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NBS SPECIAL PUBLICATION 604

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Energy Dispersive X-Ray Spectrometry

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no. 604

1981

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NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

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provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering2 — Mechanical

Engineering and Process Technology2 — Building Technology — Fire Research —

Consumer Product Technology — Field Methods.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234.

2 Some divisions within the center are located at Boulder, CO 80303.
Energy Dispersive X-Ray Spectrometry

NATIONAL BURBA

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JUL 2 0 1981

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Proceedings of a Workshop on Energy Dispersive X-Ray Spectrometry

Held at the National Bureau of Standards, Gaithersburg, MD 20760

April 23-25, 1979

Editors:

K. F. J. Heinrich

D. E. Newbury . R. L. Myklebust

Center for Analytical Chemistry

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

C. E. Fiori

Biomedical Engineering Instrumentation Branch

National Institutes of Health

Bethesda, MD 20205

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1981
Library of Congress Catalog Card Number: 81-600047

National Bureau of Standards Special Publication 604

Nat. Bur. Stand. (U.S.), Spec. Publ. 604, 441 pages(June 1981) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D C 20402 - Price $9.

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FOREWORD

The Center for Analytical Chemistry of the National Measurement Laboratory,

National Bureau of Standards, seeks to develop new techniques of chemical

analysis and to constantly improve existing analysis methods. Part of the

mission of NBS is to disseminate knowledge in the scientific and technical

community. To aid in achieving this objective, the Center for Analytical

Chemistry has sponsored a series of workshops on various topics in analytical

chemistry. The workshop topics are chosen to fulfill current needs for detailed

discussions on well defined subjects in a wide variety of specialist areas of

interest. The objective of these workshops is to bring together specialists

from throughout the world to concentrate intensively on a particular subject in

order to advance the state-of-the-art. It is often very difficult to achieve

this goal at large international meetings where the size and diversity of

topics presented often limit detailed discussion of specialist subjects. Past

topics of these workshops and the published proceedings include: Quantitative

Electron Probe Microanalysis (NBS Special Publication 298), Aerosol Measurements (NBS Special Publication 412), Secondary Ion Mass Spectrometry (NBS

Special Publication 427), Use of Monte Carlo Calculations in Electron Probe

Microanalysis and Scanning Electron Microscopy (NBS Special Publication 460),

and Characterization of Particles (NBS Special Publication 533). These proceedings are available from the Superintendent of Documents, Government Printing

Office, Washington, DC 20402. Further information on the workshops can be

obtained by writing to the Center Office, Center for Analytical Chemistry,

National Bureau of Standards, Washington, DC 20234.

This volume contains the proceedings of a Workshop on Energy Dispersive

X-ray Spectrometry. The three-day meeting involved participants from the

United States and Europe. The workshop format consisted of invited talks on each topic followed by contributed papers and extensive discussions.

The extensive development of electron beam instrumentation for microanalysis and microscopy has been greatly stimulated by the development of the

energy dispersive x-ray spectrometer based on the silicon (lithium) detector.

This publication provides a detailed view of the state-of-the-art of energy

dispersive x-ray spectrometry, from the basics of the detection process to applications of the technique to quantitative x-ray microanalysis. The papers

should be of wide interest to workers in many fields of science which utilize

x-ray microanalysis.

John D. Hoffman, Director

National Measurement Laboratory

i i i
PREFACE

The scanning electron microscope, electron probe microanalyzer, and

analytical transmission electron microscope have had great impact on the

characterization of specimens in the physical and biological sciences as well as

in the analysis of technological processes. A substantial part of this impact

is derived from the capability of these instruments to provide quantitative

elemental compositional analysis to complement the images obtained with the

electron beam. Compositional analysis has been greatly stimulated by the

development of energy dispersive x-ray spectrometry (EDS) based on the silicon

(lithium) solid state x-ray detector and the simultaneous development of the

laboratory minicomputer. The capability of the EDS system to provide a rapid

qualitative analysis of a sample in seconds and a quantitative analysis in minutes in the laboratory has made it extremely attractive to a wide variety of

workers. As a result, modern electron beam instruments are frequently equipped

with a complete EDS system.

The purpose of this meeting was to bring together workers who are actively

participating in the development of EDS detectors, signal processing systems,

and multichannel analyzers and in the application of EDS systems to x-ray

microanalysis and x-ray fluorescence analysis. We have witnessed a period of

rapid development in EDS systems over the past five years. Recent advances in

detector materials, and laboratory computer systems suggest that further

advances can be expected in the near future. It thus seemed to be an appropriate time to review the current state of development in the EDS field. We hope that this report will serve the scientific community as a useful reference

in this important area.

The editors gratefully acknowledge the excellent work of Joy Shoemaker and

the members of the Text-Editing Facility of the Center for Analytical Chemistry

in preparing this manuscript.

K. F. J. Heinrich

D. E. Newbury

R. L. Myklebust

Gas and Particulate Science Division

Center for Analytical Chemistry

C. E. Fiori

National Institutes of Health

Disclaimer

Certain commerical equipment, instruments, or materials are identified in

this paper in order to adequately specify the experimental procedure. Such

identification does not imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

iv
ABSTRACT

This book is the formal report of the Workshop on Energy Dispersive X-ray

Spectrometry held at the National Bureau of Standards, April 23-25, 1979. The

papers included in these proceedings cover a wide range of topics in the field:

(1) Physics of detectors — Properties of silicon (lithium), germanium

(lithium), intrinsic germanium, and mercuric iodide detectors are described and

compared. (2) Windowless x-ray detectors ~ The application of windowless

energy dispersive x-ray detectors to the measurement of soft x-rays below 1 keV

is described, including special electronic techniques for processing pulses

produced by low energy x-rays. (3) Electronics for energy dispersive x-ray

spectrometry — The signal processing chain is discussed with special emphasis

on state-of-the-art multichannel analyzers. (4) Spectral processing and

statistics ~ Fundamental statistical considerations in spectral processing are described. A variety of curve fitting techniques is discussed, including

sequential simplex and least squares fitting with a digital filter. (5) Artifacts in energy dispersive x-ray spectrometry in electron beam instruments ~

Artifacts arising from the basic detection and amplification processes and

instrument-related artifacts observed in electron beam instruments operating

in the range 10 - 1000 keV are described. (6) Applications — Applications

of energy dispersive x-ray spectrometry are given in quantitative x-ray

microanalysis and x-ray fluorescence analysis.

The papers include broad overviews of the subjects as well as specific

applications and should be of interest to workers in the electron beam microanalysis and x-ray fluorescence analysis fields.

Key Words: Electron probe microanalysis; energy dispersive x-ray spectrometry; germanium (lithium) detector; scanning electron

microscopy; silicon (lithium) detector; spectral deconvolution;

x-ray analysis; x-ray fluorescence.

v
TABLE OF CONTENTS

PARE

FOREWORD 1 i 1 PREFACE iv

ABSTRACT v

THE DEVELOPMENT OF ENERGY-DISPERSIVE X-RAY ANALYSIS 1 Kurt F. J. Heinrich

PROPERTIES OF SILICON AND GERMANIUM SEMICONDUCTOR DETECTORS FOR X-RAY SPECTROMETRY 5

R. W. Fink

A COMPARISON OF SILICON AND GERMANIUM X-RAY DETECTORS . . 35

N. C. Barbi and D. B. Lister

ROOM TEMPERATURE ENERGY DISPERSIVE X-RAY SPECTROMETRY WITH MERCURIC

IODIDE DETECTORS 45

A. J. Dabrowski, M. Singh, G. C. Huth, and J. S. Iwanczyk

STANDARD TECHNIQUES FOR MEASURING WINDOW ABSORPTION AND OTHER EFFICIENCY

LOSSES IN SEMICONDUCTOR X-RAY ENERGY SPECTROMETERS 61

F. J. Walter, R. Stone, D. H. Blackburn, and P. A. Pel 1 a

USE OF WINDOWLESS DETECTORS FOR ENERGY DISPERSIVE LIGHT ELEMENT X-RAY ANALYSIS 71

J. C. Russ and A. 0. Sandborg

PROPERTIES AND APPLICATIONS OF WINDOWLESS Si (Li) DETECTORS 97

R. G. Musket

THE EFFICIENCY OF Si (Li) DETECTORS AT VERY LOW PHOTON ENERGIES 127

P. J. Statham

ELECTRONIC TECHNIQUES FOR PULSE-PROCESSING WITH SOLID-STATE X-RAY DETECTORS 141

P. J. Statham

UNDERSTANDING MULTICHANNEL ANALYZERS 165

Eric Lifshin and Steven R. Hayashi

STATISTICAL CONSIDERATIONS OF DETECTABILITY LIMITS AND DECONVOLUTION IN ENERGY DISPERSIVE X-RAY SPECTROMETRY 177

P. L. Ryder

CURVE FITTING TECHNIQUES AND THEIR APPLICATION TO THE ANALYSIS OF ENERGY

DISPERSIVE SPECTRA 193

Frederick H. Schamber

SEQUENTIAL SIMPLEX: A PROCEDURE FOR RESOLVING SPECTRAL INTERFERENCE IN ENERGY DISPERSIVE X-RAY SPECTROMETRY 233

C. E. Fiori, R. L. Myklebust, and K. Gorlen

vi
Table of contents continued.

PAGE

LEAST-SQUARES FIT WITH DIGITAL FILTER: A STATUS REPORT 273 Jon J. McCarthy and Frederick H. Schamber

MULTIPLE LEAST SQUARES FITTING FOR SPECTRUM DECONVOLUTION 297

John C. Russ

ARTIFACTS OBSERVED IN ENERGY DISPERSIVE X-RAY SPECTROMETRY IN ELECTRON BEAM

INSTRUMENTS - A CAUTIONARY GUIDE 315

C. E. Fiori, D. E. Newbury, and R. L. Myklebust

ARTIFACTS ENCOUNTERED IN ENERGY DISPERSIVE X-RAY SPECTROMETRY IN THE ANALYTICAL ELECTRON MICROSCOPE 341

D. B. Williams and J. I. Goldstein

UNCOLLIMATED FLUORESCING RADIATION IN AN ANALYTICAL ELECTRON MICROSCOPE

(100-1000 KEV): SOURCES AND SOLUTION 351 Nestor J. Zaluzec

SPECTRAL PROCESSING TECHNIQUES IN A QUANTITATIVE ENERGY DISPERSIVE X-RAY MICROANALYSIS PROCEDURE (FRAME C) 365

R. L. Myklebust, C. E. Fiori, and K. F. J. Heinrich

THE EFFECT OF ELECTRON INCIDENCE ANGLE ON QUANTITATIVE ELECTRON PROBE

MICROANALYSIS 381

J. D. Brown

ENERGY AND WAVELENGTH DISPERSIVE X-RAY FLUORESCENCE ANALYSIS OF HEAVY

ELEMENTS USING RADIOACTIVE, SECONDARY FLUORESCENCE AND X-RAY TUBE

SOURCES, AND WITH GERMANIUM, SILICON, AND SCINTILLATION DETECTORS 391

M. A. Short and S. Bonfiglio

ENERGY DISPERSIVE DETECTORS - A BIBLIOGRAPHY (1981) 417

C. E. Fiori and C. R. Swyt

LIST OF ATTENDEES 441

vii

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

THE DEVELOPMENT OF ENERGY-DISPERSIVE X-RAY ANALYSIS

Kurt F. J. Heinrich

National Bureau of Standards

Center for Analytical Chemistry

Washington, D.C. 20234

Abstract

Energy-dispersive x-ray analysis with gas proportional detectors

has been used widely in x-ray spectrochemical analysis. It was particularly useful in the detection of low-energy x-ray emission,

before the commercial availability of the Blodgett-Langmuir pseudocrystals. With the development of solid-state x-ray detectors of

high resolution, and improvement in associated electronics and

computers, the interest in energy-dispersive x-ray analysis has

increased enormously, to the point where it has replaced Bragg

crystal diffraction spectrometry in many instances.

In the early years after the discovery of x-rays by W. C. Roentgen [I] 1

, x-rays of

differing energy were distinguished according to their power of penetration through materials. After Haga and Wind demonstrated in 1899 that x-rays can be diffracted, and a practical method was proposed by W. H. Laue and W. L. Bragg obtained the first diffracting

x-ray spectrum [2]. The x-ray spectra of many elements were obtained by Moseley [3], and

from that time on x-ray lines were predominantly characterized in Bragg crystal spectrometers according to their wavelengths.

1. Spectrometers for X-ray Spectrochemical Analysis

In the fifties and early sixties, Geiger counters were preferred in x-ray spectrometric

analysis, due to their simplicity and to the strong pulses emitted from these detectors.

Geiger counters were later abandoned because of their slow speed, and of the fact that

pulses generated by x-ray photons of different energy are indistinguishable. Therefore,

they could not be used for spectrometric purposes except in conjunction with Bragg crystal

spectrometers.

The subsequent widespread use of gas proportional detectors [4] generated interest in

the analysis of x-rays according to their photon energies (pulse-height analysis). In these

detectors, the energy of the x-ray photon is used to form pairs of electrons and negative

ions. These are accelerated in the electrostatic field of the detector, and collisions of

the electrons with other gas molecules produce an "internal amplification".

The total charge of the electron avalanche reaching the central anode wire determines

the height of the resultant pulse, and the statistical dispersion of heights of pulses

produced by photons of equal energy determines and limits the detector resolution. Several

schemes were developed for the use of a single-channel analyzer to exclude high-order

reflections of x-ray lines from wavelength spectra obtained with a Bragg spectrometer [5-

7]. Birks et al., have advocated the use of proportional detectors for energy-dispersive

analysis [8]. This approach was particularly useful for the separation of x-rays of long

wavelength, such as OKa and CKa bands, and was followed by Dolby [9]. In principle, this

1 Figures in brackets indicate the literature references at the end of this paper.

1
resolution can be predicted [10], and experimental results obtained with carefully designed

and prepared detectors come close to the predicted values [11] for an argon-filled

detector, it was found that

a

v/V = 0. 15 E _1/2

in which the photon energy in keV is E, V is the mean height of the pulse distribution, and

o\. its standard deviation.

Dirt and irregularities on the surface of the detector wire as well as other sources

of local electrostatic charges, considerably degrade this resolution. When the observation

of x-rays of low (1-3 keV) photon energies rendered necessary the use of very thin windows

made of organic films, further complications arose. To avoid electrostatic charges on these windows their metallization was required. Another problem arose from impuri f ication

of the detector gas, which had to be continually refilled in replacement of seepage through

the window. Oxygen or residual impurities (which may accumulate when a gas cylinder is

almost empty) cause a severe distortion of pulse height. Only when these difficulties were

overcome, and practical solid-state electronic pulse height analyzers were developed, could

pulse height selection and analysis be used to full advantage.

Energy dispersive analysis of y-rays and very energetic x-rays by means of sodium

iodide scintillators and photomultipl iers was widely practiced in the measurement of radioactive materials, but these devices had little energy resolution to be useful in the domain

of x-rays habitually used for spectrochemical analysis. Solid-state detectors became more interesting for this purpose with the advent of the lithium-drifted silicon detector

[12]. Once the effectiveness of the Si(Li) detector as an ancillary tool in electron probe

microanalyzers was demonstrated, it was soon tried in SEM's [13] and was shown to be of

great value both in qualitative and quantitative analysis.

In general, the energy-dispersive mode of analysis is preferable where high effectiveness of the detector system is more important than high resolution. In fact, even at the

present state-of-the-art in detector resolution (145 eV FWHM for Mn Ka), it is possible to find unsolvable interferences, such as a specimen having arsenic, sulfur, and lead (see

figure 1). In practice, however, the occurrence of such situations is very infrequent.

On the other hand, obtainable signal intensities (in terms of photons per primary electron)

can be - depending on the respective systems - 102 to 103 times stronger with energy

dispersive detection. For this reason, this mode has become prevalent in the analysis of

thin films and small particles. A drawback of the Si(Li) device is the large volume of

the Dewar flask used to store the liquid nitrogen required for cooling both the detector

and the preamplifier. If solid-state detectors of adequate resolution were developed

which will not require liquid-nitrogen cooling, this would considerably extend the useful

range of solid-state x-ray detection.

2. Conclusions

In view of their complementary advantages, it is best, for general purposes, to provide

instruments with both wavelength and energy dispersive spectrometers. In those analytical

situations in which the x-ray intensities emitted from the specimen are weak, the energydispersive detector is mandatory. It can be used for quantitative applications and is

attractive for simultaneous measurement of several x-ray lines. However, it is limited in

resolution and, hence, background levels are comparatively high. Hence, in spite of the

apparent simplicity of energy-dispersive spectra, a careful treatment and complex data

handling routine are required for quantitative applications.

2
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Figure 1. This log- log plot shows FWHM values for:

a) tabulated widths of the Ka x line

b) distances between Ka, and Ka2 c) resolutions obtained in a conventional electron probe microanalyzer with curved

crystals of lead stearate (PbSt), rubidium acid phthalate (RAP), NH4 -H2 P04 (ADP), and

lithium fluoride (LiF)

d) the natural widths of bands from K emissions of elements of low atomic number (circles)

e) the theoretical (dashes) and practical resolution of a Si (Li ) detector (SD), and

f) the theoretical (dashes and dots) and practical resolution of a Ar-CH4 gas proportional

detector. The high values marked with full triangles (for Si (Li) detector) are due to overlap of L lines. The bottom scale indicates photon energy.

3
3. References

[1] Roentgen, W. C. , Ann. d. Physik, 64, 1 (1898); Science 3, 227 (1896), and 3, 726

(1896).

[2] Bragg, W. H. and Bragg, W. L. , Proc. Roy Soc. (London) /A/88, 428 (1913).

[3] Moseley, H. G. J., Phys. Mag., 26, 1024 (1913); 27, 703 (1914).

[4] Fink, R. W. , Chapter 5 of Atomic Inner-Shell Processes, Craseman, B. , Ed., Vol. II,

Academic Press, New York, 1975.

[5] Heinrich, K. F. J., Adv. X-ray Analysis, 4, 370 (1961).

[6] Riggs, F. B. , Jr., Rev. Sci. Instr. , 34, 912 (1963).

[7] Weber, K. and Marchal , J., J. Sci. Instr., 4J_, 15 (1964).

[8] Birks, L. S. , Labrie, R. J., and Criss, J. W. , Anal. Chem. , 38, 701 (1965).

[9] Dolby, R. M. , Proc. Phys. Soc, 73, 81 (1959); J. Sci. Instr., 40, 345 (1963).

[10] Campbell, A. J., Norelco Rept. 14, 103 (1967).

[11] Mulvey, T. and Campbell, A. J., Brit. J. Appl. Phys., 9, 406 (1958).

[12] Bowman, H. R. , Hyde, E. K. , Thompson, D. G. , and Jared, R. C. , Science, J_5_[, 562

(1966).

[13] Russ, J. C, SEM-1971, Part 1, 65.

\

4
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

PROPERTIES OF SILICON AND GERMANIUM SEMICONDUCTOR DETECTORS FOR X-RAY SPECTROMETRY 1

R. W. Fink

School of Chemistry

Georgia Institute of Technology

Atlanta, Georgia 30332

Abstract

The characteristics of silicon and germanium semiconductor x-ray

detectors of various designs (disk, top hat, grooved, and ion implanted

types) are reviewed, including discussion of sensitive area and depth,

metallic- and dead-layers. The effects of electronic operating conditions and timing requirements on spectral peak shapes (tailing), resolution, and efficiency in singles and in coincidence measurements are reported. The response to monoenergetic photons is discussed, including

second-order x-ray spectral effects, such as the low-energy plateau,

escape peaks, the continuum around the full-energy peak, summing and sum

peaks, and effects of collimation. Various methods of measuring relative

and absolute detector efficiencies for photons from about 200 eV to 140

keV in energy are summarized, and a useful table is presented of longlived radioactive sources suitable for calibration of energy and

efficiency curves of x-ray detectors, in which the current best values of

halflife, photon energies, and the absolute number of photons emitted per

100 decays are tabulated. Finally, some recently reported advances in

detectors, electronic systems, and a new method of unfolding partlyresolved doublets (e.g. Ka x and Ka 2 lines) in coincidence spectra are summarized.

1. Introduction

The purpose of this article is to update earlier studies and reviews [1-5] 2 on the

properties and characteristic response of silicon and germanium semiconductor detectors for

x- and y-ray spectrometry in the energy range below about 30 keV for silicon and 140 keV for

germanium. Aside from some mention of efficiency and deadlayer determinations in silicon

for K x-rays as low as 185 eV (boron), the scope of this review also is limited to the

region of energies above about 3 keV, since there appears elsewhere in these proceedings a paper by Musket [6] dealing with essentially windowless Si(Li) x-ray detectors specializing

in ultra- low-energy measurements from 2 keV to 185 eV. Excluded from the present review is

any detailed discussion of fabrication techniques [7] or the fundamental physics of semiconductors. Moreover, detailed descriptions of electronic systems are not included,

although some discussion of recent advances and performance considerations is given.

Supported in part by the U.S. Department of Energy under Contract No. De-AS05-76-ER0-

3346.

2 Figures in brackets refer to the literature references at the end of this paper.

5
2. Characteristics of Various Detector Configurations

2.1 Resolution and Peak-to-Continuum Ratio

High resolution and a high peak-to-continuum ratio are of utmost importance. Resolution may be defined as the root-mean-square value of the electronic system line width, as determined with a randomized electronic pulser3

, and the intrinsic line width of the

detector:

where R is the resolution in eV, expressed as the full-width at half-maximum, FWHM, of a single photon line, such as the 14.4 keV y-ray from a 57 Co source or the 5.9 keV Ka x-ray

doublet from a 55 Fe source; F is the well-known Fano factor, which accounts for deviations

of the primary ionization from a Poisson statistical distribution, and is approximately

0.10 for Si and 0.1 to 0.129 for Ge; e is the average energy to create an electron-hole

pair and is about 3.58 ± 0.10 in Si and 2.94 ± 0.15 in Ge; and E is the photon energy in eV deposited in the sensitive volume.

A resolution of 146 eV FWHM at 5.9 keV at 1000 c/sec, for example, will resolve com- pletely the K x-ray peaks of Si (1.74 keV) and Al (1.49 keV), whereas a resolution of 165 eV FWHM produces considerable overlapping of these two peaks. The best system resolution so far reported is 139 eV FWHM at 5.9 keV at 1000 c/sec for a 10 mm2 x 3 mm Si (Li) detector.

Values of resolution below 100 eV have been obtained at 1.49 keV (Al Ka). It is clear that

for low energy x-ray experiments and for quantitative x-ray fluorescence analysis of low-Z

adjacent elements, the highest possible resolution is crucial, consistent with a reasonable

compromise on detector size in order to obtain sufficient solid angle, since one of the

principal sources of experimental error arises from deconvolution of such partially-resolved

peaks.

In addition to good resolution, the ratio of the full-energy peak to the continuum and

the full-width at 1 /1 0th maximum are important criteria of detector performance. Typically,

the peak-to-continuum ratio should exceed about 1500:1 for Si(Li) at 5.9 keV, depending on detector size, quality, and collimation; and the full-width at tenth maximum should be less

than about 1.85 the FWHM value. A low-energy "tail" extending from the full-energy peak, or a poor ratio of peak-to-continuum indicates a low quality or defective detector or an electronic system that is not properly tuned for optimum performance. The leading cause

of such poor detector performance is trapping and recombination effects in the detector

itself.

Charge trapping and recombination of electron-hole pairs, and variations in the depths

of the traps, give rise to a wide distribution of lifetimes of trapping and a wide distribution of slow risetime pulses, which lie mainly in the low-energy continuum below the peak

[3] owing to the incomplete collection of charge. For example, precipitated lithium in

Si(Li) and Ge(Li) detectors constitutes a hole trap, whereas a lithium-deficient impurity

site becomes an electron trap. The principal cause of asymmetric peak shapes ("tailing")

is a non-uniform distribution of traps in the intrinsic (sensitive) volume. Other degrading

effects which throw pulses into the tail include ohmic contact noise, microphonic problems,

infrared (blackbody) radiation, and crystal lattice imperfections arising either from the

original ingot or from radiation damage, e.g. from exposure to fast neutrons.

For photons in the energy range of interest, detectors range in size from 10 to as large as 750 mm2 active area and from 2 to 5 mm in sensitive depth; generally, the highest

resolution is achieved, the smaller the detector. To reach the best performance, it is

necessary to have fast collection times, as well as the lowest possible detector capacitance, in order to reduce the incidence of slow-rising pulses due to deadlayers, dead edges,

trapping, and recombination effects, since the largest proportion of slow pulses occurs in

the low-energy tail [3]. Thus, the detectors should be operated at a very high reverse-bias

voltage, from 1000 to 2000 volts, and to achieve such high electric fields without excessive

eV (1)

2.2 Detectors of Advanced Design

6
leakage currents (<10~13 amp at 77 °K) and to reduce the detector capacitance, the simple

disk-type planar detector configuration was soon replaced with other designs [8,9], such as those with concentric single- or double-grooves to form guard rings, or a top-hat design.

Sophisticated designs have been reported [10] using double-grooved guard ring detectors, in

which an anticoincidence signal is taken from the guard ring to reject events in the central

detector that trigger both simultaneously. This effectively eliminates partial collection

from escaping events in the peripheral regions of the central detector. A double guard ring

[8,10] can even be used, the outer ring being grounded to reduce the edge leakage current

while the inner ring serves as the anticoincidence signal discriminator to detect and reject

even very small pulses.

The anticoincidence guard-ring detector can achieve very substantial improvement of

peak-to-continuum ratios, as demonstrated by Landis, et al. [8]; for example, a 40-fold

reduction in the continuum from Cd K x-rays (23 keV) is observed with no sacrifice in the

resolution. The use of physical collimation is not at all as effective. Unfortunately,

these sophisticated guard-ring detectors with reject circuitry are not routinely available

commercially, although modern integrated NAND circuits could readily be incorporated into

the preamplifier to provide rejection of small pulses from events escaping the center

counter.

The properties of lithium-drifted Si (Li) and Ge(Li) detectors for low-energy photon

and electron spectrometry have been thoroughly reviewed in the literature cited above.

The production of ultra-high purity Si(HP) and Ge(HP) crystals (resistivities up to 10 5 ohm-cm) permits the fabrication of PN junction (rectifying) detectors which are free of

drifted lithium with all of its problems. The first of such devices were known as "surface

barrier" detectors because of their thin (200 A, 40 ug/cm2

) gold metallic front window.

These produce depletion layers under high reverse-bias voltage up to several mm in depth

[11]. For example, N-type SI (HP) with a gold surface barrier and guarded N-type lithiumdiffused (but not drifted) N+ contacts has been used to fabricate detectors with depletion

layers 4 mm thick at 800 volts which can operate as high resolution x-ray spectrometers at liquid nitrogen temperature (77 °K) [12]. Similar results with gold surface-barrier Ge(HP)

detectors have been achieved [13] and an extensive review has appeared [14]. A detailed

investigation of the performance and efficiency of such a surface-barrier Ge(HP) x-ray

detector has been published by Hansen, et al [15].

At the present time, it appears that such gold surface-barrier (Si(Au) and Ge(Au)

detectors are being supplanted by much more rugged ion-implanted ultra-high purity detectors. These can be stored at room temperature indefinitely without deterioration and are

far less sensitive to atmospheric leaks, moisture, and other impurities in the vacuum

system, and which are not totally destroyed in the event of breakage of the beryllium

window. They are also expected to have a longer useful life of laboratory use. For

example, ion-implanted Si(HP) detectors with sensitive thicknesses from 20 urn to 5 mm are commercially available. Typically these may have a front contact of only 20 ug/cm2 nickel

or aluminum on an ion- implanted boron layer about 200 K deep, and a rear contact of vacuum evaporated aluminum. The silicon deadlayer typically is in the range of 800 to 2000 A\

These Si(HP) detectors can be operated at room temperature for charged particle detection

or may be cooled to 77 °K for low-energy electron, x-ray, or y-ray spectrometry.

