of the slight contamination of the Al target with surface SiO2

was resolved by implementing a new polishing technique, described in detail in Appendix A.

However, due to complications with the accelerator operation, a new set of measurements of

the Al target were not performed. Still, the current Al results are not expected to detract

from the accuracy of this measurement.

The possible reactivity of pure Mg to water vapour in atmosphere was considered

as a source of sample surface contamination. The production of a substantial layer of

Mg(OH2)/MgO, as a by-product of the oxidation process, would ultimately change the

matrix to an unknown mixture of pure Mg and Mg hydroxide/oxide and could produce

additional systematic uncertainties in the yield measurement. Fortunately, the oxidation

layer produced is expected to be very thin (<5 nm) [102] when compared to the 20 𝜇m

X-ray production depth that contributes 90% of the observed X-ray signal. Therefore, ox-

idation is not expected to be of concern. Furthermore, care was taken to have the metal

targets carbon-coated immediately following polishing and to store them in a humidity-free

environment at all times as a means to slow the oxidative process.

83

4.4.2 PIXE measurements

The Guelph micro-PIXE facility that was used for the measurements has been described

elsewhere [156]. Targets were positioned at 45∘ relative to both the incoming 3 MeV proton

beam and outgoing X-ray direction. They were viewed by an optical microscope with a

magnification factor of 300, and an invariable focal length. Target position was controlled

by computer-driven stepping motors. Bringing the target into focus ensured that the target-

detector solid angle was maintained for all measurements. The beam spot on the target had

dimensions of about 5×10 𝜇m2. In order to ensure that errors did not arise from positioning

the beam on an inclusion or a defect, the beam was rastered during data acquisition over

an area 125× 125 𝜇m2.

X-ray acquisition was performed using a Ketek AXAS-VITUS SDD with a resolution of

144 eV (FWHM @ 5.9 keV). The SDD has a 9 𝜇m Be window, a Zr collimator with 3 mm

aperture, and a crystal thickness of 450 𝜇m. The detector was located 29 cm from the target

with the entire distance to the detector face held under high vacuum. A shaped linear pulse

is generated by the SDD in response to a detection event. The SDD default shaping time

of 2 𝜇s was used to generate pulses

All measured X-ray yields were normalized to acquired beam charge, which was measured

using a Red Nun charge integrator coupled to the in-house MCA. To provide conduction of

beam charge from the insulator material targets, thin carbon layers were deposited using

an Edwards Auto 360 carbon-coating device. Uniformity of carbon layer thickness, between

element and oxide target for each pair, was achieved by coating both targets simultaneously.

Layer thicknesses of 60 to 80 nm were applied, and the uniformity of the coating technique

between target pairs was validated to ±5 nm using atomic force microscopy (Agilent Tech-

nologies N9 451A). Additional details on the application of carbon layers to targets and AFM

measurement techniques are summarized in Appendix B. The target locations of each PIXE

measurement were randomly selected and were changed between each measurement. The

proton bombardment of targets produces secondary electron scattering at the surface of the

sample. If these electrons are able to escape the surface, the integrated charge may become

compromised. As a preventative measure, the original PIXE chamber design was made to

84

incorporate a suppressor shield held, at a fixed negative voltage, positioned about 1 cm in

front of the target. Secondary electrons scattered outward from the target are thus repelled

backwards and are collected by the carbon coating on the target surface. Small holes for

each of the incoming proton beam, microscope view and outgoing X-rays are present in the

supressor shield.

4.4.3 Magnetic assembly used to deflect scattered protons

A concern with the PIXE method is the entry of target-scattered protons into the detector.

This can cause degradation of spectrum quality, during acquisition, and impart long-term

damage to the detector crystal. [113] To eliminate this problem, a static magnet assembly was

constructed to deflect scattered protons from the path to the SDD. This assembly consisted

of an arrangement of two 5 cm long Nd-Fe-B bar magnets, purchased from Dexter Magnetic

Technologies. These were held at 6 mm separation, inside a 5 cm long C-shaped yoke of

Supranhyster 50 (50% Ni, 50% Fe) metal. This device was constructed with a design similar

to the C-shaped device built by Calligaro et al. [17]. The C-shaped design was adopted as it

is thought to help reinforce the natural field of the magnets. The Supranhyster 50 material

was used due to its very high natural magnetic permeability (𝜇 = 104 − 105) [11] which

introduces a very high field strength in the material. This is thought to assist in stabilizing

the field as Supranhyster 50 has a tendency to remain strongly magnetized even after the

Nd-Fe-B magnets are removed [11]. As a comparison, the permeability of a common ferrite

magnet is on the order of about 100–600 [11].

The magnetic field strength present between the magnets at 6 mm separation was about

0.7 Tesla. This field can laterally deflect protons moving with a velocity of about 0.1c

by several mm in the 5 cm travel distance through the magnet. The front end of the

magnetic assembly was placed at a distance of about 4–5 cm from the target surface. The

magnetic field changes the trajectory of the protons emerging from the device such that in

the remaining ∼ 20 cm of travel toward the detector, they are deflected well outside of the

solid angle incident on the detector face. A charged particle detection test revealed that

implementation of the filter reduced the possible scattered proton flux from 1800 to 0.1 s−1.

The details of this test are described in Appendix C.

85

X rays

3 MeV H+ beam

SDD

Scattered H+

-25V

+25V

SDD linearamplifier

Ortec 473A fast negative logic

Lecroy 222 NIMextended logic

B. FischerHV amplifier

In-houseMCA

Targetblock

Micr

osco

peNd-Fe-Bmagnets

Upstream beam deflection platesElectron suppressorguard

Ortec 474 timingfilter amplifier

45o

ND 579ADC(1)

(2)

Figure 4.1: Basic schematic diagram of PIXE chamber setup and SDD-electronics flow chartsshowing modules involved in (1) spectra acquisition by the MCA and (2) electrically-drivenbeam deflection.

4.4.4 Beam blanking system

An electronic beam deflection system was implemented to eliminate the need for electronic

dead time corrections and to minimize pulse pile-up in the spectra [156]. This system

functions to turn off the beam immediately upon the arrival of a photon event in the SDD.

This provides both the detector and ADC the time required to process the first pulse fully

before accepting another. A schematic of the PIXE chamber setup and electronics system

used to deflect the beam is illustrated in Figure 4.1. The SDD outputs both a shaped

amplified pulse, fed into the ADC, and a pre-amplified pulse that is sent to a sequence of

electronics used to charge plates upstream from the chamber. The pre-amplified pulse is fed

into a linear amplifier (Ortec 474) which is fed into the constant fraction discriminator (Ortec

473A) used to produce a fast, short duration, negative logic pulse. This negative pulse is

used to trigger an extended logic pulse from a Lecroy 222 nuclear instrument module. Its

pulse extension duration is set to the time required for both the SDD and ADC to finish

processing a pulse. The Lecroy unit has a 5V output. This output is fed into a high voltage

86

1 2 3 4 51 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

W i t h d e f l e c t i o n N o d e f l e c t i o n

Coun

ts

P h o t o n E n e r g y ( k e V )Figure 4.2: Spectral effect produced by incorporation of a beam deflector. Grey dots rep-resent a spectrum acquired in the absence of beam deflection. Black dots represent thespectrum acquired with the beam deflector activated.

(HV) amplifier that was set to generate two 25 V logic pulses, each having opposite polarity.

Each of the pulses were sent directly to two plates housed inside the upstream beamline

tube. This drives the deflection of the beam for the time required.

The effect of deflecting the beam to eliminate pile-up is illustrated in Figure 4.2. Both

spectra in the figure were produced via PIXE measurements performed on a pure Si target

and have been normalized to the same maximum peak height. The primary Si K X-ray

peak is seen at 1.74 keV. When the pile-up prevention system was not active, a plateau was

produced (grey dot spectrum) on the high energy side (1.74 < plateau < 3.48 keV) of the

Si K X-ray peak. The plateau corresponds to two X-ray events being counted as one larger

energy pulse. If the two X-ray events have only a slight overlap in their arrival time at the

detector, then the superposition of the two pulses produces a new pulse that has less than

double the energy of a Si K X ray. This had the effect of producing a count in the plateau

region, at less than 3.5 keV. If however, the detection of both occurs much closer in time, the

87

count may be registered at double the Si K X-ray peak energy. This count is placed in the

first-order pile-up peak centred at 3.5 keV. This peak is present in both the deflected and

non-deflected spectra. The beam deflection device can respond fast enough to shut down

the beam to elminate the plateau region but cannot eliminate the counts appearing in the

pile-up peak at 3.5 keV as these events occur too close in time to be resolved by the existing

system.

A lag time of approximately 600 ns from the pre-amp pulse production to 90% of the full

25 V height existed, during which time pulses following the intial pulse could not be rejected

from the spectrum. Based on Poisson statistics, using an average experimental count rate

of 2000 s−1, this represents a 0.12% chance of recording a pile-up event, which was deemed

to be an acceptable level for the present measurements. However, it appears likely that the

total lag time is longer than 600 ns since this estimate does not account for the time needed

to produce the pre-amp pulse upon arrival of the photon in the detector wafer. A slightly

longer duration of 1.26 𝜇s was calculated for the present system’s peak resolving time (see

Section 5.3.2 for details.). Some of this calculated resolving time likely includes this extra

pre-amp processing event.

Despite these precautions, which in principle should render the X-ray yield per unit

charge independent of counting rate, a small count rate dependence of the experimental

yield was observed. This count rate dependence was observed by plotting total counts from

a Si spectrum (primary peak, including tail + 2×pile-up peak) against the count rate for a

set of spectra collected in the range of 1000–5000 s−1. In principle, since beam deflection

is used, the count rate dependence should be flat but, the yield was found to vary in a

continuous trend that differed by 3% end-to-end. The count rate dependence on yield is

displayed in Figure 4.3 A 4th order polynomial was used to describe the trend. In an earlier

attempt to characterize the count rate dependence, a similar trend was observed near 2000

cps in which the data produced a plateau similar to that observed in the present figure.

Therefore, a polynomial with the complexity necessary to produce a plateau was chosen in

this fit. Due to the presence of this count rate dependency, all yield data collected from the

six targets were normalized to the same count rate.

88

1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 08 . 1 5 x 1 0 5

8 . 2 0 x 1 0 5

8 . 2 5 x 1 0 5

8 . 3 0 x 1 0 5

8 . 3 5 x 1 0 5

8 . 4 0 x 1 0 5

8 . 4 5 x 1 0 5

X - r a y y i e l d d a t a p o l y n o m i a l f i t

X-ray

yield

(cou

nts)

C o u n t r a t e ( s - 1 )Figure 4.3: PIXE yield results from pure Si measurements taken at varying count rates, plot-ted to derive a correction relationship for residual system count rate dependency. The fol-lowing approximate relationship was found: 𝑌 (𝑍) = (6.954×10−10)𝑐𝑟4−(7.971×10−6)𝑐𝑟3+(3.066× 10−2)𝑐𝑟2 − (5.125× 10−1)𝑐𝑟 + 8.684× 105, 𝑅2 = 0.971.

4.4.5 Spectra acquisition and fitting

Seven or more spectra from each target were recorded with approximately half a million

counts in the main peak and an energy FWHM resolution of 70–80 eV for 1–2 keV X rays.

A comparison of pure element and oxide spectra for Si is shown in Figure 4.4. The only

significant difference is in the relative contributions of Bremsstrahlung background.

The 0.3–9 keV region of each spectrum was fitted using the in-house software fitting

routine (GUFIT). Greater details on the fit are included in Appendix D but key points are

provided here in brief. The main component of the response function for each of the Kα

and Kβ X-ray lines, was a Gaussian function. The incomplete charge collection region on

the immediate low energy side of the primary peak was fitted with an exponential tail. A

DE function was used to provide a skewed Gaussian-like fit of the very small KLV radiative

Auger (RAE) feature that sits on the low energy tail. The skewed shape of this empirical

89

1 2 3 4 5 6 7 8 91 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

S i S i O 2

Count

s

P h o t o n e n e r g y ( k e V )Figure 4.4: Spectra comparison of Si and SiO2 normalized to the same primary peak height.The largest peak corresponds to the Si Kα and Kβ lines and the peak at 3.5 keV representsthe pile-up of two Si K X-ray lines.

function, toward lower energy, provides a reasonable approximation of the shape identified

with this feature in the early spectroscopy work by Åberg and Utriainen [2] and Siivola et

al. [145]. The background continuum, due mainly to secondary electron Bremsstrahlung,

was fitted with a 4- or 6-parameter exponential. Finally, a decreasing exponential function

was included at extreme low energy to represent events resulting from the injection of photo-

and Auger electrons into the detector wafer from both the Si contact and any dead layer.

Details of the functional forms of the lineshapes fitted to these spectra and treatment of the

KLL RAE feature is covered in Appendix D.

The individual components of this model were first fitted manually to achieve a visually

acceptable initial fit, and this preliminary step was followed by an automatic non-linear

least-squares fit. All these components are shown in Figure 4.5. Reduced 𝜒2 values were in

the range of 0.8 to 1.5. The characteristic X-ray yield of the target element was then taken

as the sum of the counts in the primary Gaussians, the exponential tails, the RAE feature

90

1 2 3 4 5 61 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

S i K α, K β a n d 1 s t o r d e r p i l e - u p B r e m s s t r a h l u n g c o n t i n u u m s u r f a c e e l e c t r o n s d i f f u s i o n t a i l R A E X r a y s d a t a f i t

Co

unts

P h o t o n e n e r g y ( k e V )

- 505

Re

sidual

s

Figure 4.5: Fitted Si K X-ray PIXE spectrum showing data, fit functions and total fit(𝜒2 = 0.86). Fit residuals (𝜎) are indicated in the top panel.

and double that in the pile-up peak.

4.5 Results

Because apparent changes in the detection efficiency 𝜖a(ℎ𝜈) caused by change of target were

the focus of this study, it was imperative to ensure that, for a given target, 𝜖a(ℎ𝜈) was

highly reproducible. Calibration measurements were performed on the same Si target for all

target pair measurements, whether on Mg, Al or Si. Over a six-month period, the Si counts

per unit charge exhibited a standard deviation of 0.6%, which may be compared with the

expected statistical standard deviation of repeat calibration measurements of 0.14%, based

on the square root of the total peak area. For the data presented in this section, variations

did not exceed 0.4% at one standard deviation. Presumably, some of this spread is due to

the inherent fit uncertainty which, for the Kα peak, did not exceed 0.2%.

The values of 𝜖a(ℎ𝜈), based upon use of XCOM attenuation coefficients, are presented

91

𝜖a(ℎ𝜈) (10−5 sr)

Z Element Oxide Δ𝜖a(ℎ𝜈) (%)

XCOM 12 6.00± 0.02 5.68± 0.03 5.5± 0.613 8.56± 0.03 8.12± 0.03 5.3± 0.614 10.50± 0.04 10.12± 0.03 3.7± 0.5

FFAST 12 5.33± 0.02 5.31± 0.03 0.4± 0.613 7.79± 0.03 7.59± 0.03 2.6± 0.614 9.56± 0.04 9.47± 0.03 0.9± 0.5

Table 4.3: Average 𝜖a(ℎ𝜈)-values of elements and oxides and the percentage differences(Δ𝜖a(ℎ𝜈)) calculated using the XCOM and FFAST MACs.

Relative uncertainty (%)

𝜖a(ℎ𝜈) (±1𝜎) 𝑆(𝐸) carbon (±5 nm) 𝜎Z(𝐸)ωK quadrature sum

Mg 0.30 0.20 0.18 0.40MgO 0.37 0.20 0.18 0.18 0.53Al 0.28 0.20 0.14 0.37Al2O3 0.27 0.20 0.14 0.19 0.44Si 0.33 0.20 0.07 0.40SiO2 0.19 0.20 0.07 0.18 0.33

Table 4.4: Summary of propagated uncertainties used to derive 𝜎(Δ𝜖a(ℎ𝜈)). The finalcolumn represents the quadrature sum of the other columns.

in Table 4.3 as average values over seven or more measurements. These numbers have been

corrected for the chemical effects, as outlined in Subsection 4.3.2, and the uncertainties

represent the combination of the statistical spread with that of the stopping power, carbon

coating (±5 nm) thickness, and chemical effects on the KL1 yield. These uncertainties are

summarized in Table 4.4 with the last column representing their quadrature sum.