Single-crystal germanium with a residual active impurity concentration of <10 10 atoms/cm3

(1 part in 10 12) is currently used for ion implantation of boron (^200 R thick)

as a replacement for the evaporated metallic P-type front contact layer. This has two main

advantages: (1) there is no fluorescent excitation of characteristic x-rays (e.g., gold,

palladium) from the metallic layer, the Ge deadlayer being as thin or thinner than formerly,

and (2) the ion-implanted device is more rugged and reliable, since the implanted atoms are embedded in the crystal lattice, rather than being deposited as a thin delicate surface

layer. The rear N+ contact layer is made by diffusing lithium at a concentration of

1020-22 atoms/cm3

. The behavior of diffused lithium in a crystal should not be confused

with that of drifted lithium. It is the latter that is unstable at room temperature. For example, lithium, being an interstitial impurity, can move about a mm a day in room temperature germanium, but only a few Angstroms a day at 77 °K. Thus a lithium-drifted Ge(Li)

detector can suffer serious deterioration if warmed to room temperature even for a few

minutes. Moreover, the movement of drifted lithium in Ge(Li) and Si C Li ) detectors, either

through aging at 77 °K or by cycling to room temperature, can cause changes in the volume

of the sensitive lithium-drifted region and thus changes in efficiency and deadlayer

7
thickness [16-17]. We have observed such efficiency changes in several Si(Li) and Ge(Li)

x-ray detectors in our laboratory (e.g., see reference 15). We also observed a decline in

efficiency of an early Ge(HP) gold surface-barrier detector with an N-type front layer,

because of the fact that the depletion layer thickness in this type of detector is highly

dependent on reverse-bias voltage (equation 2 below). Owing to increasing leakage current

with age of this detector, the maximum bias voltage that could be used declined from over

100 volts to 25 volts, causing the efficiency that could be achieved to decline proportionately; i.e., the depletion depth is given by

W = (2eV/qN) 1/2

(2)

where s is the dielectric constant, V is the applied voltage, q is the collected electron

charge, and N is the concentration of doping impurities in the semiconductor material.

Figure 1, courtesy of ORTEC, Inc., compares the spectra in the energy range 60 to 87

keV produced by f 1 uorescently exciting each of the two detectors with a

57 Co source, whose

122 and 136 keV y-rays can excite gold K x-rays (Ka 1 68.79 and Ka2 66.97 keV). The upper curve is for a Ge(Li) detector with a gold front layer [490 mm2 x 7 mm depth, having reso- lution of 385 eV FWHM at 5.9 keV and 617 eV FWHM at 122 keV], and the lower curve is for an ion-implanted boron Ge(HP) detector [490 mm2 x 10 mm depth, with resolution of 317 eV FWHM

at 5.9 keV and 543 eV FWHM at 122 keV]. The lower continuum and complete absence of fluorescent gold x-rays makes it clear that ion implantation is superior.

The effect of risetime discrimination on resolution and especially on tailing and the

peak-to-continuum ratio was investigated in 1969 by Freund, et al. [3] on Ge(Li) detectors,

when it was found that an improvement in the peak-to-continuum ratio of a factor of about 3 could be attained by rejecting slow pulses of 1 to 2 usee risetimes without affecting reso- lution of the 14.4 keV y-ray peak from 57 Co. In contrast, our 1979 ion-implanted Ge(HP)

detector shows less than five percent improvement with similar risetime discrimination, and

its peak-to-continuum ratio is about seven times better than that of the 1969 Ge(Li)

detectors. This can be attributed to the presence of considerable trapping effects in the

older Ge(Li) detectors. The enormous decrease in trapping in the modern Ge(HP) detector

means that it is no longer necessary to impose risetime discrimination in order to improve

the performance of the detector.

It might be of interest to indicate the observed performance of our 1979 ion- implanted

Ge(HP) detector. It is a 30 mm2 x 3 mm depth, top-hat type, with physical collimation, as shown in figure 2. It was operated at 1000 volts reverse-bias and is fitted with an ORTEC

dynamic charge restoration preamplifier and operated with a Canberra Model 1412 amplifier

in the symmetric mode. After optimizing all electronic parameters, the observed resolutions

are: 173 eV FWHM at 5.9 keV at 1000 c/sec with a 6 usee shaping time constant, the FW at 1/10M being 1.95 the FWHM at 5.9 keV; 205 eV FWHM at 14.4 keV, and 475 eV FWHM at 122 keV,

where the FW at 1/1 0M is 1.8 FWHM. The peak-to-continuum ratio and resolution of this

detector exhibit an essentially flat response with bias voltage between 500 and 1000 volts,

both of these characteristics deteriorating significantly below 500 volts. Typically,

Ge(Li) and Si(Li) detectors have exhibited sloping performance of these two characteristics

with changing bias voltage.

It is clear that the advent of ion-implanted Ge(HP) detectors, free of the problems of

drifted lithium and of delicate metallic front contact layers, represents a major advance

in the technology, both for the small detectors used in x-ray and electron spectrometry and

for large volume Ge y-ray spectrometers. It is quite likely that the coming years will see

a complete phase-out of the lithium-drifted detector.

8
9
COLLIMATOR

Figure 2. Cross sectional view of the mounting and internal collimator of

typical ion-implanted Ge(HP) detectors (courtesy ORTEC, Inc.).

2.3 Measurement of Deadlayers

The presence of metallic and/or semiconductor entrance windows or deadlayers depends

on the fabrication techniques used to produce the detector and are very undesirable, because

they cause a cutoff of the lowest energy events, degrade the energy resolution, linearity,

peak-to-tail ratios, and affect the detector efficiency at low energies. Typical entrance

windows of most silicon and germanium detectors consist of 200 A thick gold (40 ug/cm2

) and

a relatively-insensitive "deadlayer" of the order of 0.01 to 0.5 pm in Si [18,19] and 0.1 to 0.3 urn in Ge [20,21]. Even worse, the deadlayer thickness varies with reverse-bias voltage

[18]; e.g., the deadlayer of a Si(Li) detector was found to be 0.2 urn at 500 volts. The

gold layer has been observed under the electron microscope [3] to have granular structure,

with large ragged open regions, and surface roughness with whiskers of gold creating

islands and bridges up to 2000 A in size. Other metals, such as Al , Ni , or Pd, can be used

instead of gold for the surface barrier, but it is possible that similar inhomogeneities

occur with these as well. That the metallic layer is inhomogeneous in thickness as well as

in a real distribution is demonstrated by the fact that carbon K x-rays (277 eV) can be

detected, even though a 40 pg/cm2 thickness of gold should absorb them completely. Owing

to the higher Z and density, the deadlayer in Ge is much more serious than one in Si of

similar thickness [66].

o As pointed out above, the use of boron ion-implantation [22-24] to about 200 A in depth

in N-type material as a replacement for the metallic front surface barrier in Si(HP) and

GE(HP) detectors results in deadlayers as thin or thinner than in Si (Li) and Ge(Li) detectors, and of course the fragile metallic layer with its inhomogeneities is eliminated.

Another technique of solid-phase regrowth (recrystal 1 ization processing [25]) also has been

used to construct thin P+ contacts, and a window thickness of about 0.2 pm was found.

Musket [6,26] has used grooved Si(Li) detectors to measure ultra-soft K x-rays of B, C, 0, Al , and La+p x-rays of Cu (from 185 to 1487 eV). These x-rays were produced essentially

free of bremsstrahlung by irradiating these elemental targets with 350 keV protons. The

detector efficiency and fraction of the x-ray energy detected for carbon K x-rays (277 eV)

all decreased with decreasing bias voltage, and this finding is consistent with an increasing silicon deadlayer as the bias voltage is reduced.

10
Approximate measurements of deadlayer thickness may be made by comparing the pulse

heights obtained from irradiation of the detector surface with alpha-particles at 90° and

at, say, 45° incident angles, or by comparing the pulse heights from alpha-particles at

normal incidence and from collimated photons. Thicker deadlayers may be estimated purely

with photons. However, even with low energy photons, such as 3 keV, the low value of the

silicon K-shell fluorescence yield [28] (u>„ "v 0.045) does not permit convenient measurement

9 of silicon deadlayers thinner than about 10,000 A (1 urn), whereas typical deadlayers are currently less than 100 ft (0.01 urn) thick [27].

To determine the deadlayer thickness, for example, in Ge detectors a convenient source

of y-ray photons is used with energy above the 11.1 keV K-absorption edge of germanium (or

if a gold layer is present, above the 11.92 keV gold L3 -absorption edge). Typically the

14.4 keV y-ray from 57 Co, or Sr K x-rays from an 88Y source, is used to measure deadlayers

above about 3000 ft thick in Ge detectors. The deadlayer thickness X« is then estimated

from the formula

XGe

= — AA) (3)

where (C

K/C^) is the observed ratio of Ge K x-rays and the exciting y-rays, (e^/e^) is the

ratio of intrinsic detector efficiencies for the exciting y-ray and Ge K x-rays, is the mean K-shell attenuation coefficient in Ge for the exciting y-rays, and = 0.55 for

germanium [28].

A new table of mass absorption coefficients for K and L x-rays in low-Z materials

(carbon to arsenic) has appeared recently [29] which will be useful for estimating the

absorption of x-rays in silicon, germanium, and other low-Z elements. For example, the

absorption of 3 keV photons in a uniform 5000 A (0.5 urn) thickness of Ge or Si is 22 percent

and 10 percent, respectively. Llacer, et al [66] found that Ge(HP) x-ray detectors have a relatively thick Ge entrance window that renders them practically useless below approximately 2.3 keV, and that this effect is due to a basic property of electron transport near

a surface. We have also observed such a low-energy cutoff near 2 keV in our (1979) Ge(HP)

detector.

2.4 Detector Mounting and Cryostats

Methods of mounting detectors within vacuum cryostats of various configurations, with

and without beryllium windows, have been described in the literature [5-7,10,26]. Figure 2,

courtesy of 0RTEC, illustrates a typical mounting arrangement of a Ge(HP) detector. One

should note that the low-Z collimator may be transparent to photons above a certain energy,

e.g., 60 keV or so, so that the effective geometry (solid angle) is larger for higher energy

photons than at low energies, and this can affect the shape of the efficiency curve in the

energy region where transparency sets in. Very recently a mini-cryostat that is cooled by

a miniature Joule-Thompson liquifier inserted into the cryostat and fed by compressed

nitrogen gas at pressures above 100 atmos. has been developed [30]. This eliminates the

large Dewar vessel and the need for liquid nitrogen. It produces a minimum available

temperature of 77° K at the detector (same as liquid nitrogen). The cryostat is used with

a Ge(HP) detector and thus can be stored normally at room temperature until used. About 10

min time is required to cool to operating temperature, and a 50 liter commercial tank of N2 gas supplies the device for about 40 hours. The typical resolution of a 20 mm Ge(HP) detector in this cryostat is reported at several energies to be 221 eV FWHM at 5.9 keV ( 55 Fe),

297 eV at 17.74 keV ( 241 Am), and 544 eV at 122 keV ( 57 Co). Such a miniature x-ray detector

can be used as a portable field instrument or on scanning electron microscopes, and it is

clear that ion-implanted Si(HP) detectors could as well be used with such a cryostat.

11
3. Electronic Systems

The contribution of the electronic system noise to line broadening is shown in equation

(1), Low-noise, DC-coupled, cooled, field-effect transistor (FET) preamplifiers have been

discussed extensively in the literature [5,7,8]. To maintain the high resolution at higher

counting rates (up to 20,000 c/sec) the use of pul sed-optical feedback preamplifiers was introduced in 1969 [2-33] and is available commercially in Kevex Corp. detectors. Unfortunately, reset pulses are introduced by this method which are very difficult to eliminate

completely in electronic systems designed for x-ray coincidence spectrometry, and in addition, the shorter timing requirements of coincidence circuitry (typically, less than 2 usee)

do not take advantage of the improvement in resolution at higher counting rates produced by

the use of pulsed-optical feedback preamplifiers with very long time constants (8 - 17 usee) [8]. For these reasons, pulsed-optical systems should be avoided in coincidence

spectrometry.

An alternate choice is the dynamic charge restoration preamplifier [34-36] which has

continuous feedback and produces equivalent resolution with much shorter time constants

(e.g., 6 usee instead of 17 usee at 104 c/sec) and without introducing the reset pulses

which trouble coincidence circuitry.

Very recently, a pulsed drain feedback preamplifier has been introduced [37] which can reach 60,000 c/sec with only 10 percent noise degradation and a 4 percent deadtime without

use of pulse pileup rejection.

Finally, a new pulse pileup rejection circuit has been reported [38] which gives good

resolution performance at ultra-high counting rates (up to 260,000 c/sec) at 5.9 keV; without it, such counting rates completely obliterate the spectral resolution.

Since no two detector-preamplifier systems are identical, it is always necessary to optimize carefully the state-of-the-art main amplifier for such variables as time shaping

constant, pole-zero adjustment, low-level threshold setting, and selection of a suitable

mode of baseline restoration. It is often overlooked, but the input requirements of the

amplitude-to-digital converter (ADC) of the multichannel analyzer may restrict the choice

of amplifier time constant which may be used; for example, the use of an output amplifier

pulse having a time constant appreciably longer (e.g., 8 usee) than the length of the

normally-closed linear gate input of the ADC (e.g., 2 usee) results in failure to achieve

optimum resolution. Therefore, the time length of the linear gate in the ADC should be

adjusted to accommodate the amplifier output pulses, but not so long as to introduce un- necessary deadtime.

Goulding [70] has presented some lesser known but potentially important aspects of

x-ray detectors and electronics, including deadlayer effects, time-dependent continuum and excess continuum, noise parameters of field-effect transistors, and time-variant pulse

shaping.

4. Spectral Response to Monoenergetic Photons

For incident photon energies below 25 keV for silicon and below 60 keV for germanium,

the interaction is primarily photoelectric, the ratio of photoelectric/scattering cross sections being greater than 10. For such low energies, the tailing arises primarily from

slow charge collection due to trapping, and this was a serious problem in early detectors

[3,5,7,15]. More recently, there have been dramatic improvements in quality for both Si and Ge detectors, and the tailing appears to have been reduced to a value not much larger

than predicted by the ratio of photoelectric/Compton cross sections; e.g., the peak/tail

counting- rate ratio in Si (Li) exceeds 1500/1 at an energy ratio of 5.9/1 keV; and in our recent ion-implanted Ge(HP) detector, it exceeds 90/1 at 5.9/2 keV.

With incident photons above about 25 keV in Si and above 60 keV in Ge, the ratio of

photoelectric/Compton cross sections is less than 10, and thus Compton scattering becomes

important and is manifested in two ways: (1) Compton scattering within the sensitive

volume of the detector with escape of the scattered photon, and (2) Compton scattering out- side of the detector with detection of the scattered photon. The former process appears as

a rising tail at very low energies [3] or as a low-energy plateau [27], while the latter

12
effect appears as a bump or shoulder on the low-energy side of the full-energy peak [3].

The importance of Compton backscattering in giving rise to such a shoulder, or even a spurious "peak" below the full-energy peak in a tightly-coll imated geometry [39] should not

be overlooked. Calculation of the magnitude of the low-energy plateau from the first

process has been published [27].

In addition to the above processes, Si and Ge K x-rays are generated in the detector

from the primary interaction of the incident photons of energy above the K-absorption edge

at 1.838 and 11.103 keV, respectively. A fraction of these K x-rays will escape the sensitive detector volume, removing 1.74-1.84 and 0.85-11.10 keV of Ka and Kp energy from Si and

Ge detectors, respectively, to give rise to escape peaks located below the full-energy peak

by exactly these energies, respectively. These Si and Ge K-escape peaks are very similar in

FWHM to the full-energy peak. The magnitude of the escape, expressed either as a ratio R of

the escape/main peak heights, or as the fraction of K x-rays which escape the sensitive

volume, corrected for the underlying continuum, as a function of incident photon energy, has

been carefully studied [3,15,19,27,40,41,67,69]. At the K-edge, for example, the K x-ray

escape probability can be as large as 12 percent in Ge and 3 percent in Si detectors,

depending on size and configuration. For photons in the energy range 3-20 keV in Si detectors, experimental measurement [27] of the escape/main peak height ratio R was fitted

by a formula R = a(E + b) , where E is the incident photon energy in keV, and a, b, and c, are constants. The value of R is about 10" 2 at 3.5 keV, decreasing with energy to a negligible value at 25 keV for a Si (Li) detector 50 mm2 x 4 mm deep [27]. It is clear that

the effect is much more serious for Ge than for Si detectors for photon energies above the

Ge K edge (11.103 keV) and represents a serious complication, especially in multielement

x-ray analysis, where the escape peaks can be mistaken for trace elements not really

present, or can be hidden under peaks of other elements, the intensities of which have to

be suitably corrected. A particular example in which an error could be introduced is in

the calibration of detector efficiency with Np M x-rays from an 241 Am source. These M x-rays are spread from about 3 to 6.5 keV, but have their main intensity in a broad

multiplet centered at 3.3 keV. Unfortunately, in Ge detectors, the Ka and KB x-ray escape

peaks from the 13.9 keV La x-rays in 241 Am decay fall into the M x-ray region and must be

taken into account in determining the Ge detection efficiency at 3.3 keV. This correction

can easily be evaluated [15] by attenuating the M x-rays with a suitable absorber and

observing the ratio of the escape/main (La) peak.

The low-energy continuum extending from the very low-energy Compton region to the fullenergy peak is quite horizontal in both Si and Ge detectors. In a detailed investigation

[27] of a Si (Li) detector of dimensions 50 mm2 x 4 mm, this continuum region declined

rapidly from about 1.5 x 10 -5 to about 9 x 10"6 of the main peak intensity for photons from

3.5 to 12 keV, and then it increased to about 2 x 10"5 at 50 keV.

Sum peaks and the pulse pileup in the continuum lying above the full -energy peak can become serious problems, since they are dependent not only on counting rate, but also on the presence of more than one incident photon energy. For example, incident monoenergetic

photons of two different energies entering the detector simultaneously can add to give

three sum peaks, two of which are simple sums of the two incident radiations, and the third

is a mixed sum peak caused by addition of two photons of different energies. Complex

multipeak spectra can thus exhibit a multiplicity of both simple and mixed sum peaks, and

in addition, the superimposed continuum between the peaks can give rise to a summed

continuum as well. A detailed analysis of the statistical probability of summing has

appeared [27], together with an analysis of the improvements which can be realized with the

aid of a pulse pileup rejection circuit. The probability of a pileup pulse is approximately

It, where I is the rate of incidence of monoenergetic photons per sec and t is the deadtime

of the system. Provided the counting rate I is not so high as to give appreciable threephoton summing, the probability of summing of two incident photons is approximately I 2 x, and relative to the incident counting rate of single pulses, the probability of summing

increases linearly with counting rate.

13
includes the electronic pulse processing time, then varies, typically, from about 1.1 usee

at 2.5 keV to 0.1 usee at 50 keV [27]. Recently, a spectacular new pulse pileup rejection

circuit has been reported [38] which at 5.9 keV gives good resolution at ultrahigh counting

rates up to 260,000 c/sec. The remaining sum peaks can be stripped from the spectra in the course of computer analysis of the spectral data [27].

The use of critical x-ray absorbers can also reduce summing of very strong peaks in

spectra containing weaker peaks of other energies. For example, the 6.40 keV Ka and 7.06

keV Kp x-ray peaks of iron, which are of overwhelming intensity in spectra of samples of

steels, iron-ores, and whole blood, may be attenuated selectively by critical absorption in

a thin foil of chromium, while affecting neighboring x-ray peaks only slightly.

5. Methods of Energy and Efficiency Calibration

5.1 Energy Calibration

The accuracy of energy determinations based on interpolation or extrapolation from

peaks of accurately known energies depends on the overall system linearity and gain stability. An excellent discussion of this is given for larger volume germanium detectors used

in y-ray spectrometry [42], and similar principles apply as well to detectors used in the

low-energy region below 140 keV. Experimentally determined non-linearities can be

incorporated into computer programs (such as PEAKFIND, GAUSS, SAMP0, etc.) which search,

locate, and fit peaks in a spectrum to determine their energies and intensities [43]. The

correction for non-linearity is measured with radioactive sources of accurately known

energy, such as those listed in Table 1. The computer program can then correct each peak

position for non-linearity. This correction is of course more important in the analysis

of low-energy \-rays, because the latter are known from crystal spectrometry far more accurately than is required for identification in a semiconductor spectrum, since non- linearities of modern systems are too small to shift x-ray energies by amounts that would

cause them to be mis- identified and assigned to the wrong element.

The linearity of Si (Li) and Ge(Li) detectors for low-energy photons has been studied in

the range of 5 to 120 keV [44,45] and deviations below ±0.2 percent were found, the largest

being at the lowest energies where the cause is ascribed to inefficient charge collection in

the gold and detector deadlayers. With Si(Li) detectors having very thin deadlayers, better

than 1 percent linearity was observed in the 500 eV to 2 keV region [18], and with a 30 mm2 grooved Si C Li ) detector [8] and using collimated beams of very low-energy x-rays excited by

350 keV protons [26] the linearity was found to be better than 1 percent in the region of

carbon K to Al K x-rays (277 eV to 1.5 keV). The linearity of non-ion-implanted Ge(HP) gold

surface-barrier detectors has been studied [54] and a maximum deviation of ±0.2 percent was found in the energy range 6.4 to 122 keV, and no abrupt change across the Ge K-edge at 11.1

keV was found, contrary to a previous report [44]. Equal or superior linearity should be

exhibited by the new ion-implanted Ge(HP) and Si(HP) x-ray detectors at low energies, owing

to the absence of a metallic front contact layer and deadlayers as thin or thinner as those

of older types of detectors, but there are no published studies reported so far.

5.2 Efficiency Calibrations

Experience has shown that no two semiconductor detectors exhibit exactly the same

spectral response or efficiency, even though they may have identical nominal specifications

as to active area, depth, resolution, and peak-to-continuum ratio. A theoretical model

[15] can, however, describe the detection efficiency with an accuracy comparable to that of

the experimental measurements in the region from 5-60 keV, provided that all of the

parameters are carefully measured for each detector (e.g., source-to-detector distance;

thicknesses of the Be window, gold contact layer, if any, deadlayer, and sensitive depth;

determination of the sensitive area and collimation, if any, by photon scanning [15,52];

and charge collection efficiency. The theoretical calculation [15], once these detector

parameters are determined, requires a knowledge of the photoelectric and Compton absorption

and attenuation coefficients [29] as a function of photon energy for the absorbing media

between the source and detector sensitive volume, as well as the escape peak probability vs. photon energy.

14
The full-energy peak detection efficiency for photons of energy E can be written

as

£

E = eR G(E) (4)

where is the fraction of the 4n solid angle subtended (fi/4n) by a plane parallel to the

detector surface and located at a mean interaction depth for photons of energy E. The

relative detector efficiency is the product of the intrinsic detector efficiency e a and

several correction factors

eDRaaAudecs = e f f. f . f f e (5)

where f , f. , and f . correct for the attenuation in materials in front of the sensitive

a' Au' d volume, such as air and Be window, gold layer if present, and the deadlayer, respectively;

f and f correct for the escape of Ge (or Si) K x-rays from the sensitive volume and for

the effects of collimation, if any, respectively; and e

g is the efficiency of charge

collection for the full-energy peak. The details of evaluating these factors are given in

reference 15. The results of scanning the detector with highly collimated photon beams,

in order to probe the sensitive area and depth, are reported [10,15,19,41,52]; again, the

efficacy of guard rings [10] in minimizing the dependence of the sensitive volume on photon

energy should be noted. The evaluation of deadlayer thickness and uniformity has been

discussed above (section 2.3) and the problem of escaping Ge (or Si) K x-rays is treated in

section 4. 0 above.

An alternate, and much easier, procedure is to fit an analytical expression to the

experimentally measured efficiency points [19,46,47], in order to facilitate interpolation

without graph-reading error. This method depends entirely on the absolute calibration of

the radioactive standard sources and does not offer a check based on calculations from the

theoretical model

.

Experimental calibration of detection efficiency with radioactive sources has been

extensively discussed [3,15,19-21,41,45-50]. Recently, however, Campbell, et al. [46]

demonstrated that the use of the usual drop-evaporated standard radioactive sources in the

energy region 5 to 40 keV can result in values of efficiency points from sources of 54Mn,

57 Co, 65 Zn, and 241 Am that are low by amounts varying from 3 to 16 percent, when compared

with vacuum-evaporated sources. This is a very serious situation when one is attempting

to establish an efficiency curve to 1 - 2 percent. Unfortunately, the metrology laboratories which supply absolutely calibrated low-energy photon sources normally do not offer

vacuum-evaporated sources, but in view of this finding, they should.

The accuracy which can be reached with this method, using vacuum-evaporated standardized radioactive sources, can reach about 1 percent from 6.4 to 18 keV, increasing to about

2 percent at 30 keV in Si(Li) detectors [46]; with drop-evaporated sources, checked by

calculations of a theoretical model, accuracies of about three percent in the region five

to 60 keV were obtained [15], and at higher energies, the accuracy decreases to as low as seven percent.

In addition to errors caused by self-absorption in crystallites in drop-evaporated

sources described above [46,47], there are two other principal causes of error in efficiency determinations with radioactive sources. The rather long amplifier time-constant

used to achieve the best energy resolution gives rise to pulse pileup at higher counting

rates, and correction for the deadtime is required. This means that the efficiency calibration for the detector should be performed at the same counting rate that the detector

will observe in experiments for which the calibration is being done; otherwise, the errors

involved in determining an accurate deadtime correction will enter. The second cause of error is produced by small variations in source position, not only in distance from the

detector, but also in lateral deviation from the axial center [65]. The latter may be

caused by poor centering of the detector itself inside the vacuum shield. A 5 percent

error in a 2 cm source-to-detector distance for a 20 mm dia x 3 thick detector causes an

1 5
efficiency shift by about 8 percent [5], and a 2 mm shift off-center causes a 1.5 percent

drop in efficiency at the same distance. The longer the source-to-detector distance, the

smaller are these kinds of errors.

Table 1 gives a compilation of long-lived radioactive nuclides suitable for energy and

efficiency calibration of low-energy photon detectors. Only a few nuclides from this long

list are presently available as absolutely calibrated photon emission standards in the x-ray

region from the National Bureau of Standards [51], the Physikalisch-Technische Bundesanstalt, Braunschweig, West Germany, or the Laboratoire de Metrologie des Rayonnements

Ionisants, Gif-sur-Yvette, France, but is is likely that more will become available in the

future.

Changes in detector response characteristics (resolution, peak/tail ratio) and in the

efficiency have been observed to occur with age [15,53]. Rather substantial efficiency

changes in Si (Li) detectors (up to 20 percent increase) were observed on recycling to 77 °K

after warmup to room temperature, and slowly decreasing efficiency with time for Ge(Li)

detectors [15] and Si(Li) detectors [47] is regularly found. The ion-implanted Ge(HP) and

Si(HP) detectors in principle should have constant efficiency at constant bias-voltage, but

no long-term observations have so far been reported. We have noted that a Ge(HP) gold

surface-barrier detector purchased in 1970 exhibited a decline in efficiency with age, because of the lower and lower bias voltage that could be imposed due to the increasing

leakage current with age. Thus, it is clear that the need to repeat the efficiency calibration exists in long experiments, and one should routinely recalibrate the detectors at regular time intervals.

Typical efficiency curves for Si (Li) and Ge detectors are shown in figures 3-5,

determined with calibrated radioactive sources.

Campbell, et al. [46] have pointed out that new intensity data around 45 keV are desirable, as are better values of the x-ray/y-ray intensity ratios in the 20 - 25 keV

region. From Table 1 it can be seen that the radioactive nuclides 145 Pm, 109 Cd, 113 Sn, and

119m

Sn would be suitable sources for these energy regions, should they be made available

from the meteorology laboratories as point-source (preferably vacuum-evaporated) absolute

photon emission-rate standards.