The data show an increase with increasing Z, which reflects increasing detector efficiency

with energy. The fractional difference between the 𝜖a(ℎ𝜈)-values, derived from each element

and oxide pair, at 4–6%, is clearly significant relative to the uncertainty.

When the FFAST MACs for oxygen, Mg, Al and Si were substituted in place of the

XCOM values, the fractional difference in 𝜖a(ℎ𝜈)-values between elements and oxides is

significantly reduced. Since the FFAST coefficients are generally smaller than those of

XCOM, the FFAST-based X-ray yield involves a lower calculated attenuation of outgoing

92

X rays suggesting that the predicted X-ray yield is larger. This results in smaller 𝜖a(ℎ𝜈)-

values. The apparent energy-dependence of these values indicates an energy-dependence

of the detector’s intrinsic efficiency. This reflects possible incomplete charge collection of

events absorbed near the detector’s contact as well as attenuation in the dura-Be detector

window. The internal 30 mbar nitrogen atmosphere column (length = 1 mm) between the

dura-Be window and Si active crystal also contributes a small amount of attenuation.

4.6 Discussion

4.6.1 Present data vs. theory

The goal of this work was to examine differences between the XCOM and FFAST theories,

on the order of a few percent, and care was exercised to remove the influence of systematic

uncertainties. Furthermore, the methodology adopted in comparing differences in absolute

efficiencies prevents any uncertainties associated with most of the fundamental parameters

used in Equation 4.1 from factoring into the total error in the results. The uncertainties that

remain are due to the spread in repeat measurements, reported as one standard deviation,

and those associated with the three remaining parameters not shared commonly: stopping

power, X-ray production cross-section and the MAC. The stopping power uncertainty was

set as (0.20%). The uncertainty in 𝑟XT from Table 4.1 cancels since this uncertainty represents

the Kavčič method-based prediction of satellite intensity that was applied equally to both

targets. However, the uncertainty associated with 𝑟XT ·Δ𝑟X applies to the difference in the

results from the two targets since this accounts for relative differences corresponding to

chemical effects. As this value concerns only the relative differences in the satellite yield due

to chemical effects, this uncertainty was therefore applied to only the oxide target data. It

presents only a small additional level of uncertainity (<0.20%) on the total oxide yield. Each

of these uncertainties have been added in quadrature with the statistical spread (±1𝜎) to

produce the final uncertainty reported in the results. Since the stopping power uncertainty

has been shown to be small and corrections for the increase in X-ray yield, due to increases

in the cross-section have been applied, one can infer that the 𝜖a(ℎ𝜈)-value differences seen

in Table 4.3 are due to the MACs and thus to the L and higher shell 𝜎PE theory being used.

93

1 . 6 8 1 . 7 0 1 . 7 2 1 . 7 4 1 . 7 6 1 . 7 8 1 . 8 0 1 . 8 2 1 . 8 42 5 0

3 0 0

3 5 0

4 0 0

4 5 0

X C O M F F A S T P T B

MAC (

cm2 /g)

P h o t o n e n e r g y ( k e V )Figure 4.6: Experimental [75] Si MAC values plotted with XCOM and FFAST MACs belowthe Si K edge.

Since the element oxide difference in the 𝜖a(ℎ𝜈) results are much less overall when using

FFAST over XCOM, these results show greater support of the FFAST predictions of 𝜎PE.

4.6.2 Comparison of light element experimental works with theory

In light of these results, it is helpful to compare the two theories with modern existing data

for light elements. The focus is on the low energy side of the K edge for Mg, Al and Si where

XCOM exceeds FFAST by 10–11% in all cases. This trend is also true for heavier elements,

i.e. 3d transition metals [61, 89, 126, 159], though for these the discrepancy drops to around

5% below the K edge. The Si MAC data from PTB, presented in Figure 3.3 of Chapter 3

are reintroduced in Figure 4.6 with specific emphasis shown on data existing below the

K edge. This figure shows data that agree to within 2% with the predictions of FFAST,

but are about 8% less than XCOM. The stronger agreement of the present PIXE results

(Δ𝜖a(ℎ𝜈) = 0.9±0.5) for Si when using FFAST MAC’s is supported by the observation that

PTB MAC data agree more closely with FFAST, below the K edge.

94

1 . 4 4 1 . 4 6 1 . 4 8 1 . 5 0 1 . 5 2 1 . 5 4 1 . 5 63 0 0

3 5 0

4 0 0

4 5 0

5 0 0

X C O M F F A S T P T B

MAC (

cm2 /g)

P h o t o n e n e r g y ( k e V )Figure 4.7: Experimental [8] Al MAC values plotted with XCOM and FFAST MACs belowthe Al K edge.

A similar, earlier dataset [8] from Bessy exists for Al, with experimental data extending

down to 1.440 keV. A comparison of the two theoretical predictions with these data is shown

in figure 4.7. Unlike the Si data, this set shows closer agreement to the XCOM predictions,

though the data dip towards the FFAST predictions just above the Al Kα X-ray energy.

This effect is similar to a trend observed below the K edge for four previously mentioned

experiments: Ag [159], Mo [89], Cu [61] and Zn [126]. In each case, the data in this energy

region agree with XCOM but drop to partway between the two theories, into slightly closer

agreement with FFAST, in the vicinity of the Kα energy. However the PTB Al data were

acquired using a very thin foil, one that was a full order of magnitude smaller than that used

for the PTB Si MAC measurements. The corresponding (𝜇𝑡) of their Al foil was 0.07 which

is well outside the extended range criterion (0.5 < 𝜇𝑡 < 6). This may have caused issues

with statistical accuracy of the data collected in the energy region. Alternatively, there is

the possibility that slight errors in determining the Al foil thickness, given the challenges

outlined in Section 4.3, have manifested as an overall systematic error.

95

4.6.3 Attenuation by oxygen in oxide targets

The discussion so far has focused upon the attenuation coefficients for photons having energy

less than the K edge energies of the elements Mg, Al and Si. However the oxygen constituent

in the oxide samples accounts for over 75% of the total attenuation of the material. In the 1–

2 keV range the XCOM 𝜎PE’s for oxygen are consistently 5–6% higher than those of FFAST,

but there is a lack of modern data to compare with theory. Ultimately, this work is a test

of the two theories, for not only the three cations, but also for oxygen. The specific method

used in this study does not permit a direct measurement of either the element of interest or

the oxygen MAC as it relies on the combination of both simultaneously to derive a result.

Indeed, the values of the coefficient for both elements could conceivably be selected such

that both are in error but nevertheless conspire to produce a zero difference in the efficiency

constant. However, in light of the lack of accurate data in this region, the method used

here allows for the comparison of the two available theories, and shows that the FFAST

values provide superior agreement with the PIXE results. Furthermore, the preference for

the FFAST coefficients, over XCOM, is supported by the high quality PTB data for Si.

4.6.4 Comparison of Scofield, Cromer, and other theoretical treatments

In this work, only the XCOM and FFAST databases were analyzed for accuracy. Several

other theoretical or semi-empirical MAC databases do exist and were investigated for pos-

sibile use in this work. A collection of databases was provided by Mr. Pierre Caussin of

the Bruker Corporation (P. Caussin private correspondence with C. M. Heirwegh). The

databases of Elam et al. [52], Cullen et al. [123], Ebel et al. [50], De Boer [12], Henke et

al. [72] and Leroux and Thinh [101] were provided in an interpolation program, called Ab-

sorptionExplorer. This program provides the option to interpolate values of MACs of various

materials, from each of the six databases, using inputted photon energies. The Henke et al.

database may also be accessed from The Center for X-ray Optics (CRXO) online interface,

supported by the Lawrence Berkeley National Laboratory (LBNL) [65]. The MAC values

for Mg Kα1 in Mg, Al Kα1 in Al and Si Kα1 in Si have been calculated from this program

for each of the six databases. These results are presented in Figure 4.8 for comparison with

96

M g A l S i3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

5 5 0

F F A S T X C O M E l a m C u l l e n E b e l D e B o e r H e n k e L e r o u x & T h i n h

Kα1 M

AC (c

m2 /g)

E l e m e n t M a t e r i a lFigure 4.8: MAC values reported for the Mg, Al and Si Kα1 energies in the respective Mg,Al and Si materials using the database values of FFAST, XCOM, Elam [52], Cullen [123],Ebel [50], De Boer [12], Henke [72] and Leroux and Thinh [101]. Slight differences in thetheoretical treatments used in database construction are indicated in the figure legend.Bolded text indicates the use of Scofield [142] 𝜎PE’s and italicized text indicates pre-Scofieldformalism.

the corresponding values derived from the XCOM and FFAST databases.

The differences between the six databases, shown in Figure 4.8, are subtle, if present at

all. The Elam, Cullen and Ebel databases show MACs that are largest in value and do not

show any divergence from the Scofield values represented by XCOM. This is expected given

that these three databases draw their MACs directly from Scofield [142]. The three sets

of data that sit very slightly under the top lines correspond to the databases of De Boer,

Henke and Leroux and Thinh. Each of these draw their estimates from various theoretical

compilations that pre-date Scofield’s calculations. As well, De Boer and Leroux and Thinh

draw some of their data from Henke who manufactured his database from a combination

of theoretical and experimental compilations. However, for the light element, low energy

region represented in Figure 4.8, the experimental data summarized by Henke is minimal.

97

Therefore, the values of these three databases are represented primarily by the pre-Scofield

theories. The database that shows the greatest divergence from each of AbsorptionExplorer

databases is FFAST. The difference represented here in many ways draws attention to the

uniqueness of Chantler’s approach. His exclusive use of Cromer’s approach may be respon-

sible for these large differences.

The similarity between the presented MACs of the six Caussin databases with XCOM

suggests that little would be gained in using their values in the analysis in this work. Only

the FFAST database shows significant difference from the rest of the values. Therefore, the

present analysis was restricted to the comparison of XCOM and FFAST.

On a fundamental level, the theories of Scofield and Cromer are similar in that they

are both constructed using DHS formalism. However, several differences in their treatment

may be responsible for the differences seen in their tabulations. Among these differences

are Cromer’s use of Bearden’s tabulated binding energies [7], while Scofield relied on cal-

culated energies. As noted in Section 4.1, different exchange potentials were used; Scofield

made use of a Slater’s potential while Cromer used the Kohn-Sham potential. Additionally,

each author supplied a unique treatment of perturbation terms [C. T. Chantler, private

correspondence with C. M. Heirwegh].

Scofield recognized the potential for high accuracy in a full HF approach. In his original

DHS work [142] he provided HF/HS renormalization factors that allow conversion of his

HS tabulations to approximate HF values. This has been denoted as a “restricted HF”

treatment [139]. The difference between these renormalized and un-renormalized values is

generally 5–15%, for lower photon energies [31]. It is of some interest to investigate the

impact that a full HF treatment might have on the results from this work. Application of

the HF renormalization factors to each of his calculated DHS subshell cross-sections have the

effect of reducing the overall attenuation of Mg, Al and Si, below their K edges. For Si, the

magnitude of this reduction is about 8% at 1.5 keV. This puts the Scofield renormalized value

to within less than 2% of the FFAST value. This is a significant reduction when compared

to the 10% difference that otherwise exists between the un-renormalized value and FFAST.

The HF value also finds agreement with the PTB Si data, as it is only approximately 1%

different. Unfortunately, it might be difficult to interpolate the exact MAC value for specific

98

lines, as only 2–3 data points are provided by Scofield below the K edges of Mg, Al and Si.

4.7 Conclusion

The experimental approach used here does not focus on one specific element. Rather, it en-

ables a comparison of the HS predictions of Chantler and Scofield in a specific circumstance.

For Mg, Al and Si, the theories have been tested in a situation where photo-absorption does

not occur in the K shell whereas for the associated oxygen, photo-absortion occurs in all

shells. Care was exercised in this work to remove the influence of systematic uncertainties

and to minimize the influence of other FPs. The results from this study strongly favour

the DHS 𝜎PE’s developed by Chantler. This outcome can be compared with the now well-

established picture for Kα1 energy locations of Cu, Zn, Mo and Ag in Cu, Zn, Mo and Ag,

respectively. At these locations, partial to full agreement with FFAST theory is apparent

and warrants its replacement over XCOM for the three elements, Mg, Al and Si, below their

K edges locations.

99

Chapter 5

Quantification of major light elementsin silicate glass and mineral standardsusing PIXE

5.1 Introduction

In a comparison of PIXE and EPMA methods of materials analysis, PIXE has one advan-

tage of having a much reduced Bremsstrahlung background. Detection of minor or trace

quantities of certain elements is thus more likely with PIXE, than with EPMA. The Guelph

PIXE microprobe can be used to detect, and in some cases, quantify minor and trace ele-

ments (3d transition metals and heavier) in geological materials on the order of several tens

to hundreds of parts per million. This is often accomplished using a thin filter that sup-

presses the lower energy K X-rays emitted by the matrix’s major elements (11 6 Z 6 14).

This is done to prevent the large number of light element X-rays from consuming all of the

detector signal processing time which, in turn, suppresses the already small signal emitted

by trace elements. At present, high accuracy PIXE analysis of major light element and trace

element composition in geological materials has not been fully realized. Determination of

major element constituents, using a calibrated method such as EPMA, is often used prior to

performing minor element quantification from PIXE spectra. It is therefore desirable to seek

a simplified approach that would allow both steps to be performed by PIXE alone. Based

on the results obtained in Chapter 4, calibration of PIXE systems for light element quan-

tification might be attainable by reassessing MACs used in the calculation of composition

in geological materials.

100

This chapter presents a brief overview of the older EPMA method including some of

its characteristics, its longstanding role in the geochemistry community and the calibration

approach of EPMA systems for geological materials analysis. Focus will then be turned

to the PIXE method, its characteristics and current attempts to continue calibration to-

ward quantification of major light element constituents. To this end, PIXE measurements

performed on well-characterized silicate standards will be reported. The data from these

measurements have been analyzed using the comparative efficiency constant approach of

Chapter 4 to determine if quanitification might be improved by using one MAC database

over the other.

5.1.1 A brief history on electron-probe micro-analysis

Several materials analysis methods exist that make use of charged particles as a means of

ionization. These include EPMA, proton beam PIXE analysis and heavy-ion beam PIXE

analysis. Heavy-ion beam PIXE is similar to proton-PIXE but has been less well character-

ized than the latter and beyond acknowledgement of its existence, does not require further

mention.

Following its introduction by Castaing [30], EPMA was widely adopted in the 1960’s

by the geochemistry community. The ability of the EPMA probe to provide focused com-

position measurements of localized homogeneous zones, in heterogeneous samples, is one

of the strengths of this method. Like most X-ray emission analysis techniques, EPMA is

non-destructive. Also, EPMA equipment does not require large amounts of space, when

compared to other ion-beam techniques, and is thus relatively easy to implement in the lab

setting. For these reasons, among others, EPMA systems became a useful tool in geochem-

istry labs.

In his attempt to quantify EPMA measurements, Castaing realized that quantification,

using a theoretical FP approach could be simplified by adopting a reference based approach.

The fundamental idea was to compare X-ray intensities, emitted by a sample, to X-ray

emissions from a known standard [70]. This permitted several FPs, used in the theoretical

approach, to cancel which in turn transformed measurement quantification into a reality.

Development of this method led to the reference-based approach becoming the conventional

101

technique for elemental quantification.

Even with the reference-based calibration approach, EPMA measurements were found

to produce a wide variation of results when attempting to quantify light elements. J. V.

Smith found that efficiency-of-generation calibration curves produced from elements Mg, Al

and Si in pure form differed markedly from those in oxides [149]. This suggested that no

single curve, for a given element could be used for quantification of a measurement of the

same element in any given matrix. Smith could only speculate at the factors responsible for

this observation but he suggested that if the low-energy MACs used were incorrect, then

this could introduce the wide variations seen in his results [149].