An accurate method for determining detection efficiency with uncal ibrated radioactive

sources is based on measurements of K-conversion electrons-K x-ray coincidences [15,55],

known as the "CEX" technique. The detection efficiency e^(E) for K x-rays of given energy

E is given by

£ (E) = CK(e

K

} (6)

C e

K w

K

e

c

where C^e ^ is the number of coincidence counts of K x-rays gated by K-conversion electrons

K from the same -y-ray transition, C is the number of counts in the K-conversion electron

e

K gate, corrected for underlying continuum and L-conversion electron tailing [15,55], is

the K-shell x-ray fluorescence yield [28] of the K x-rays of energy E, and e

c

is the coincidence efficiency, which should be adjusted to unity by using a sufficiently wide time-gate

in the time-to-amplitude (TAC) unit of the coincidence system.

The CEX method has been used with sources of 113 Sn (E^ = 393 keV), 137 Cs (662 keV),

203 Hg (279 keV), 243Am (277.6 keV transition in 239 Pu), and 249 Cf (388 keV transition in

245 Cm). With either a Si(Li) or Ge detector for K x-rays, depending on energy, and a Si(Li) detector for conversion electrons having energy sufficient to pass through the Be window, a coincidence resolving time of Zn = 1.0 usee was sufficient to give a 100 percent

coincidence efficiency (e =1.0) [15]. Since values of w„ are known [28] to accuracies of

C l\

1 to 2 percent, depending on the region of Z involved, the detection efficiency can be

16
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17
3 5 10 20 30 50 100 200 300

PHOTON ENERGY (keV)

Figure 4. Detection efficiency on a log-log plot of an N-type gold surface-barrier

Ge(HP) detector, ca. 1970. The solid line is a prediction from the

theoretical model of ref. 15, and the dashed curve is drawn through the

experimental points. (Figure from ref. 15.)

18
19
determined to about 1.5 to 3 percent accuracy (2a) [15]. The efficiency found at 32.0 keV

in a 50 mm2 x 3 mm (Ge(Li) detector by the CEX method was 2 percent higher than that from

drop-evaporated IAEA calibration standards, and at 24.1 keV, 3.6 percent higher [55].

This seems to confirm the results quoted above [46,47] that efficiencies determined from

drop-evaporated radioactive calibration standards are several percent lower than those

from vacuum-evaporated sources.

The principal advantages of the CEX technique are that it does not depend on absolute

calibration of radioactive sources by someone else, nor is it dependent on knowledge of

such radioactive decay parameters as halflife, conversion coefficients, or branching fractions. Moreover, having determined the detection efficiency in this way, the K x-ray

singles counting rates from the sources of 113 Sn, 137 Cs, 203 Hg, 243Am, 249 Cf, etc. used in

the CEX method then give the absolute K x-ray emission rates from these sources, so that

they can be used in turn to calibrate other detectors in the singles mode.

Another method for calibrating radioactive sources in the low-energy region, relative

to a primary standard source, over a wide range of energies not exceeding about 50 keV, has

been reported by Chew, et al . [56]. The source to be calibrated is placed external to the

Be window of a large-volume, wall-less multiwire proportional counter, the gas detection

efficiency of which can be varied by changing the gas pressure. A maximum occurs in the

gas counting efficiency with variation of the pressure. Thus, if the counting rates of

low-energy photons from different sources are maximized by changing the pressure, then the

detection efficiency of the center counter is the same , and its absolute value depends

only on the ratio of the two geometrical distances defining the center- and ring-counters,

as shown in figure 6. Because the true effective geometry is not identical with that based

on the actual dimensions, this method can be used only relative to a primary standard

source and not as an absolute standardization method [56]. With this method, several

different sources may be standardized to better than about 3.5 percent (2a + linear addition of systematic errors) against one or more primary x-ray standards [56].

An accurate determination of the K-edge discontinuity at 11.103 keV in Ge efficiency

curves (and at 1.838 keV in Si) is of interest, because the magnitude of this K-jump is

strongly dependent both on the deadlayer thickness and on the probability of K x-ray

escape. Detailed experimental studies of the Ge K-edge discontinuity have appeared [57,

58]. These are based on comparisons of the detected ratios either of La/LB or Ka/KB x-ray

intensities with known theoretical or experimental values from elements with emitted x-ray

energies near and above the K-edge, or which straddle it (e.g. Ka and KB peaks from

arsenic). The La,B x-rays may be excited in foils of heavy elements of Ta, W, Ir, Pt, Au,

Hg, Tl , Pb, and Bi (covering the energy range 8.088 to 13.021 keV) either by low-energy

protons or by an 241 Am source, which excites few, if any, K x-rays in these elements

[57]. The Ka and KB x-rays can be excited by an 241 Am source in thin samples of Ga, Ge,

As, Se, Br, and Rb (covering the region 9.252 to 14.951 keV) [58]. These methods cover

the region of the Ge K-edge in fine steps each smaller than 500 eV. A convenient fast

way, however, to estimate the depth of the K-jump is with a radioactive source of 75 Se,

whose Ka and KB x-rays of arsenic straddle the Ge K-edge [15,21].

The efficiency of Ge(Li) and Si (Li ) detectors in the energy range of 1 to 5 keV can

be determined by a simple method [59,60] in which an element or a chemical compound, such

as CaCl 2 , is f luorescently excited by 55 Fe (e.g. a 100 mCi annular source). From the

ratio of Ca and CI fluorescent K x-ray peak intensities at 3.69 and 2.62 keV, respectively,

the deadlayer thickness may be determined [59]. The deadlayer is the major factor in the

drop of efficiency in the region below 5 keV, other than the beryllium window. The energy

region of 3-4 keV, which usually is calibrated with Np M x-rays from 241 Am decay [15], has

a high continuum coming from the strong 59.6 keV y-ray in the decay of 241 Am, but fluorescent excitation with 55 Fe leaves a clean background all the way down to noise level

(<185 eV). Targets of elemental Mg, Al , Si, S, and compounds CaF2 and KC1 have been used

for this region [59,60]. The accuracy in this region in the absolute detection efficiency

curve is better than about 5 percent.

A convenient and simple method [61] for obtaining the efficiency curve of a Si(Li)

(or Ge) detector up to about 50 keV utilizes a comparison of its response with that of a small Nal(Tl) scintillation detector (e.g. 13 mm thick x 38 mm dia) fitted with a 5-mil

(0.125 mm) Be window. The intrinsic efficiency of the Nal(Tl) detector is 100 percent

20
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gZZZZZZZZZZZZZZZZZ22 $

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£ if) s s 3SZ

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21
from 60 to 12 keV, below which a correction for attenuation in the Be window is necessary.

An 241 Am source (100 mCi) is used to excite K x-rays from 6.40 to 48.8 keV in targets

ranging from iron to erbium [61]. Since the Nal(Tl) detector does not resolve the Ka and

KB peaks, a correction must be applied for known Kp/Ka ratios to obtain the net Ka x-ray

intensity for a comparison with that from the semiconductor detector. Also, the iodine

K and L x-ray escape peaks must be included in the K-peak intensity from the Nal(Tl)

scintillator, although the latter correction is small in the case of L x-ray escape. The

overall uncertainty is estimated to ±3 percent from 5 to 30 keV, and increases to a maximum of ±10 percent at 50 keV.

The absolute sensitivity of a ruggedized Si(HP) surface-barrier type detector having

a 50 ± 2 ug/cm2 aluminum front electrode (25 mm2 active area x 500 urn depletion depth) was investigated in the DC (current) mode in the energy range from 185 to 932 eV [62]. Protons were used to excite K x-rays from boron (185 eV) and carbon (277 eV) and L x-rays from

titanium (452 eV), chromium (574 eV), iron (704 eV), and copper (932 eV). These collimated

x-rays were intensity-calibrated with a gas proportional counter. A small discontinuity

in the sensitivity between 452 and 574 eV was observed, which was tentatively ascribed

to oxygen. It would be desirable to have further detailed x-ray studies of ion-implated

Si(HP) detectors with depletion layers 3 to 5 mm deep, as these become available com- mercial ly.

Finally, a detailed study of the performance of a 50 mm2 x 4 mm thick Si (Li) detector

having a 150 A gold front contact layer and fitted with a 1 -mi 1 (0.025 mm) Be window has

been reported by Keith and Loomis [63]. This detector was found by them to have a deadlayer no thicker than 100 A, and to exhibit resolutions of 327 eV FWHM at 5.4 keV,

decreasing to 490 eV FWHM at 46 keV, using a relatively short pulse-shaping time constant

of one usee and a deadtime of about 10 percent at 104 c/sec. The efficiency of this

detector was calibrated in the range 2.5 to 50 keV with a claimed accuracy of ±0.1 percent

(la) by comparison with a flow-gas proportional counter in the range 2.6 - 11.2 keV and

with a Nal(Tl) scintillation counter in the range 5.9 - 50.4 keV, using monoenergetic

x-rays generated by a copper-anode x-ray tube and selected by a crystal monochromator

operating in a helium atmosphere. The quoted accuracy of the efficiency values, however,

is not realistic. The geometry of the solid angle of the x-ray beam passing through a defined, but not accurately measured, collimator introduces much larger uncertainties

than quoted; it must be obtained from accurate physical dimensions and checked with an independently-calibrated radioactive source, the continuum from which also introduces

appreciable uncertainties. For example, the use of proportional and Nal(Tl) counters with

defined solid angle geometry to calibrate radioactive sources at low energies at the

National Bureau of Standards [51] results in an overall accuracy that is of the order of ±1 percent [64] (la). Thus, the overall accuracy of the efficiency determination of Keith

and Loomis is in fact of the same order of accuracy as other measurements of efficiency

reported on the usual basis of 2a statistical error plus linear addition of systematic

error, or a similar method of combining statistical and systematic error, such as the

student-t distribution.

6. Considerations in the Choice of Ge or Si Detectors

We have already pointed out that there are strong reasons to prefer ion- implanted

high-purity Ge and Si detectors over the lithium-drifted types, and that in the future

the lithium-drifted Ge detector, and perhaps even the Si (Li) detector as well, will disappear. As to the choice between Ge and Si detectors, the following considerations should

be reviewed.

For photons above 3 keV, the Ge detector has an order-of-magnitude greater peak-tocontinuum response than Si, owing to its higher-Z; in addition, because the mean energy to create and electron-hole pair in Si is about 26 percent larger than in Ge, the intrinsic

resolution which can be achieved is, in principle, better in Ge detectors. On the other

hand, the smaller bandgap in Ge increases the detector noise problems, and this, together

with the Ge deadlayer, severely degrades the performance of Ge detectors below 3 keV.

2.2
With photons of energies above about 30 keV, however, the efficiency of Si detectors,

even with depletion depths as thick as 5 mm, falls off drastically, and the Ge detector

becomes the one of choice from 30 to 140 keV. Unfortunately, the presence of such higher

energy photons in a Ge detector being used to investigate lower energies not only gives

rise to a continuum which rises steeply (e.g. at 45°) with decreasing energy, but also can cause amplifier overload by the largest pulses, whereas a Si detector will have a relatively

lower and more horizontal continuum, being as it is much less sensitive to the higher

energy photons.

Generally, for photons below about 25 keV, the Si detector clearly is the superior

choice, because its thinner and less opaque deadlayer is a major consideration at the very

lowest energies (<3 keV). Moreover, in the region above the 11.10 keV Ge K-edge, the Si

detector is free of the fairly intense Ge K x-ray escape peaks, which cause spectral

complexity, especially when investigating weak peaks, as in the analysis of a mixture of

elements, some of which are at trace levels while others are at minor or major concentrations. For this reason, many workers prefer to analyze the L x-rays of heavy elements

with Si detectors, in spite of the more complex nature of the L x-ray spectrum and the

lower x-ray fluorescence yield [28] of the L-shell compared with the K-shell. The escape

peaks from Si detectors are resolved only at very low energies (<2 keV) and are in any

case much smaller than those from Ge.

There are cases, however, in which K x-rays or y-rays need to be detected at energies

above about 30 keV, as in the K x-ray - L x-ray coincidence experiments indicated in the

following section, and for these, the use of a Ge detector is indicated.

7. Coincidence Spectrometry

The requirements of a long amplifier pulse-shaping time constant (>8 usee) for the

best detector resolution and of a short logic gate-width (~2 usee) in the coincidence

circuitry requires a compromise; namely, in order to achieve the maximum ratio of true/

chance coincidences, the ultimate resolution is not utilized because the amplifier time

constant is reduced approximately to the same duration as the gate-width. The choice of

2 psec for this gate-width is determined by a reasonable compromise between having a high

ratio of true/chance coincidence events while maintaining a reasonable counting rate, and

it also matches the requirement of the ADC units in our Nuclear Data ND-4420 multiparameter

analyzer that all input signals (analog and logic) arrive within an interval of 2 psec, in

order to be recognized and stored as coincidence events. In our current coincidence

system [68] with a Ge(HP) and a Si(Li) x-ray detectors, the typical coincidence resolving

time is 70 to 110 nsec, and the true/chance ratio is typically about 99/1 for singles

counting rates of the order of 1200 c/sec. Under these conditions the resolutions of our detectors are approximately 90 percent of their ultimate capabilities.

In experiments involving x-ray coincidences with semiconductor detectors, it is

frequently necessary to resolve an x-ray multiplet, such as the and Ka 2 doublet, in

partly-resolved spectra. Previously, the singles shape of a calibration y-ray in the same energy region has been normalized to the higher energy member of the multiplet, e.g., the

Ka x peak, in order to subtract its tail contribution from the lower energy members, e.g.,

the Ka2 peak. However, in recent work from our laboratory [68], it was observed that the

low-energy tail of a photon line in a singles spectrum is somewhat broader than it is in a coincidence spectrum; and even more importantly, in a coincidence gate set, for example,

on a Ka2 peak, the Ka2 events originate from the Ka2 peak , whereas the Ka x events arise

from the tail of the Ka x peak. Since the tail contains a larger proportion of slow-rising

pulses [3] than the full-energy peak, this implies that the tailing correction depends on the coincidence timing. Therefore, we developed a new method [68] for the evaluation of

the net intensity of each component of a partially-resolved multiplet in a low-energy

coincidence spectrum. The method requires only that all members of the multiplet have the

same peak shape which need not be known and does not require any additional measurements

other than the coincidence runs and is independent of timing problems.

23
The author thanks Dr. R. A. Braga, of the School of Chemistry, Georgia Tech, for making

available the results of his measurements on the performance characteristics of the new

(1979) Ge(HP) ORTEC x-ray detector and of the somewhat older Kevex Si(li) x-ray detectors

used in coincidence x-ray studies, as well for helpful discussions. He also thanks

Mr. Rex Trammell of ORTEC, Inc., Dr. R. G. Musket of Kevex Corp., and Nuclear Semiconductor

Corp. for technical information on their latest detectors.

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[62] Gaines, J. L. , Kuckuck, R. W. , and Ernst, R. D. , Proc . 4th Ann . Conf . on the Use of

Smal 1 Accelerators (Institute of Electrical & Electronic Engineers, New York, 1976),

p. 234; and ERDA Symposium on X^ and y- ray Sources and Appl . (Univ. of Michigan, Ann Arbor, 1976), p. 6.

[63] Keith, H. D. and Loomis, T. C. , X-ray Spectrom . 5, 93 (1976); Appl . Spectros . 29, 316

(1975).

26
[64] Hutchinson, J. M. R. , private communication (1976).

[65] Alfassi, Z. B. and Nothman, R. , Nucl . Instr . Meth . 143 , 57 (1977).

[66] Llacer, J. , Haller, E. E. , and Cordi , R. C. , IEEE Trans . Nucl . Sci . NS-24 , 53 (1976).

[67] Dyson, N. A., Nucl . Instr . Meth . 114 , 131 (1973).

[68] Gnade, B. E. , Braga, R. A., Western, W. R. , Wood, J. L. , and Fink, R. W. , Nucl . Instr . Meth . (in press, 1979).

[69] Reed, S. J. B. and Ware, N. G. , J. Phys . (London) E5, 582 (1972) and Statham, P. J.,

J. Phys. (London) E91023 (1976).

[70] Goulding, F. S. , Nucl . Instr . Meth . 142, 213 (1977).

27
Table 1. Long-lived radioactive ources suitable for energy and efficiency calibration

of x-ray detectors.

Photon Energy Absolute No. of Photons

Nucl ide Half life (keV) emitted per 100 decays

/I A 47. 3 ± 1.4 y 4.0861 (Ka

2

) 5.8 ± 0. 6

4.0906 (Kap 11.7 ± 1. 1

4.46 (Kp) c. lb ± u. «£ 1

C7 Qt; + n r\i /

- ^,^ o / . oo i U. Uo ^y J 07 7 O/. / ± 1

.

b

7Q /in + n nc /m'n /O.'fU I U.UO \y) Q/l 7 J. 1 I 1

.

0

49

v 330 ± 20 d 4.5049 (Ka

2

)

"7 1C ± u. 2

4 . b 1 Uo (. K.0I

-J J 14. 6 ± 0. i

b.UU vl\pj o en Z . bU ± U. 1

U

54Mn 312. 6 ± 0.8 d 5.4055 (Ka

2

) 7.43 ± 0. 21

5.4147 (Kap 1/1 7 14./

D . UU ^ l\p ) 2. 94 ± 0. 10

o34.oi ± U.UI (.yj 99. 9760 ± 0. 0002

55 rFe 2. 72 ± 0.03 y 5.88765 (Ka

2

) 8. 2 ± 0. 7

5.8975 (Ka-j) 16. 3 ± 0. 12

0 . f3 V. I\p J

O. 'J 0. 0 ± U

0

. 0

b/

Co 271 .23 ± 0.63 d 6.3908 (Ka

2

) 16.5 ± 0 .5

6.4038 (Ka,) 32.5 ± 0 .8

7.06 (Kp) 6.56 ± 0 .20

14.4147 ± 0.0025 (y) 9. 54 ± 0 . 13

too nco -i- n ( IZ^.Ubo ± U.UUo (.y; ob. 59 ± u . 19

100.4/0 x U. UUo \%) ID. 0 1 ± u . Io

DO-,Zn 244 .52 ± 0.07 d 8.0278 (Ka

2

) 11.5 ± 0 .3

8.0478 (Ka^ 22.6 ± 0.,5

8.94 (Kp) 4.61 ± 0..13

1115.52 ± 0.3 (y) 50.75 ± 0..10

continued

28
Table 1 continued.

Photon Energy Absolute No. of Photons

Nuclide Half life (keV) Emitted per 100 Decays

73

As 80 .30 + 0 .6 d 9 .8553 (Ka

2

) 30.5 ± 1.6

9 .8864 (Ka

} ) 60 ± 3

11 .0 (Kp) 13.3 ± 0.7

53 .437 ± 0.009 (y) 10.5 ± 0.3

75

Se 120 + 1 d 10 .5080 (Ka

2

) 16.3 ± 1.0

10 .5437 (Ka,) 31.7 ± 1.8

11 .79 (Kp) 7.3 ± 0.5

66 .05 ± 0.01 (y) 1.011 ± 0.024

96 .733 ± 0.002 (y) 3.20 ± 0.06

121 .115 ± 0.003 (y) 16.72 ± 0.19

136 .000 ± 0.005 (y) 56.8 ± 0.7

85 cSr 64 .68 + 0 .23 d 13 .3358 (Ka

2

) 17.

1

± 0.4

13 .3953 (Ka,) 33.0 ± 0.7

15 .0 (Ka^ 8.67 ± 0.24

513 .990 ± 0.010 (y) 98.0 ±1.0

88

y 107. 2 + 0. 7 d 14. 0979 (Ka

2

) 17.6 ± 0.8

14. 16500 (Ka,) 33.9 ± 1.4

15. 9 (Kp) 9.1 ± 0.4

898. 020 ± 0.020 (y) 93.4 ± 0.7

1836. 040 ± 0.020 (y) 99.35 ± 0.03

109

Cd 453 + 2 d 2. 98 (La))

1

1

± 4 3.24 (Lp)j

21. 9903 (Ka

2

) 29.1 ± 1.0

22. 1629 (Ka^ 55.1 ± 1.8

25. 0 (K

p

) 17.8 ± 0.7

88. 037 ± 0.005 (y) 3.72 ± 0.11

continued

29
Table 1 continued.

Nucl ide

113

Halflife

Sn 115.1 ± 0.3 d

Photon Energy

(keV)

Absolute No. of Photons

Emitted per 100 Decays

3.28 (La)

3.6 (LB)

24.0020 (Ka

2

)

24.2097 (Kap

27.67 (KB)

255.115 ± 0.015 (y)

6.7

20.7

39.0

13.0

± 2.3

± 0.8

± 0.14

± 0.5

2.12 ± 0.07

119m

Sn 293.0 ± 1.3 d 3.44 (La) ) 3.77 (LB) | 23.875 ± 0.101 (v)

25.0440 (Ka

2

)

25.2713 (Kap

28.8 (KB)

12 ± 4

16.2 ± 0.4

7.8 ± 0.4

14.7 ± 0.6

4.96 ± 0.21

125, 60.14 ±0.11 d 3.77 (La))

4.16 (LB)|

27.2017 (Ka

2

)

27.4723 (Ka-,)

31.3 (KB)

35.46 ± 0.03 (y)

15

39.8

74.2

25.8

± 6

± 1.4

± 2.5

± 1.0

6.67 ± 0.22

129

I (1.97 ± 0.14) x 10

7 y 4.105 (La)j

4.57 (Lp) J

29.4580 (Ka

2

)

29.7790 (Ka] )

34.0 (KB)

39.58 ± 0.03 (y)

8.2 ± 2.5

20.0 ± 0.6

37.1 ± 0.9

13.2 ± 0.4

7.5 ± 0.20

continued

30
Table 1 continued.

Nucl ide Half life

133

Ba 10.74 ± 0.10 y

Photon Energy

(keV)

4.28 (La))

4.48 (Lp)

I

30.625 (Ka

2

)

30.973 (Ka^

35.4 (Kp)

53.155 ± 0.016 (y)

79.621 ± 0.011 (y)

80.997 ± 0.005 (y)

Absolute No. of Photons

Emitted per 100 Decays

17 ± 5

33.4 ± 1.1

61.7 ± 1.8

C

22.9 ± 0.6

C

2.18 ± 0.04C

2.67 ± 0.07c

34.7 ± 0.8

C

137

Cs 30.02 ± 0. 15 y 4.46 (La)

| 4.99 (LB)

J

31.8171 (Ka

2

)

32.1936 (a,)

36.4 (Kpp

37.3 (Kp

2

)

661.661 ± 0.003 (y)

1.1 ± 0.3

2.11 ± 0.07

3.90 ± 0.12

1.13 ± 0.10

0.29 ± 0.03

85.3 ± 0.4

139

Ce 137.66 ± 0.05 d 4.64 (La))

5.21 (Lp){

33.033 (Ka

2

)

33.4413 (Ka.,)

37.80 (Kpp

38.73 (KB

2

)

13 ± 4

23.7 ± 0.8

43.7 ± 1.4

16.0 ± 0.6

145

Pm 17.7 ± 0.4 y 0.98 (M)

5.22 (La)

5.9 (Lp)

36.8474 (Ka

2

)

37.3610 (Ka^

42.269 (Kpp

43.298 (Kp

2

)

67.20 ± 0.10 (y)

72.40 ± 0.10 (y)

14.4 ± 2.0

21.1 ± 0.7

38.6 ±1.2

14.7 ± 0.5

0.65 ± 0.03

2.18 ± 0.15

31 continued
Table 1. continued.

Photon Energy Absolute No. of Photons

Nuclide Half life (keV) Emitted per 100 Decays

181 W 120.95 ± 0.02 d 1.71 (Mc^)

1.77 (Mp)

6.21 ± 0.03 (y) 0.97 + 0.22

8. 12 (La)

j 9.5 (Lp) j

22 + 3

56.2770 (Ka

2

) 19.0 + 2.0

57.5320 (Kc^) 33 + 4

65.210 (Kpp

66.999 (Kp

2

) i

14.0 + 1.5

203,

,

Hg 46.59 ± 0.5 d (note d)2.71 (Ma.,)

2.362 (Mp

2

)

10.22 (La) ) 12.24 (Lp)j

5.9 + 0.7

70.8319 (Ka

2

) 3.73 + 0.12

72.8715 (Ka^ 6.32 + 0.19

82.558 (Kpp 2 1 + 0.

1

84.904 (Kp

2

) U. DO n or

279.188 ± 0.006 (y) 81 .55 + 0. 80

207

Bi 33.4 ± 0.8 y 2.346 (Ma^

2.443 (Mp

2

)

--

10.5 (La) 35.0 + 2.0

12.62 (Lp) L£: La: Lp: Ly = 5 .5 43.5:42.4:8.6

14.76 (LYl )

72.8042 (Ka

2

) 22.8 + 0.5

74.9694 (Ka

] ) 38.6 + 0.7

84.922 (Kp

2

) 87.343 (Kp

2

)

17.1 + 0.4

569.670 ± 0.020 (y) 97.8 + 0.5

1063.620 ± 0.020 (y) 74 + 3

1442 ± 10 (y) 0.147 + 0.020

1770.22 ± 0.04 (y) 7.3 + 0.4

32 continued
Table 1. continued.

Photon Energy Absolute No. of Photons

Nuclide Half life (keV) Emitted per 100 Decays

241

Am 431 ± 4 y 3. 30 (Ma)

e

6. 35 ± 0.60 f

13.

13.

945 (La.|)

758 (La

2

)

13. 3 ± 0.4

16.

17.

837 (Lp

2

) 740 (LP,)

19. 3 ± 0.7

20. 77 (Ly-j) 4. 93 ± 0.21

59. 536 ± 0. 001 ( Y ) 35. 5 ± 0.3

The data represent current "best values" assembled from the following sources:

Martin, M. J., Nuclear Decay Data for Selected Radionucl ides (Nuclear Data Project),

Oak Ridge National Laboratory Report ORNL-51 14 (1976).

Nuclear Data Sheets, current issues (Academic Press, Inc. , New York).

Lederer, C. M. and Shirley, V. S. , Table of Isotopes , 7th Edition (Wiley-Interscience,

Inc. , New York, 1978).

Fink, R. W. , current isotope files, based on current literature.

u Calculated using data from Freund, H. U. and McGeorge, J. C. , Z. Physik, 238, 6 (1970).

cValues are those from the Physikal isch-Technische Bundesanstalt, Braunschweig,

West Germany, from which absolutely calibrated sources of 133 Ba may be purchased.

^Some chemical or photochemical instability with age of 203 Hg sources has been observed.

Owing to this and the relatively shorter halflife, the use of 203 Hg sources as absolute

calibration standards is not especially recommended.

M x-rays from 241 Am decay are spread over a range of about 3 to 6.5 keV. Special caution

should be exercised with Ge detectors, since the Ka and Kp x-ray escape peaks from the

13.945 keV La x-ray fall in the region 2.65 to 6.64 keV which is within the M x-ray

region.

fValue from Karttunen, E. I., Freund, H. U. , and Fink, R. W. , Phys. Rev. A4, 1695 (1971).

33
Remarks concerning Table 1.

Not all of the sources listed in Table 1 are available commercially, and only a few of

them at present can be obtained as absolutely calibrated photon emission standards from the

National Bureau of Standards, the Physikalisch-Technische Bundesanstalt of Braunschweig,

West Germany, or the Laboratoire de Metrologie des Rayonnements Ionisants, Gif-sur-Yvette,

France. Some additional calibrated low-energy photon emission sources may become available

from these metrology laboratories in the future.

Other long-lived radioactive nuclides which could become suitable standards for energy

and efficiency calibration of x-ray intensities are 391.1 ± 1.2 d 235 Np (which exhibits

approximately 2.95 K x-rays per 100 EC decays and 49 L x-rays per 100 EC decays [1],

(1.03 ± 0.04) x 10 s y

41 Ca, 1 x 10 s y59 Ni [2], ion-implanted 2.1 x 10 5 y8l9 Kr [3],

16.4 ± 0.4 y

93m

Nb, 3000 y

939Mo, 3.0 ± 0.4 y

10l9Rh, 90 y

151 Sm, 234 d 153Gd, 144 d 159 Dy

[4], 33 y

16s9Ho, which might be suitable as a very low energy calibration standard, since

it emits essentially only 1.3 keV M x-rays, 588 ± 10 d 179Ta, 115 d 182 Ta, which is a good

relative photon intensity standard for -y-rays from 84.68 to 264.1 keV, but which needs

absolute K and L x-ray photon emission intensity measurements, and 50 y

1939 Pt, which emits

only M and L x-rays at 2.0 and 9.1 - 10.8 keV, respectively. The calibration and use of

7370 y

243 Am [5] and of 350.6 ± 2.1 y 249 Cf [6] for x-rays and y-rays might be useful for some laboratories.

[1] McGeorge, J. C. , Nix, D. W. , Fink, R. W. , and Landrum, J. H. , Z. Physik, 255, 335

(1972).

[2] Chew, W. M. , Xenoulis, A. C. , and Fink, R. W. , Nucl. Phys. A218 , 372 (1974).

[3] Chew, W. M. , Xenoulis, A. C. , Fink, R. W. , Schima, F. I., and Mann, W. B. , Nucl.

Phys. A229, 79 (1974).

[4] Rao, P. V., Z. Physik, A260, 47 (1973); McGeorge, J. C. , Freund, H. U. , and

Fink, R. W. , Nucl. Phys., A154 , 526 (1970).

[5] Pate, J. C. , Baker, K. R. , Fink, R. W. , McClure, D. A., and Kendrick, Jr., N. S. , Z. Physik, A272 , 169 (1975).

[6] Schmidt-Ott, W. D. , Fink, R. W. , and Rao, P. B. , Z. Physik, 245, 191 (1971).

34
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

A COMPARISON OF SILICON AND GERMANIUM X-RAY DETECTORS

N. C. Barbi and D. B. Lister

Princeton Gamma-Tech

Princeton, New Jersey 08540

Abstract

In recent years, semiconductor detectors (germanium and silicon)

have become widely used in the analysis of x-rays. Si C Li ) detectors have

been applied predominantly in the analytical range of 1 to 30 keV, while

Ge(Li) and intrinsic germanium (IG) detectors have been applied in the

analysis of higher energy photons. This devision of detector application

with x-ray energy is a consequence of the inherent characteristics of the

two materials.

Germanium, due to its lower band gap energy, can theoretically

exhibit superior resolution compared to silicon. For example, resolution at 5.9 keV for a commercially available IG detector with an active

area of 25 mm2 would be comparable to a Si (Li) detector with half the

active area. However, the resolution advantage in germanium detectors is

often countered by other material characteristics, and is therefore not

realized throughout the spectrum. In particular, for photon energies

less than 3 keV, peaks from the germanium detector exhibit severe

tailing, due to incomplete charge collection, rendering the germanium

detectors of little use in the analysis of Na through S.

As the photon energy increases from 3 to = 20 keV, the IG perfor- mance becomes comparable to that of the Si (Li) detector, with the

exception of the more complex escape peak effects in germanium. Above =

30 keV, the IG detector has the advantage of better detection efficiency

than the Si (Li).

The choice of detector for a particular application should be based

on resolution and efficiency parameters for the analytical range of

interest. A secondary, but practical, consideration is that IG detectors

can be stored indefinitely at room temperature without degradation.

Practical applications of the IG detector in x-ray analysis will be

discussed and a comparison of the detector performance to that of Si (Li)

for these applications will be presented.

1. Introduction

In recent years, semiconductor detectors (silicon and germanium) have become widely

used in x-ray spectroscopy. Si(Li) detectors have been applied predominantly in the analytical range of 1 to 30 keV, while Ge(Li) and intrinsic germanium (IG) detectors have been

applied to the analysis of higher energy photons. This selection of detector type based on x-ray energy is a consequence of the inherent characteristics of the two materials.

An empirical comparison of the performance of IG and Si (Li) detectors in the x-ray

region is presented below. Ge(Li) detectors are not explicitly discussed since they have

been essentially supplanted by IG detectors for this application, although the general

description of the characteristics of IG detectors will also apply to Ge(Li) detectors.

Before presenting the results, a brief description of the two main detector parameters,

resolution and efficiency, is given in the light of the germanium and silicon comparison.

35
1.1 Detector Resolution

The resolution of a semiconductor x-ray detector is taken as the measured FWHM of a MnKa peak (emitted from a Fe-55 radioactive source). Modern IG and Si (Li) detectors

exhibit comparable resolution on this basis, although germanium has an advantage at medium

and high energies due to its lower band-gap energy. At lower x-ray energies (<3 keV), the

inherent superior resolution of germanium is completely overcome by a broadening of the low

energy side of the peak due to incomplete charge collection. This degradation in peak

shape, called tailing, is severe for x-rays in the 1.2 to 2 keV range in germanium

detectors.

1.2 Detector Efficiency

The beryllium window through which x-rays must pass to reach the detector in the

cryostat is the dominant factor in determining the efficiency of a Si (Li) spectrometer for

low energy x-rays. A typical spectrometer may be nearly 100 percent efficient for a CaKa

x-ray (3.69 keV), but only 40 percent efficient for a NaKa x-ray (1.04 keV). This loss of

efficiency is due primarily to the relatively high absorption of Na x-rays in the beryllium

window itself. Although the beryllium window presents an identical problem in germanium

detectors, the usefulness of the 16 detector in the low energy region is limited first by

poor resolution due to incomplete charge collection, as discussed above. Peak tailing

severely limits the use of these detectors below 3 keV, where window absorption is just

beginning to significantly degrade detector efficiency. A conventional Si (Li) spectrometer,

which is generally useful down to 1 keV is therefore superior in performance, from a resolution standpoint, to a Ge detector in this low energy region.

In the high energy region (>30 keV), however, the IG detector is clearly superior.

Because germanium is a higher atomic number material than silicon, it is inherently a stronger x-ray absorber. X-rays which might pass right through a silicon crystal may still

be absorbed (and, therefore, detected) in germanium.

Figure 1 shows the efficiency for photoelectric absorption of 5 mm thick Si (Li) and

germanium detectors, emphasizing the medium and high energy region. The dotted line in the germanium curve represents the loss in efficiency due to the rather significant escape

peak interaction for medium energy photons. At £30 keV, the efficiency of germanium

becomes superior to that of a Si (Li) detector of equal thickness, with its advantage

becoming more pronounced as x-ray energy increases beyond that value.

i

36
I

1 1 1 1 1 1 1

1

1 10 20 30 40 50 60 70 80 90 100

KeV

Figure 1. Photoelectric efficiency (5 mm thick detector).

2. Experimental Procedure

For purposes of collecting low energy x-ray spectra, a 5 mm thick IG and a 3 mm thick

Si (Li) detector of sl60 eV resolution were interfaced to a scanning electron microscope.

The accelerating voltage was either 28 or 18 kV, as noted on the individual spectra.

To compare high energy performance, 5 mm thick detectors were used to analyze radiation emitted from, or fluoresced by, radioactive sources.

3. Results

3.1 Low Energy Performance

Figure 2 shows a comparison of the two detectors used for low energy spectra at

MnKa. Note that tailing is already becoming apparent in the spectrum (in dots) collected

using the germanium detector.

37
Figure 2. MnKa (5.9 keV) and Kp (6.3 keV)

collected using the IG detector

(dots) and Si (Li) detector

(bars) interfaced to the SEM

for this study.

Figure 3 illustrates a stainless steel spectrum acquired at 18 keV using the IG detector. Its appearance is essentially identical to that expected from a Si(Li)

spectrometer.

STPIMLESS STEEL 18 KU ELECTRONS IG

Figure 3. Stainless steel spectrum collected using IG Detector on SEM.

Performance differences between the detectors become significant only in the regions

below 3 keV or above 30 keV. Figure 4, however, shows a spectrum from pure molybdenum,

illustrating two of the more subtle characteristics exhibited in the germanium detectors

in the medium energy region: prominent escape peaks and a clearly defined efficiency jump

at the germanium K- absorption edge.

38
enamy in kcu Figure 4. Mo Spectrum, collected using IG Detector on SEM,

showing escape peaks and K absorption edge.

Figures 5a and 5b illustrate the effects on peak shape of the incomplete charge

collection in germanium when low energy x-rays are absorbed. Figure 5a shows MoLa at 2.29 keV, SiKa at 1.74 keV, and GeLa at 1.19 keV, as analyzed with a germanium detector.

The tailing is apparent for the molybdenum peak, and becomes more pronounced as the energy

decreases to the SiKa energy. It will continue to worsen until the L absorption edge for

germanium is reached (=1.2 keV). Below this energy, peak shape is partially restored, as seen for the germanium L line. For K lines of light elements, this means that the peak

shape will be the poorest for a MgKa line (1.25 keV), but will be at least partially

restored for the NaKcr line (1.04 keV). Figure 5b compares silicon peaks analyzed using

the Si (Li) and IG detectors, showing the magnitude of the peak shape degradation in

germanium.

LOU EMER&Y PERK SMPPE-I& DET

> 2 3

Figure 5a. Low energy peak shape degradation due to incomplete

charge collection in the IG detector.

30
SJ 2BK.U ELECTROINS I & US 51 ILI!

SB

Figure 5b. Comparison of Si peak from IG and Si (Li) Detectors.

To illustrate the effects of the tailing problem exhibited at low energies in germanium

detectors, several materials containing light elements were analyzed. Figure 6a is an iron-3 percent silicon material, analyzed with the IG detector (the iron peaks are off-scale). Although this spectrum by itself does not look particularly unusual, the

degradation in peak-to-background ratio for the silicon peak (and therefore sensitivity for

silicon) in the IG spectrum becomes apparent when compared to the Si(Li) spectrum of the same material, shown in figure 6b. The difference in silicon P/B between the two spectra

is nearly an order-of-magnitude.

FE-35SSI 1SKU ELECTRONS IC DET

> Z34SE7B9 1B

Eftemiir in keu

Figure 6a. Fe-3 percent Si spectrum collected using IG Detector on SEM.

40
FE-33SSI 1SKU ELECTRONS SI (LI) DET

ertcnct in kcu

Figure 6b. Fe-3 percent Si spectrum collected using Si(Li)

Detector on SEM.

Figures 7a and 7b compare spectra from anorthite (=16% Al , 24% Si, 10% Ca, 0.4% Fe,

remainder oxygen) using the two detectors. Not only is the separation between the aluminum

and silicon peaks severely degraded in the IG spectrum, but the small Na peak is lost

completely in the aluminum and silicon tails.

PINORTH I TE 28 KU ELECTRONS IG DET

enencr im kcu

Figure 7a. Anorthite spectrum collected using IG Detector on SEM.

41
RINORTH I TE Z8KU ELECTRONS SI (LI) DET

C 3

z a « s 7

Figure 7b. Anorthite spectrum collected using Si (Li)

Detector on SEM.

3.2 High Energy Performance

High Energy Performance

Figure 8a illustrates a high energy spectrum using an IG detector. The spectrum is

the Pb K-series, excited by a Co-57 radioactive source. Figure 8b is a comparison of the

Pb K spectra obtained using the IG and Si (Li) detectors. The spectra were normalized at the PbLa line (10.6 keV), at which energy the efficiency of the two detectors is about

equal. The superiority of the IG detector in the higher energy region is clearly

demonstrated.

PB SPT CO-S^ SOURCE I G DET

ENERGY IN KEV

Figure 8a. Pb K spectrum excited by Co-57, collected

using IG and Si (Li) Detectors.

42
PB CO-57 SOURCE I G US SI (LI)

ENERGY IN KEV Figure 8b. Comparison of Pb K spectra collected

using IG and Si (Li) Detectors.

Figure 9 compares Si(Li) and IG spectra from Eu K x-rays (40-50 keV), emitted from a GD-153 radioactive source. Note that not only is the efficiency better for the IG detector, but the resolution (quality of peak separation) at the Ka x -Ka2 lines is also

superior.

CD- 153 SOURCE CIUING EU XRPYS IG DET

Figure 9. Comparison of Eu K spectra emitted from Gd-153 source, collected using IG and Si (Li) Detectors.

43
4. Discussion and Conclusions

The reasons governing the application of Si(Li) detectors to low energy x-ray

spectroscopy (<30 keV) and germanium detectors to high energy x-ray spectroscopy have been

illustrated. The limitation on using germanium detectors for low energy spectra is placed

by the resolution degradation due to incomplete charge collection for x-rays less than

3 keV in energy. The limitation on using Si (Li) detectors for high energy spectra is

placed by poor efficiency.

The phenomenon of incomplete charge collection is discussed in detail by Llacer

et al., [I] 1

. X-rays absorbed near the surface of the germanium detector (and to some extent in silicon as well) produce charge which is not always fully collected. At about

3 keV in germanium, the resulting peak tailing becomes significant, and continues to worsen

as x-ray energy (and therefore penetration into the detector) decreases. As the x-ray

energy decreases below the germanium L absorption edge, the mass absorption coefficient

drops significantly, penetration increases, and peak shape is at least partially restored.

With further decrease in energy, one would expect tailing to increase again.

Incomplete charge collection is therefore responsible for poor resolution at low

energies in germanium detectors. However, at medium and high energies, germanium detectors

exhibit better resolution than Si (Li) detectors due to the lower band-gap energy for

germanium compared to silicon. For the higher energy region, therefore, germanium will

exhibit better resolution than a Si (Li) detector for a given detector area, or for a given

detector resolution, a germanium detector of larger size can be applied. For example, a premium Si(Li) detector of 145 eV resolution will typically have an active area of 10 mm2

. A premium germanium detector of the same resolution will typically have a 25 mm2 active

area. In fact, IG detectors with resolutions better than 185 eV and an active area of

100 mm2 are routinely fabricated.

One practical advantage of IG detectors is their ability to be stored indefinitely at room temperature. Although the detector must be used at cryogenic temperature to limit

thermal and electronic noise, the liquid nitrogen level in the cryostat need not be

maintained when the detector is not in use.

On the other hand, it is generally recommended that the Li -drifted detectors be

stored at cryogenic temperature at all times in order to limit the possibility of Li redistribution in the crystal. The IG detector is high purity germanium, and does not require lithium-drifting; therefore, room temperature storage is normal procedure. For some industrial or remote process monitoring applications, this feature may be a significant

advantage.

In summary, the following conclusions can be stated:

1. The Si(Li) detector outperforms germanium for low energy x-ray analysis

based on superior resolution for x-rays <3 keV.

2. The germanium detector outperforms Si (Li) for higher energy x-ray

analysis, based on superior efficiency for x-rays >30 keV.

3. The IG detector can be stored at room temperature, whereas it is generally recommended that Si(Li) detectors be maintained at cryogenic

temperature at all times.

4. IG detectors typically exhibit better resolution for medium and high

energy x-rays than Si (Li) detectors with equal active area.

Reference

[1] Llacer, J., Haller, E. E. , and Cordi , R. C. , "Entrance Windows in Germanium Low-Energy

X-Ray Detectors", IEEE Trans on Nuclear Sci. , NS-24 , 53-60 Feb. 1977.

The figure in the bracket indicates the literature reference at the bottom of this page.

44
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ROOM TEMPERATURE ENERGY DISPERSIVE X-RAY SPECTROMETRY WITH MERCURIC IODIDE DETECTORS

A. J. Dabrowski 1

, M. Singh, G. C. Huth, and J. S. Iwanczyk 1

University of Southern California

Medical Imaging Science Group

4676 Admiralty Way, Suite 932

Marina del Rey, California 90291

Abstract

The compound semiinsulator mercuric iodide is under development as

a detector for room temperature operable, energy dispersive x-ray

spectrometry. Energy resolution values approaching those characteristic

of cryogenical ly cooled semiconductor detectors have been achieved. A miniature spectrometer consisting of an Hgl 2 detector and preamplifier

and having a total volume of 300 cm3

, has been fabricated. This does not even closely approach reduction in spectrometer size that is possible and

much smaller designs are possible in the near-term future. Portable

x-ray spectrometers are currently being designed for on-site, extralaboratory applications such as detection of uranium in mines, or for

microanalysis in space. An energy resolution of 420 eV is currently

obtained with a 7 mm2 Hgl 2 detector for the 5.9 keV x-rays of Fe-55. An improvement in resolution by 50-100 eV is expected in the near future.

The various sources which contribute to noise and background in the

x-ray spectra have been studied and compared to corresponding features of

cooled Si (Li) and germanium detectors. X-ray spectra of many targets

have been measured with mercuric iodide detectors. These spectra were generated either by charged-particle excitation of the sample, or by the

method of x-ray fluorescence. Several potential applications have been

outlined which are particularly suited to mercuric iodide x-ray

spectroscopy.

1. Introduction

The success of silicon and germanium in fabrication of nuclear detectors for a wide

range of applications has stimulated consideration of other materials that might offer

special advantages in specific applications. In particular the development of high energy

resolution, ambient temperature solid state x-ray detectors has been highly desirable.

Because of the high level of noise associated with leakage current at room temperature,

high bandgap materials had to be taken into consideration for this purpose.

Numerous solid state crystals have been exploited for this purpose. The most work has

been done with three compounds: gallium arsenide [1,2] 2

, cadmium telluride [3,4], and

mercuric iodide [5,6]. A comparison of some properties of these materials is shown in

Table 1. Gallium arsenide and cadmium telluride are semi-insulators with a bandgap of

1.4 eV and 1.5 eV respectively, which leads to a room temperature leakage current in the

order of nanoamperes or higher.

1 0n leave from the Institute of Nuclear Research, Swierk, Poland.

2 Figures in brackets indicate the literature references at the end of this paper.

45
Table 1. Comparison of some physical and electrical properties of three

solid state compounds.

Compound

GaAs

CdTe

Semi-Insulator

Semi -Insulator

Insulator

Band-Gap

(eV)

1.4

1.5

2.1

Leakage Current

Nanoamperes

Nanoamperes

Picoamperes

(Mt)

h

2 (cm /V) (cm

2/V)

<io" 3 <5xl0~ 5

< 5x10"

2 <5xl0~ 4

-3

<10 J <io" 5

As we have shown elsewhere [7], a leakage current of nanoamperes produces noise levels

which limit the energy resolution to >1 keV (FWHM). Mercuric iodide has a bandgap of 2.1 eV which leads to the leakage current at room temperature in the region of picoamperes.

Because of that, electronic noise of the Hgl 2 spectrometer could be reduced to the values

close to those obtained by cryogenical ly cooled systems. This feature combined with

exceptionally good charge carrier transport properties (for insulators), makes Hgl 2 a particularly attractive material for room temperature low energy x-ray spectroscopy.

2. Crystals and Detectors

Some physical properties of mercuric iodide are presented in Table 2. ormercuric

iodide is a bright red tetragonal crystal which above 127 °C undergoes a phase transformation into the yellow colored orthorhombic p-HgI 2

. The low- temperature tetragonal crystalline structure is the basic form of mercuric iodide used for nuclear detector fabrication.

This is a lamellar crystal cleaving easily along the iodine-iodine planes.

The raw a-HgI 2 has such impurities like carbon, silicon, iron, aluminum, potassium,

and boron. It is purified by zone refining, extraction from solvent or repeated sublimation. Crystals are grown using initially purified raw materials by either a static or dynamic pulling vapor-growth or solution-growth method. Large crystals weighing as much as 100 grams are common with the growth period being of several weeks duration [8,9].

Taking advantage of the lamellar nature of the crystal, "plates" are carefully cleaved

to be used in the fabrication of x-ray detectors [7]. These plates can be up to a centimeter in diameter. Subsequent x-ray detector fabrication is exceptionally simple.

The detector plate is first etched to the required thickness (200-500 urn). The actual

detector area is then defined by painting on contacts of emulsified carbon ("aquadag") to both sides of the crystal. Evaporation of thin metal films, usually Pd, is also used.

Contact wires are embedded in the carbon contacts for remote attachment. Finally, the

crystal is sprayed with a common commercial electronic encapsulant ("humiseal"). This

effectively eliminates moisture or vacuum interaction with the Hgl 2 surface and possible

degradation of detector leakage current and noise behavior. These detectors perform

equally well under high vacuum (10-6 torr) or ambient pressure. In use, the detectors are biased in the 50-500V range, depending on the thickness of the plate and transport

properties of the crystal. Leakage current less than 5 pA is common and is required for

high energy resolution.

46
Table 2. Physical

Atomic Numbers

Crystal Structure

Lattice Parameters

Molecular Weight

Atomic Density

Melting Point

Phase Transition

Density

Dielectric Constant

Index of Reflection

Band Gap

Band Gap Change

with Temperature

Resistivity

Atoms/Cell

Electron Mobility

Hole Mobility

(Mt)

e (Mt) h Energy per Electron-hole

ies of mercuric iodide.

53, 80

Tetragonal (low T.

)

Orthorhombic (high T. )

a = 4.368 A

c = 12.435 A

454.45 g/mole

2. 54x1

0

22 at/cm3

259 °C

127 °C

3 6.3 g/cm

e =8.3

n = 2.89

2.13 eV

-7xl0~

4

eV/K

range (100K - 300K)

vlO 13

ficm at 300K

6MOO cm

2/Vs at 300K

•^250 cm

2/Vs at 100K

^10 cm

2 /Vs at 300K

-v.30 cm

2 /Vs at 100K

<103 cm

2/V

<10 5 cm

2/V

4.15 eV at 300K

47
3. Energy Resolution

Various factors which have led to the presently available level of energy resolution

of mercuric iodide detectors, will be discussed here.

In the solid state energy dispersive x-ray spectrometers the energy resolution is

mainly affected by the first two stages of the system: detector and preamplifier (we do not consider here high counting rate).

To discuss the energy resolution of the solid state spectrometers it is convenient to distinguish three values of this parameter.

1. Electronic noise (in FWHM) of the electronic system without the detector - this gives the lower possible limit of the energy resolution of the

spectroscopic system.

2. Electronic noise (in FWHM) of the spectrometer (detector and electronic

system) measured as the width of the line of a pulser (injection of a definite amount of charge by an external pulser through a small capacitor

into the input stage of the system).

3. Energy resolution (in FWHM) of the observed x-ray line - usually it is

referred to the Mn K x-rays at 5.9 keV.

Energy resolution values obtained by different systems with Hgl 2 detectors are summarized in Table 3.

The upper line of Table 3 shows the first published spectrum of low energy x-rays

obtained using a Hgl 2 detector in 1974 [5]. The detector was 4.2 mm2 x 210 |jm. The energy

resolution of the Fe 55 peak was 850 eV (FWHM) and for the pulser 760 eV (FWHM). By selection of components of the input stage of the electronics, it was possible to improve the

electronic noise level and consequently the energy resolution of the detector, as is shown

in the second and third line of Table 3 [6,7]. To further reduce the noise level of the

preamplifier, the input FET was cooled using a small thermoelectric Peltier cooler

[10,11] (fourth row of Table 3).

In all hitherto applications, commercially available components have been used in the

input stage of the preamplifier. As it is known, the package of the commercially available

FET's cause additional degradation of their noise characteristics. Consequently the next

step in the evolution of the input stage of the low noise room temperature preamplifier

was taking the FET chip out of the "noisy" case and mounting it on a low noise ceramic

substrate.

The results are shown along the fifth row in Table 3 and in figure 1. These results,

presented for the first time here, show an energy resolution of 380 eV (FWHM) for Fe 55

peak, and 300 eV (FWHM) for the pulser. The electronic noise of the system without

detector is 240 eV (FWHM) for Hgl 2 (which corresponds to a value of 205 eV (FWHM) for

silicon). This value was obtained when both Hgl 2 detector and preamplifier were at room temperature.

For comparison, at the bottom of Table 3 the results for the best cryogenical ly cooled

Si C Li ) systems are shown. It is worth saying that the results both for proportional and

scintillation counters for Fe 55 line, would be situated outside the right border of this

Table.

Referring to Table 3, further improvement of the spectrometer means that the value of

the electronic noise of the system without the detector, E, has to be further diminished,

and the distances between E and P as well as P and X on this diagram have to be possibly

smal 1

.

48
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Further improvement of the electronic noise of the system, E, is expected, when better

selection of the input FET will be made. Also the FET chips on the ceramic substrate will

be cooled down to about -50 °C using small thermoelectrical Peltier coolers. Reduction of

the value of E to below 200 eV (FWHM) for Hgl 2 is expected after these procedures.

To shorten the distance between E and P on the discussed diagram, the smallest values

of the leakage current and the capacity of the Hgl 2 detector should be achieved. Improvement in fabrication of Hgl 2 detectors, to achieve these goals, is in progress.

Factors which contribute to the spread between E and X in the diagram of Table 3 are mainly transport properties of the charge carriers in the Hgl 2 cyrstal (trapping), statistical effects in the charge generation process, nonuniformity of the crystal, which causes dispersion of trapping parameters and distortions of the electric field through the active

volume of the detector, and detector edge and window effects. Additional effort in fabrication of mercuric iodide crystals and designing of the detectors which could mainly improve

the energy resolution for x-rays and bring it closer to the pulser line width, is now under

investigation.

4. Low Energy X-Ray Spectra

The behavior of mercuric iodide spectrometers for detecting very low energy x-rays

from different targets has been studied using as an excitation source, an alpha particle

beam obtained from a Van de Graaff accelerator at the California Institute of Technology

[10]. A 7 mm2 x 310 urn Hgl 2 detector masked down to an exposed entrance window of 3 mm2 has been used for this experiment. Characteristic lines below 10 keV of different elements

(such as sodium, magnesium, aluminum, silicon, iron, nickel, copper) have been measured.

The energy resolution of the spectrometer at 5.9 keV was 450 eV (FWHM). The input FET

(in its original case) was thermoelectrically cooled.

The spectrum obtained from a magnesium target using a 6.0 meV orexcitation beam is

shown in figure 2. The K line at 1.25 keV has a FWHM of 390 eV and is well separated from

the noise.

The low-energy x-ray peaks obtained by Hgl 2 detector exhibit excellent symmetry as demonstrated by the Mg peak on figure 2. This is in contrast to the behavior of cooled

high-purity germanium (HPGe) detectors for energies below about 2.3 keV, where a large

asymmetry in the form of a tail on the low-energy side of the peak is observed. This

behavior of HPGe has been studied by Llacer et al. [12]. During the experiment in Caltech

mentioned above we have also observed such asymmetrical low-energy x-ray peaks with a cryogenical ly cooled HPGe detector. As an example of this effect the x-ray line from a Si- target measured by Hgl 2 spectrometer (450 eV FWHM for Fe 55) and a cryogenically cooled

HPGe spectrometer (175 eV FWHM for Fe 55) under exactly the same experimental conditions,

is shown in figures 3 and 4.

51
1000

_ 800

co

o 600

CO

CD

400

200

0

1 1

. Mg-K a

A 1.25 keV

•

* •

•

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1

•

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l 1^-—..I 80 160

Channel Number

240

Figure 2. K x-ray line from a magnesium target using 6.0 MeV a-excitation beam,

52
1000

to

czCD

100

Channel

180 260

Number

Figure 3. K x-ray line from a silicon target measured by Hgl^ spectrometer.

53
4000 HP cooled Ge detector

X-RAY SPECTRUM FROM Si

SlKa= 1.74 KeV

s

3000

>-

icoz

LU 2000

1000

500

100 200

Channel Number

300

Figure 4. K x-ray line from a silicon target measured by a cryogenical ly cooled HPGe spectrometer.