In a hypothetical scenario, Smith proposed that if a plethora of calibration curves for

various combinations of light element matrices were formed, then this would allow for com-

position determination with high accuracy. However, such a task would have been an ex-

haustive undertaking. He then proposed a more feasible approach of producing calibration

curves of only primary minerals. In a large undertaking, Smith et al. gathered a collec-

tion of minerals of which the composition had been determined using chemical analysis. In

constructing calibration curves from EPMA measurements performed on each mineral class:

olivine [150], alkali feldspar [152], plagioclase feldspars [128], orthopyroxenes [76], clinopy-

roxenes [151] and garnets [95], Smith et al. demonstrated that the EPMA method could be

applied to geochemical measurements with a 1–5% accuracy.

5.1.2 Micro-probe use in geochemistry

For geological composition analysis, the need to calibrate EPMA systems was the impetus

for the production of a number of geological material reference standards in the 1960’s.

Naturally occuring glasses and minerals were harvested by researchers and members of var-

ious institutions, such as the United States Geological Survey (USGS), for the purpose of

constructing microprobe calibration standards that could be distributed to EPMA labs.

Characterizations of standards have generally been performed using a variety of techniques

including chemical or flame analysis, neutron activation analysis, X-ray fluorescence spec-

troscopy, mass spectrometry and EPMA. The results of these methods, and many others,

used in the characterization of one glass standard (BHVO-1) that has been measured in this

102

work, have been tabulated elsewhere (Table 7, p. 287 of Ref. [59]).

Among the many standards measurements produced, none are perhaps as highly regarded

for accuracy as those of Jarosewich et al., formerly of the USGS [85, 86]. They recognized the

possible danger in measuring a standard’s composition using the same analysis method as

that for which the standard is intended to provide a means of calibration. In particular, they

attempted to avoid producing and compiling measurements incorporating EPMA methods.

Each of the standards reported in their work was therefore measured using chemical analysis

and other non-microprobe techniques. Also, their work was published as a response to a

growing problem in which a number of standards were being classified without rigorous

testing for composition. At times, the nominal chemical formula was used to describe only

the major components. In other cases, the composition may have been measured in detail

but, it was then attributed to other samples of the same type harvested from different

locations. The disadvantage in this approach is evident as the two samples may differ

substantially in major and trace element composition. Jarosewich et al. [86] recognized and

avoided these pitfalls in the construction of their database. As a result, their collection of

standards became a valued commodity in the microprobe community.

5.1.3 PIXE and geological applications

By the early 1980’s the PIXE method was being examined in Heidleberg [13], Los Alamos [130]

and Guelph [15, 16, 66], for its application in geological quantitative work. A major advan-

tage of this method over EPMA is the much reduced Bremsstrahlung background [87] which

improves the sensitivity of the method for measuring minor and trace element quantities.

The original design of the Guelph PIXE micro-probe [23, 27, 156] incorporated many of

the features described in Section 4.4 including pulsed beam deflection, microscopic imaging,

thick-target correction analysis and micro-focused beam capabilities. It did not, however,

contain a magnetic deflection device to eliminate scattered proton flux. In collaboration

with G. K. Czamanske of the USGS in 1993, the Guelph micro-probe was evaluated for

its performance in detecting and reproducing measurements of minor and trace element

quantities found in a suite of silicate glass and mineral standards [43]. A number of these

standards were from the Jarosewich collection. The standards were typically prepared by

103

a glassification melting process to remove mineral zones, common to naturally occurring

geological samples. Given the challenges already present in calibrating the PIXE system,

the presence of a homogenous medium simplifies the task of performing matrix corrections

for proton stopping and X-ray transmission. It also eliminates complexity added when

composition differs depending on measurement location.

In this collaboration, Czamanske et al. [43] determined the detection limits and repro-

ducibility of measuring trace amounts of elements Z > 20 that were not major constituents

in the sample. Major element characterizations were provided by earlier analyses, as is done

in this work. Their method utilized an Al foil to suppress both scattered protons and high

intensity light-element contributions and therefore could not be used to determine major el-

ement constituents. The high degree of reproducibility in their results for elements (Z > 20)

emphasized the potential of the PIXE method for use in the geochemical community. Still,

it was recognized that external methods would be needed to provide major light element

quantification.

In response to this, the Guelph-USGS collaboration assessed the feasibility of major

light element quantification. Modification of the earlier PIXE chamber setup incorporated

a Noran Si(Li) detector designed to be resistant to damage from scattered proton entry [22].

This provided the necessary ability to detect light element K X-rays in the absence of a

filter. Many of the samples from the 1993 study were once again used. Their compositions

were derived from numerous sources including those of Jarosewich. The approach involved

solving for the H-value efficiency constants of the major light elements. For the H-value

calculations of Mg, Al and Si, Campbell et al. included additional PIXE measurements on

pure targets of these elements and subsequent calculation of the pure target H-values. When

comparing the pure element H-values to those of the same elements in the complex standard

targets, they found that the results differed by an average of 6–8%. No clear explanation for

this effect was offered other than that the presence of the discrepancy pointed to problems

with one or more of the FPs used in analysis.

104

5.2 Objectives and approach

This chapter aims to expand upon the work from the 1997 Guelph-USGS collaboration [22]

by utilizing the current PIXE methodology, described in Chapter 4, along with the knowledge

gained from the results of that study. Using the efficiency constant comparative approach,

the aim will be to compare results obtained when either of the XCOM or FFAST MAC

databases are incorporated in analysis.

PIXE measurements were performed on some of the silicate and mineral standards that

have been used in the earlier USGS collaboration. The criteria applied to the selection of

silicate standards from the available collection, consisted of two main considerations. The

first requirement was to obtain a broad variety of standards, some of which are fundamental

minerals that represent the basic constituents of more complex minerals and geological

materials. To this group, some silicate glasses, namely those that have been very well

characterized, were added.

The second criterion was the preferential selection of those standards that have under-

gone the highest quality composition determination. The standards compiled and measured

by Jarosewich were given the highest rank (#1). Those ranked second (#2) included stan-

dards that have undergone characterization from a multitude of methodologies. For these,

composition measurements made by numerous methodologies are expected to produce an av-

erage that is relatively free from errors introduced by any one method. The lowest rank (#3)

was assigned to compositions determined using older methods having uncertain reliability.

Each standard possesses varying major compositions of the light elements Mg, Al and

Si in oxide form, though for some, Mg, Al or both may be absent. Using PIXE spectra

acquired from the silicate standards, efficiency coefficients were calculated and compared to

those associated with the Mg, Al and Si pure element target measurements. This was done

separately for XCOM and FFAST MAC databases in order to assess their accuracies.

105

5.3 Materials and methods

5.3.1 Selection and PIXE measurement of silicate standards

The silicate glass and mineral standards used in this work have been professionally prepared

by the USGS. This preparation has included glassification, target mounting, polishing and

carbon-coating for electrical conduction. Further preparation for their measurement in the

PIXE chamber has only required the precise co-ordinate determination of the positions of

each grain as viewed under the chamber microscope camera.

The standards selected for these measurements have been summarized in Table 5.1. The

corresponding compositions used in analysis of this work are provided in Table 5.2 and

information in these two tables are linked by the given short name. Concentrations of each

target sum to 100 wt%. Due to the input requirements of the program used in the analysis

of this work, all elemental concentrations were separated from their oxygen constituents.

A large amount of effort was dedicated to locating the most reliable reports of the

mineral composition. In cases where multiple concentration values were available for a given

target, selection was based on those values that matched the unreferenced concentrations

values used by Dr. G. K. Czamanske during the earlier USGS collaboration [22]. This was

done keeping in mind the initial criteria described for ranking the choices of selection. Of

the targets selected, there were four basalt glasses, two rhyolitic glasses, two olivines, two

feldspars, an albite and a pyrobole.

Three or four PIXE measurements were acquired of each selected target. When the

silicate grain was large enough, spot location was moved between measurements. Raster-

scanning was used in all measurements. The most common setting was the 125× 125 𝜇m2

raster area. For smaller grains, the 63 × 63 𝜇m2 or 31 × 31 𝜇m2 setting was selected to

prevent either the proton beam from missing the target or X-rays from exiting from the side

of the grain rather than from the surface. Either situation here would produce an undesired

alteration of the observed X-ray yield.

Measurements were performed using an integrated beam charge of 0.5 𝜇C with an average

beam current of 2.0–3.0 nA. For major elements in these spectra, the integrated charge used

106

Cla

ssSh

ort

nam

eLo

ngna

me

and

harv

est

loca

tion

Gue

lph

Ran

kR

efer

ence

Num

ber

Bas

alt

glas

sB

HV

O-1

Bas

alt

Haw

aiia

nV

olca

nic

Obs

erva

tory

1−1

2G

ladn

ey[5

9]In

dian

Indi

anO

cean

;USN

M11

3716

1−3

1Ja

rose

wic

h[8

6]V

G-A

99M

akao

puhi

Lava

Lake

,HI;

USN

M11

3498

-11−4

1Ja

rose

wic

h[8

6]V

G-2

Juan

deFu

caR

idge

;USN

M11

1240

-52

1−5𝑎

1Ja

rose

wic

h[8

6]O

livin

eSa

nC

arlo

sSa

nC

arlo

s,G

ilaC

o.,A

Z;U

SNM

1113

12-4

442−7

1Ja

rose

wic

h[8

6]C

ape

Ann

Faya

lite,

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kpor

t,M

A;U

SNM

8527

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1Ja

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h[8

6]P

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Libb

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ch[6

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2R

hyol

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Tul

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Mex

ico;

RLS

-132

2−5

2M

acD

onal

d[1

05],

p.15

4Fe

ldsp

arO

r-1

Adu

lari

aO

rtho

clas

e,V

alC

rist

allin

a,Sw

itze

rlan

d;O

r-1

2−9

3G

oldi

ch[6

2]La

keC

o.P

lagi

ocla

se,L

ake

Co.

,OR

;USN

M11

5900

3−1

1Ja

rose

wic

h[8

6]A

lbit

eA

b-1

Alb

ite,

Am

elia

Co.

VA

;Ab-

14−11

3M

orey

[111

]

Tab

le5.

1:Su

mm

ary

ofsi

licat

est

anda

rds

used

inP

IXE

anal

ysis

,so

rted

acco

rdin

gto

clas

s.C

olum

nspr

ovid

edre

pres

ent,

left

tori

ght,

clas

s,sh

ort

nam

e,lo

ngna

me,

the

Gue

lph

mic

ropr

obe

lab

targ

etnu

mbe

rs,t

hera

nkin

gap

plie

dto

the

targ

etac

cord

ing

tore

liabi

lity

crit

eria

and

the

refe

renc

eso

urce

ofth

eco

mpo

siti

on.

107

Shor

tna

me

Na 2

OM

gOA

l 2O

3Si

O2

P2O

5K

2O

CaO

TiO

2C

r 2O

3M

nOFe

2O

3Fe

OH

2O

Oth

er

BH

VO

-1[5

9]2.26

7.23

13.8

49.94

0.273

0.5211.4

2.71

0.168

2.82

8.58

0.21

0.03

6(C

O2),

0.00

9(C

l),0

.038

(F),

0.01

(S)

Indi

an[8

6]2.48

8.21

15.39

51.52

0.12

0.0911.31

1.3

0.17

1.12

8.12

0.18

VG

-A99

[86]

2.66

5.08

12.49

50.94

0.38

0.82

9.3

4.06

0.15

1.87

11.62

0.02

VG

-2[8

6]2.62

6.71

14.06

50.81

0.2

0.1911.12

1.85

0.22

2.23

9.83

0.02

San

Car

los

[86]

49.42

40.81

0.05

0.16

0.25

0.14

9.55

0.37

0.37

(NiO

)C

ape

Ann

[86]

29.22

0.04

2.14

1.32

66.36

Px-

1[6

2]0.2416.93

0.66

53.94

24.55

0.26

0.21

0.07

1.13

1.91

0.03

0.03

5(Sr

O),

0.00

6(B

aO)

Per

u21

5[1

05]

4.2

0.01

15.65

72.6

0.42

3.6

0.18

0.04

0.09

0.11

0.45

0.29

0.05

(Cl)

,1.3

7(F)

Mex

ico

132

[105

]5.25

0.05

11.44

75.7

0.01

4.53

0.12

0.21

0.15

1.86

0.45

0.07

0.19

(Cl)

,0.1

9(F)

Or-

1[6

2]1.14

18.58

64.39

14.92

0.03

(Rb 2

O),

0.03

5(Sr

O),

0.82

(BaO

)La

keC

o.[8

6]3.45

0.14

30.91

51.25

0.1813.64

0.05

0.01

0.34

0.15

0.05

Ab-

1[1

11]

11.49

20.00

68.06

0.15

0.15

0.04

0.02

Tab

le5.

2:M

ajor

elem

ent-

oxid

eco

mpo

siti

ons

ofsi

licat

est

anda

rds,

asre

port

edin

the

corr

espo

ndin

gso

urce

ssu

mm

ariz

edin

Tab

le5.

1.In

form

atio

nin

Tab

les

5.1

and

5.2

are

linke

dth

roug

hth

eco

lum

n“S

hort

nam

e”.

108

corresponded to peak areas of magnitude 103 − 105. Some target grains from standards not

analyzed here were found to be too small for practical measurement. In these situations, the

incident beam introduced enough heat to cause the surrounding resin to liquify. This was

observed by the appearance of grain migration, analogous to floating, during measurement.

These data were discarded. For the data presented here, no such behaviour was observed

during acquisition.

5.3.2 Fitting of silicate standards spectra

The GUPIX program can handle automatic assigning of peak tails and the fitting of repeat

measurement spectra with greater ease than the GUFIT program. For the collected silicate

spectra, there exists both a greater complexity of elemental peaks, when compared to the

pure element spectra, and a large number of spectra to process. As such, the GUPIX program

was chosen for this fitting task. Using GUPIX, the Kβ peak and KLL RAE features are

treated as single Gaussian functions with intensities held fixed, relative to the Kα line. The

original GUPIX database prescription of the KLL RAE intensity was too intense for the

acquired spectra and a new intensity was defined. The justification for this modification is

described in Appendix D.

A sample of one of the fitted spectra is provided in Figure 5.1. This spectrum is repre-

sentative of the typical complexity found in processing a basalt glass standard. The X-ray

peaks from major element quantities are identified in this spectrum.

Exponential functions were applied by the program to the tails of each peak. The tail

parameters (height and slope) used in their description were supplied by the user as inputs

to the program. These parameters were derived from individual custom fits applied to PIXE

measurements performed on pure targets of light elements Mg, Al and Si and each of the 3d

transition metals (21 6 Z 6 30). An Astimex metal mount was used to supply the elemental

spectra from each of the 3d transition metals. The light element (Mg, Al, Si) tail parameters

were derived from the spectra fitted using GUFIT, as described in Chapter 4. Custom fitting

of the 3d elements was also carried out using the GUFIT program. A sample treatment of

one of these pure element spectra (Sc) is illustrated in Figure 5.2. In this spectrum, as in

109

01

23

45

67

89

10100101102103104

Ti Kβ

Ti Kα

Ca Kβ

K KαCa

Kα

Mn Kα

Fe Kβ

Fe Kα

PNaMg

Al Da

ta Fi

t

Counts

Photo

n Ener

gy (ke

V)

Si-505

Residuals

Fig

ure

5.1:

Sam

ple

PIX

Esp

ectr

umof

fitte

d(𝜒

2=

7.33

)Si

licat

eB

HV

O-1

glas

sst

anda

rd.

Maj

orel

emen

tlin

esar

eid

enti

fied

inth

efig

ure

and

fitre

sidu

als

(𝜎)

are

show

nin

the

top

pane

l.Fit

incl

udes

the

mai

nlin

es,

pile

-up

peak

san

des

cape

peak

sbu

tth

edi

gita

llyfil

tere

dba

ckgr

ound

isno

tdi

spla

yed.