54
5. Portable Spectrometer

We have fabricated a prototype of room temperature mercuric iodide x-ray spectrometer

(figure 5). The spectrometer is based on a Tennelec Model 161D preamplifier, which has

been modified to improve room temperature noise level. The Hgl 2 detector is incorporated

over the first stage of preampl if ication for optimum coupling. The preamplifier case has

been modified to achieve a light-tight environment for the Hgl 2 detector. Total volume of

the spectrometer is 300 cm3

. An energy resolution of 380 eV (FWHM) has been obtained with

7 mm2 Hgl 2 detector for 5.9 keV x-rays from FE 55.

One of the major applications of energy-dispersive mercuric iodide spectrometry lies

in room temperature XRF analysis. The detector-preamplifier combination and the excitation

source can all be contained within a small package, thereby producing a portable and

convenient high energy resolution XRF system without requiring the cumbersome paraphernalia

associated with cryogenic cooling.

In typical XRF applications, the excitation source (a radioisotope or an x-ray tube)

is chosen to produce characteristic x-rays under 30 keV from the sample. Although this

energy region is usually sufficient to identify a large number of elements from their

characteristic K or L emissions, it is sometimes desirable to record higher energy x-rays

from some samples. A typical Hgl 2 detector approaches 100 percent detection efficiency

below 30 keV, and, because of its high Z, it can also be used to record higher energy

x-rays with ~80 percent efficiency for energies up to 100 keV. Although room temperature

energy resolution of Hgl 2 detectors is still poorer than for cryogenical ly cooled silicon

and germanium detectors, it has potential possibilities to be used both for low and high

energy x-ray measurements. This provides an advantage over silicon detectors which are not suitable for energies above about 20 keV, and over germanium detectors which show a large

"tailing" effect below 2.3 keV [12].

In addition to the characteristic x-rays, a typical XRF spectrum recorded with a Hgl 2 detector consists of several peaks ("scatter" peaks, "escape" peaks) and a continuum background. The "scatter" peaks produced by a coherent or incoherent scattering event between

the exciting radiation and the sample matrix, are quite large, and are usually responsible

for producing most of the background in an XRF spectrum. The "escape" peaks in Hgl 2 arise

from production and subsequent escape of characteristic x-rays of mercury and iodine from

Figure 5. Front end of a prototype portable x-ray mercuric

iodide room temperature spectrometer.

6. Analysis by X-Ray Fluorescence (XRF)

55
the detector material. The continuum background can be produced by several mechanisms such

as escape of Compton scatter photons, escape of photoelectrons, incomplete charge collection, etc. Similar background effects are seen with all energy-dispersive semiconductor

detectors. We have computed the probabilities for "escape peak" production and "continuum

background" generation in a Hgl 2 detector. Details of these calculations and experimental

verification will be published in a separate report [13].

XRF spectra from several standard and unknown samples have been obtained using Fe 55 or Cd 109 as the excitation source [13]. A few typical results will be mentioned here.

A Cd 109 source was used to excite L x-rays of uranium, and XRF spectra were obtained

from uranium ores containing different concentrations of uranium. The Hgl 2 detector used was about 500 urn thick and had an active area of ~9 mm2 which was then masked down to about

3.4 mm2 to avoid edge effects. An angle of 110° was formed between the exciting source radiation and the 1 ine-of-sight of the detector. The detector to sample distance was about

1 cm, and the source to sample distance about 2 cm. The Cd 109 source was ^10 mci.

A series of spectra obtained from ores containing different concentrations of uranium

is shown in figure 6. The characteristic L x-rays of uranium observed are the La peak at

13.6 keV, Lp! and Lp2 at 17.22 and 16.4 keV, and Ly x at 20.16 keV. Also seen are the

coherent "scatter" peaks at 24.94 and 22.16 keV (silver x-rays from the Cd 109 source),

and the incoherent scatter peaks corresponding to an angle of 110° at 23.4 and 20.9 keV. These peaks can be easily separated in the spectrum recorded from a 4 percent uranium

sample (see figure 6a). For a lower concentration of uranium, the Ly peak becomes obscured

by the scattered source radiation. However, the La and Lp peaks remain visible down to

0.01 percent uranium as shown in figure 6d. The energy resolution obtained for the 13.6 keV

La peak was about 850 eV (FWHM) in this measurement.

In applications involving Cd 109 excitation, the escape peaks mainly arise from the

escape of mercury La and Lp x-rays (9.98 keV and 11.85 keV) from the "scatter" peaks, and

have an intensity of about 1-2 percent of the parent peak (see figure 6). They are best

seen in figures 6c and 6d on the left of the 13.6 keV uranium peak.

Another example of XRF analysis is shown in figure 7. Figure 7a is from a sample

containing 50 percent Fe and 50 percent Ni , while figure 7b is from a sample which had 90

percent Fe and 10 percent Ni. A Cd 109 source was used. As seen from these spectra,

although the Kp peak of iron is obscured by the Ka peak from nickel, the Ka peak of iron

(6.4 keV) and the Ka and Kp peaks of nickel at 7.47 and 8.27 keV are well separated. By

using peak stripping techniques, it should be possible to subtract the contribution of

iron Kp peak from the nickel Ka line. An accurate analysis of iron-nickel ores is therefore

possible with a Hgl 2 system.

Room temperature mercuric iodide spectrometry has several other potential applications,

some of which are listed below. - In mining, to determine the amount of specific elements in ores. - In industrial production and manufacturing, to check on levels of some

key elements (on-line).

- In electron microscopy, where a gain of about two orders of magnitude in

geometrical detection efficiency due to the small size of the detector

housing could be achieved relative to other detectors.

- In space applications, to determine the chemical composition of

planetary bodies on space missions.

56
57
a)

149 980 10913

Figure 7. Fluorescence x-ray analysis of a sample containing a) 50% Fe and

50% Ni; b) 90% Fe and 10% Ni, measured by Hgl 2 detector.

7. Conclusions

A portable, room temperature x-ray spectrometer has been developed consisting of a "single crystal" mercuric iodide detector coupled to a low-noise room temperature preamplifier. An energy resolution of 380 eV (FWHM) for 5.9 keV x-rays has been obtained.

Further improvement in energy resolution and reduction of the peak to background ratio is

expected upon development of guard-ring structure mercuric iodide detectors as well as thermoelectric cooling of the input FET chip of the preamplifier. A hybrid low noise

preamplifier is being designed to achieve microminiaturization of the spectrometer.

58
References

[1] Eberhardt, J. E. , Ryan, R. D. , and Tavendale, A. J., "Evaluation of Epitaxial n-GaAs

for Nuclear Radiation Detectors", Nucl. Instr. Meth. , 94, 463 (1971).

[2] Kobayashi , T. , Kuru, I., Hojo, A., and Sugita, T. , "Fe-Doped High Purity GaAs as a Room Temperature Gamma-Ray Spectrometric Detector", IEEE Transactions on Nuclear

Science, NS-23 , No. 7, 97 (1976).

[3] Iwanczyk, J. and Dabrowski, A. J., "The Effect of Charge Trapping on the Spectrometric

Performance of n-Type CdTe Surface-Barrier Detectors", Nucl. Instr. Meth., 134 , 505

(1976).

[4] Dabrowski, A. J., Iwanczyk, J., Szymczyk, W. M. , Kokoschinegg, P., Stelzhammer, J.,

and Triboulet, R. , "Advantages and Limitations of n-Type Low Resistivity Cadmium

Telluride Nuclear Radiation Detectors", Nucl. Instr. Meth., 150, 25 (1978).

[5] Swierkowski, S. P., Armantrout, G. A., and Wichner, R. , "Recent Advances with Hgl 2 X-ray Detectors", IEEE Transactions on Nuclear Science, NS 2_[, No. 1, 302 (1974).

[6] Slapa, M. , Huth, G. C. , Seibt, M. , Schieber, M. , and Randtke, P. T. , "Capabilities of

Mercuric Iodide as a Room Temperature X-ray Detector", IEEE Transactions on Nuclear

Science, NS-23 , No. 1, 101 (1976).

[7] Dabrowski, A. J. and Huth, G. C. , "Toward the Energy Resolution Limit of Mercuric

Iodide in Room Temperature Low Energy X-ray Spectrometry", IEEE Transactions on Nuclear Science, NS-25 , No. 1, 205 (1978).

[8] Schieber, M. , Schnepple, W. F. , and Van Den Berg, L. , "Vapor Growth of Hgl 2 by

Periodic Source or Crystal Temperature Oscillation", Journal of Crystal Growth, 33,

125 (1976).

[9] Van Den Berg, L. and Whited, R. C. , "Recent Improvements in Hgl 2 Detectors", IEEE Transactions on Nuclear Science, NS-25 , No. 1, 395 (1978).

[10] Dabrowski, A. J., Huth, G. C. , Singh, M. , Economu, T. E. , and Turkevich, A. L. , "Characteristic X-ray Spectra of Sodium and Magnesium Measured at Room Temperature

Using Mercuric Iodide Detectors", Appl. Phys. Lett., 33 (2), 211 (1978).

[11] Huth, G. C. , Dabrowski, A. J., Singh, M. , Economu, T. E. , and Turkevich, A. L. , "A new X-ray Spectroscopy Concept - Room Temperature Mercuric Iodide with PeltierCooled Preampl if i cation" , Proceedings of the 27th Annual Conference on Applications

of X-ray Analysis held in Denver, August 1-4, 1978.

[12] Llacer, T. , Haller, E. E. , and Cordi , R. C. , "Entrance Windown in Germanium Low-Energy

X-ray Detectors", IEEE Transactions on Nuclear Science, NS-24 , No. 1, 53 (1977).

[13] Singh, M. , Dabrowski, A. J., Huth, G. C. , Iwanczyk, J. S. , and Clark, B. C. , "X-ray

Fluorescence Analysis at Toom Temeprature with an Energy dispersive Mercuric Iodide

Spectrometer", Proceedings of the 28th Annual Conference on Application of X-ray

Analysis held in Denver, July 30-August 3, 1979, p. 249, in Vol. 23 , Advances in

X-ray Analysis.

59

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

STANDARD TECHNIQUES FOR MEASURING WINDOW ABSORPTION AND OTHER EFFICIENCY LOSSES IN SEMICONDUCTOR X-RAY ENERGY SPECTROMETERS

F. J. Walter and R. Stone

EG&G Ortec

100 Midland Road

Oak Ridge, Tennessee 37830

and

D. H. Blackburn and P. A. Pella

National Bureau of Standards

Washington, D. C. 20234

Abstract

A conceptual technique for measuring the relative window and deadlayer thickness on semiconductor x-ray energy spectrometers by

monitoring the transmitted intensity of a series of lines from a standard

target has been described previously. This technique, which utilizes

secondary x-rays fluoresced from a standard glass by the Mn x-rays from

decay of Fe-55, was never incorporated in a measurements standard because

there was no suitably stable glass target whose composition was traceable

and certifiable. Recently, the National Bureau of Standards has

developed a glass, SRM-477, specifically for this application.

Results of tests on the sensitivity, repeatability, and reliability

of the technique will be presented. The various contributions to dead

and partially dead layers will be discussed. Other techniques used to verify the efficiency of semiconductor x-ray detectors which are also

proposed for the new IEEE/ANSI/IEC standard will also be described.

1. Introduction

The efficiency of any semiconductor detector depends on many physical parameters. For x-ray measurements, especially below 5 keV, window thickness is one of the most significant

factors affecting detector efficiency. The window transmission losses may include a metal

(e.g., beryllium) window in the vacuum enclosure, a possible dead-layer of semiconductor

material and any other material which might be present between the source of the x-rays and

the active region of the semiconductor detector. Because of the importance of window

absorption, a standard measuring technique is needed to characterize this parameter. The

purpose of this work is to describe measurements used in the development of such a standard

technique and to describe the standard as it appears in the latest working draft of

"Standard Test Procedures - Semiconductor X-Ray Energy Spectrometers" which was prepared by

the Nuclear Instruments and Detectors Committee of the IEEE Nuclear Science Group.

There are several characteristics which are desirable for a window attenuation standard. Window transmission loss characteristics should be specifiable and measurable without

detailed knowledge of the internal structure of the spectrometer. This allows ready comparison of spectrometers made to different designs or by different manufacturers. It should

also allow a user in his own laboratory to verify the window transmission losses of his

detector independently of specific data provided by the detector manufacturer. In addition,

it is desirable to be able to specify window attenuation at any of several energies

61
appropriate to the type of detector and its intended use. Finally, it is advantageous to have a standard technique which does not require complex and expensive equipment.

These desirable characteristics are realized by basing window attenuation measurements

on the relative transmitted intensities of x-rays fluoresced from a standard glass. The

National Bureau of Standards has provided several samples of a glass (designated SRM-477)

which was designed specifically for this purpose. We have made measurements to demonstrate

the applicability of this technique and to determine any needed constraints (such as limitations on geometry) which should be incorporated into a standard technique.

Methodology The construction of the apparatus used in making the evaluation measure- ments is shown in figure 1. The glass target, a disk 25 mm in diameter and 3 mm thick, is

held in a plastic lid which covers the top of the chamber. An annular 55 Fe source irradiates the glass with 5.9 and 6.4 keV Mn Ka and Kp x-rays. Fluorescent x-rays from the

glass then pass through the hole in the source and are incident on the detector. Mn x-rays

from the 55 Fe, which are coherently backscattered from the glass target, are also incident

on the detector. Several spacers are used to vary the geometry, and an adapter is provided

to fit the detector endcap. Additional thicknesses of beryllium may be inserted over the

detector to test the sensitivity of the technique. The chamber is evacuated to prevent

x-ray absorption by air.

Standard Glass, NBS SRM-477

Figure 1. Apparatus for window thickness

measurements with a glass fluorescent source.

The construction of the annular 55 Fe source is shown in figure 2. It is necessary that

this same geometry be used to make measurements under this standard. Source to glass

target distances are measured from the glass surface to the Be window of the source. The source strength used was 22 mCi. Although the standard is independent of source strength,

an activity much below 20 mCi may result in inconveniently long counting times needed to achieve suitable statistics.

62
NICKEL PLATED

BRASS CAPSULE

HI TEMP

EPOXY SEALS

.004 in. BERYLLIUM

WINDOW

ANSI N5.10-1968 CLASS C32212

Figure 2. New England nuclear 55 Fe annular source registered with the

NRC as Model NER-461A.

It is proposed that window attenuation characteristics be specified by a window

attenuation index. This index is the ratio of the area of a specific fluorescent peak to the area of the backscattered 5.9 keV Mn peak as measured by this method. It was decided

not to express this index in terms of a Ge, Si, or Be thickness eguivalent in order to keep

the standard simple and equally meaningful for all detector and window materials.

Measurement of this index does not involve a detailed knowledge of the system's

structure and may be made independently with simple equipment. It may be made at any of

several available energies and does not rely on elaborate calibration and correction.

Despite the apparent simplicity of this technique, limitations on the geometry used for the

standard measurement are necessary, largely as a result of self-absorption in the glass

target.

Experimental Results A characteristic spectrum obtained with this technique is

displayed in figure 3 on a relative intensity log scale. A Si C Li ) detector 6 mm in

diameter and 5 mm thick with a resolution of 165 eV FWHM at 5.9 keV was used for this

measurement. The glass target is composed of the oxides of Si, Ba, Ca, Li, Mg, Zn, and B. The energies of the peaks range from 1.0 to 6.4 keV. For this spectrum, the front contact

of the detector itself and 8 urn of Be are the only materials between the active region of

the detector and the glass. Especially prominent are the coherently backscattered Mn lines,

L lines of Ba, Ka, and Kp lines of Ca, and Ka lines of Si, and Mg. The window attenuation

index may be measured at the energy of any of the fluorescent lines.

The spectrum in figure 4 was taken under the same conditions as figure 3, except that

an extra layer of 130 um of Be was inserted over the detector. Comparing this with the

previous spectrum, the much greater attenuation of the lower energy lines is evident,

illustrating the basis of this technique. The counting statistics here are not the same

as for the previous spectrum, but the relative size of the lower energy peaks with respect

to the 5.9 keV peak is obviously much less.

Table 1 quantitatively illustrates the effect of increasing window thickness on the

window attenaution indices. This measurement and those which follow were made with a

Si (Li) detector 4 mm in diameter and 4 mm thick with a resolution of 160 eV FWHM at 5.9

keV. Even small changes in Be window thickness produce quite measurable effects in the

normalized intensities of the 1.25 keV Mg or 1.74 keV Si lines. The Mg line or the 1.0 keV

Zn line (not shown) would probably be useful in characterizing so-called windowless

systems. However, with increasing window thickness, the Mg and Zn lines soon lose their

usefulness because of the rapid decrease in count-rate.

63
(0 o> O «°

CO

OE

s- +jo

<u >

.- ^

O) in

5? s- +Jo

(0

5-(0o

S64
I

: . *

65
Table 1. The effect of window thickness on relative peak intensity normalized

to the 5.9 keV Mn Ka peak.

Be Window

Thickness

(Mm)

13

25

64

140

190

270

Mg Ka (1.25 keV)

(Iv u /I M ) X 100

Mg Mn y

1.80

.97

.18

Si Ka (1.74 keV)

(i

si /iHn

) X 100

28.8

22.2

12.1

2.35

.90

Ca Ka (3.69 keV)

(i Ca/i Mn

) X 100

51.5

51.3

49.4

44.4

41.1

35.1

In order to determine the sensitivity of this technique to the source-target-detector

geometry, relative intensity measurements were made at several different source to glass

target distances and several different source-to-glass target distances and several different target to detector distances. Maximizing count-rate is useful because runs of

several hours are needed to produce accurate results. This may obviously be accomplished

by moving the 55 Fe source or the detector closer to the glass target.

The effect of source to target distance on count- rate is shown by curve b in figures 5,

6, and 7. Count rate increases as this distance decreases until a maximum is reached at

7 mm. As the distance decreases beyond this point, the rate drops rapidly because of the source structure which begins to block the radiation path.

The effect of source to target distance on the intensity of the Ca, Si, and Mg Ka lines as normalized to the Mn Ka line is shown by curve a in figures 5, 6, and 7. This

value, of course, is the actual window attenuation index times 100. Since the index is

sensitive to the source to target distance, this distance needs to be specified in the

standard, and a value of 8 mm has been proposed. Nevertheless, the rate of change of the

index with this distance is not so great that small errors in source position would

appreciably affect the results.

The effect of changing the glass to detector distance is shown in Table 2. Within the

range of distances used, there is very little effect on the normalized intensities of the

various peaks. Of course some effect here is actually expected, but if the solid angle

which the detector makes with the glass source is kept smaller than it is for any case

shown in Table 2 (where the detector diameter is 4 mm), no significant effect on the

measured window attenuation index is evident. An appropriate limitation is a specification

that the glass to detector distance be at least 10 times the active diameter of the

detector.

66
0 5 10 15 20 25

Source-to-Glass Distance (mm)

Figure 5.

a) Intensity of Ca Ka Line as Normalized to the Mn Line

and Plotted as a Function of Source-to-Glass Distance.

b) Count Rate of Ca Ka Line as a Function of Source-to-Glass

Distance.

67
5.0 10.0 15.0

Source-to-Glass Distance (mm)

20.0 25.0

Figure 6.

a) Intensity of Si Ka Line as Normalized to the Mn Line

and Plotted as a Function of Source-to-Glass Distance.

b) Count Rate of Si Ka Line as a Function of Source-to-Glass

Distance.

68
(A I 1.0 c

4.0

2.0

10 15

Source-to-Glass Distance (mm)

20 25

Figure 7.

a) Intensity of Mg Ka Line as Normalized to the Mn Line

and Plotted as a Function of Source-to-Glass Distance.

b) Count Rate of Mg Ka Line as a Function of Source-to-Glass

Distance.

Table 2. The effect of glass to detector distance on relative peak intensity

normalized to the 5.9 keV Mn Ka peak.

Glass to detector

Window Distance

(mm)

60.2

52.8

46.3

39.8

33.1

Mg Ka (1.25 keV)

(I M /I M ) X 100

_ Mg Mn y

1.80

1.81

1.93

1.89

1.89

Si Ka (1.74 keV)

(I C v -/I M ) X 100

Si Mn y

28.8

27.9

28.8

28.5

29.1

Ca Ka (1.69 keV)

dCa

/i Mn ) X 100

51.5

52.0

52.9

52.8

51.9

Probable Error ±.20 ±.4 ±.5

69
Conclusions A "window attenuation index" based on the relative detected intensities

of x-rays from NBS SRM 477 fluoresced by an annular 55 Fe source is a sensitive, costeffective, reliable and simple method for characterizing the low energy efficiency roll-off

from window attenuation in semiconductor x-ray energy spectrometers. This technique does

not require detailed knowledge of the interior of the spectrometer and does not require

corrective calculations for source age (i.e., it is independent of the activity level and

age of the source), internal absorptions, etc. Because of the straightforward nature of

the measurement, there should be relatively little debate about how the results should be

interpreted or compared.

Although this technique is in principle sensitive to the geometry of the measurement,

the cylindrically symmetrical configuration and relatively small solid angles employed

result in a configuration that is satisfactorily insensitive to small errors in geometry.

The SRM 477 glass target is very uniform, very chemically stable in normal ambients, and

highly reproducible.

2. Summary

A standard technique for measuring window absorption and other efficiency losses in

semiconductor x-ray spectrometers is described. This technique has been proposed as an IEEE and IEC standard. Measurements are reported which have been used to establish limits

on the geometry of the technique and to establish the effectiveness of the method.

This study was made possible through the continued support of L. Costrell and

K. F. J. Heinrich of the National Bureau of Standards in the development of SRM-477.

The authors would like to thank T. Raudorf and M. Martini for their assistance with

this manuscript.

70
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981

)

USE OF WINDOWLESS DETECTORS FOR ENERGY DISPERSIVE LIGHT ELEMENT X-RAY ANALYSIS

J. C. Russ and A. 0. Sandborg

EDAX Laboratories

P. 0. Box 135

Prairie View, Illinois 60069

Abstract

Energy-dispersive Si(Li) detectors can be used to efficiently

detect soft x-rays in the 0.2-1.5 keV (C K - Al K) range by removing the

Be window normally used to isolate the detector from the microscope

vacuum. This requires some hardware changes to protect the detector from

contamination, electrons, and light. Normal high energy performance of

the same detector is nor compromised in most cases. Analysis of elements

in this range in concentrations from about 0.5 percent up is practical

for many samples; examples and exceptions are noted. The spectrum

appearance in this range is not easily handled by the techniques

routinely employed at higher energies, and in any case conventional

quantitative (ZAF) methods are nor readily used for these elements.

1. Introduction

The energy-dispersive method of x-ray analysis on the electron microscope has generally

been limited in three aspects, in comparison with the wavelength-dispersive electron microprobe: resolution, count rate, and ability to analyze the light elements. The resolution

of energy dispersive detectors has been gradually improved, but will never challenge the

diffracting crystal; fortunately computer stripping methods have become widely used and in

many cases give satisfactory separation of peaks not resolved by the detector itself.

Similarly, improvements have been made in count rate handling capability, particularly in

the prevention of spectral distortion at high rates; the continuing use of smaller beams

and thin sections has shifted much of the emphasis from the ability to handle high count

rates, to the use of very large detector solid angles to get acceptable results in inherently low count rate situations.

At the same time, work has progressed in the detection of the light elements: carbon,

nitrogen, oxygen, and fluorine. First, detection was demonstrated with a windowless detector [l] 1

. Then a commercially available detector was offered that could be used to detect

and even analyze these elements semiquantitatively [2,3]. Further progress has now refined

the detector design so that good routine operation is possible with minimum sacrifices in

other desirable features, while at the same time a body of experience has been accumulated

that can be used to assess the practical capabilities that such a detector provides.

2. Mechanical Design

After some evolution, the design of current detectors (see figures 1-4) is compact,

positive in operation, and interlocked for safety. The windows are carried on an outer

cylinder, concentric with the vacuum cylinder housing the detector and FET preamplifier.

The outer tube is rotated to one of three detent positions to align the standard beryllium

window, an aluminum-coated Formvar window, or an open hole, into place in front of the

detector. When the beryllium window is in place, it is sealed by an 0-ring around its

edge, by retracting the outer tube about 1 mm. The mechanism to rotate or seal the windows

Figures in brackets indicate the literature references at the end of this paper.

71
is external to the vacuum and positive in operation. When a positive seal is made, it is

locked to prevent accidental leaks. Only then does the remote indicator (normally mounted

next to the microscope vacuum system controls and/or electrically connected to them) show

that it is safe to vent the chamber.

Because the unit is compact in size, the mounting flange can be adapted to many SEM's

(minimum port size 5 cm). The mounting bracket is extensible (figure 2) so that the front

end can be moved close to the sample. The magnetic electron trap and cryo-trap against

detector contamination take up some space in front of the detector, placing the front

surface of the silicon 24 mm behind the outer end of the tube. For a 10 mm2 detector, this

gives a solid angle of 0.00824 steradians, for a sample 3 mm from the end cap. The internal

hardware is designed to assure excellent collimation and eliminates stray radiation from

the pole pieces, the specimen stage, etc.

Figure 1. Rotating outer barrel holds windows, and is

moveable from outside the vacuum chamber.

Figure 2. Extensible mounting provides close specimen-to-detector

positioning. Window controls are readily accessible.

72
Figure 3. Organic window mounted on an aluminum ring, held

in place with an easily removed c-clip.

Figure 4. Complete detecting unit, with

10 liter Dewar and unitized

ampl if ier.

The aluminized Formvar window is easily replaced by the user, while the cryostat

remains sealed by the standard beryllium window. Depending on the particular SEM design

and the access to the chamber, in many cases window replacement is possible without removing

the detecting unit from the microscope. The window is mounted on an aluminum ring, held in

place by a "C" clip (figure 3). The user can replace the window either to use a different

organic polymer (to minimize absorption for a particular light element, or minimize permeability when samples outgas water heavily) or to use different coating thicknesses (as

dictated by the cathodoluminescence of samples). In any case these organic windows age in vacuum and should be replaceable by the user whenever needed. The organic window will not

withstand a one-atmosphere pressure differential.

73
3. Low Energy Efficiency

Removing the beryllium window from the front of the detector eliminates a major

absorber of low energy x-rays, but not all the absorption. Figure 5 shows the efficiency of

the detector as a function of energy, for the standard 7 urn beryllium window and without a window. The gold contact of the surface, and the inactive silicon near the surface, still

produce a drop at low energies. An additional loss in detection efficiency at very low

energy that limits the sensitivity for light elements is due to absorption of the x-rays

within the sample itself. Accordingly, this absorption should be minimized by the use of

as high an x-ray takeoff angle as practical. By comparing spectra measured with the window

removed, with the aluminized Formvar, and the beryllium window in place, we can directly

compare the improvement in light element sensitivity. Figures 6 and 7 show spectra measured

on cryolite (predominantly Na3 AlF6 , with a small amount of Si02 ). The increase in sensitivity for the lower-energy peaks is apparent; it can be more readily compared in terms of

the estimated minimum detection limit. Neglecting the effects of matrix absorption or peak

interferences, we can express this as,

CMDL

= 3 ^ C STD

/P

where P and B are the net peak intensity and background intensity, C^jp is the concentration

of the element in the sample and the 3 indicates that a peak three times greater than the

standard deviation of the background is required for detection. These values are not always

good guides to actual detection limits in a given sample, but show the relative variation

due to the window removal. Using data measured on the cryolite sample for 200 seconds at

6 kV, a 30 degree surface tilt, and 57 degree takeoff angle, the following calculated

results were obtained. The beam current gave 700 c/s in the total spectrum with the beryllium window in place; this rose to 2600 c/s with the window removed. (Figure 8.)

Figure 5. Low energy efficiency (100% full

scale) for standard 7 urn beryllium

window and window! ess detecting

units.

113SEC

5 0 '0 H S

" ec/s

18EV/CH

Figure 6. Cryolite with standard beryllium

window (solid) and no window

(dots).