110

2 . 0 2 . 5 3 . 0 3 . 5 4 . 01 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

F i t D a t a T a i l B a c k g r o u n d D E S c K L V R A E G a u s s i a n f u n c t i o n s

Co

unts

P h o t o n e n e r g y ( k e V )

- 505

Re

sidual

s

Figure 5.2: PIXE spectrum of a pure Sc target (𝜒2 = 5.98) custom-fitted using the GUFITprogram. The Bremsstrahlung background was fitted with a single DE. The GUFIT librarydescription of the element K X-ray lines and escape peaks were also used. The tail is apparenton the low-energy side of the Sc Kα peak. The Sc Kα peak was truncated to remove the ScKβ contribution. Fit residuals (𝜎) are indicated in the upper panel.

the others, only the region surrounding the Kα peak, down to its escape peak, were fitted.

The GUFIT library description of the K X-ray lines were incorporated in these fits. The

four Gaussians in Figure 5.2 correspond, left to right, to the Si K X-ray escape from an

Sc Kα photon, Si K X-ray escape from a Sc Kβ X-ray, Sc KLV RAE and the primary Sc

Kα peak. An additional DE function was added to better fit the asymmetry of the RAE

feature. The fitted exponential tail on the primary peak is also indicated. Using this basic fit

routine, two spectra from each 3d element were fitted. From these spectra, the relative height

and slope parameters of each tail were plotted separately yielding relationships showing a

primary peak energy-dependence. These were fitted with linear and polynomial functions.

The fitted slopes, tail heights and tail areas have been provided in Figure 5.3. From these

functions, parameters were interpolated over a spread of photon energies and entered into

the GUPIX program detector efficiency file (seen in Appendix E.1). This is a file that is

111

read by the program that uses the energy-dependent tail parameters to assign tails to peaks

present in a spectrum.

The GUPIX program corrects for X-ray yield appearing in the tail by utilizing the

energy-dependent tail area relationship, plotted in the bottom panel of Figure 5.3. The tail

area parameters derived from the pure elements are also shown in Appendix E.1.

In each pure element and silicate standard spectrum, the Bremsstrahlung continuum

is present as background under the peaks. When fitting using the GUPIX program, this

contribution was subtracted from the peak fits using the GUPIX digital filter option. Peak

pile-up fitting was automatic, as was Si K X-ray escape. The GUPIX program applies pile-

up fits using various possible combinations of coincident events ie. each photon of a specific

energy may combine with a photon from any other element. The pile-up correction factor

used in analysis is provided by the program output. For some spectra, the pile-up peaks

could not be discerned from fitting. In this situation, the program defaults to using the

detector-system resolving time (𝑡𝑟) for calculating the pile-up correction factor. Since the

typical detector count rates (𝑟𝑐) were small such that 𝑟𝑐𝑡𝑟 ≪ 1, the expression:

𝑟𝑝𝑢 = 𝑟2𝑐 𝑡𝑟 (5.1)

was used to calculate the resolving time [94]. Both 𝑟𝑐 and the pile-up peak production rate

𝑟𝑝𝑢 were used in this calculation. A series of pure Si spectra, collected at 𝑟𝑐’s of 1000, 1500,

2000 and 2500 𝑠-1 were fitted. Both rate parameters were extracted from the primary peak

and pile-up peak fits and used to calculate 𝑡𝑟. An average value of 1.26 𝜇s was derived as the

detector-system 𝑡𝑟 value. This was inputted into the GUPIX detector efficiency file for use

by the program. Generally, pile-up correction factors calculated for the silicate standards

spectra were small (∼ 1.002− 1.004).

Due to the subtle differences of this fitting procedure over that of Chapter 4, the pure

element spectra were re-fitted using this procedure. This was expected to eliminate possible

systematic errors that may be introduced by any subtle differences existing between the

GUFIT treatment in last chapter and the GUPIX treatment in this chapter. A sample

of one of the GUPIX-fitted Si spectra is shown in Figure 5.4. The differences between

112

1234567

Invers

e slop

ey = 7 . 8 5 7 x - 7 . 1 3 8

y = - 0 . 0 5 7 9 x 3 + 1 . 0 6 1 x 2 - 5 . 8 8 1 x + 1 1 . 6 2

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

y = - 0 . 0 2 3 1 x + 0 . 0 4 6 0

y = - 4 . 9 4 2 E - 0 6 x 3 + 1 . 8 0 8 E - 0 3 x 2

- 2 . 9 2 4 E - 0 2 x + 1 . 2 6 5 5 E - 0 1

Relat

ive tai

l heig

ht

1 2 3 4 5 6 7 8 9

0 . 0 1 20 . 0 1 40 . 0 1 60 . 0 1 80 . 0 2 00 . 0 2 2 y = - 7 . 8 5 1 E - 0 5 x 3 + 2 . 0 5 9 E - 0 3 x 2

- 1 . 7 8 0 E - 0 2 x + 6 . 3 0 5 E - 0 2

y = - 9 . 3 7 4 E - 0 3 x + 3 . 2 0 9 E - 0 2

Relat

ive tai

l area

P h o t o n E n e r g y ( k e V )Figure 5.3: Tail parameters from light element and 3d transition metal spectra. Of thethree panels, the top displays the inverse slope parameter, the middle shows the tail heightrelative to the primary peak and the final panel displays the integrated area. Individual fitsto each data trend are illustrated in each panel.

113

1 2 3 4 5 6 71 0 0

1 0 1

1 0 2

1 0 3

1 0 4

1 0 5

F i t D a t a

Co

unts

P h o t o n e n e r g y ( k e V )

- 505

Re

sidual

s

Figure 5.4: PIXE spectrum of pure Si target (𝜒2 = 1.35) fitted using the GUPIX softwareprogram with background subtraction using a digital filter. Fit residuals (𝜎) are indicatedin the top panel.

the two fits of Si as seen in Figure 4.5 and Figure 5.4, are rather subtle, other than the

apparent absence of a function fitted to the background in the latter figure. The GUPIX

program applied a fit to both the first and second order pile-up peaks found at the locations

corresponding to double and triple the energy of the main peak, respectively. Given the

already low count rates, it is not clear that there is any contribution to the spectrum from

the second order pile-up peaks. The presence of the fit at 5.22 keV is merely and artefact of

the program calculation.

A noticeable variation in the fit residuals shown in Figure 5.4 indicate a slight misfit

of the Kβ peak intensity. In the GUPIX fit routine, the intensity of the Kβ peaks of each

element were linked to the corresponding Kα intensity. The program adjusts the Kα:Kβ ratio

as needed to account for any absorbing layers such as the detector window and front contact.

These layers are inputted by the user and are specified in the same detector efficiency file

illustrated in Appendix E.1. In this analysis set, the window was set in accordance with

114

design specifications, at 9 𝜇m of Be, and the electrode contact was suspected to be about 80

nm of Si. Despite correction for absorbing layers on the Kβ intensity, a small misfit remains

among most of the Kβ’s of Mg, Al and Si. This is not surprising given the susceptibility of

low energy X-rays to errors in estimation of all detector absorption layers. This slight misfit

is not expected to have a significant impact on the results here. Each of the Kβ, Kα2 and

RAE lines are fixed relative to the Kα1 line and therefore analysis using only the dominant

Kα1 line is sufficient. The GUPIX program provides, as output, the Kα1 intensities of the

primary peaks fitted to each element. The analysis in this chapter was restricted only to

working with the peak Kα1 intensities, for all elements.

5.3.3 Calculation of light element efficiency constant

Calculation of the efficiency constant from the yield expression (Equation 4.1), in Sec-

tion 4.3.1 was performed for major elements Mg, Al, Si and Fe, when present. Fe was

included here as its reported yield is less sensitive to attenuation variations and can be used

as an overall consistency check.

For each target, a subroutine of GUPIX called GUYLS (Guelph X-ray yields per stera-

dian) was used to compute the FP calculation of Equation 4.1. This program accepts the

thick target conditions of the experiment as input, including the full breakdown of the sample

elemental concentrations. The program also accounts for secondary fluorescence contribu-

tions to X-ray production. This is a necessary consideration in multi-element targets as the

higher energy K X-rays emitted by the heavier elements may induce X-ray fluorescence of

lighter elements. The final output represents a predicted X-ray yield for a given K X-ray

line, normalized to absolute efficiency and concentration, as follows:

𝑌𝑙(Z)𝐶Z𝜖a(ℎ𝜈)

= GUYLS result(︂

𝑐𝑜𝑢𝑛𝑡𝑠

𝑠𝑟 · 𝑐𝑜𝑛𝑐.

)︂(5.2)

where the variables have the same meaning as before and are a rearrangement of Equa-

tion 4.1. Rearranging this expression to solve again for 𝜖a(ℎ𝜈) allows for input of remaining

variables that include the peak area X-ray yield (Y𝑙(Z)), extracted from fitted spectra, and

the fractional concentration of element Z in the sample. An alternative way to solving this

115

is to use the GUPIX program directly, as explained in Appendix E. The resultant efficiency

constants of Mg, Al and Si from each sample were then compared directly to the efficiency

constants derived for each pure element target.

5.3.4 Error analysis

The uncertainties involved in calculation of silicate mineral 𝜖a(ℎ𝜈)’s are similar to those

made in the previous chapter. Here, many FPs used in calculation cancel when efficiency

constants for the pure element and silicate targets are compared. Those that do not cancel

include the stopping power, the proton-ionization cross-section and the MACs. Additionally,

the chemical composition of targets used in analysis is an uncertainty that was not present

in the study from the previous chapter. The standards also have a carbon coating layer of

unknown thickness which needs to be corrected for in terms of X-ray absorption.

It was necessary to make some assumptions as to the magnitude of some of these

uncertainties. For stopping power, the agreement of experimental stopping data with theory,

for the heavier elements (eg. Ca and Fe) found in the silicate targets, is very good [165].

Therefore, it is reasonable to expect that an uncertainty for proton stopping in the silicate

targets will be similar to the 0.2% uncertainty applied in Chapter 4. This is the value applied

to the H-value measurements seen here. The uncertainty corresponding to the X-ray pro-

duction cross-section parameter requires consideration for the nearest-neighbour bonding of

the light elements in the silicate targets. In these targets, the major elements are all bonded

immediately to oxygen and are less likely to be bound to other major elements. As such, the

chemical bond consideration was restricted to the same argument put forth in Section 4.3.2

in which the σZ(𝐸)ω𝐾 parameter is compared between the major element in pure form ver-

sus oxide form. For these calculations, the same correction factor numbers (𝑟XT ·Δ𝑟X), as

found in Table 4.1, were used to correct the changes in yield due to oxide-bonding in the

silicate targets.

The targets from Chapter 4 were coated with carbon layers not greater than approx-

imately 70 nm. Based on the tint level apparent between these targets and the silicate

targets, the coat thickness appear to be similar, if not less. In this approximation, a ±25

nm uncertainty was applied to measurements, assuming a 50 nm thickness. This value is

116

multiplied by a factor of√2 to account for the 45∘ angle of X-ray transmission through the

carbon layer. The uncertainty correction is thus√2 · (50 ± 25) nm and is applied to the

transmission of each Kα1 X-ray energy of Mg, Al and Si.

Additional uncertainty is introduced by the statistical spread of repeat measurements.

Most measurements were performed four times. This was enough to give an idea of the spread

in repeat measurements. The statistical spread was calculated as one standard deviation of

the four data points. It was also calculated separately for each fit by the GUPIX program.

The final statistical uncertainty for a given element, in a given target, was taken as the larger

of the standard deviation spread or the average GUPIX-calculated statistical uncertainty.

The uncertainty associated with the efficiency constants of the silicate targets is the

quadrature sum of the statistical spread and the uncertainties applied to the stopping power,

the X-ray production cross-section and the carbon coat thickness difference.

The majority of the uncertainty not accounted for in the analysis is attributed to the

MAC parameter and to a lesser degree, the compositions reported in the literature and

summarized in Table 5.2. On the latter, it is not feasible to assign an estimate to the

uncertainty associated with each major element. Therefore, the results presented below

may be subject to error imposed by both the MACs and the reported composition. The

compositions reported for the higher ranked targets should be considered the most reliable.

Overall, the bulk of the unaccounted uncertainty applies to the MACs.

5.3.5 Implementation of a GUPIX MAC database of FFAST coefficients

The GUPIX database package includes XCOM MAC coefficients, for all elemental K, L and

M X-ray lines above 1 keV, in all elements (1 6 Z 6 94). To permit analysis using FFAST

MACs, in place of the XCOM MACs, these values were taken directly from the NIST website.

The GUPIX program requires coefficients at the energies corresponding to the principal lines.

These are not supplied directly in the FFAST database and had to be interpolated for all of

the elements found in major and minor quantities in the silicate standards. These elements

include Z = 8, 11–17, 19–30. To acquire a complete MAC database for each element, it

is necessary to interpolate MACs for all photon energies involved. This means that MACs

must be calculated for every K X-ray emitted by the elements: Z = 11–17, 19–30 as they are

117

absorbed in each element Z = 8, 9, 11–17, 19–30. As an example, a few of the combinations

include: attenuation of Si K X-rays in O, Na, Mg, Al, Si,... Zn and Fe K X-rays in O, Na, Mg,

Al, Si,... Zn. A typical GUPIX X-ray line description includes five to nine individual lines

including: Kα1, Kα2, KLL, KLM, Kβ and KMM for the element Si. The new interpolated

FFAST database thus consists of nearly 3000 MAC data points. Included in this dataset are

some additional data points interpolated for attenuation of major elements (11 6 Z 6 14) K

X-rays in some of the minor elements (𝑍 = 1, 9, 37, 38, 56) present in the silicate standards.

Piecewise fitting of the FFAST datapoints using either a multi-parameter polynomial

or inverse-power function was used for interpolation. Choice in the selection of these two

functions was made according to the results of the fit quality. Generally, if the fit was better

than R2 = 0.99990, then it was considered acceptable for interpolation. The polynomial was

also used on data near edges where the MAC data tends to deviate from a power function

trend. The fitting was done piecewise because of the complex nature of the shape of the

plotted data. No single function could be used across a broad energy range (eg. 1.84–9

keV in Si) to yield a satisfactory fit to the data. With the use of a piecewise fit treatment,

there was concern that the piecewise functions may not provide reliable interpolation at

the ends. Therefore, data were interpolated from the piecewise functions avoiding edges.

Specifically, interpolation was terminated before the last two or three data points fitted

within the function, on either ends.

The interpolated FFAST database has been made accessible to users online through

the University of Guelph’s scholarly communications website (The Atrium) [157]. FFAST

coefficients for each of the element materials mentioned above have been saved individually

to this site in comma separated value (.csv) format. The FFAST MACs were inputted

into the GUPIX database using the GUPIX subroutine GUKLUP. This subroutine saves

the inputted values in a KXUP.DAT file that is called upon by the GUPIX program. Two

separate KXUP.DAT files, one for each of the XCOM and FFAST databases were retained

as back-ups. Use of either in analysis is made possible by the placement of desired database

KXUP.DAT file in the same folder as that holding the GUPIX program.

A third MAC database was produced for use by the GUPIX program. This one was

formed using the XCOM database as a basis and inserting FFAST MACs into energy

118

2 3 4 5- 1 5

- 1 0

- 5

0

5

Perce

ntage

differ

ence in

µ/ρ (

%)

P h o t o n e n e r g y ( k e V )

X C O M F F A S T P T B

Figure 5.5: Percentage difference of XCOM and FFAST theories with respect to Al PTBdata. The dotted line through zero represents the PTB data and the individual dots repre-sent the 𝑁 = 326 individual measurements in the energy range of 1.44–5 keV.

locations supported by data-theory agreement trends seen in the Al [8] and Si [75] PTB

experimental data. The reader may refer back to the Si PTB data presented in Figure 3.3

as reference to this discussion. The Al PTB data in the energy range of 1.44–5 keV are

included in Figure 5.5.