74
0 266SEC 28183 I NT

VS: 580 HS: 10EV/CH

ED flX

Figure 7. Cryolite with no window (dots)

and aluminized Formvar window

(solid).

Figure 8. Spectra used for M.D.L. calculation (see text).

Al

Na

F

Be Window

0.10%

0.18%

2.50%

Formvar

0.08%

0. 13%

0. 18%

No Window

0.08%

0.11%

0.15%

For the 1.5 keV Al K x-rays, the removal of the window does little to increase sensitivity, while for the 0.67 keV F K x-rays, more than an order of magnitude improvement is

observed.

For the light elements 0, N, and C, detection limits can be calculated in the same way

from samples such as minerals, metals, carbides, or other stoichiometric compounds.

Generally these numbers are not very accurate when extrapolated to low concentrations, but

serve as a rough guide to practical expectation and illustrate the effect of the Formvar

window.

Formvar No Window

0.2%

0.5%

0.6%

0.2%

0.5%

0.5%

4. Problems of Peak Identification

The energy-dispersive x-ray peaks at these low energies present a few problems for

routine identification that are not normally encountered at higher energies. Line identification is complicated by the presence of interfering peaks from other elements. Figure 9 shows a spectrum from an alloy steel, with the window in place and open. The large Cr L and Fe L peaks are evident (.57 keV and .71 keV, respectively) and the cursor marks the

smaller Ni L peak (.85 keV). It is likely that this particular sample is partly oxidized

on its surface, but no oxygen K peak (.53 keV) can be seen as it is completely obscured by

75
the other peaks. The shape of these peaks is too complex for practical stripping, both

because the detector response for these low energies is not perfectly Gaussian due to incomplete charge collection (the low energy incident x-rays are absorbed very near the

detector surface, where imperfect lithium compensation of the silicon produces an inactive,

or partially inactive layer of material), and due to the presence of many overlapped

individual lines.

• 47SEC $C/$

VS: 580 HS: HEV/CH

Figure 9. Stainless steel (see text) with

beryllium window (solid) and without

window (dots).

An additional complication in the low energy spectrum is the presence of escape peaks

from other, higher energy x-rays. Escape peaks result from the removal of energy from the

detector in the form of a silicon K x-ray. The probability of exciting the silicon x-ray

near the surface, where it can escape, is greatest for incident x-rays with energy just

greater than the silicon K edge [4]. This produces an escape peak (energy equal to the

incident energy minus the 1.74 keV silicon K energy) in the .2 - 1.5 keV range we are interested in. For example, figure 10 shows the escape peak from the analysis of pure

sulfur (2.32 keV - 1.74 keV = 0.58 keV). The peak is easily mistaken for oxygen K, suggesting that some of the sulfur might be an oxide, but this spectrum was measured with

the window closed! With the window open (figure 11) the background shape changes, but no additional peaks are seen. The small silicon peak (marker) is due to fluorescence of

the inactive surface layer of silicon on the detector [5].

Figure 10. Escape peak from sulfur K x-rays Figure 11. Sulfur "escape peak" and fluo-

(Be window in place). resced silicon peak - window

open.

76
Another artifact peak that can cause confusion can arise from pulse pile-up. If two

x-rays enter the detector close together in time, their pulses add together in the amplifier

producing a single pulse with a height that is the sum of the individual pulses. This

produces a sum, or pileup peak in the spectrum. At higher energies these pileup peaks are suppressed by a special circuit in the amplifier that senses pulses and rejects ones that occur so close in time that they cannot be individually measured. The rejection circuit

uses a very "fast" (short time constant) amplifier, which therefore has poor energy

discrimination and misses the pulses from these ultra-low energy x-rays. Consequently, at

all but the lowest count rates, pileup will occur at energies below 1 keV.

This is illustrated in figures 12 and 13 for the carbon peak. The two spectra shown in

comparison were stored at total count rates of 200 and 1500 cps, to the same height of the

K peak. The pileup peak (0.5 keV) is evident in both spectra, but is 2.5 times larger at the higher count rate. Figure 14 shows the same effect for the copper L lines (0.93 keV)

at count rates of 200 and 1500 cps. The table below summarizes data for carbon (0.28 keV),

oxygen (0.53 keV), and copper (0.93 keV) as a function of count rate, and emphasizes the

need to keep total count rates quite low when the best low energy results are required.

C K

0 K

Cu L

Ratio-sum peak to main peak

200 cps 600 cps 1500 cps

1.4% 1.9% 3.4%

1.0% 1.4% 2.6%

0.6% 1.0% 1.7%

0

VS : 277SEC 5070SINT

500 HS: 10EV/CH

Hi - LL

•

|01

EDftX

Figure 12. Carbon peak from pure graphite.

Spectra recorded at different

rates to same peak height.

Figure 13. Size of carbon sum depends on count rate (see text).

77
Figure 14. Pileup peak for copper L lines

at 200 (solid) and 1500 (dots)

cps. (same height main peak).

5. Sample Contamination

Another problem that may be encountered in trying to analyze light elements i_n samples

is their presence as surface contaminants. These can be present before the sample is put

into the microscope, or build up during analyses. For example, most metals have a thin

layer of oxide that will give a significant peak. Some, like magnesium, oxidize so rapidly

it is nearly impossible to prevent it from forming. Figure 15 shows the spectrum from

freshly polished pure magnesium - the oxygen peak is evident.

Figure 15. Oxygen peak from surface oxide

on pure magnesium.

The polishing of samples itself can cause contamination as well. Figures 16 and 17 show pure copper. The small Si and C peaks are from the silicon carbide polishing compound

used to remove oxide or other surface contaminants. The sample was cleaned before analysis

in an ultrasonic cleaner, but some Si C particles remained. Similar problems with oil,

fingerprints, bits of adhesive, etc., become far more evident when light element analysis

is attempted.

78
Figure 16. Trace oxygen is very small on Figure 17. Silicon and carbon peaks on freshly polished copper. freshly polished pure copper,

due to polishing compound.

Even assuming the sample makes it into the microscope "clean", contamination can build

up with time under the electron beam. This is partly a function of the microscope vacuum

system - the degree of trapping to prevent oil backstreami ng from pump, etc. But a more significant effect is usually sample-related - the outgassing or volatilization of the

specimen produces contaminant molecules near the surface that are readily deposited on it

and largely independent of the SEM vacuum system. In its simplest case, this type of

contamination is usually carbon rich. Figure 18 shows two spectra measured on a pure

aluminum stub, one when it was first introduced and the second after the beam had been on

it, scanning in a small raster, for 15 minutes. The buildup in C is apparent. Other types

of complex samples, or the adhesives used to attach them, can produce exotic combinations

of contaminant elements.

Figure 18. Carbon contamination peak on pure

aluminum after exposure (compared to original spectrum).

6. Removing Background at Low Energies

A critical step in any quantitative analysis or trace element analysis is accurately

subtracting the spectrum background. In conventional spectra the high energy region, above

about 3 keV, is generally simple, while the region from 1-3 keV where the background varies more rapidly presents greater problems. Since the use of a windowless detector "straightens

out" this region so that it continues to rise, this might be expected to simplify things.

Two questions need to be answered, 1) is there any change in the use of background

79
subtraction methods in the 1-3 keV range? and 2) do these methods also work at energies

below 1 keV (where the background is partly from Bremsstrahlung generated in the sample and

partly electronic noise)?

Figures 19-20 show spectra measured on a graphite stub containing a small amount of

sulfur (and traces of Si, P, and V). With the window closed, standard spectra were obtained

with the characteristic sharp drop in background. Good results in fitting background,

detecting the trace peaks, and getting the net peak areas, are provided by calculation

methods (figure 19) [6,7].

Figure 19. Calculated background, normal Figure 20. Calculated background, no window. window.

When the detector window is removed the shape of the background changes, continuing to rise to below 1 keV and then dropping. The calculation method (figure 20) ignores the

electronic noise under the carbon peak and errs on the continuum in the 0.5-1 keV

region. This is probably due to inadequacies in the present models for continuum generation, particularly at these very low energies. (The models assume the absorption of

continuum x-rays is the same as for characteristic x-rays, or in other words that their

depth of generation is the same. For very low energies this is clearly not the case since

the continuum x-rays are generated deeper in the sample, toward the end of the electron

path and are therefore subject to more absorption.)

Since the calculation method is widely used for handling general quantitative calculations, it is important to see how well it works for light element peaks. Figure 21 shows

spectra measured on the cryolite sample used previously for illustration. Figures 22 and

23 show the calculated backgrounds, illustrating the large absorption edges that become

increasingly significant at low energies. Figures 24 and 25 show that the calculated

backgrounds fit quite well to the spectra and give far more accurate net peaks (figures

26 and 27) than any interpolation or fitting method can achieve. Examples of quantitative

results for oxygen using this approach will be shown below. For carbon, a separate fitting

and subtraction of the zero-energy noise peak should be made first, or other more empirical

background fitting methods employed.

80
488SEC143887INT

188 HS: 18EV/CH

EDftX

81
8 0SEC 1 32t?mT t 8SECifi3?l3i WT

VS : 5008 HS : 10EV/CH VS 5888 HS : 10EV/CH

EDftX - EDAX

Figure 26. Net peaks for standard Figure 27. Net peaks for windowless

window. detector.

7. Background from High Energy Electrons

In a conventional system, the beryllium window serves to stop electrons and light

photons from reaching the detector. A standard 7 urn window can stop electrons up to about

25-30 keV. Higher-energy electrons still can penetrate to the detector, and cause high

background in the spectrum at high energies [8]. This is usually prevented in the TEM by

the magnetic field around the sample that prevents the scattered electrons from passing in

a straight line through the collimator to the detector. In the SEM the sample is in fieldfree space and no such protection is available, but high voltages are rarely used with

bul k material

.

When the window is removed, we substitute a magnetic field as an electron trap, to deflect scattered electrons so they cannot reach the detector. This trap is also useful at high voltages even with the normal Be window in place, since some high energy electrons

may not be completely stopped by 7-8 urn of Be. Figure 28 shows the effect of the trap in

reducing electron background.

0 800SEC1097111NT

VS.2500 HS : 188EV/CH

Figure 28a. Spectrum from solder (Pb-Sn

alloy) at 50 kV. Dot spectrum

shows electrons penetrating

window, solid spectrum shows

effect of magnetic electron

trap.

Figure 28b. Expansion of 30a around tin

Ka line.

82
8. Electron Spectroscopy

As a side note, it is interesting to use Si C Li ) detectors as electron energy spectrometers as well as for normal x-ray work. By removing the magnet, the spectrum of x-rays

plus electrons is recorded. When the trap is in place, the x-rays only are detected. The

difference gives the energy distribution of the backscattered electrons. Figure 29 shows

superimposed, the spectra (for a copper target) of x-rays plus electrons (upper curve) and

x-rays only (lower curve), the latter obtained with the use of a magnetic electron trap in

front of the detector. Subtraction then gives backscattered electron spectra as shown in

figures 30a-30f. In each spectrum the caption shows the atomic numbers of the elements. In all cases the higher curve corresponds to the higher atomic number. The shape of the curves

is in good qualitative agreement with the various published curves of predicted backscattered electron intensity.

Figure 30. Backscattered spectra from: a) uranium, magnesium; b) gold, copper;

c) copper, aluminum; d) molybdenum, titanium; e) gold, tin; and

f) aluminum, magnesium.

83
From the figures, it is evident that there is the expected trend of increasing total

backscattering (total area under the curve) with increasing atomic number. This can be

discerned even for a difference of one Z number, as shown for magnesium and aluminum in

figure 30f. The peak energy of the curve also rises with Z, since higher Z materials can turn electrons back through the surface with fewer scattering events, a shorter total path

and less energy loss. Figure 31 summarizes the data for these measurements. The peak

energy was determined from a polynomial fit to each spectrum. Both curves have the general

trend expected from theory; that is, they rise with increasing atomic number.

tl-ll..l.llllU-|ll»l

Figure 31. Total B.S.E. intensity (bottom)

and peak energy (top).

These curves were measured with 24.8 keV electrons incident on a sample inclined at 20

degrees, and the detector at an angle 20 degrees above the horizontal (a total takeoff angle

of 40 degrees). To study the effect of orientation, a copper target inclined at an angle of

45 degrees was rotated so that the detector was in the direction of tilt and 65 degrees from

that direction. The backscattered electron spectra (figure 32a) show that the total backscattering is less at the 65 degree angle but the peak energy is higher. A, similar effect

is seen with perpendicular electron incidence on the copper target and changing takeoff

angle (by raising and lowering the SEM stage). At a takeoff angle of 45 degrees there is more total backscattering than at 10 degrees, but at 10 degrees the peak energy is higher

(figure 32b).

Figure 32. B.S.E. spectra from copper geometries

described in text. (a) lateral angles

(b) vertical angles.

84
Studies with other geometries, targets, and electron energies will be useful to supplement existing data on backscattered electron energies, and to confirm Monte Carlo

models for electron scattering [9-11], as well as to develop optimized geometrical arrangements for backscattered electron detectors.

9. Cathodoluminescence

The detector must also be protected from light photons. Figure 33 shows what happens

when light can reach the detector - the electronic noise in the preamplifier causes resolution to get worse, and peaks to shift. The sample is BeO, and the light is emitted under

the electron beam. To keep these cathodoluminescence photons from reaching the detector,

the aluminized Formvar window is moved into place. This window can also be kept in place

to reduce the rate of contamination molecules entering the detecting unit when samples are being examined that may outgas.

Figure 33. Spectrum from BeO with window open

(dots) and with aluminized Formvar

window (bars).

10. Analytical Results

Ultra- low energy peaks can be used in the same way as other peaks for x-ray mapping on the SEM. Figure 34 shows an example, maps for carbon and oxygen on an area of grey cast

iron ingot mold. The oxygen around the graphite flake resulted from internal oxidation

during service. The requirements for adequate peak/background ratios and numbers of dots

must still be met for acceptable pictures.

85
Figure 34a. Secondary electron

image.

Figure 34b. Carbon Ka x-ray

image.

Figure 34c. Oxygen x-ray map.

86
Figure 34d. Spectrum taken from scanned

area on page 1

.

A practical application for semiquantitative oxygen analysis is the measurement of

oxide coating thicknesses (for instance, oxygen on semiconductors). The use of the

intensity ratio of oxygen to silicon normalizes minor variations in surface finish or orientation, and gives a sensitive calibration curve as shown in figure 35. Ten percent

accuracy is readily achieved using this simple approach [3].

Figure 35. Calibration curve for oxide layer

thickness.

3 .4 ,S 6 7 8 9 1.0 t I I 2 I J 1.4 I 9 IK 17 SiO; LAVfcrt THICKNESS I .)

11. Examples of Application

Another example of comparative analysis is the oxygen content of man-made Courtelle

fibers [12,13]. Areas 2.5 urn square in the shell and core regions of 10 sections of

fibers gave spectra such as shown in figure 36. The background was subtracted by fitting

a Gaussian noise peak plus a constant. The ratio of net peak intensity to background under

the peak was used as a measure of concentration. This ratio was 2.66 ± 0.52 in the shell

regions, and 1.88 ± 0.25 in the core of the fibers. A similar measurement on a fiber with

a uniform structure, independently analyzed as having an 8.4 percent oxygen content, gave

a P/B ratio of 2.50. Using this as a standard gives a calculated concentration of

6.3 ± 0.8 percent in the core and 8.9 ± 1.7 percent in the shell.

87
Figure 36. Comparison to show difference in oxygen

between core (bars) and shell (dots) of

Courtelle fibers (series A).

Another example of oxide analysis, showing ability to accurately discriminate

different oxide forms, was carried out with oxides Fe2 03 and Fe3 04 (see table below). By

analyzing the iron L and oxygen K lines, the two oxides can be distinguished (figure 37).

The ratio of the two peaks, shown in the table, agrees closely with the calculated values

using a full ZAF calculation [14]. The net intensities were obtained using the calculated

background method (figures 38 and 39).

6 288SEC ec/s

VS:25ee HS: 18EV/CH

Figure 38. Spectrum and calculated back- Figure 39. Spectrum and calculated background for Fe2 03 . ground for Fe304 .

88
Oxide Element

Iron Oxide Analysis

Cone. Total

Intensity

B.G. (calc.) Net

Fe 2 03 Fe fiQ Q4 OJtf 83?4 1984 6340

0 30.06 28581 2858 25723

Fe304 Fe 72.36 8073 2136 5937

0 C / . OH 29427 2888 26539

Intensity Ratio Fe2 03 /Fe 3 04

Element Measured Theoretical

Fe 1.068 1. 063

0 0.969 0. 970

A fully quantitative oxide analysis was performed on a complex oxide containing Si02 and spinel (MgO • A1 2 03 ). The spectrum (figures 40 and 41) shows the peaks to be clearly

resolved. Again, the background was calculated (figure 42) as part of the iterative

quantitative analysis routine, simultaneously with the ZAF factors. The intensities are listed below.

Intensities on Oxide Specimen

Intensity (400 sec. ) Element Actual Cone. Total B.G. (calc.) Net

Mg 15.07 19723 2782 16941

Al 33.43 37083 2395 34688

Si 10.16 10063 2024 8039

0 41.34 48044 2349 45695

89
e 480SEC

VS:5808 HS: 16EV/CH

Figure 40. Complex oxide (silica and spinel) Figure 41. Oxygen, magnesium, aluminum,

m

and

used for quantitative analysis. silicon peaks.

Figure 42. Calculated background for quantitative analysis.

SEC 26?1

HS: 18E

As a standard for the analysis, pure Si02 was used. In this case the background was calculated directly since the concentrations were known. Then the ZAF factors were computed

(again using the known concentrations) and used to calculate the pure intensities that would

be obtained from theoretical pure element standards of silicon and oxygen.

Intensities on Si

0

2 Standards

Intensity (400 sec.

)

Element Cone. Total B.G. (calc.) Net Calculated Pure Int.

Si 46.72 44934 2078 42856 97560

0 53.28 55057 2417 52640 73453

90
Then the theoretical pure intensities (P) of aluminum and magnesium were calculated using

the expression

Pa (U-l) 5/3 R(W L) T F/A

u = overvoltage

R = backscatter factor

(WL) = fluorescence yield and line intensity

T = spectrometer efficiency

F = self absorption correction

A = atomic weight

Using simple algorithms for these terms (which are in fact also needed in the calculation of the ZAF factors), and for the analysis conditions used (6 kV accelerating voltage,

sample surface inclination 30 degrees, x-ray takeoff angle 40 degrees) we obtain the

ratios,

PA1 /P

si =1.1263

PMg

/P

s

. = 1.3206

which gives us theoretical pure intensities for aluminum and magnesium (109876 and 128828

respectively).

Dividing these pure intensities into the net intensities gives K-ratios that are used

in a conventional ZAF calculation, giving the results shown below. The estimated error

shown for each element is based on the total statistical error introduced by background

subtraction on both unknown and standard. Normalizing the total concentration to 100

percent further improves the results. Each of these various steps, shown here separately

for clarity, are in fact carried out automatically by the program.

Quantitative Results

Concentration- Percent

Element K-•Ratio Calculated Normal ized Actual

Mg 0. 1315 13.83 ±.24 14.06 ± .25 15.07

Al 0. 3157 35.25 ± .50 35.83 ± .51 33.43

Si 0. 0824 9.24 ± .17 9.57 ± .17 10.15

0 0. 6221 39.89 ± .32 40.54 ± .33 41.34

Other examples of quantitative oxygen analysis with oxide standards have also been

published. In all these cases of ZAF calculations for light elements, special care is

needed to assure that the program used has models, particularly for energies, which extrapolate well to low energies. In the ultimate case, it may be necessary to directly enter

from tables the best available parameters for mass absorption coefficient, etc., for the

light elements.

A good example of this is the analysis of carbon in Fe 3 C, measured in a spheroidized

steel. The spectrum, shown in figure 43 (dots) had well defined Fe L and C K peaks, but

there was also a small oxygen peak, due to oxidation of the polished steel by the metal lographic etchant. This peak was removed by stripping away a manually generated oxygen peak

(figure 44 bars).

91
Figure 43. Removal of oxygen peak by

stripping.

Figure 44. Subtraction of background

from spectrum.

To obtain good net intensity values, a peak-generating routine was used to construct

peaks for the iron and carbon (figure 45) which matched the measured peaks as closely as possible. It should be noted that these peaks may not lie at exactly the precise energies

indicated by charts nor are they symmetrically shaped, and the generated peaks should be

adjusted to fit the actual data, as judged by the remains after subtraction (figure 46).

The net intensities are then taken as the areas of the generated peaks. The values, with

statistical (one sigma) uncertainties, are:

Fe: 10272 ± 163

C: 1307 ± 75

2 8 0 S E C 1 8 8 9 H S

Figure 45. Constructed Fe and C peaks. Figure 46. Background left after peak

removal

.

92
Standards consisting of pure electrolytic iron and a small diamond were analyzed under

identical conditions (in fact they were mounted in the same holder and the stage controls

used to bring them into view and into focus with no readjustment of the electron gun or

lens settings). The standard spectra (figure 47) were used to obtain net intensities in

exactly the same way, by generating peaks for iron (figure 48) and carbon (figure 49) that

matched the measured peak, and then taking the total area of the generated peak. Again

with statistical uncertainty, these standard intensities are:

Fe: 10454 ± 159

C: 13621 ± 184

Figure 47. Carbon and iron standard Figure 48. Fe standard spectrum with

spectra. net pure Fe peak.

Figure 49. C standard spectrum with net pure C peak.

This gives K ratios, uncertainties, and relative errors for iron and carbon:

Fe: 0.98259 ± 0.02160 (2/2%)

C: 0.0959 ± 0.00566 (5.9%)

93
A modified ZAF program with tables of parameters for ultra light elements gives:

RUN

KV=? 6 ELEM(Z)

? FE/ 26

? C / 6

? 0

S=? 16 T=? 20 LINE

? L/ 2

? K/ 1

K- RAT 10

? 0.99259

? 0.09574

ELEM(Z)

26

Z-CORR.

0.976144

1.27816

A-CORR.

1.07194

1.15084

F-CORR

1.00592

6 1 . 00058

ELEM(Z)

26

6*READY

%

93.0295

6.50372

These answers, with the uncertainty due to the statistical errors above, are 93.0 ± 2.04 percent Fe, 6.5 ± 0.38 percent C which are quite close to the actual composition

of Fe2 C (93.3% Fe, 6.7% C). This is partly due to choosing analyzing conditions which keep

the Z, A, and F factors relatively near to 1.0.

The second-generation design of this removable-window detector assures reliable routine

operation, and minimizes operating limitations by providing protection against electron,

light photons and contamination molecules entering the detecting unit. The extensible

design allows practical optimization of detector position.

The inherent limitations of ultra-low energy x-ray analysis must be understood by the

user. They include problems of peak identification and sample preparation, but can be

dealt with by intelligent operation and data interpretation. Count rate in the low energy

region must be limited to avoid pile-up effects.

Practical analytical results, both qualitative and quantitative, can be obtained.

Calculated backgrounds, possibly supplemented with an "instrumental noise" background, and

either synthesized or measured and stored peak shapes can be used to deconvolute a spectrum

provided severe overlaps are avoided. ZAF methods using appropriate standards and

parameter values can yield good results.

12. Conclusion

94
References

[1] Jaklevic, J. M. and Goulding, F. S. , "Detection of Low Energy X-rays with Si(Li)

Detectors", Trans. Nucl. Soc. , 187-191 (Feb. 1971).

[2] Barbi, N. C. , Sandborg, A. 0., Russ, J. C. , and Soderquist, C. C. , "Light Element

Analysis on the Scanning Electron Microscope using a Windowless Energy-Dispersive

X-ray Spectrometer", Proc. 7th SEM Symposium, 151-158 (1974).

[3] Barbi, N. C. and Russ, J. C. , "Analysis of Oxygen on a SEM using a Windowless Energydispersive X-ray Spectrometer", Proc. 8th SEM Symposium, 85-91 (1975).

[4] Reed, S. J. B. and Ware, N. G. , "Escape Peaks and Internal Fluorescence in X-ray

Spectra Recorded with Lithium-drifted Silicon Detectors", J. Phys. E: Sci. Inst., 5,

582-584 (1972).

[5] Russ, J. C. , "Erroneous Peaks in Energy Dispersive X-ray Spectra", 24th Annual

EPASA (1972).

[6] Ware, N. G. and Reed, S. J. B. , "Background Corrections for Quantitative Electron

Microprobe Analysis Using a Lithium-Drifted Silicon X-ray Detector", J. Phys. E: Sci. Inst., 6, 286-288 (1973).

[7] Fiori, C. E. , Myklebust, R. L. , and Heinrich, K. F. J., "Prediction of Continuum

Intensity in Energy-Dispersive X-ray Microanalysis", Anal. Chem. , 48, 172-176 (1976).

[8] Russ, J. C. , "X-ray Microanalysis at High beam Voltages", Proc. 9th Annual SEM

Symposium (ed. by 0. Johari), 134-140 (1976).

[9] Darlington, E. M. , Ph.D. Thesis, Univ. Cambridge, 1971, cited in [10].

[10] Love, G. , Cox, M. G. C. , and Scott, V. D. , J. Phys. D. , Appl. Phys. , 10, 7-23 (1977).

[11] Heinrich, K. F. J., Newbury, D. E. , and Yakowitz, H. (ed.), Use of Monte-Carlo

Calculations in Electron Probe Analysis and Scanning Electron Microscopy, NBS Special

Publication 460, Dec. 1976.

[12] Love, G. , Cox, M. , and Scott, V., "Electron Probe Microanalysis of Oxygen in Heattreated Orion Fibres", Mat. Res. Bull., 10, 815-818 (1975).

[13] Specimens courtesy of Dr. G. Love, University of Bath, School of Materials Science,

Bath, U.S.

[14] Yakowitz, H. , Myklebust, R. L. , and Heinrich, K. F. J., "FRAME-an on-line Correction

Procedure for Quantitative Electron Probe Microanalysis", NBS Technical Note 796,

1973.

95

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

PROPERTIES AND APPLICATIONS OF WINDOWLESS Si (Li) DETECTORS

R. G. Musket

Kevex Corporation

Foster City, California 94404

Abstract

Windowless and ultra- thin-windowed Si (Li) detectors are useful for

the energy-dispersive spectrometry of x-rays with energies Ex above

approximately 200 eV. Properties of such detectors are discussed with

regard to the limits they impose on performance. In particular,

considerations of detection efficiency, output linearity, energy

resolution, peak shapes, vacuum conditions, and general requirements are presented. Recent developments have greatly increased the actual and

potential applications of these detectors. Examples of applications to bulk, thin film, and surface-layer analyses using ion and electron

excitation will serve to demonstrate the advantages and limitations of

these detectors in contrast to beryllium-windowed detectors.

1. Introduction

The development and applications of windowless (WL) Si (Li) detectors have been progressing steadily since the first documented results in the early 1970' s [1.2] 1

. Such detectors

are useful for the spectrometry of x-rays with energies greater than about 200 eV. In contrast, Si(Li) detectors with beryllium windows have a useful x-ray analysis range above

1 keV. This low-energy limit applies, even for the thinnest commercially available window

of approximately 8 urn, because of strong absorption in the beryllium of the x-rays with

energies below 1 keV. Recently ultra-thin-windowed (UTW) Si (Li) detectors have been

employed successfully in applications where the sample chamber vacuum has not been adequate

for WL operation. Since the UTW has reasonable x-ray transmission and an integral light

barrier, a UTW Si (Li) detector can often provide performance superior to a pure WL detector.

The main purpose of this document is to discuss several properties of WL and UTW Si(Li)

detectors with regard to the limits they impose on performance and applications. Considerations of detection efficiency, output linearity, energy resolution, peak shapes, vacuum

conditions, and general requirements permit definition of the limits. This will be accom- plished by (a) describing the above properties and procedures for their determination and

(b) presenting results of applications to bulk, thin film and surface-layer analyses using

ion and electron excitation. Although much of the material discussed here has been

previously published, it is reviewed and combined with some new results to yield a more complete understanding of WL and UTW detectors and their applications.