A common theme observed in much of the data is that above the K edges, in the energy

region below 5 keV, the MAC data do not appear to favour strongly one treatment over

the other. For this reason, it is not clear which database should be chosen for above-edge

MACs of all pertinent light elements (11 6 Z 6 14). If the trends seen in the 3d element

and higher Z MAC data [60, 61, 89, 110, 159, 160] are any indication of what might be

expected for the light elements, they suggest that the XCOM database is more accurate in

representing above-edge MACs. In the energy range below 5 keV, the Al data of Figure 5.5

do not agree very well with either theory but are closer to XCOM above the edge. This

combined evidence here suggests that there is merit to maintaining XCOM coefficients above

119

Δ𝜖a(ℎ𝜈) (%)

Database Mg Al Si

XCOM 7.35± 0.51 6.83± 0.22 8.50± 0.09FFAST 0.32± 0.55 3.35± 0.24 2.17± 0.10Mixed -0.43± 0.55 2.09± 0.25 0.83± 0.10

Table 5.3: Inverse variance weighted means of the differences in 𝜖a(ℎ𝜈) constants of Mg,Al and Si in silicates relative to pure elements, calculated using the three MAC databases.A positive difference indicates that the pure element 𝜖a(ℎ𝜈)’s are greater than those fromsilicate targets. The uncertainty given is the error in the mean calculated using the inversevariance weighting method.

the light element K edges but adopting the FFAST values below the light element edges,

given the support demonstrated in the previous chapter.

In this third database, a KXUP.dat file of purely XCOM coefficients was used as a base.

The FFAST MACs were then substituted in for the K X-ray lines of: Na in Na; Na and Mg

in Mg; Na, Mg and Al in Al; Na...Si in Si and Na...Si in oxygen. Each of these coefficients

appear below the K edges of the associated light elements. One additional MAC that was

substituted for FFAST was the Si K X-rays in Al, lines that appear above the K edge of Al.

The PTB Al data of Figure 5.5 show a stronger support for FFAST MACs at this energy

which has been used to justify this selection. In this database approach, the K X-rays of

the light element majors are still considered to be attenuated by the XCOM MACs of the

other major elements such as K, Ca, Ti and Fe.

5.4 Results

Analysis was carried out using the three MAC databases introduced in the previous section.

The first is that of exclusively XCOM values, denoted as XCOM. The second is that in

which the major elements found in the silicate standards have been replaced with FFAST

MACs, denoted as FFAST. Lastly, the database incorporating FFAST coefficients that lie

primarily below the light element K edges, is denoted as Mixed. The efficiency constants of

Fe were calculated from six of the standards containing Fe as one of the major constituents.

These were found to vary by 1% (±1𝜎) about the mean and indicated that, for these six

120

BHVO

-1Ind

ianVG

-A99

VG-2

--------

-San

Carlo

sCa

pe An

n----

-----

Px-1

--------

-Per

u 215

Mexic

o 132

--------

-Or

-1La

ke Co

.----

-----

Ab-1

- 8- 6- 4- 202468

1 01 2

X C O M F F A S T M i x e d

Mg %

diffe

rence

Figure 5.6: Efficiency constant differences between pure Mg targets and Mg in silicatetargets. Results are identified by the short names of each targets and groups of targets areseparated by vertical dotted lines, according to their classification.

standards, PIXE measurement and analysis consistency was satisfactory.

The 𝜖a(ℎ𝜈) constants calculated for elements Mg, Al and Si in the silicate targets have

been compared to the 𝜖a(ℎ𝜈) values that correspond to the respective pure element targets.

These comparisons are presented, in Table 5.3, as weighted averages of the percentage dif-

ferences in the efficiency constants (Δ𝜖a(ℎ𝜈)), calculated for each standard. Δ𝜖a(ℎ𝜈) values

have been calculated for each of the three MAC databases.

The Δ𝜖a(ℎ𝜈) values obtained using the XCOM database were comparable to the average

differences apparent from the results in the 1997 study of Campbell et al. [22]. This observa-

tion demonstrates that these measurements have been reproduced with reasonable accuracy.

The Δ𝜖a(ℎ𝜈) values from each of the silicate target PIXE results, have been plotted for Mg

(Figure 5.6), Al (Figure 5.7) and Si (Figure 5.8). Each plot shows the individual results

found using each of the three MAC databases.

121

BHVO

-1Ind

ianVG

-A99

VG-2

--------

-San

Carlo

sCa

pe An

n----

-----

Px-1

--------

-Per

u 215

Mexic

o 132

--------

-Or

-1La

ke Co

.----

-----

Ab-1

- 2

0

2

4

6

8

1 0

1 2

X C O M F F A S T M i x e d

Al %

diffe

rence

Figure 5.7: Efficiency constant differences between pure Al targets and Al in silicate targets.Results are identified by the short names of each targets and groups of targets are separatedby vertical dotted lines, according to their classification.

The quality of the new data has been checked, in part, by comparing the XCOM ob-

servations with the 1997 measurement results of Campbell et al. [22]. Any Mg, Al or Si

Δ𝜖a(ℎ𝜈) deviation that was more than a few percent from the corresponding data in the

1997 study was interpretted as a potential problem existing in the present data.

In one example, analysis of measurements performed on the Springwater Olivine sample

revealed efficiency constant differences of approximately 15% for Mg and Si, using the XCOM

database. The 1997 results suggested that these differences should not be more than about

3%. To analyze this effect appropriately, several points are reviewed. The first is to recall

that a positive difference in Δ𝜖a(ℎ𝜈) means that the light element 𝜖a(ℎ𝜈) constants are

smaller for the silicate targets relative to the pure element targets. The second point is

that from Equation 4.1, it is evident that 𝜖a(ℎ𝜈) is proportional to the observed X-ray yield

(𝑌l(Z)). The combination of these two points suggests that an incomplete production of X-

rays has occurred which has resulted in a reduction of the observed X-ray yield. If this were

122

so, then this would translate into an apparent decrease in the efficiency constants leading to

the marked increase in the Δ𝜖a(ℎ𝜈) values (15%) that were observed for Springwater. Such

an effect would be possible if the rastered beam was periodically missing a small portion

of the target. However, this would also manifest a similar X-ray yield reduction in the Fe

𝜖a(ℎ𝜈) results. Therefore, the Fe 𝜖a(ℎ𝜈) value of Fe for Springwater was compared to the

average from the six targets. The Springwater value of 1.232×10−4 sr was very close (within

1.3%) to the average Fe value of (1.248± 0.021)× 10−4 sr (±1𝜎). Since the present Mg and

Si results differed from the 1997 results by 15− 3 = 12%, the Fe would also be expected to

differ by approximately 12%, if incomplete X-ray yield production was the problem. The

negligibly small 1.3% difference calculated here does not support this hypothesis. No other

explanation of this effect could be offered. Due to the unreliability of the Springwater results,

its analysis was removed from this work.

The ablite target (Ab-1) also yielded values that were initially surprising. The Al and Si

Δ𝜖a(ℎ𝜈) values were both 14–15%. Although this target was not reported in the 1997 study,

the large discrepancy, compared to the 6–8% average expected, suggested a problem. This

target falls under the rank of #3 given in this work, meaning that its composition may not be

entirely credible. In the original reference [62], the authors mention that their composition

measurement of Ab-1 differs from the numerous earlier reports summarized by Deer et

al. [44]. Each of the nine analyses of Ab-1 summarized by Deer agree well with each other in

their measured major element compositions of (Na2O, Al2O3 and SiO2). Their agreement is

typically within less than one percent of the average. The compositions reported by Goldich

for SiO2 and Na2O differed from the Deer data by 3.6% and 2.3%, respectively. This larger

discrepancy has suggested that the Goldich composition of Ab-1 is less accurate than the

older compositions reported by Deer. As such, the composition from the most recent Ab-1

measurement cited by Deer et al., the work of Morey and Fournier [111], was adopted for use

in this work. Substitution of the Morey composition data in this analysis yielded efficiency

constant results that were similar to the other standards. Furthermore, since the Morey

concentrations show strong agreement (<1% variation about the mean) with the other eight

analyses, confidence in the accuracy of their values is justified.

123

BHVO

-1Ind

ianVG

-A99

VG-2

--------

-San

Carlo

sCa

pe An

n----

-----

Px-1

--------

-Per

u 215

Mexic

o 132

--------

-Or

-1La

ke Co

.----

-----

Ab-1

- 2

0

2

4

6

8

1 0

1 2

1 4

X C O M F F A S T M i x e d

Si %

differ

ence

Figure 5.8: Efficiency constant differences between pure Si targets and Si in silicate targets.Results are identified by the short names of each targets and groups of targets are separatedby vertical dotted lines, according to their classification.

The Δ𝜖a(ℎ𝜈)’s plotted in Figures 5.6, 5.7 and 5.8 were grouped according to their

mineral class. These are separated by dotted lines between the x-axis descriptors in these

three figures. This was done to help visualize any data trends that might be apparent

from one class to the other and will be discussed further in the next section. From the

three Δ𝜖a(ℎ𝜈) plots and the results summarized in Table 5.3 it is apparent that the XCOM

coefficients produce the largest discrepancy at an average of 7–9%. This result appears to

agree closely with the 6–8% discrepancy found by Campbell et al. [22] and may only be

slightly higher as a result of the additional correction for chemical bond effects that was

added in this work. The additional correction would increase the observed discrepancy by

about 1%, as reported in the final column of Table 4.1.

This 7–9% discrepancy was reduced to 0.5–3.5% with the substitution of all FFAST

MACs used in matrix corrections. For this dataset, the Mg results closely approach 0%.

The overall 0.5–3.5% discrepancy was further reduced to within −0.5–2% when the Mixed

124

MAC database was used. These results suggest that the use of the Mixed database produced

the most accurate correction for X-ray absorption. The best agreement, in this database, was

found with the Mg and Si Δ𝜖a(ℎ𝜈) values; they had an average discrepancy of −0.43±0.55%

and 0.83±0.10, respectively. These numbers are similar to the Mg (0.4±0.6) and Si (0.9±0.5)

Δ𝜖a(ℎ𝜈)’s calculated in the comparison of pure versus oxide target PIXE measurements

(Table 4.3). As was found in the previous chapter, the Al results show the greatest average

discrepancy at 2.09 ± 0.25%. This is again consistent with the Al result of Table 4.3 at

2.6± 0.6%.

5.5 Discussion

5.5.1 Silicate standard efficiency constant trends

The collection of plotted Δ𝜖a(ℎ𝜈) values for Mg and Al is less extensive than that of Si as

not every standard contains Mg or Al. Still, their plots can be used to indicate the accuracy

of the MAC databases used in this analysis. For Mg, the only strong deviation apparent is

seen in VG2. For this target, the XCOM value is nearly identical to the trend found in the

1997 study, which suggests that this is not an error in the approach of this measurement.

The only speculation offered is that there is a flaw in its reported composition. For Al, there

are more data points. Of these standards, the Ab-1 shows the greatest difference relative to

the other standards at 4.7± 1.0 using the Mixed MAC database. This too may represent a

slight imperfection in its quantification. In the Si collection, the result for the Px-1 shows the

greatest discrepancy. Its composition was determined by Goldich [62], likely using the same

methods of analysis used for Ab-1. As was mentioned earlier, the Goldich SiO2 composition

was found to differ from the values tabulated by Deer et al. [44]. It is therefore possible that

the Px-1 target SiO2 composition was also under-estimated and therefore has produced the

inflated Δ𝜖a(ℎ𝜈) result seen in Figure 5.8.

The magnitude of difference between Δ𝜖a(ℎ𝜈) values imparted by the substitution of one

MAC database with another warrants further analysis. The values in Figures 5.6, 5.7 and 5.8

reveal the greatest differences when switching from the XCOM database to the complete

FFAST database; the Δ𝜖a(ℎ𝜈)’s are reduced by 4–7%. Upon substitution of the FFAST

125

database by the Mixed database, the discrepancies are further reduced by about 0.5–1.5%.

The Cape Ann fayalite standard is a notable exception as it shows a decrease of 3.4% when

switching from XCOM to FFAST and a greater reduction (4.5%) upon substitution of the

Mixed database.

The Cape Ann fayalite sample is unique in that it contains a very high percentage of

FeO (66% by wt.) relative to the other Fe-bearing standards(∼ 10% by wt.). It is also

unique in that its only other main constituent is SiO2. The MnO in the Cape Ann sample,

at 2% composition, plays only a subsidiary role in the overall attenuation. Each of the

other standards bear more than two element-oxides as their primary constituents. With

this sample, exchange of the FFAST MAC database, with the Mixed database, produces a

difference in the MAC of Fe at 1.74 keV (Si Kα energy). All of the other relevant MACs

remain unchanged. Therefore, the Δ𝜖a(ℎ𝜈) results may be considered a rigorous test of the

MAC of Fe at the Si K X-ray energy.

The results of the Cape Ann sample indicate that the Mixed database is more accurate

for Fe at 1.74 keV than exclusively FFAST. In particular, it suggests that the XCOM value

of Fe at this energy is more accurate than the FFAST value and should be adopted in

future analyses. The 1993 publication of Henke et al. [71] shows one experimental datum

value [146] that exists in the 1–2 keV range, corresponding to the Al Kα energy in Fe. This

value shows close agreement with Scofield MACs [142], as plotted by Henke. Due to the

less-sophisticated measurement design of the single Fe MAC measurement [146], this value

may be considered unreliable. However, it does suggest that support for XCOM is warranted

for MACs of Fe at low energies. It is still not clear if the Fe agreement with XCOM could

translate to other nearby elements such as K, Ca or Ti given that the targets measured here

are composed of more than two major element-oxides. Any improvements in the accuracy

of one MAC end up being averaged with decreases in the accuracy of other element MACs.

On the whole, however, the results from the Mixed database suggest that XCOM MACs

should be adopted for the other major element (K, Ca, Ti, Cr and Mn) silicate constituents.

The Al Δ𝜖a(ℎ𝜈) results observed in the previous chapter (2.6%) and in this chapter (2.0%)

represent the largest remaining overall discrepancy. This may be due to slight disagreement

between the true values and the FFAST prescription of Al and oxygen at 1.487 keV. If the

126

Al PTB data of Beckhoff [8] are completely accurate at 1.487 keV, then it suggests that the

MAC is situated between that of XCOM and FFAST. If this experimental value were to be

adopted, then the Al Δ𝜖a(ℎ𝜈) results found in this work would change by increasing. From

analysis of Equation 4.1, one can show that 𝜖a(ℎ𝜈) is proportional to 𝜇/𝜌, which supports the

suggestion that this effect would cause an increase in pure Al 𝜖a(ℎ𝜈). The 𝜖a(ℎ𝜈) value for Al

in the Al2O3 target or in silicates would also rise but not as far given that the composition

of Al here is only one quarter or less than that in the pure target. In order to produce an

Al Δ𝜖a(ℎ𝜈) result equal to 0%, the oxygen MAC at this energy must then also be increased

accordingly.

5.5.2 Theoretical database assessment and future directions

The overall results show that full FFAST is better than full XCOM for geological materials

measurements, but that a database of Mixed MACs is best. Application of the Mixed

database to analysis has demonstrated strong support for the use of XCOM MACs below K

edges of elements 20 6 Z 6 30 in the 1–2 keV energy range. It has also demonstrated, that at

this energy, FFAST MACs should be used below the K edges in light elements 12 6 Z 6 14

and above the K edge of oxygen. The PTB Al data have further shown support of adopting

FFAST in the immediate vicinity above the K edge, the location of the Si K X-ray energies.

This work has shown that between the photo-ionization cross-section datasets of Scofield

(XCOM) and Chantler (FFAST), the Scofield 𝜎PE’s show superior agreement with exper-

imental data across the majority of the database. When considering that the Chantler

coefficients have been calculated using a slightly more sophisticated approach, one might

expect that FFAST would show stronger overall agreement. Still, there are regions where

the Chantler database appears to be superior. One of these regions is immediately above

the binding energy edges of light and 3d elements. This superiority has been demonstrated

by the stronger agreement of the modern data with FFAST over XCOM, as was the case of

the PTB Al study and other 3d elements (Cu [34, 61] and Zn [126]) immediately above their

edges. In each of these references, the trend just above the edge is the same; the FFAST

values are higher than XCOM, but the experimental data are higher still. Therefore, the

FFAST values are, at least, a more accurate representation of the data over XCOM. Within

127

the Zn reference [126], a minor error was discovered in Figures 1 and 10 which was brought

to the attention of the prinicipal investigator (C. T. Chantler). Some of their illustrated

databases had been mismatched when plotted. Allowing for their correction, the trends in

the Zn data agree well with those from their other MAC measurements of Ag [159], Mo [89]

and Cu [34, 61]. Regions of superiority of the Chantler treatment are also prevalent below

the K edges of elements at low energies, as mentioned in Section 4.6.4. One further region

where the Chantler database is superior is in energies below 1 keV. Values below 1 keV

are not supplied by Scofield or XCOM and therefore the FFAST database supplements this

region in the absence of additional data. However, the accuracy attributed to the FFAST

coefficients in this region is beyond the scope of this work.

The analysis in this work has aided in the selection of MACS from the XCOM or FFAST

databases, for use in geological materials analysis. The selection of XCOM and FFAST

MACs, inserted into the Mixed database, were found to reduce the pure element to silicate

element-oxide efficiency constant discrepancy to within 2.0%. For Mg and Si, the agreement

was less than 1%, and implies that the accuracy of this approach has been greatly improved

in comparison to the 6–8% discrepancy observed in the earlier quantification attempt by

Campbell et al. [22]. This suggests that the goal of using PIXE to quantify major light

elements, with only a few percent variation in accuracy, has been realized.

For Al and to a lesser degree, Mg and Si, there is room for improvement in determining

the precise values of MACs at low energies. New MAC measurements of pertinent geological

elements such as: oxygen, Mg, Al, K, Ca, Ti and Fe in this range might assist in refining

the selection of MACs used in a FP PIXE approach. If new measurements can be produced

with the level of quality seen in the recent MAC publications [34, 60, 61, 88, 89, 110, 126,

160], then these could be used to supplant the current reliance of selecting from either

theory. As well, if only a few good MAC measurements are acquired, these could be used

in a PIXE approach as “anchors” to determine MACs of other elements. This might be

done by keeping the efficiency constant as a fixed parameter and treating the MAC as an

unknown. For example, if a metal alloy, consisting of only two elements (Fe and Mg), were

measured with PIXE, and one of these element’s MACs were known with high accuracy, the

other element’s MAC parameter could be treated as the unknown (or measurement result).

128

Composition of the alloy, reported by the manufacturer, could be checked using RBS.

Perhaps a similar treatment might be used to determine the MAC for oxygen. For

example, PIXE measurements performed on targets of Si, SiO, SiO2 and SiO4 might be

used to derive oxygen MAC at 1.74 keV. While this technique may not be as refined as a

conventional attenuation measurement, it can be used in the immediate future to provide a

rough estimate as to the value that should be adopted.

5.6 Conclusions

PIXE measurements were performed on a collection of silicate standards as a means to

evaluate the XCOM and FFAST MACs used in a FP calculation approach. This experiment

was motivated by the calibration results of an earlier PIXE study [22]. In their work,

Campbell et al. discovered a 6–8% discrepancy between the calculated efficiency constants

of light elements in pure targets relative to those in silicate standards. Of the silicate targets

measured in the earlier study, many of those selected have been analyzed thoroughly for

composition and were professionally prepared for microprobe analysis by the USGS. Most

of the error assocated with the efficiency constant comparative approach used here could

therefore be attributed to the MACs used in analysis. This approach was a stringent test of

MACs of light elements at low energies.

The present results from the twelve analyzed silicate targets revealed an average discrep-

ancy range of 7–9% in the pure element to silicate efficiency constant comparisons of Mg,

Al and Si, using the XCOM database. This discrepancy was similar to that observed by

Campbell et al.. A new MAC database for GUPIX was constructed for selected elements

using FFAST MACs downloaded from NIST. Using this FFAST database, the efficiency

constant discrepancy was reduced to 2–5%. A third MAC database, denoted as Mixed, was

formed using a combination of XCOM and FFAST MACs. The selection of MACs for this

database was supported using available experimental MAC data. Using the Mixed database,

the discrepancy was reduced further to an average of −0.5–2% suggesting that this database

is superior to either only XCOM or only FFAST.

These data have provided support for the use of XCOM values over FFAST for photons

129

energies of 1–2 keV, in elements heavier than Si. For these heavier elements, this energy

range is exclusively below the elemental K edge. This observation was demonstrated in

particular for Fe using the PIXE results from the iron-rich Cape Ann Fayalite standard.

130

Chapter 6

Concluding Remarks

This work incorporated several studies that were performed within the scope of improv-

ing light element FPs for materials analysis, using PIXE, or other X-ray emission analysis

techniques.

The first study focused on high accuracy spectral fitting techniques applied to low energy

(1–8 keV) monochromatic spectra. Monte Carlo [25] predictions and recent literature were

used to identify non-Gaussian features that arise in a semi-conductor detector spectrum.

Detailed spectral fitting treatments were applied to all major features in spectra produced

using monochromatic X rays. MC predictions were used to diagnose a downward energy

shift of electron escape features. The source of this effect was linked to the process(es)

responsible for producing the primary peak tail. Data from the fitted non-Gaussian features

were examined for feasibility of using the Monte Carlo approach to construct an automated

fitting routine. While this approach appeared satisfactory in the higher incident photon

energy range (4–8 keV), the agreement of simulation with data at lower energies was poorer.

As well, the MC approach using the Goto diffusion and reflection model does not fully

account for the primary peak tail observed in the data. The MC approach is therefore

not quite ready for use in forming automated fitting routines. Still, the present fit results

represent one of the most sophisticated fit treatments of low energy spectra to date. Future

work might entail improvements made to the MC program in modelling both interactions

at low energies (<1 keV) and charge losses that contribute to the primary peak tail.

The fitted higher energy (4–8 keV) spectra were used in Chapter 3 to derive a new

estimate of Si ωK. While the escape peak ratio derivation method has been used in previous

131

Si ωK measurements [14, 20], this work represents an improvement for two main reasons. The

first is that the non-Gaussian features underlying the escape peak are now well understood

and were fitted to produce an accurate description of the peak background. The second

is that new high accuracy Si MAC data exist [75, 160] which allowed for the selection of

only one MAC at each X-ray energy. The new Si ωK estimate of 0.0504 ± 0.0015 is in

very good agreement with the semi-empirical Bambynek result [4, 78]. The magnitude of

the present uncertainty is similar to those from earlier measurements. In light of the more

refined treatment used to derive this parameter, this suggests that previous measurements

may have underestimated their uncertainty level.

The work in Chapter 4 was motivated by a 6–8% discrepancy observed in an earlier

attempt to quantify major light elements in silicate minerals and glasses using PIXE [22].

The present work involved PIXE measurements performed on a set of light element pure

targets and a set of light element-oxide targets as a means to investigate MACs used in

light element quantification. This was done using an “efficiency constant approach” that

compared pure element vs. element-oxide results using either XCOM or FFAST MACs. In

this approach, most FPs cancelled and uncertainties corresponding to the remaining FPs

were accounted for fully. Using the XCOM database, an average discrepancy of 4–6% was

observed in the comparison of light element to oxide targets. In substituting the FFAST

database in place of XCOM, the discrepancy was reduced to 0.5–2.5% which suggested that

FFAST MACs are superior for light elements below the K edge and for oxygen above the K

edge.

The light element study was expanded in Chapter 5 to incorporate PIXE measurements

performed on well characterized silicate targets. Using a comparison of efficiency constants

of light elements in pure targets to the same elements in silicate standards, three MAC

databases were examined for accuracy. With the XCOM database, a 7–9% discrepancy

was observed which is similar to the 6–8% discrepancy oberved in the previous study of

Campbell et al. [22]. This discrepancy was reduced to 0.5–3.5% using a database of FFAST

MACs. Lastly, a database consisting of mostly XCOM and a few selected FFAST MACs

was examined for accuracy. These results revealed a −0.5–2% discrepancy for the three

elements, which indicated that this Mixed MAC database produced the highest accuracy in

132

PIXE analysis.

Of the many FPs used in PIXE analysis of geological materials the MAC is the main

determinant of uncertainty. This work has helped in selecting MACs from one of the two

primary MAC theoretical compilations, XCOM and FFAST. Presently, this work has demon-

strated that a FP approach to PIXE quantification of light element major constituents can

be carried out with a 2% variation in accuracy. This represents an improvement over the

6–8% discrepancy observed previously [22]. Further improvement in PIXE analysis on the

-0.5–2% discrepancy found with the Mixed database, might be obtained by the development

of new MAC measurements of relevant light elements and 3d transition metals in the 1–2

keV range.

133

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Appendix A

A contaminant-free polishingprocedure for soft metal targets usedin PIXE

Polishing of metal targets for PIXE is a rather simple process but several steps are required

to avoid surface contamination of targets with elements that might produce additional flu-

orescence, or introduce additional attenuation of outgoing X-rays. Soft metals, such as Al

are highly susceptible to small amounts of materials deposits on their surface, specifically,

in any troughs created by the passing of grains over the surface during the polishing pro-

cess. To eliminate elemental contamination that may impact a measurement, the process

described here is designed to perform the necessary target smoothing with a finishing process

designed to remove remaining contaminants that may have been deposited by the preceeding

smoothing process.

In the first step, a rough target may be polished using a successive set of sand-paper pads.

In the first step, the 200 grit pad might be a good place to start. This should be followed by

the successive available grits (400, 600, 800, 1000, 1500, 2000, 3000). Grit numbers should

not be skipped as this will only prolong the polishing process in the next grit step to achieve

the same finish quality that would otherwise be obtained. For metals of Al or even Mg,

water may be used with the pads during polishing. After the last polishing grit has been

used, the target should be washed.

At this point, the target may contain surface trace amount of residual silica. It is desire-

able to remove this trace. This has been done using polishing paste containing successive

144

levels of diamond grit. The use of diamond grit is expected to provide the advantage in that

it can make the target smoother than that achieved with even the 3000 grit sandpaper, it

can possibly dislodge the trace Si contaminants, and any traces of itself left behind, being

carbon-based, will not be registered using PIXE.

The diamond pastes may be applied starting with the roughest grits and working down

to the smallest. The paste may be applied by hand or using pieces of linen, cotton or

leather. If by hand, latex gloves are recommended. If the pastes start to dry, they may be

reconstituted with water as most commercially available diamond pastes are water based.

In between grits, the target should be washed with water (dish detergent may be used at

this stage but is not necessary) and further buffed with diaper-grade cotton. The buffing is

quite successful in removing residue from the paste. De-ionized water might provide superior

results in the final stages as it may reduce oxidation introduced by exposure to tap water.

After the last paste was applied, target surface should be washed with detergent and

water, rinsed thoroughly, and buffed with cotton until it shines. A near mirror finish should

be attainable. The target should next be dehydrated to remove any residual water on the

surface. This may be done using a drying agent such as a pure alcohol or acetone. After

thoroughly soaking in the drying agent for a few minutes, the target may be dried using a

clean piece of cotton and stored in a dessicator.

During this process, care should be taken to avoid dropping or scratching the target

surface. For softer metals such as Al, fingernail contact may potentially mar the surface.

145

Appendix B

Procedural details of carbon coatingapplication and layer thicknessmeasurement

B.1 Details on the application of carbon layers

This section describes several key points of the procedure adopted for carbon coating mi-

croprobe targets using the Edwards Auto 306 system housed at the University of Guelph

Electron Microscopy Unit. Operation of the Edwards Auto 306 unit might be performed by

experienced personel or by individuals trained to follow all safety protocols. The basic steps

to run the machine are not covered here. Rather, the technique of adjusting the dial to get

a maximum effectiveness in the application of the coat is described in detail. Responsibility

is left in the hands of potential operators to understand all other steps necessary to operate

the machine safely.

Samples intended to be carbon coated should be cleaned to remove any residue or par-

ticulate matter and dried thoroughly. The carbon filament gun of the Edwards Auto 306

carbon coating machine can be positioned at a distance that permits multiple targets to be

encompassed inside the field of view of the sputtered carbon. Positioning the gun well above

the targets increases the coating diameter region. However, the further away that the gun

is positioned, the longer the time that is needed to apply the coat. The carbon emission is

collimated such that if the filament gun is set at 20 cm above the target plate, a beam of

about 8–10 cm diameter will be emitted at the plate. This is wide enough to fit two 1 inch

targets, placed side by side, so that an even coat might be applied to both simultaneously.

146

Coating of targets may commence when the targets have been positioned and the cham-

ber has been pumped down to a satisfactory vacuum level (4 × 10−5 mbar). The operator

must first ensure that the dial on the control panel is set to zero. The switch on the control

panel next to the dial may then be turned to the “LT” position. The dial is then turned

slowly from zero to five or six, over the course of about 15 seconds. The filament will begin

to glow white-hot at which point the dial setting should be held for 15–20 seconds more,

alowing the filament to heat up. The field of view in which carbon is deposited will be

illuminated on the plate holding the targets. If targets have been positioned outside of this

beam, they will not be coated and must be re-adjusted before continuing. Carbon will begin

to sputter slowly at a setting of five or six. When the glow from the filament appears to

darken slightly the dial setting should be increased, very slowly over the next minute or

more, up to a setting of 9 or 10. Small dial adjustments along the way will be necessary at

every point that the light begins to dim slightly. As the current (dial setting) is increased,

the gun will get hotter and sputter carbon more efficiently onto the target. However, if the

current is set too high, too early, the filament may break apart thereby ending the coating

process prematurely.

Optimal coating is often applied when the the dial setting is between 7.5–9 or 10. The

coating process may take up to two or more minutes from the moment that current is first

passed through the carbon filament. Eventually, the carbon filament will fail to continue

conducting electricity due to the stress of the applied heat. This marks the end of the

process. The operator must turn the dial back to zero and then turn the switch back to the

central position. The gun will require up to 30 minutes to cool before it may be re-built

for subsequent coatings. Generally, a successful coating of two minutes will introduce a

coat thickness of approximately 40 nm. While this thickness may be sufficient for electrical

conduction, the standard used by the Guelph microprobe facility is to apply two coats to

achieve at least a 60 nm thickness.

A simple test with a household battery and a multimeter has shown that for an 80 nm

coat, electrical conduction is quite strong across the surface of a coated insulator target.

Generally, for a coating of this thickness, very little drop (less than 10%) in the applied

potential (V) can be observed. A well-applied coat will tend to have a dark grey appearance

147

when observed on a white background. Coatings that appear brown in colour do not tend

to possess very strong conductive properties and may indicate that the coat is either too

light or of poor quality. If this is observed, an additional coat may be necessary to achieve

satisfactory conduction.

B.2 Techniques in measuring the thickness of carbon layers

Atomic force microscopy (AFM) measurements may be performed on hard target surfaces

only. In this procedure, a gentle scratch is applied to the carbon with a sideways angled

scalpel. The scratch is intended to remove all carbon down to the surface of the target

without causing further removal of the target material itself. For materials as hard as pure

Si and SiO2, this process is satisfactory and additional etching of the target material itself

can be avoided. The scratch provides the sharp “cliff-like” edge necessary to perform an

AFM measurement in tapping mode.

If it is necessary to know the thickness of the carbon coating on a soft material targets

such as an epoxy, metal or geological sample then the measurement should not be made

on the target directly. A scalpel scratch in this case might carve out some of the target

material in addition to the carbon layer. This would lead to an erroneous AFM estimate of

layer thickness. A solution to this problem is to place an etch-resistant material, such as a

glass microscope slide, within the field of view of the sputtered carbon during the coating

process. In principle, the part of the slide located closest to the target will have received

the same thickness of coat as the target. The scalpel scratch may then be applied gently

to the glass and the resultant AFM measurement of the glass carbon layer may be used to

infer thickness of the carbon layer on the target.

B.3 Carbon layer removal

Should it become necessary to remove old layers of carbon from non-volatile target materi-

als, this can be done in several ways. Some of these methods have been extracted from the

Astimex mineral and metal mount manuals supplied by Dr. John Rucklidge of the Depart-

ment of Earth Sciences, University of Toronto, Toronto, Ontario. Carbon layers can have

148

varying levels of adhesion to target surfaces and the difficulty present in removing these lay-

ers cannot always be predicted. As such, it is best to start with methods that are minimally

invasive. This includes use of a solvent such as acetone or rubbing alcohol (isopropyl). The

target may be submerged in an ultrasonic bath containing these solvents but for very short

times (6 5 sec) only. The ultrasonic process has the tendency to wear away at etched labels

and may enlarge any existing pits or scratches on the target surface.