2. Properties

2.1 Detection efficiency

Detection efficiency will be defined as the probability of detecting an x-ray incident

upon the outer surface of the Si (Li) crystal. A Si (Li) crystal consists of a 2 - 3 mm thick

sensitive region with an x-ray entrance surface consisting of layers due to contamination,

the gold contact for application of bias (i.e., an electric field) and silicon characterized

by incomplete charge collection (i.e., the dead-layer). Assuming the detection efficiency

Figures in brackets indicate the literature references at the end of this paper.

97
is 1.0 for all x-rays that penetrate these three layers and interact in the sensitive

region, the efficiency n,(E) of a WL detector for x-rays of energy E can be expressed as q(E) = exp(-M

c (E)m - P Au (E)£ - Psi (E)t)

^ (])

where p^CE), p

A(j (E), and P$.j(E) tne mass absorption coefficients, in cm2 /g, for x-rays

of energy E and m, SL, and t are the thicknesses, in g/cm2

, of the contamination, gold, and

dead layers.

Although the exact nature of the layers is complex and probably indeterminate, an assumption of discrete and uniform layers with appropriate "effective" thicknesses permits

assessment of the efficiency for various layer combinations and conversely, the layer thick- nesses from measure efficiencies.

For a WL detector with a negligible amount of contamination the efficiency can be

expressed as [3,4]

ri(E) = exp(-MAu (E)£ - P

si (E)t)

(2)

Thus, if the efficiency is experimentally determined for two x-ray energies E and E. , 9 D

SL and t can be evaluated from simultaneous solution of the following two equations:

and

n(E

a

) = exP (-uAu

(E

a

)£ - M

s1 (Ea)t)

n(E

b

) = exp(-p

Au

(E

b )£ - Ms -(Eb )t);

(3a)

(3b)

The x-ray energies E and E. best suited for this purpose are those for which n(E.) and

n(E

b

) are strongly dependent on SL and t through large values of p Au and [i^ , as tabulated in

appropriate tables [5].

This approach has been applied to C(K) and 0(K) x-rays excited by protons to yield

effective layers SL = 42 ug/cm2 and t = 21 pg/cm2

, which translate to thicknesses of 220 A Au and 0.09 urn Si for bulk densities of 19.3 and 2.33 g/cm3

, respectively [4]. The experimentally measured efficiencies for C(K) and 0(K) were 0.26 and 0.52. Figure 1 is a plot of

Equation (2) using the above gold layer and silicon dead-layer thicknesses and tabulated

values for M

A(J (E) and (E). From this energy dependence of the detector efficiency q» the detector efficiency for B(K) x-rays should be 0.14, which is quite significant. In fact, the detection of proton-induced B(K) x-rays has been demonstrated (figure 2) [4]. The

minimum at approximately 130 eV in the q(E) function of figure 1 is a consequence of the

maxima in the energy dependencies of the p A and p^..

i.o

0.8 -

0.6

0.4

0.1

1 1 1 1 1 1 1 1 1

Au M. , EDGES / Si K EDGE

1 1 1—i i—i i i i

Si(Li) DETECTOR

2

10 mm , 2 mm thick

220 A Au layer

0.09 tim Si dead layer

-

'

Figure 1. Energy dependence of windowless

Si (Li) efficiency for indicated gold

layer and silicon dead layer thick- nesses using tabulated mass absorption coefficients [4].

0.2 0.5 1.0 2.0

X-RAY ENERGY IkeV)

5.0

98
16,383 B(KJ

1\ (10 mm Si (Li ) detector)

Figure 2. Proton- induced x-ray spectrum from

polished boron [4]. Proton energy

was 350 keV with an analysis time

of 200 seconds at approximately 50%

dead time.

0.8 1.2

X-RAY ENERGY (keV)

Since the effective silicon dead-layer is a function of the applied bias [1], the

relative efficiency for any x-ray energy will depend on the bias. This dependence for C(K)

x-rays in two different 3 mm thick Si(Li) detectors is presented in figure 3 [6]. The

absolute efficiencies at 1000 V bias were 0.010 and 0.0079 for the 14 mm2 and 30 mm2 detectors, respectively [6].

Figure 3. Relative Si (Li) detector efficiencies for C(K) x-rays [6].

DETECTOR BIAS V b(volts)

Although exact values of n.(E) are required in Equation (3) to determine £ and t for a given detector, only the efficiency ratio n, m

(E)/ri 0 (E) for contaminated and clean crystals is

required to evaluate m, the effective thickness of the contamination layer. This may be

expressed as

nm (E)/n

0 (E) = exp(-u(E)m)

(

Unfortunately even less is known about the exact nature of the contamination layer than was

known about the gold contact and silicon dead layers, for which the essential elemental

composition is known. Evaluation of M

C (E) in Equation (4) requires knowledge of the

composition of the layer. This knowledge may be obtained from a variety of sources including (a) the partial pressures and sticking coefficients of gases impinging on the

crystal, (b) identification of the K-shell binding energies corresponding to a discontinuity in the x-ray spectrum from a bremsstrahlung source, and (c) testing assumed layer

compositions for several x-ray energies until self-consistent results are obtained.

All three ways were employed to assess the type and amount of removable contamination

present on a detector used almost daily for one year at pressures usually less than

2 x 10~7 torr. The relative efficiency of the detector for C(K) and F(K) x-rays was determined using the ALPHA-X [7] spectrum from Teflon. Since water vapor was the major gas

present during the year of operation, it was reasonable to assume that there was an ice

layer on the crystal. Figure 4 compares the Teflon spectrum obtained after the year of

99
TEFLON 2 = 00

PR= 400S 400SEC 0 I NT

U = 1 6 K H=10KEU 3<4Q AQ=10KEU 36

Figure 4. Comparison of the ALPHA-X spectrum of Teflon for a Si(Li)

detector iced and clean.

operation (i.e., iced detector) to one subsequently obtained following a temperature-cycle,

in an oil-free vacuum, to room temperature and back to normal operating temperature. The

relative increase in sensitivity for F(K) is greater than that for C(K). This would be

consistent with the removal of oxygen atoms from the surface of the crystal because oxygen

absorbs F(K) x-rays more strongly than C(K) x-rays. Assuming the removable contamination

was purely ice (i.e., H2 0) and using tabulated mass absorption coefficients [5], Equation

(4) yielded the effective values of m: 82 pg H2 0/cm2 and 62 pg H2 0/cm2 from the relative

efficiencies of F(K) and C(K), respectively. The average value of 72 pg H 2 0/cm2 is

equivalent to 0.72 urn of ice (assumed density of 1 g/cm3

).

Of course, such a thick ice layer should also be manifested as a large discontinuity

in a bremsstrahlung x-ray spectrum at the 0(K) edge. For the crystal conditions corresponding to those of figure 4, the beta- induced x-ray spectra obtained from an erbium film

hydrided with deuterium and tritium to form ErDT on a molybdenum disk are shown in figure

5. Although the ALPHA-X instrument was used to obtain the spectra of figure 5, the alphainduced contribution to the spectra was negligible compared to that from the beta particles

emitted by the decay of tritium in the film. The discontinuity due to absorption at the

0(K) edge is the only strong discontinuity present. Even the clean detector shows some evidence for the 0(K) edge, presumably from ice formed as a consequence of the partial

pressure of water vapor present when the crystal was initially cooled or an oxide on the

silicon crystal. Using Equation (4) for x-rays with 0.58 keV, corresponding to the large

dip just above the edge, and the measured relative intensities, the removable oxygen

contamination was calculated to be 57 pg 0/cm2

, which is equivalent to 65 pg H2 0/cm2 or 0.65 pm of ice. Even though this procedure is less precise than that employing the

efficiency for characteristic x-rays, the effective thickness of the ice layer is within

about ten percent of that determined using characteristic x-rays.

100
< 0 . 09KEU XES 1 . 28KEU)

Figure 5. Comparison of the beta-induced x-ray spectrum of a ErDT film on molybdenum

for a Si(Li) detector iced and clean. The spectra were normalized at the

large Er(M) peak, (a) 0-2.56 keV and (b) 0 - 1.28 keV.

101
Although the properties of the surface layers on Si(Li) crystals cannot be specified

exactly, these studies have shown that the efficiency of a WL Si (Li) detector can be characterized and understood in terms of the "effective" layers on the surface of the sensitive

volume of the crystal.

2.2 Detector linearity

For the present discussion, a detector with good linearity can be defined as one exhibiting a nearly linear relationship between x-ray energy and the output pulse

amplitude. In a practical sense, the entire system consisting of (a) the Si(Li) crystal,

(b) the field-effect-transistor (FET) with pulsed optical feedback, (c) the preamplifier,

(d) the pulse processor (i.e., linear amplifier, pulse pileup rejector, and live time

corrector), and (e) the energy-to-digital converter must be considered in any linearity

discussion. Linearity of complete commercial systems is quite good for x-rays with

energies above 1.5 keV, but documented results for energies less than 1.5 keV have

suggested quite different linearity curves [1,4,6].

Jaklevic and Goulding [1] reported that the C(K) and N(K) peaks had system pulse

heights of 0.7 and 0.9 of the values expected from extrapolation of the linear relationship

from higher x-ray energies. Since pulser measurements of the linearity of their system

electronics did not exhibit any non-linearity, they attributed the non-linearity to charge

collection effects in the 20 mm2 Si C Li ) crystal. However, they did not resolve the C(K)

and N(K) peaks from the steep low-energy background and, hence, the peaks only appeared to have a much lower pulse-height than expected. Measurements of proton-induced x-rays with

a 30 mm2 detector colli mated down to the center 17.7 mm2 coupled with computerized fits to the digital data provided the pulse height for the resolved C(K) Gaussian peak [6]. This

latter study yielded an x-ray vs. pulse-height relationship within two percent of linear for

C(K), 0(K), Cu(L), and A1(K) x-rays. Subsequent measurements with a 10 mm2 detector

extended this linearity relationship down to an energy of 183 eV using B(K) x-rays resolved

from the steep noise tail (figure 6) [4]. These latter studies have shown that Si(Li)

detector systems can be linear for x-rays with energies greater than 183 eV. Typically,

complete commercial detector systems have resolved C(K) and 0(K) pulse heights of about 0.9

and 0.97 of the values expected from linearity considerations. Although these deviations

are not desirable, unambiguous identification of the x-ray peaks can still be made.

800

600

E5COs

r>z

—i 400 /:uiL a ^i

l/slope - 1.99 ± 0.02eV/ch /0(Ka

)

/B(Ka) (10 mm

2

Si(Li) detector)

Figure 6. Relationship between detected pulse

height (i.e., channel number) and

x-ray energy for a Si (Li) detector

[4].

0.8 1.2 2.0

X-RAY ENERGY (keVI

Isolation of the component causing the non-linearity is quite difficult because of

problems associated with accurately simulating the pulses, including noise, for such low

energy x-rays. However, the bias applied across the Si (Li) crystal is one parameter that

can be changed without affecting the remainder of the system. This approach has yielded

spectra such as that presented in figure 7. The detector was a 30 mm2 Si (Li) crystal (3 mm

thick) collimated down to the center 10 mm2

. A study of the dependence of the relative,

resolved pulse-heights on detector bias provided the data displayed in figure 8 [6]. The

14 mm2 and 30 mm2 detectors were collimated down to the center 12.5 mm2 and 17.7 mm2

. The

pulse height is a measure of the ability of the detector to collect the energy deposited

into electron-hole pairs and, as normalized, can be considered to be the fraction of the

102
x-ray energy that was detected. Normalization was made with respect to the values at V^=

1000 V because this bias yielded a C(K) pulse height that fitted the linearity relationship. Note the asymptotic nature of the curves indicates that the C(K) pulse height would

not be increased significantly for bias values exceeding 1000 V (i.e., electric fields of

333 V/mm). Determination of the proper operating bias to obtain the best linearity from

each Si(Li) crystal and collimation arrangement must be made on an individual basis.

GRAPHITE

PR = 180S 10 ivi c c O V O u

U=16K H=18KEU 3 ' 4Q

***** iL = u W

9 I NT

AQ=10KEU 3Q

< 8.08KEU

. C<K>

ALPHAXES

DOTS- 1 000U BIAS

*. BARS"200U BIAS

I I I ft 1 1 M t l I I I I t i II I I t I < 0 . 64KEU>

Figure 7. Effect of applied bias on the pulse amplitude of C(K)

x-rays detected by a Si (Li) detector.

103
Figure 8. Relative pulse heights for C(K)

x-rays in 3 mm thick Si (Li) detectors [6].

DETECTOR BIAS Vb

(voltsl

2.3 Energy resolution

Energy resolution is a measure of the capability to distinguish clearly x-ray peaks

with similar energies. For energy dispersive x-ray spectrometry the resolution is normally

expressed as the full-width at half-maximum (FWHM) for the Mn(Ka) x-ray peak, with lower

FWHM values corresponding to better resolution. A good approximation is to assume that

isolated x-ray peaks are Gaussian-shaped with a standard deviation o, yielding FWHM =

2.355a.

Contributions to the FWHM can be estimated from [8]

(FWHM) 2 = A

2

.| + A

2 (5)

where is the width associated with the electronic noise including leakage currents and

microphonics and A^ is the width intrinsic to the detector and includes pair creation

statistics. If pair-creation effects dominate A^, then A^ is given by

A

2 = (2.355)2 FeE

(6)

where F is the Fano factor (i.e., the fraction of the total photon energy detected in the

pulse of electron-hole pairs), e = 3.81 eV [9] is the energy required to form an electronhole pair in silicon near liquid nitrogen temperature, and E is the total photon energy in

eV. Combining Equations (5) and (6),

A^ = (FWHM)

2 (2.355rFeE

(7)

Determination of the FWHM for two appropriate characteristic x-rays with the same detector

system (i.e., same A

g

^ and F) permits evaluation of A

g

^ and F. This has been demonstrated

using resolved C(K) and A1(K) x-rays to yield F = 0.11 ± 0.04 [4], where the uncertainty

reflects the uncertainties in the FWHM's. This range of 0.07 - 0.15 for F contains all of

the tabulated experimental F values for Si(Li) detectors [1,8], but does not include the

theoretically projected value of F = 0.05 [8,9]. It is important to note that not every

Si C Li ) detecto system satisfies Equation (7) over a wide energy range. However, figure 9 displays the energy dependence of the resolution calculated from Equation (7) for detectors

having Mn(Ka) resolutions of 146, 155, and 165 eV. This plot shows that a 9 eV difference

in Mn(Ka) resolution leads to a 13 eV difference in C(K) resolution. Such considerations

combined with the fact that the K x-rays are more closely spaced for low-atomic number Z elements emphasizes the importance of using the highest resolution detectors for low energy

x-ray spectroscopy.

104
i7o

i

, r

90 1 1 1 ' ' 1 1 0 I 2 3 4 5 6

X-RAY ENERGY E ( keV)

Figure 9. Calculated energy dependence of the resolution

of Si(Li) detectors with specified Mn(Ka)

resolutions of 146, 155, and 165 eV.

Although increasing the shaping time constant of the pulse processing electronics above

8 us does not significantly change the resolution of x-ray energies above 5.9 keV (Mn(Ka)),

such increases have a marked effect on the resolution for the low energy x-rays because the

electronic noise term of Equation (5) becomes dominant at the lower energies. These points

are illustrated by the data of figure 10. Thus, for low count rates the use of higher time

constants can significantly improve the resolution for low energy x-rays. The reduced

electronic noise corresponding to higher time constants will also be reflected in the

reduced noise tailing observed in the spectrum. This noise tailing is due to a large peak

located at essentially zero energy. If the noise tail is minimized then the interference

between the tail and low-energy x-rays (e.g., C(K)) would be minimized and elemental detection limits would be improved. Typically, = 95 eV corresponds to an electronic noise

cutoff energy of < 200 eV. Analysis of x-rays with energies less than 200 eV can only be

performed under special conditions (i.e., low-energy x-ray count rate comparable to noise

count rate at the energy of interest).

The major sources of noise are (a) various forms of electronic noise intrinsic to the crystal and design of the detector system, (b) microphonics due to the environment of

the application, (c) leakage currents due to surface contamination on the crystal, and

(d) photoconductivity induced in the crystal by visible and ultraviolet (UV) light.

Although source (a) is not within the control of the user of the Wl detector system, sources (b) - (d) can be minimized by appropriate attention to vibration, proper vacuum

procedures (see Section E below), and exclusion of light from the detector region. Sources

(c) and (d) are not significant in a beryllium-windowed detector/cryostat because the

beryllium window insures both the quality and light tightness of the detector vacuum environment. Since proper low-energy x-ray spectrometry imposes much greater resolution and

noise tail constraints on a detector system than does that for higher energy x-rays, the

importance of minimizing the noise sources must be emphasized.

105

2.4 Peak shapes

Although single x-ray peaks in energy dispersive x-ray spectra have shapes that are essentially Gaussian for x-ray energies above 1.5 keV, the observed shapes for peaks at

lower x-ray energies have been varied. Jaklevic and Goulding [1] found increased peak tailing (on the low-energy side of the peak) at lower energies from measurements of the

full-width at 1/10 maximum and at low bias voltages. They interpreted these results as dead- layer related charge collection problems. Unfortunately they obtained the results from

peaks that were not resolved from the background and adjacent peaks. From the nature of

their spectra they must have drawn most of the results from the somewhat clean 0(K) and F(K)

peaks. The F(K) spectrum was obtained from electron-excitation of Teflon, which is hydroscopic, and was not corrected for either the background or the 0(K) peak due to water in the

Teflon.

Musket and Bauer [2,10] found that the resolved 0(K) x-ray peak could be approximated

by a Gaussian shape of the form

where the energy E Q

is the Gaussian centroid, a is the standard deviation width, and H is

the peak height. Subsequently, Musket and Bauer [3,4,6] presented data that clearly showed

C(K) peaks that were quite symmetric and, by least-squares curve-fitting, Gaussian. However, they observed that the resolved 0(K) peak, although approximately Gaussian, did not

fit a Gaussian form as well as the C(K) peaks. The 0(K) peaks had a longer tail on the low

energy side of the peak. In one study [11], the shape of the resolved 0(K) peak was found

to be well-fit by a modified Gaussian G of the form

where E is the energy (below the centroid) at which the function Gm changes from Gaussian

to exponential. This modified Gaussian is similar to the B-ray response function of scintillators [12]. Musket and Bauer [11] thought this type of function would provide some

physical basis for describing the effects of photoelectrons from x-rays absorbed near the

"boundary" between the sensitive and volume and the dead layer of the Si (Li ) crystal.

Figure 11 shows the symmetrical (and Gaussian) shape of a C(K) peak when the influence

of the low-energy noise tail is negligible. The spectrum in figure 12 clearly shows the

apparent inconsistency of tailing on the 0(K) peak and no tailing on the C(K) peak. Refer

to figure 4 and note that the F(K) peak from thin Teflon tape is more symmetric than an 0(K) peak. ALPHA-X spectra of thick Teflon always show an 0(K) peak on the low-energy side

of the F(K) peak due, presumably, to the presence of water vapor in the Teflon.

Although no detailed study of the low-energy peak shapes has been reported, a plausible

explanation, to account for the apparent inconsistencies in the detected shapes can be based

on the influence of solid state effects on the true x-ray spectrum. Since the resolution

for these low energy x-rays is about 100 eV, such solid state effects in the sample (width

of 10 - 20 eV) may influence the shape and width of the resultant peak. The most direct

way to test this hypothesis is to eliminate solid state effects by determining the x-ray

spectrum from gaseous oxygen. Indirectly the hypothesis may be tested by adding, in quadrature, contributions from electronic noise, charge-collection statistics, and true C(K),

0(K), and F(K) spectra. The resultant peaks would then be compared to resolved peaks

obtained from appropriate samples. If the results were in agreement, then the explanation

would seem plausible. Ultimately this approach could be used to determine some solid state

properties by removing the contributions of electronic noise and charge collection statistics from the peak shape. In any case, low-energy x-ray peak shapes remain a topic for

future investigations.

G(E, E Q

) = H exp(-((E - E Q

) 2 /2a2)) (8)

Gm(E, E Q

) = H exp(-((E - E Q

) 2/2a2 )), E > E Q

- E c

Gm(E, E Q

) = H exp(-(E

c/a2 )(Eo

- E - 0.5E

c )), E > E Q

- Ec

(9a)

(9b)

107
UITREOUS CARBON- 1 5KU Z = 00

PR= 110-S 118SEC 9 I NT

U = 3 3 K H = 18KEU 3

> 4Q AQ-1'BKEU 3Q

KEUEX/6UTW

C<K>

FWHM-102EU

< 0 . 00KEU XES

C + C

t ! f M » 6 . 64KEU)

Figure 11. Electron-excited x-ray spectrum from vitreous carbon.

108
BED WITH 50UG C/CM2 Z^0 4 BE

U = 4 8 9 6 H = 1 0 K E U 1 • 2 Q A Q a 1 0 K E U 1 Q

ALPHA-X

0.00 K E

U

C< K )

X E S

0<K)

iliii

@ . 6 4 K E U >

Figure 12. ALPHA-X spectrum of a BeO crystal covered with

a carbon film (50 yg/cm2 ).

2.5 Vacuum considerations

The essential requirement for operation of a WL Si (Li) detector is that the vacuum environment be clean and oil-free. Such an environment can be achieved by combining proper

vacuum design and materials with sputter- ion and cryogenic pumping. Ideally both these

pumping capabilities would be integral components of the detector/cryostat to insure the

proper vacuum environment even when the detector is not in use. Initial evacuation of the

detector with the detector at room temperature must be to the lowest possible pressure to minimize the formation of layers on the crystal after the liquid nitrogen is added (pres- sures below 1 x 10"6 torr are acceptable). Since the major condensable gas in such vacuum

systems is usually water vapor, the discussion on ice layer formation and removal in Section

A above is applicable. For long-term, high efficiency operation of detectors, the maximum

recommended operational pressure in a clean, oil-free system is 1 x 10~7 torr. The time

required to form an ice layer of a certain thickness will be longer for lower operational

pressures. In contrast, oil-vapor contamination generally cannot be removed by simply warming the crystal to room temperature under a good vacuum, and the oil may actually render the

crystal unusable for high resolution purposes. These constraints place extreme restrictions

on the sample and sample chamber environment for application of a WL detector and excludes

use of WL detectors on diffusion-pumped sample chambers.

An attractive alternative to the WL detector is the UTW detector, which employs an UTW

mounted with an o-ring to make a vacuum seal and completely isolate the detector vacuum from

the sample chamber vacuum. The UTW is an ultra-thin parylene-N film (i.e., C 8 H 8 ) coated

with a similarly thin aluminum coating of about 2000 A. Typical transmission probabilities

are 0.2 and 0.5 for C(K) and 0(K), respectively. Since the UTW cannot support pressures

exceeding about 10~~3 torr, the UTW detector must be treated as though it was a WL detector

(i.e., it must be retracted and its gate valve closed before venting the sample chamber).

109
The restrictions on the sample chamber vacuum for UTW operation are (a) clean, oil-free

sample chambers with pressures less than 1CT3 torr and (b) only diffusion-pumped systems

with liquid nitrogen traps and operational pressures less than 5 x 10~6 torr. Restriction

(a) insures that the UTW doesn't rupture and restriction (b) minimizes contamination of the

UTW by oil vapor from the diffusion pump.

2.6 General considerations

In addition to the vacuum requirements outlined above, there are several other

considerations to observe for proper energy-dispersive measurements of low-energy x-rays.

Included are (a) exclusion of light (visible and UV), electrons, and other energetic particles from impinging on the Si(Li), (b) collimation using apertures not tubes, and (c) pulse

pileup of low energy x-rays.

Light photons can be eliminated using a light barier (LB) such as the window material

described above for the UTW detector. A pinhole-free aluminum coating of 1000 - 2000 A is more than adequate to reduce the light photon flux to an acceptable level. The difference

between the LB and UTW versions of such windows is that UTW versions provide vacuum isolation as well as "dark" detector environments. If electrons are not prevented from bombarding the Si (Li) crystal then an anomalously high bremsstrahlung background will be observed

due to the interaction of the electrons with the crystal. Even the thinnest beryllium

window (8 urn) will stop electrons with energies below 25 keV and preclude their bombardment

of the crystal. Thus elimination of electrons from a beryllium-windowed detector needs only

to be considered if the application includes scattered electrons with energies above

25 keV. However for a WL, LB, or UTW detector system an integral magnetic trap in front of

the window and crystal can serve to remove the backscattered electrons from the x-ray

axis. Other energetic particles can also give rise to peaks and background not associated

with the true x-ray spectrum and their flux should be reduced to an acceptable level by

procedures appropriate to the nature of the particles.

For beryllium-windowed detectors, collimation is usually accomplished using an aluminum

collimator with a carbon-coated tubular aperture. A1(K) x-rays generated by x-rays, electrons or other particles bombarding the collimator are absorbed in the carbon coating and

C(K) x-rays are absorbed by the beryllium window. Consequently, there is no evidence of the

collimator in the detected x-ray spectrum. For detector/cryostat systems capable of detecting C(K) x-rays, such a tubular collimator approach is not acceptable. Replacing the carbon

coating with a beryllium coating is plausible but the thickness of beryllium required to absorb an appropriate fraction of the A1(K) x-rays is quite large and the potential health

hazards associated with beryllium coatings are great. However, collimation using a series

of knife-edged apertures has proven to provide the needed collimation without introducing

extraneous x-rays into the detected spectrum.

Although commercially available pulse-pi leup rejection systems perform quite well for

pulses from x-rays with energies above about 1 keV, performance decreases with decreasing

x-ray energy. Thus, pileup or sum peaks are not uncommon in spectra containing low energy

x-rays. Figure 11 shows the main and sum C(K) peaks for electron excitation of graphite.

The total C(K) count rate was about 900 cts/sec. In some cases a sum peak due to a lowenergy x-ray peak and a higher energy x-ray peak can be obtained (e.g., 0(K) + A1(K) from

A1 2 03 ). In the absence of appropriate pulse-pileup rejection circuitry true low energy

spectra can only be obtained at count rates low enough to assure negligible sum peaks. Note

that this restriction on count rates is consistent with using longer time constants for

optimum resolution. Thus, in general, the most correct low-energy x-ray spectra will be

obtained using low count rates and long shaping time constants.

3. Applications

Examples of applications of WL and UTW detectors to bulk, thin film, and surface-layer

analyses using ion and electron excitation will illustrate several detector capabilities.

For most analyses using energy-dispersive x-ray spectroscopy, the penetration of the ion or electron defines the maximum depth of analysis. However, in analyses for the light elements, the escape of the low energy x-rays limits the depth probed. This effect is shown in

figure 13 where the x-ray yield for a layer of thickness L has been normalized to that for

an infinitely thick layer under the assumption of uniform ionization of the atoms in the

110
E

=1

CO

CO

111 o

or

LU

-i

9A/\ Q13IA AVU-X Q3ZnVWdON

111
layers and a 90 ° takeoff angle. A reasonable measure of the thickness probed is the layer

thickness giving a normalized x-ray yield of 0.9 that of the infinitely thick layer. The

ninety percent thicknesses for C(K) in Fe, 0(K) in Si02 , and C(K) in C are 2000 A, 2.2 urn, and 3.8 urn, respectively. Of course the ninety percent thicknesses would be reduced considerably for smaller takeoff angles due to increased x-ray absorption. Such considerations

serve to highlight the critical importance of a high takeoff angle for the analysis of low-Z

elements. The applications discussed below were undertaken with either a 90° takeoff angle

(ion excitation) or a 45° takeoff angle (electron excitation).