Gentle scrubbing with these solvents or with dish soap and water with a soft cloth

may also be enough to remove the carbon coat. In this manner the process may be slow

for coatings that have strong adhesion. Use of household scrubbing pads may hasten the

removal process but any microscopic hardening agents (such as alumina or silica grains)

present could introduce etch marks on the surface. A compromise might involve the use

of scrub pads that are free of abrasive materials. Additional polishing may be necessary

following carbon removal to return the target to its original smooth finish.

149

Appendix C

Examination of proton deflection byNd-Fe-B magnet assembly using asurface barrier detector

C.1 Introduction

This section summarizes a small experiment performed to test if the Nd-Fe-B magnet as-

sembly, introduced in Chapter 4, was functioning to deflect scattered protons from entry

into the SDD. A charged particle detector was first calibrated using a radionuclide source

and then put in place of the SDD on the PIXE chamber to measure proton yield. Results

indicated that the assembly deflected nearly 100% of the scattered 3 MeV protons and was

thus performing adequately.

C.2 Materials and methods

C.2.1 Detector energy calibration

The electronics used in both the energy calibration and the PIXE deflection test included

an Ortec® TU–012–050–100 Ultra Ion-Implanted-Silicon Charged-Particle Detector (SCPD)

(12 keV FWHM @ 5.486 MeV). This was connected to an Ortec® 142B preamplifier model.

This SCPD had a window size of 50 mm2. A +50V bias voltage was delivered to the

SCPD using an Ortec® 428 Bias Supply. Pulses were fed into an Ortec® 672 Spectroscopy

Amplifier, set to a unipolar amplification factor of 50, with a 3 𝜇s shaping time. The

amplifier was connected to an ND 575 ADC with a 2K conversion gain setting. The ADC

150

was fed into a in-house MCA.

The energy calibration source was a weak (700 Bq) 244Cm alpha emitter. This nuclide

was electro-deposited on a steel backing to ensure stability. The short range of the emitted

alpha particles necessitated the use of vacuum conditions. The source and SCPD were

housed in a six-way sealed beamline cross that functioned as a vacuum chamber. Vacuum

was supplied by a portable vacuum roughing pump. A portable alpha particle filter was

connected to the pump air exhaust to prevent recoil Pu nuclei from entering into the lab

atmosphere. The SCPD was connected to the underside of one of the six-way cross portals

in a plastic mount. Plastic was used to minimize detector leakage current and isolate the

detector from environmental electronic noise.

For the calibration run, an acquisition time of 18 hours was necessary due to the small

count rate of the source. The calibration spectrum from this measurement is seen in Fig-

ure C.1. The tail on the low energy side of the alpha peak at 5.805 MeV is due to incomplete

charge collection, primarily due to the short penetration range of the alpha particles into

the detector front face. An approximate energy calibration relation was derived (Equa-

tion C.1) using the two known peaks appearing in this 244Cm decay spectrum. These peaks

are identified in Figure C.1.

𝐸𝑛𝑒𝑟𝑔𝑦(MeV) = 0.004826 ·𝐶ℎ𝑎𝑛𝑛𝑒𝑙 + 0.5401 (C.1)

C.2.2 PIXE chamber deflection test

The SCPD was situated at the approximate location corresponding to the front face of the

SDD. The SCPD was housed in a plastic mount situated in a 90∘ vacuum beamline elbow,

seen in the bottom right corner of Figure C.2. The preamplifier unit was connected directly

above, seen held by a clamp in the top right corner of Figure C.2.

The experimental conditions and components used in the detector energy calibration

with the 244Cm source were repeated during the PIXE run for measurement of proton flux.

Three different spectra were acquired using an integrated beam charge of 1.0 𝜇C. An Al

151

3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 51 0 0

1 0 1

1 0 2

1 0 3

1 0 4

2 4 4 C m α

2 4 0 P u α

Intens

ity

E n e r g y ( M e V )Figure C.1: Calibration spectrum acqured using an SBD detector adjacent to a low activity244Cm source. The largest peak corresponds to the energy of the alpha-emission of 244Cm(5.805 MeV). The smaller peak appearing on the tail of the larger peak corresponds to theslow α-decay of the daughter product (240Pu, 5.168 MeV) of 244Cm.

standard was trivially chosen as the incident target, set at 45∘ with respect to both the

incident and scattered detected beam. The first spectrum was acquired with the beam off

as a means to get a reading of the background noise. The next spectrum was acquired with

the beam on and with the magnetic assembly in place. The last spectrum was collected with

the beam on but with the assembly removed. For this final run, the collimation geometry

introduced by the magnet assembly in the previous run, was reproduced using a simple Ta

collimator having a diameter indentical to that of the magnet assembly. This step was done

only to maintain approximately, the same solid angle view to the detector.

C.3 Results and discussion

The ADC low level noise discriminator on the ADC was set such that a small amount

of electronic noise was allowed in spectra at around the 1 MeV energy mark. The “noise”

contribution can be seen at the low energy side of the spectrum in Figure C.3. The spectrum

152

Figure C.2: Photograph of the arrangement housing the SCPD detector connected to thePIXE chamber. The elbow connector at the bottom of the right lower quadrant housesthe SCPD such that it faces the PIXE chamber. The insulated wire running to the SCPDconnects to the pre-amplifier unit seen in the top-right quadrant.

taken with the beam turned on and the magnet in place, denoted as “magnet” in the figure,

demonstrates a similar trend in the noise present at very low energy. However, it also

shows several counts in the range from about 1 to 3 MeV. In this range, about 100 non-

noise counts appear. They are interpreted as representing a very small amount of scattered

proton fluence (∼ 0.1s-1) entering into the detector field of view. The spectrum “no magnet”

taken with the magnet assembly removed has shown that in the absence of the assembly, the

scattered proton fluence is substantial (∼ 1800s-1). The most intense region of the spectrum

acquired with the magnet removed is found in Figure C.3 between about 1 and 3 MeV. This

corresponds to single scattering events entering the SCPD. The continuum of events in this

region might be explained by the possibility of proton energy loss upon entry into the Al

standard before scattering in the direction of the SCPD. In the region beyond 3 MeV, the

intensity likely represents pile-up due to multiple particles arriving at the same time.

153

2 3 4 5 61 0 0

1 0 1

1 0 2

1 0 3

n o i s e m a g n e t n o m a g n e t

Coun

ts

P r o t o n e n e r g y ( M e V )Figure C.3: SCPD spectra showing detector background signal (grey line), and aluminumtarget scattered proton flux with the Nd-Fe-B magnetic assembly in place (white dots) andabsent (black dots).

The SCPD active diameter is 8 mm and the SDD collimator is only 3 mm. Therefore,

about one seventh of the fluence observed in these spectra are expected to enter into the

SDD. With the deflector in place this represents only a very small proton contamination

which indicates that the Nd-Fe-B magnetic deflector is working satisfactorily and that the

few events observed are likely due only to secondary scattering. This secondary scattering

would manifest as protons that have undergone primary scattering off the target and have

further interacted (through secondary scattering) with either the walls of the bellows or with

the Ta collimator such that they are ultimately incident on the SDD face.

154

Appendix D

Fitting light element spectra withconsideration for the radiative Augereffect

D.1 On the radiative Auger effect

The radiative Auger effect is mentioned in brief in this chapter since its presence, manifested

in light element spectra, has been found to complicate the matter of fitting the tail region of

the primary peak. In this chapter, the RAE process is reviewed and followed by a detailed

description of the fitting of the light element spectra acquired in Chapter 4. A comparison

of light element RAE intensities, reported in the literature, will be discussed to lend support

to the fitting treatment applied to light element spectra.

The photoelectric effect involves photon or particle induced ejection of an inner shell

electron from an atomic cloud leaving the atom in an excited state. This instability is

followed immediately by a return to stable ground state through emission of either an Auger

electron or characteristic X ray. These are the two well understood principal and competing

modes of de-excitation that an atom may take. However, in the late 1960’s a third mode of

K shell de-excitation was identified by Åberg et al. [1, 2, 145] using WDXRS. Observations

of photons appearing at energies lower than the Kα primary peak were described as being

emitted by a radiative Auger effect or (RAE) which incorporates both the emission of a

photon and a KLV Auger electron simultaneously. The energy of de-excitation is shared

between the two principal modes [2] and is described by the energy-conservation relation:

155

~𝜈 + 𝐸kin(𝑉 ) = 𝐸(𝐾𝐿𝑉 ) (D.1)

where ~𝜈 is the RAE photon energy, 𝐸kin(𝑉 ) is the energy of the emitted KLV Auger

electron and 𝐸(𝐾𝐿𝑉 ) is the total energy of the standard KLV Auger electron emission.

𝐸kin(𝑉 ) has an energy of any value in the range of 0 to 𝐸(𝐾𝐿𝑉 ) which permits the emitted

photon ~𝜈 to carry the remaining energy of 𝐸kin(𝐿) to 0. In a light element spectrum, the

feature that manifests from this process peaks in intensity just below the K X-ray peak and

falls rapidly with decreasing energy. The peak energy location corresponds to the maximum

energy that ~𝜈 can take. For Si, this is approximately 1.61 keV [2] which agrees closely

to Seigbahn et al.’s tabulated Si KL2,3L2,3 Auger electron energies [143]. Åberg’s WDXRS

work, using light elements [2], demonstrated that observed RAE peaks correspond fairly

well with Seigbahn et al.’s measurements of the dominant Auger electron emissions.

D.2 Fitting light element SDD acquired spectra

The high purity Mg, Al and Si spectra acquired through PIXE measurements were fitted

using an in-house University of Guelph PIXE group fitting routine (GUFIT). This an older

software version of the present GUPIX package but possesses the manual fit routine and

additional fitting functions not present in the newer package. These additional features

have been made of use in the fitting task presented here. The spectra acquired using the

Ketek SDD consists of a peak containing characteristic K X rays from Mg, Al or Si sitting

on top of a Bremsstrahlung continuum background. The energy resolution of the SDD is

between 70 to 80 eV FWHM for K X rays emitted by light elements. At this resolution, the

light element Kβ peak appears as a low-intensity shoulder on the high energy side of the Kα

peak. Both of the Kα and Kβ peaks were fitted each with a single Gaussian function with

the Kα:Kβ intensity ratio set as a free parameter. The Gaussian function has the form [28]:

𝐺(𝑖) = 𝐻𝐺 exp

[︂−(𝑖− 𝑖0)

2

2𝜎2

]︂, (D.2)

where 𝐻𝐺 is the peak height, 𝑖0 is the centroid channel number, 𝜎 is the peak standard

156

deviation and the Gaussian primary peak is a function of channel number 𝑖. In all spectra an

additional independent Gaussian function was fitted to the first order pile-up peak situated

at double the energy of the primary peak. Further Gaussian fits were applied to the trace

impurities seen in the Bremsstrahlung continuum in certain spectra.

On the low energy side of the main peak, there exists a tail feature as discussed in

Chapter 2. The tail has an inherent exponential shape and is appropriately fitted with the

exponential function (D.3). This function includes a convolution to an error function to

permit smoothing at the edges in the format of:

𝑇 (𝑖) =1

2𝐻𝑇 exp

(︂𝑖− 𝑖0𝛽

)︂erfc

(︂𝑖− 𝑖0

𝜎√2

+𝜎

𝛽√2

)︂(D.3)

where 𝐻𝑇 is the height of the tail and 𝛽 is the slope, and these are the only two free

parameters of the fit.

The RAE feature appears in all spectra, identifiable as a low energy shoulder sitting

on top of the exponential tail. The RAE differs from principal X-ray lines in that the

distribution has a slight mound shape on the tail with a skew towards lower energies. This

is due to the numerous possible combinations of KLV radiative Auger energies that may be

emitted during de-excitation. As a completely empirical solution to fit this feature, the DE

function introduced in Chapter 2 was implemented (denoted here as 𝐷𝐸(𝑖)) and found to

produce a satisfactory fit. A close-up of the KLV RAE fit region from Figure 4.5 is shown in

Figure D.1. This function was made to resemble a wide, skewed Gaussian shape by inputting

a very steep slope parameter in the DE fit, one that decreases with decreasing energy.

The original GUPIX description of the KLV RAE feature for Mg, Al and Si was a single

Gaussian function having intensity about 1% the intensity of the primary peak. This was

found to produce a mis-fit of the RAE region as the fitted RAE intensity exceeded the

data in all element spectra. As well, the use of a Gaussian did not quite match the skewed

distribution, apparent in Figure D.1 in the energy region of 1.4 to 1.6 on the tail. The reason

for the difference in intensity has been investigated here by comparing light element KLV

RAE data found in the literature, summarized in Table D.1.

157

1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1 . 6 1 . 7 1 . 8 1 . 9 2 . 0

1 0 2

1 0 3 K L V R A E X r a y s S i K α, K β p e a k s d a t a t a i l f i t

Co

unts

P h o t o n e n e r g y ( k e V )

- 303

Re

sidual

s

Figure D.1: Close-up image of fitted tail and RAE region on low energy side of Si K X-rayprimary peak. The RAE feature may be seen as centered at 1.57 keV. Fit residuals of thedisplayed energy region are included.

The first two columns of data in Table D.1, show the theoretical intensity values of

Scofield [139] and Åberg [1]. Their values are clearly much larger than the experimental

intensity values reported by other authors in the remaining table columns. This observation

was realized by Åberg [1]. It is somewhat surprising that Scofield’s HF calculated intensity

predictions of the KLL features disagree with the experimental data. When considering that

the same calculations have been confirmed to agree very well with experimental Kβ/Kα

X-ray intensity ratios and KMM RAE intensities, as mentioned in Section 3.7, the large

disagreement with measured KLL intensities is quite inconsistent.

The experimental intensity values of Åberg [1] and Siivola [145] were given as only

preliminary approximate estimates. The later measurements of Richard et al. [129] and

Oltjen et al. [115], using charged particle ionization (H and He) produced estimates that are

almost a factor of ten less than the theoretical calculations. These intensities correspond to

the dominant K-L23L23 RAE emission lines which represent about 70% of the total RAE

158

KLV RAE Intensity (%)

Element Calculated Experimental

Mg 2.3 2.5 <1 0.4 0.23± 0.05Al 1.8 1.9 <1 0.4 0.16a 0.15± 0.03Si 1.3 1.5 0.7 0.18a 0.2a 0.19± 0.01

Ref. Åberg Scofield Åberg Siivola Richard Oltjen This work[1] [139] [1] [145] [129] [115]

a Intensity of the major K-L23L23 line only

Table D.1: Integrated RAE intensities relative to Kα1,2 lines. Uncertainties in the intensitiesderived from this work represent a standard deviation spread in the fitted spectra.

intensity. [163].

The integrated intensities of the DE(i) functions fitted to the RAE features in the Mg,

Al and Si spectra of Chapter 4 were used to derive new estimates of the total RAE intensity

for the light elements. Average intensities computed using the 7 spectra for each element,

are presented in the final column of Table D.1. The average intensities show reasonable

agreement with the intensities reported by Richard et al. [129] and Oltjen et al. [115] and

justify the fitting treatment to the RAE feature that was adopted in this work.

While the similarity of the estimates in this work to those in the literature is encouraging,

some caution is warranted given that the tail structure of the SDD used has not been

completely characterized. Some amount of error is likely produced from the competition of

functions 𝑇 (𝑖) and 𝐷𝐸(𝑖) being fitted in this region. This likely explains why the error from

one standard deviation spread, reported in Table D.1, is somewhat high. Measurements

of the RAE feature using metal oxide spectra were even less reliable, ie: had a greater

spread. This is due likely to the greater intensity of the Bremsstrahlung feature sitting

under the primary peak. Full characterization of the RAE contribution to the tail might be

accomplished using synchrotron radiation to predict the shape of the tail in the absence of

the KLV RAE feature.

The DE(i) function appears to provide a very good fit of the RAE feature. Still, a

Gaussian function, with an intensity similar to that reported by this work, can provide a

satisfactory fit to the entire tail region, as was found in Chapter 5.

The fitting treatment of the Bremsstrahlung continuum is described as follows. This is

159

a mound-like shape, extending from about zero to six keV, that was present in all spectra

acquired using the present PIXE system. This feature was fitted using the exponential

function:

𝐵(𝑖) =𝑚∑︁𝑖=0

exp(𝑐m𝑖m) (D.4)

where 𝑖 is the channel number and 𝑐𝑚 are coefficients that are variables in the fit and

can take on positive or negative values. When the Bremsstrahlung background was of low

intensity, 𝑚 = 3 parameters provided satisfactory fits of this feature. For higher intensity

backgrounds, additional energy parameters were sometimes needed. For these, (𝑚 = 4 or

5) parameters were used. This function produced the mound-like shape seen in Figure 4.5.

The use of Equation D.4 in GUFIT manual mode can be a challege as the single loop

automatic least-squares minimization step is not always sensitive enough to force parameter

changes to agree with the spectrum shape. This function works best when a good guess

at its shape is given. Typically, specifying consecutive parameters 𝑐0, 𝑐1, 𝑐2 and so on as

alternating positive and negative values is a good start. One can build a Bremsstrahlung

function through trial and error, using mathematical plotting software, or go straight into

variable manipulation using the GUFIT manual fitting routine.

The coefficients of lower order, ie. 𝑐0 and 𝑐1 affect the region of lowest energy most

strongly. Similarily, the coefficients of highest order (𝑐3...𝑐5) affect the higher energy region

most strongly. If the desired effect is to fit a Bremsstrahlung background that has its high

intensity region, very near to 0 keV, then a recommended approach is to make one or both

of the first two coefficient large, with respect to the other coefficients, and positive in value.

The third coefficient would need to be negative to curve the feature back towards the energy

axis. If on the other hand, the desired effect is to fit a mound that has its maximum at

slightly higher energy, well beyond 0 keV, then the lowest order parameters should be kept

small and the 3rd parameter made large. For complex treatments such as this, the full

six parameter function may be needed. The complete lineshape used to fit the spectra of

Chapter 4, denoted as 𝐿(𝑖), is the sum of the primary functions described in this chapter:

160

𝐿(𝑖) = 𝐺(𝑖) + 𝑇 (𝑖) +𝐷𝐸(𝑖) +𝐵(𝑖) (D.5)

where 𝐺(𝑖) is applied three times, one each to the Kα, Kβ and first-order pile-up peaks

and i is the channel number on which all parameters are dependant.

161

Appendix E

GUPIX determination of PIXEH-value efficiency constants

An H-value is a PIXE instrumental parameter that represents the solid-angle efficiency as-

sociated with a given PIXE measurement setup. Its unit is the steradian, representing the

three-dimensional cone view that the emitted source photons traverse on their way to a

detector. Since this is only a geometrical constant, it has no dependence on matrix com-

position or photon energy. The H-value is analogous to the absorption efficiency constant

(𝜖a(ℎ𝜈)) discussed in this work as both have units of steradians. However, the 𝜖a(ℎ𝜈) param-

eters differs in that it represents the combined solid angle (H-value) efficiency and detector

intrinsic efficiency. It also includes efficiency corrections of any detector-internal or -external

absorbing layers. For the efficiency constant comparison used in this work, either the 𝜖a(ℎ𝜈)

or H-value approach could have been used to derive the same results.

For materials analysis using PIXE, it may be necessary to calculate the system H-value

using an approach such as that outlined in Chapter 4. In principle, the system’s H-value

can be determined using the characteristic X-ray yield produced by a PIXE measurement

performed on a pure element target. The X-ray peak area (yield) would be used in the PIXE

FP equation (Equation 4.1) along with the known target concentration and the GUPIX cal-

culation of the remaining parameters to derive the H-value. This approach assumes that

all other detector intrinsic efficiency and absorbing layers are accounted for accurately. In

practice however, it is often difficult to account for detector efficiency with a high degree of

accuracy. This ultimately has little effect on H-values calculated for higher Z targets due

to the lower attenuation effects of the associated higher energy X-rays. However efficiency

162

inaccuracies manifest strongly in lower Z target PIXE measurements and often result in

the calculated H-values finding poorer agreement with the higher Z values. For this reason,

the standard practice is to derive H-values at as many energy points (using multiple pure

Z targets) as possible so that any Z-dependence in this parameter can be discovered. The

resultant Z-dependant H-value relationship can be used as input in the GUPIX program

for subsequent PIXE measurements intended to derive concentrations. In this manner, the

Z-dependence allows for the correction of any inaccuracies in the consideration of intrin-

sic efficiency and absorbing layers. Therefore, efficiency correction errors will not affect

concentration results in a given measurement.

Calculation of an H-value for a given pure element measurement can be done using the

basic formula when working with the GUPIX program:

𝐻𝑡𝑟𝑢𝑒Z 𝐶𝑡𝑟𝑢𝑒

Z = 𝐻𝑓𝑎𝑙𝑠𝑒Z 𝐶𝑓𝑎𝑙𝑠𝑒

Z (E.1)

where 𝐻Z and 𝐶Z are the H-value and concentration corresponding to a particular element

in a sample. The superscripts “true” and “false” indicate the correct and incorrect values

respectively. If the goal is to determine the true H-value (𝐻𝑡𝑟𝑢𝑒Z ), then this can be done

by supplying GUPIX with a complete fixed matrix compositon, using all major element

concentration values known to be correct. 𝐶𝑡𝑟𝑢𝑒Z will be one of these input values. Also as

input, the user will specify an initial guess at an H-value. This will be the 𝐻𝑓𝑎𝑙𝑠𝑒Z value that

is applied to all elements. This can be chosen as a constant H-value in the GUPIX setup

panel. A reasonable initial guess might be 0.0001 sr. Upon running GUPIX with these

parameters, the program will use the fitted peak areas to determine concentrations of all

specified major and minor elements. In the output, these concentrations (𝐶𝑓𝑎𝑙𝑠𝑒Z ) will differ

from the initial guess, beyond differences imposed by the statistical spread of the observed

peak areas. This is expected since a false H-value was used. 𝐻𝑡𝑟𝑢𝑒Z is then derived simply

by rearranging Equation E.1 to solve for this parameter explicitly by subbing in the known

parameters.

Using this approach, the H-values for each major element can be determined in a single

fitting run with the GUPIX program. The calculated 𝐻𝑡𝑟𝑢𝑒Z s for each major element can

163

then be used as input in an energy-dependent GUPIX setup by incorporating these into a

(*.hed file). Further details of this approach may be found in the GUPIX manual and are

therefore not elaborated upon here. The validity of the calculated 𝐻𝑡𝑟𝑢𝑒Z ’s can be checked by

performing a new run with the new H-values used as input to the GUPIX program. If they

are correct, the resultant major element concentrations should appear to be equal to the

initial fixed matrix concentrations (𝐶𝑡𝑟𝑢𝑒Z ). If not, further iterations of this process might

be required until convergence of concentrations is observed.

The accuracy of the derived H-values can be confirmed if the targets examined are

of a simple and very well-known composition, such as a pure element target. Repeated

measurements might also reduce the error associated with the statistical spread. H-values

associated with the SDD-PIXE setup used in this work were calculated and are given in

Section E.1.

The efficiency constant approach was adopted over the H-value approach in this work

since the GUYLS subroutine does not allow for corrections for detector efficiency. Use of

the GUYLS subroutine over GUPIX was a choice made due to the greater freedom available

to specify matrix composition. In GUYLS, all elements may be entered according to their

fractional weight percentage in the matrix, regardless of their atomic number. In GUPIX,

the program interface does not allow certain “invisible” elements and compounds, such as

fluorine or H2O to be added to the matrix. It also does not permit entry of multiple

compound configurations of element-oxides. For example, many of the silicate standards

consisted of quantities of FeO and Fe2O3. In GUPIX, these may be entered under one

valence configuration only, either FeO or Fe2O3. In GUYLS, both configurations can be

deconstructed in terms of total Fe and total O prior to input into the subroutine. Although

the difference in the effects imparted upon the final concentration results in using the two

approaches is likely small, it appears that greater accuracy could be obtained using the

GUYLS approach.

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E.1 Detector efficiency file and H-value parameters

The following text represents the detector file content used to fit the spectra in Chapter 5.

Included in the content are the energy-dependent parameters of the tail height, slope and

relative area, given for 20 energy points. This file is named AP0DET.DAT.

$$DET_ID - used to specify that next line contains det. # and ID2 ’Silicon Drift Detector’$$DET_WIN – window information to follow1 – # of window materials followed by 1 line for each material4 0.00090 0 – Z, thickness(cm), dens(g/cm3) – if dens=0 use standard tabulated value$$DET_EL – electrode information to follow14 0.000008 0 – Z, thickness (cm), dens(g/cm3), – if dens=0 use standard tabulated value$$DET_CRYS – detector crystal info to follow14 0.045000 – Z, thickness(cm)

0 Target to crystal distance in cm, (enter 0 if collimated)1.0 Ag/At cutoff energy (keV) or if<0, Si dead layer thickness (cm)180 maximum nominal det res at 5.9 keV1260 default tau (ns)25. Pulse dead time in usF 1000 Voigtian line shape switch with cutoff1 Minimum background value for MDL calculation20 2.1 1.74 1 0 0

Energy grid (keV) for parameters below1.041 1.254 1.487 1.74 2.0 2.293 2.984 4.09 4.5089 4.95 5.4116 5.896 6.4005 6.926 7.473 8.0428.631 10 12 20

Long plateau heights0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Short step heights0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Height of tail #10.021982 0.017047 0.011745 0.005844 0.2 0.13 0.085 0.0352 0.03652 0.026053 0.019602 0.0176530.016067 0.011981 0.010971 0.011483 0.011628 0.011628 0.011628 0

Slope of tail #12.5 2.934505 4.12152 6.73533 0.5118 0.6198 0.874502 1.34149 1.29677 1.520945 1.7818051.94772 2.05735 2.512805 2.839175 2.8205 2.600575 02.600575 2.600575 0

Height of tail #20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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Slope of tail #20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Tail to Gaussian area ratios for efficiency corrections0.0205 0.019952 0.018895 0.015427 0.3 0.2 0.06 0.018852 0.018854 0.015778 0.01386 0.0136950.013179 0.012006 0.012304 0.012862 0.012046 0.012046 0.012046 0

Weighting mode & Sys. error values (Rel.Int error, Frac abs error, KMM & tail )1 0.01 0.01 F F

KMM Radiative Auger errors given by tail height & slope values (Z=20-33).100 .095 .090 .085 .080 .075 .070 .065 .060 .050 .040 .030 .020 .010 2.1 2.1 2.1 2.1 2.1 2.1 2.12.1 2.1 2.1 2.1 2.1 2.1 2.1

Low energy tailing errors (given by height, slope & cutoff values on E grid) Tail heights forweighting0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Tail slopes for weighting0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Tail cutoff energies (keV) for weighting0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

In this work it was not necessary to calculate H-values for use as efficiency corrections.

Still, this was done as a means to calibrate the present PIXE system so that any subsequent

PIXE measurements on targets could be processed to derive concentrations. To do this,

H-values were derived from PIXE measurements performed on pure element targets (Z =

12 . . . 14 and Z = 21 . . . 30) found in an Astimex metal mount. The resultant H-values are

listed aside the corresponding element Kα X-ray energy, seen in Table E.1. The format of the

numbers in the table, without the headings, can be inputed directly into a file with a (*.hed)

extension. This is called upon by GUPIX to provide the H-value efficiency corrections when

determining absolute concentrations. Column headings, left to right, are Energy (keV) and

H-value (sr), respectively.

For some of the given file energies, such as 1.041 and 1.835–3.000 keV, no elemental

data were available. Therefore, the H-values of the 3d transition metals were extrapolated

crudely to fill in these values.

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Energy (keV) H-value (sr)

1.041 0.000051.254 0.00008221.487 0.00011821.740 0.00012751.835 0.00013001.850 0.00011002.000 0.00011203.000 0.00012004.091 0.00012474.511 0.00012504.952 0.00012545.415 0.00012575.899 0.00012616.404 0.00012656.930 0.00012697.478 0.00012738.048 0.00012788.639 0.0001282

Table E.1: H-values derived from an Astimex metal mount containing elements Z = 12 . . . 14and Z = 21 . . . 30.

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Appendix F

Curriculum vitae of the author

Place of BirthSimcoe, Norfolk County, Ontario, Canada

Educational BackgroundM.Sc. Medical Physics: 2006–2008 McMaster University

Thesis Title: In Vivo Quantification of Bone Strontium Using X-Ray FluorescenceAdvisor: Professor David R. Chettle

B.Sc. (Honours) Physical Science: 2000–2004 McMaster University

List of publicationsT. L. Hopman, C. M. Heirwegh and J. L. Campbell 2014 Lineshapes characterization

of nickel contact Si(Li) detectors using 1–8 keV X-rays, X-ray Spectrom. Acceptedsubject to major revision.

C. M. Heirwegh, I. Pradler and J. L Campbell 2013 An accuracy assessment of photo-ionization cross-section databases for 1–2 keV X-rays in light elements using PIXE, J.Phys. B: At. Mol. Opt. Phys. 46(18), 185602.

T. L. Hopman, C. M. Heirwegh, J. L. Campbell, M. Krumrey and F. Scholze 2012 Anaccurate determination of the K-shell fluorescence yield of silicon, X-ray Spectrom.41(3), 164–171.

C. M. Heirwegh, D. R. Chettle and A. Pejović-Milić 2012 Ex vivo evaluation of a coher-ent normalization procedure to quantify in vivo finger strontium XRS measurements,Med. Phys. 39(2), 832–841.

C. M. Heirwegh, D. R. Chettle and A. Pejović-Milić 2010 Evaluation of imaging tech-nologies to correct for photon attenuation in the overlying tissue for in vivo bonestrontium measurements, Phys. Med. Biol. 55(4), 1083–1098.

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Conferences, presentations and workshops

C. M. Heirwegh, I. Pradler and J. L. Campbell, Comparison of attenuation coefficientdatabases used in PIXE analysis, using geochemical reference materials. (Presenta-tion, delivered by J. L. Campbell). 7th Workshop, International Initiative on X-RayFundamental Parameters, March 25–26th, 2014, Paris, France.

C. M. Heirwegh, I. Pradler and J. L. Campbell, A comparison of attenuation coefficientdatabases used in 𝜇-PIXE analysis - XCOM, Chantler or...? (Presentation). 13th

International Conference on Particle Induced X-Ray Emission, March 3rd–8th, 2013,Gramado, Brazil.

C. M. Heirwegh, R. Butler, D. R. Chettle and A. Pejović-Milić, Evaluation of MR,CT and ultrasound imaging modalities for estimation of finger soft-tissue thickness:efforts to improve normalization of in vivo strontium X-ray fluorescence measure-ments. (Presentation). 7th International Topical Meeting on Industrial Radia-tion and Radioisotopes Measurement Applications Conference, June 22nd–27th, 2008,Prague, Czech Republic.

E. Da Silva, C. Heirwegh, A. Pejović-Milić A and V. Heyd, Use of hydroxyapatite bonecomposites for the calibration of in vivo EDXRF - based systems for bone strontiumquantification. (Poster, presented by E. Da Silva). European Conference on X-raySpectrometry, June 16–20th, 2008, Cavtat, Dubrovnik, Croatia.

C. M. Heirwegh, The use of chick chorioallantoic membranes in cancer research. (Pre-sentation). October 30th–November 1st, 2003, Canadian Undergraduate Physics Con-ference, McGill University, Montréal, Québec, Canada.

Scholarships2009 Ontario Graduate Scholarship (Provincial - 3 year stipend)2006 Graduate Student Scholarship (Dept. of Medical Physics - 2 year full stipend)2006 McMaster University Entrance Scholarship (Dept. of Medical Physics - 1 year)

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