3.1 Ion excitation (ALPHA-X)

The ALPHA-X version of the alpha-induced x-ray emission technique is shown schematically in figure 14 [7]. Detailed description of the physical concepts have been previously

published [7,14]; however, a brief outline is given here to provide a basis for interpretation of the results to be presented. The ALPHA-X technique consists of the detection and

analysis of x-rays emanating from a specimen irradiated with 5.8 MeV alpha particles (i.e.,

doubly ionized helium) from a radioactive Cm- 244 source. As shown in figure 15, the ALPHA-X

probe is a compact combination of a specially designed alpha source attached to a unique

source holder and electron-trap/collimator assembly which is installed in the end of the

tubular housing of a high-efficiency WL Si (Li ) detector. Although a WL detector can usually

be employed for analysis of alpha- induced x-rays (provided extraneous light is eliminated),

alpha-induced light can be a source of noise in the case of some samples (e.g., thick

insulators such as alumina, boron nitride, beryl li a). Of course, either a LB or UTW detector could be used as a component of ALPHA-X. Figure 16 shows a typical installation of the

ALPHA-X instrument. The exterior dimensions and features are the same as those of a WL

detector. Since the alpha-particles are continuously emitted, analysis with ALPHA-X con- sists of (a) evacuating the sample chamber to <10~7 torr (for WL detector), (b) opening

the gate valve to ALPHA-X, (c) moving the probe near the sample, and (d) turning on the

detector electronics.

Figure 14. Schematic of components of ALPHA-X technique [7].
rSi(Li) DETECTOR

STAINLESS

STEL TUBE

U^2 COLD FINGER

ALPHA-X

ANNULAR SOURCE

END CAP

SPECIAL COLLIMATOR UNITJ

Figure 15. Geometry of the ALPHA-X probe [13].

10 MM

Figure 16. Typical installation of the ALPHA-X instrument.

Three types of studies will demonstrate the usefulness of ALPHA-X for analyses of low-Z

elements on various substrates. These studies are determination of oxygen surface densities

(i.e., oxide thicknesses), carbon on and in steels, and low-Z elements in aerosol samples.

The value of ALPHA-X for assessment of detector performance has been illustrated in the

section on properties.

Oxide thickness determination

Using previously published procedures and principles for oxide thickness determination

from ion-induced oxygen x-rays [2], the ALPHA-X oxygen x-ray yield I (x-rays/sec) from an

113
unknown sample can be compared to that I from a standard with known oxygen surface density

2

L

g (g 0/cm ) to permit evaluation of the unknown oxygen surface density L u using

, C I u

L = - — In 1 -

s u u

u uM \ I u \ s / (10)

where C = (l/p c )(l " exp(-p L )) and u and p<. the oxygen x-ray absorption coefficients

t\ S S S S U o (cm /g 0) for the unknown and standard oxides.

In general, the composition of the unknown oxide is required to evaluate p

u

; however,

if C

u

I up

u/Is « 1 (i.e. , for very thin oxides), then

L

u - Cs

I u/I

s , (11)

and the exact composition of the oxide is not needed. However, determination of the thickness T

u of the unknown oxide requires knowledge of both the oxide density pu (g/cm3

) and the

weight fraction of oxygen in the oxide W for use in

T

u S W. (12)

The net 0(K) yield from the oxidized silicon standard (figure 17) provided the value

for I in Equation (10). Then the oxygen surface density represented by the net 0(K) yield

2

° of figure 18 was determined to be 20.2 pg 0/cm , which corresponds to about 1280 A of Ta2 0 5 (bulk density = 8.74 g oxide/cm3

). Of course the same formalism applies to the thickness

determination of layers other than oxides.

Figure 17. ALPHA-X spectrum of high purity silicon with 2.37 yg

oxygen/cm2 (i.e., 192 A Si02 ). 114
U -4096 H = 1 Q K E U 1

: 2 Q A Q 1 8 K E U 1 Q

Figure 18. ALPHA-X spectrum of anodized tantalum.

Analysis for carbon on and in Steels

ALPHA-X studies have provided quantitative, non-destructive determinations of the

carbon content in surface layers and in the bulk [13]. For thin carbon layers, a version of

Equation (11) was applied to determine the carbon surface densities from the ratio of the

carbon x-ray count-rate for the unknown layers to that for standard free-standing carbon

films. Figure 19 compares the low-energy part of the spectrum from Steel "L" with a low

level of carbon contamination to that from Steel "H" with a higher level of carbon contamination. The levels of carbon contamination represented by these spectra were found to be

0.25 and 0.88 ug carbon/cm2

, respectively. Equivalent carbon thicknesses were 11 and 39 A,

assuming a bulk carbon density of 2.26 g/cm3. The three-sigma minimum detection limits for

twenty minute analyses were 0.1 ug carbon/cm2 (equivalent to approximately 5 A). Argon-ion

sputtering was used to completely remove the contamination from the surface of the steel.

Subsequent ALPHA-X measurements showed no evidence for C(K) x-rays, indicating that the bulk

carbon of 0.05 weight-percent was not detectable.

115
STEEL " L

P R = i 2 0 0 S 1 2 0 0 S E

U = 4 8 9 6 H = 1 6 K E U 4 : 2

" .

. ...

0 INT

6 K E I' 4 (:'! ALPHAX

C(K)

0 . 0 Pi k" F ,x

i t b

F E < L >

D 0 T 8 = S T E E L H

If If IHtiiti>ini(ii"iiiimi>i,iiiitiiiniii'ii

1 . 2 s k e u >

S T E E L " L

" Z = 0 0

PR* 1200S 1200 SEC 0 INT

U = 1024 H « 1 0 K E U 3 i 4 Q A Q * 1 8 K E U 3

Q

A L P H A - X

F E < L )

C< K )

u u i o '«.' i u im u n

3 ~" 1 4

~

< 0.80 K E

U

X E S 1.28 K E U

>

Figure 19. Comparison of ALPHA-X spectra from Steel "L" and Steel "H" : (a) 4096 CFS and

(b) 1024 CFS [13].

116
Evaluation of the detection limit for carbon in steel was made using British Chemical

Standard Steel 407 with 0.5 weight percent carbon. Figure 20 gives the low-energy spectra

of this steel before and after sputter-cleaning to remove the surface carbon. Note the

small-but-distinguishable differences at the energies of the C(K) and 0(K) x-rays. These

reductions are attributable to sputter removal of a thin oxide and light carbon contamination from the freshly polished sample. The minimum detection limit was 0.1 weight percent

carbon for a twenty minute analysis. This result verified the inability of ALPHA-X to detect the 0.05 weight percent carbon in sputter-cleaned Steel "L". BCS STEEL 487 < S P U T ) Z = 0 0

P P 1 L'"1 l/l Q 1 p ft a C; pr r a T M T

U -102 4 H = 1 0 K

E

J 4

; 3 Q A 13=10 K E U 4

Q

P 6 D O fi ki * Ch bi Ci 1,1 j

Jill HLrnM n

*

»

1

» 1

1

C

<

K

>

J

1 11 lip

°

t s * n ° spuT

< 0

! 0 0 k!II!^

^ iiiiiiiiiiiii

<ES 1.28KEU>

Figure 20. Comparison of the low-energy spectrum of BCS Steel 407

before and after sputter-cleaning [13].

Low-Z analysis of aerosol samples

The analysis of low-Z elements in aerosol samples collected on plain and ni eke I -coated

porous silver filters is important for a proper evaluation of particulate material in air

pollution studies. ALPHA-X has provided such analyses with a minimum detection limit of

less than 1 ug carbon/cm2 on a nickel-coated silver-filter in a 300 second analysis.

Assessment of the total carbon in aerosol samples collected on silver filters is not a straight-forward procedure due to the contribution from the Ag(M) x-ray peak (figure 21).

However, a coating of 1.1 urn nickel on the silver filter is more than adequate for absorption of the Ag(M) x-rays and such a filter is an excellent substrate for direct analysis of

carbon (figure 22). The Ni(L) peak does not interfere with any other peaks in the

aerosol sample; however, care must be taken to minimize the oxide on the evaporated Ni coating.

117
Figure 21. Comparison of ALPHA-X spectrum of aerosol sample collected

on plain silver filter with that of blank filter.

118
AEROSOL 2-14/16-78 L K 2^28 HI

PR- 240S 2408EC O INT

U = 1024 H-lSklU 4 = 3Q AQ=10KEV 4Q

Figure 22. Comparison of ALPHA-X spectrum of aerosol sample collected on nickel -coated silver filter with that of blank

nickel-coated filter.

3.2 Electron excitation

As discussed in the section on detector properties, low-energy dispersive x-ray spectrometry on a diffusion-pumped scanning electron microscope (SEM) can be readily accom- plished by an UTW detector with an integral electron trap. Spectra have been obtained using

such a detector that subtended 10"4 steradians from the SEM sample under bombardment by a true electron current of 10 to 15 nA. The pulse processor time constant was 8 us.

Figure 23 shows the spectrum from solid Si02 for 15 keV electrons. Note the 0 + Si

pileup peak. Figure 24 resulted from the bombardment of salt by 10 keV electrons. The

small Ni(L) peak seems to be a system peak characteristic of the SEM and appears in many of

the spectra. For Fe2 03 the low energy portion of the electron-excited spectrum is displayed

in figure 25. 15 keV electrons were used to generate the spectrum of x-rays from calcium

carbonate (figure 26).

119
... . .... .. .... , . . .... , SI 02 < 15KEU > K 2=1 4 SI

PR= 76S 76SEC 0 INT

U = 16K H = 1 9 K E U 3<3Q AQ=18KEU 3Q

< 0.60KEU XES 2.56KE.U>

Figure 23. Electron-excited spectrum from solid Si02 (Electron

energy = 15 keV).

120
NACL < 18KEU) K Z=ll HA

PR = 60S 6 0 S E C 0 INT

U=2048 H=10KEU 3=3Q AQ=10KEU 3Q

121
P P *|5

I Imm Imm 03 < 15KEU ) Z

= U 0

tn> l„l mm C| Q Cj q q C; jr £; fl I N T

U = 4@96 H=1@KE U 3 < 3 Q (VQ«10KEU 3Q

KEUEX/6UTW

|0<K>

C<K> 1

llllllllfflllllllllllllllliiiii < 0

Lllllllilllllilillllll II

. 00KEU :<' P c:

lttlflll!l!lll!ll!llll!ll!HHlllll!l!H!l

1 . 28KEV>

Figure 25. Low-energy portion of electron-excited spectrum from

solid Fe2 0 3 (Electron energy = 15 keV).

122
C A C 0 3 (15 K E U ) LK Z = 2 8 C

A

PR= 119S 119SEC e INT

U = 8192 H = 1 9 K E U 3 ; 4 Q A Q = 1 8 K E U 3 Q

< 8 . 80KEU SEQ 5.12KEU)

Figure 26. Electron-excited spectrum from calcium carbonate

(Electron energy = 15 keV).

Although the UTW detector with an electron trap permits energy-dispersive x-ray spectrometry of low-energy x-rays in the mediocre vacuum normally found with a SEM, proper

interpretation of the acquired spectra requires consideration of the processes occurring on

or in the electron-bombarded sample. Buildup of contaminants on an irradiated sample can be

particularly difficult to eliminate unless the vacuum system is better than average. A com- parison of spectra obtained from fresh (i.e., previously unirradiated) MgO to that obtained

after the electron beam was rastered over the analysis region for ten minutes is presented

in figure 27. Both spectra were obtained in 22 seconds of live time. The large increase

in the C(K) intensity and the small increase in 0(K) intensity presumably resulted from

molecular dissociation of background gases that impinge on the surface of the sample. This

presumption is consistent with the linearly increasing C(K) intensity with time (figure

28). For a liquid-nitrogen-trapped, diffusion-pumped system operating at less than 5 x 10"6 torr (as required for use of an UTW detector), this type of sample contamination can be

reduced considerably.

123
M G 0 ( 1 8 K E U ) 2 = 00

PR= 22S 22SEC 0 I NT

U-2048 H = 10KEU 3'4Q AQ=10KEU 3Q

. C<K)

KEUEX/6UTW

MG< K )

£0<K>

0 . 00KEU

BARS"FRESH MGO

DOTS-CONTAM MGO

2 . 5 6 K £ U >

Figure 27. Comparison of electron-excited spectra from MgO at the

beginning of electron irradiation and after ten minutes

of irradiation (Electron energy = 10 keV).

124
C BUIDUP ON MG0-16KU Z = 00

PR= IS 1 SW 0 I NT

U = 2048 H = 10KEU 3 = 3Q A Q = 1 9 K E U 3Q

KEUEX/6UTW

GROSS C<K> COUNTS

PER 10SEC

< 10 . 0 S SEQ 650 . 0 'S>

Figure 28. Time dependence of the carbon buildup on MgO for a SEM environment (Electron energy = 10 keV).

4. Conclusions

Windowless (WL), light-barrier (LB), and ultra-thin windowed (UTW) Si(Li) detectors

have been shown to be useful for the energy-dispersive spectrometry of x-rays with energies

above 200 eV. Efficiencies of 0.05 to 0.25 for C(K) x-rays and 0.10 to 0.50 for 0(K) x-rays

may be expected for UTW (or LB) and WL detectors, respectively. These detectors have an output linearity more than adequate to permit unambiguous identification of the basically

Gaussian peaks with an energy resolution of about 102 eV for C(K) when that for Mn(K) is

146 eV. The availability of detectors appropriate for a variety of vacuum environments

permits low-energy x-ray spectroscopy in a wide range of analysis systems, including SEM's,

TEM's, Auger systems, and ESCA systems. These detectors should play an increasingly

important role in the total analysis (i.e. , surface and bulk) of materials.

References

[1] Jaklevic, J. M. and Goulding, F. S. , IEEE Trans . Nucl . 5c

i

. , NS-18 , 187 (1971).

[2] Musket, R. G. and Bauer, W. , J. Appl . Phys . , 43, 4786 (1972).

[3] Musket, R. G. and Bauer, W. , Nucl . Instr . and Meth. , 109, 593 (1973).

[4] Musket, R. G. , Nucl . Instr . and Meth . , 117 , 385 (1974).

[5] Veigele, W. J., Atomic Data , 5, 51 (1973).

[6] Musket, R. G. , and Bauer, W. , Nucl . Instr . and Meth . , 109, 449 (1973).

125
[7] Musket, R. G. , Research/Development , 28 (10), 26 (1977).

[8] Zulliger, H. R. and Aitken, 0. W. , IEEE Trans . Nucl . Sci. , NS-17 , 187 (1970).

[9] Zulliger, H. R. , J. AjddJ. Phys. , 42, 5570 (1971).

[10] Bauer, W. and Musket, R. G. , J. Appl . Phys . , 44, 2606 (1973).

[11] Musket, R. G. and Bauer, W. , J. Appl . Phys . , 47, 353 (1976).

[12] Wohn, F. K. , Clifford, J. R. , Carlson, G. H. , and Talbert, W. L. , Jr., Nucl . Instr . and

Meth . , JjOJ, 343 (1972).

[13] Musket, R. G. , in, Advances in X-ray Analysis , Vol. 22, G. J. McCarthy et al., eds. , (Plenum, New York, 1979), p. 401.

126
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981

)

THE EFFICIENCY OF Si (Li) DETECTORS AT VERY LOW PHOTON ENERGIES

P. J. Statham

Link Systems Ltd.

Halifax Road, High Wycombe

Bucks. HP12 3SE England

Abstract

The efficiency of a Si (Li) detector for low x-ray energies is an important parameter in determining both the shape of the observed

continuum and the sensitivity factors for "standardless" techniques such

as the analysis of thin samples by the intensity ratio technique. This

paper considers the factors which determine the efficiency and examines

the possibilities for measurement and their limitations.

1. Introduction

To date, detailed studies of detection efficiency seem to concentrate on the x-ray

energy region above about 2.5 keV primarily because of the absence of "doublet" standards

with peaks at suitable energies. However, the tacit assumption is often made that the

efficiency for energies below 2.5 keV can be predicted satisfactorily using physical

formulae based on quoted thicknesses for materials constituting the detector entrance

window. There is already enough evidence in the literature to suggest that such an assumption is not always justified and this paper attempts to gather together such evidence

and present new information to demonstrate the errors which could result.

Knowledge of detection efficiency is important in a number of techniques such as measurement of absolute x-ray yields [l] 1

, prediction of the continuum (bremsstrahlung)

shape for background correction and measurement [2-6], determining efficiency factors for

the "ratio" technique for thin film quantitation [7] and for "standardless" analysis [8].

Whereas the Si (Li) spectrometer behaves in an almost ideal fashion in the energy range

5-10 keV, the elements Na, Mg, Al , Si are often of interest and fall in an energy region

where the efficiency is determined by several unrelated factors. In "windowless" systems

some of the uncertainty is removed by elimination of the Be vacuum window, but new experimental problems are introduced which detract from this apparent advantage.

If we assume that the detector is well collimated so that off-axis efficiency

variations [9] are not significant, then the geometric efficiency can be determined [1]

and the remaining fall-off in efficiency is due to absorption in the Be window, gold

conductive coating and silicon "dead layer", charge collection effects within the detector,

and to the efficiency of pulse detection by the electronics.

1 . 1 Beryl 1 i urn wi ndow

The manufacturers of beryllium foil recommend that it be stored and used in a dry

atmosphere (< 50% RH) to maintain the vacuum integrity of the foil which is extremely

sensitive to all oxidizing agents. Although detector manufacturers take great care to obey these guidelines, the user is not always aware of them and windows are sometimes

quite innocently exposed to water vapor and even caustic gases.

1 Figures in brackets indicate the literature references at the end of this paper.

127
Although the crystal and cold finger are suspended in a vacuum, there is invariably a small amount of heat transfer to the cryostat wall with the result that the window is

normally one of the coldest surfaces in the specimen chamber. Furthermore, the problem is worse when space requirements force the detector snout to be very narrow. As a consequence,

oil and water vapor condense preferentially on the window and cause an increase in effective

window thickness and thus a drop in efficiency. The problem is somewhat alleviated by the

presence of a cold finger, as would be used to reduce specimen contamination for example,

and some success has been achieved by using the heat from a miniature electric light bulb

to slightly raise the snout temperature (D. Smith, University of Alberta, private communication).

Eight urn or "1/3 mil" is the most common thickness used for x-ray work, and aluminum

foils of this thickness exhibit roll-marks and occasional pin-holes (P. Morris, Alcan

Laboratories, private communication). The SEM micrograph in figure 1 shows that surface

defects are also visible on a typical beryllium foil and furthermore the side view in

figure 2 indicates that the thickness is irregular on a microscopic scale. In practice,

the average thickness may vary by about 30 percent from the nominal value of 8 urn and

there may be some inaccuracy in values for mass absorption coefficients [10] so, even if

irregularities are ignored, direct calculation of x-ray absorption is prone to error. Since foil thickness is usually determined indirectly by weighing a specific area of

material, in effect it is the average mass thickness which is being measured so it seems fair to assume that an average mass thickness could be used in a formula for absorption.

However, absorption is a non-linear effect and the use of an average thickness tends to produce an underestimate of transmission efficiency. This is demonstrated in Table 1 where the transmission efficiency has been calculated for a composite foil where half of

the foil area has a thickness of 5 urn and the remainder is 10 urn thick. The mean thickness

of the foil is 7.5 urn and the calculated transmission efficiencies agree quite well down to 1.5 keV. However, at low energies, when the efficiency falls below 50 percent, the com- posite foil transmits a disproportionate fraction of the radiation through the thin areas

and at 0.5 keV, the transmission is almost a factor of three greater than predicted for a 7.5 urn foil of uniform thickness.

Figure 1. SEM micrograph in

plan view showing the

surface of a 1/3 mil. Be foil originally

used for a detector

window.

Figure 2. SEM micrograph as for

figure 1 but viewed

from the side and

slightly above the

foil. Some blobs of

epoxy resin, originally used to secure

the window, are visible on the

surface.

128
Table 1. Calculated transmission efficiency for a 7.5 pm Be foil compared with that for

a composite foil with equal areas of 5 pm and 10 pm thickness.

Energy Percent Transmission

keV 7.5 pm Composite

0.5 0.8 2.

1

0.6 5.8 8. 6 0.7 15.9 19.0

0.8 28. 6 31 .

1

0.9 40.9 42.7

1.0 51.6 52.9

1.5 81.3 81.5

2.0 91.3 91.4

2.5 95.3 95.3

1.2 Gold layer

A gold film is used to establish the electrical contact at the front surface of the

detector and usually has a nominal thickness of 200 A. There is evidence to suggest that

such films are extremely non-uniform in thickness. Jaklevic and Goulding [11] point out

that the resistance of the evaporated layer shows an abrupt decrease when the thickness

exceeds 80 ft and suggest this is due to the growth of islands which eventually coalesce to form a filamentary net. Films of similar thickness are used when coating specimens for

electron microscopy and do exhibit this filamentary structure [12] and large open areas may

still be present in 200 ft films [11].

When calculating absorption within this layer, it has been suggested [11] that con- ventional theory becomes inappropriate for radiations such as C Ka (A = 44 ft) where the

wavelength approaches the film thickness. However, as pointed out at this meeting

(R. Musket, Kevex Corporation), if the film is indeed segregated, the regions responsible

for absorption will be thicker than the average value of 200 ft so normal absorption

calculations should be valid. Nevertheless, the situation is still not trivial and, as shown in Table 1, a film of non-uniform thickness transmits more radiation than a uniform

film with the same average thickness. It has been suggested [11] that if the film were

uniform, then the consequent attenuation produced by 200 ft of gold would seriously affect

the usefulness for detecting low energy x-rays from carbon and oxygen.

One way of measuring the effective gold layer is to observe the absorption steps

present in the bremsstrahlung background. However, if the gold layer thickness is included

as a variable parameter in a least squares fitting routine used to determine all the

window thickness parameters, the predicted gold thickness often turns out to be much less

than 200 ft or sometimes even negative. For this reason, the gold layer is often ignored

but Au absorption steps can usually be seen in a spectrum of carbon with good counting

statistics (at least 1 OK counts per channel). Brombach [13] has suggested that the gold

edges are less visible than one would expect because the largest discontinuities

(Au Mjy y) actually involve a smooth transition in the mass absorption coefficient value.

The author provides a theoretical argument which is substantiated by the experimental

findings of Steele [14] and shows that experimental spectra can be accurately fitted in the

1.5 to 2.5 keV energy range by using a modified formula for absorption coefficients, and

assuming a gold layer thickness of 200 A and a 25 percent gold free area. The main absorption step occurs in the region of 2.2 keV and typically corresponds to a change of 5 percent

in efficiency. There is little to differentiate between the absorption effects of

beryllium and gold below 2 keV, so when window thickness parameters are determined indirectly, by least squares fitting to bremsstrahlung spectrum, the predicted gold thickness

is often misleading. A 5 percent step in the background is undetectable in typical

electron-excited spectra with roughly 200 counts per channel, so omission of the gold layer

129
term does not have any serious consequences for background corrections. However, observed

characteristic peak intensities for elements such as P, CI, S will be influenced by the

gold which therefore should not be ignored for accurate work.

1.3 Silicon dead layer

The silicon "dead" layer is a region near the surface of the detector where charge

collection is poor and not a passive zone of silicon on the front surface of the detector.

When an x-ray photon of energy E is absorbed close to the surface, some of the charge may

be lost in the energy-to-charge conversion process. As a consequence a count appears in

the spectrum at an energy less than E and is "lost" by virtue of not appearing in the

correct channel. The proportion of photo-electric events which occur close enough to the

surface to exhibit such behavior is determined by the absorption coefficient for radiation

of energy E in silicon, so counts are "lost" from a spectral peak just as if photons were absorbed in a thin layer of passive silicon. Therefore, for some applications, it is convenient to calculate the absorption due to a silicon "dead" layer which would remove counts completely from the spectrum, rather than attempt to correct the spectrum for

incomplete charge collection which displaces counts from a peak into a low-energy tail.

The "escape" effect [15] also results in counts being displaced (by 1.74 keV) from

their true position in the spectrum. SiKa photons are only produced in the detector by

incident photons with energy greater than 1.84 keV. Therefore, a step of about 2 percent

appears at the S i

K

a^ energy because there is approximately a 2 percent chance of SiKa

escape for photons with energy greater than the silicon absorption edge whereas no SiKa

escape is possible below this energy. The size of this step is consistent with a passive

layer of silicon roughly 0.03 um in thickness.

Since the "dead" layer is a consequence of poor charge collection, the thickness is

influenced by both the magnitude of the H.T. bias voltage and the state of the surface.

Jaklevic and Goulding [11] measured a variation from 0.4 urn to 0.2 urn as the H.T. bias was raised from 100 V to 1000V. In a more recent paper, Llacer et a]_. [16], suggest that

the window effect is due to a fundamental transport limitation of electrons which is only

slightly affected by the applied external field and cite the example of a high purity Ge detector where the observed peak tailing was only slightly dependent on bias in the region

300-1000V in a 4 mm detector that was fully depleted at 20V. Although the theory [16]

leads to a larger window than is observed, the general form of peak tailing is predicted

quite well and the authors [16] propose that the hot-electron transport mechanism represents a physical limit to window thickness (0.4 urn in Ge) which is not likely to be overcome because of the very high field that would be required to do so. However, they

point out that the surface potential could depend on surface treatment and metal work

function, so different detector fabrication techniques may lead to slightly different

window thicknesses.

The effect of the Si dead layer is most evident for x-rays with energy just above

the silicon absorption edge so the peak shape for phosphorus shows a marked low-energy

tail which, in the example shown in figure 3, reaches a maximum of roughly 7 percent of

the peak height. X-rays with energy just below the edge have a much greater range so very

few get absorbed in the window region and no tailing is apparent on a SiKa peak for

example. When the bremsstrahlung spectrum is observed in this energy region (figure 4),

the step in the vicinity of the Si absorption edge is enhanced because degraded events from

just above the edge are "dumped" in a region just below the edge. The phosphorus example

in figure 3 suggests that this enhances the step height by roughly 7 percent and distorts

the background shape in a region extending roughly 0.5 keV below the absorption edge. (The

distortion of the background shape is very clearly demonstrated in a spectrum of carbon

presented by Fiori and Newbury [17].) This effect has rather serious implications in the

context of efficiency since it implies that if the apparent Si dead layer is determined by

observation of the step in the background, the thickness will be overestimated by more than

0.1 um. In terms of modeling the background shape, the significant problem is the distortion below the edge but the real danger arises if one attempts to use such a "fitted" dead

layer thickness when predicting overall efficiency.

130
%of peak height

20 1

10 «

04-

1.0 ENERGY keV —

Figure 3. Low-energy tailing on the side of a phosphorus

peak.

ENERGY keV -

Figure 4. The shape of the bremsstrahl ung spectrum in the

vicinity of the silicon K absorption edge. The

dashed curve shows the shape of a phosphorus

peak with associated tail due to poor charge

col lection.

If counts that appear in the low-energy tail are included when calculating characteristic peak areas, then charge-collection defects will have no effect on the apparent

efficiency and there will be no "dead" layer to consider. In other words, if the spectrum

were corrected for charge-collection, then there would be no discontinuity in efficiency at the silicon edge. This statement requires some qualification since it only applies to a detector which has been prepared in such a way that there is no inactive silicon left on the front surface. If the lithium drifting, or subsequent surface preparation, is

imperfect, there may indeed be truly dead areas of silicon. In this case, x-rays absorbed

131
in such regions may give rise to fluorescent SiKa x-rays which travel towards the active

region of the detector and produce an "internal silicon fluorescence" peak [15]. In Figure 4, I have labeled such a peak as "spurious" since, as a purely empirical phenomenon,

it cannot be predicted from physical theory. If the peak is present it is often greater

in size than is predicted by calculations which assume a totally-dead layer of 0.1 urn of

silicon [15]. Furthermore, the spurious Si peak can be much larger if electrons are actually transmitted through the BE window [18]. Thus, although there is evidence to suggest that the "classic" silicon dead layer is a result of charge-collection processes

which may vary only slightly from detector to detector and could in principle be calculated,

there may be an amount of dead silicon on the surface which is a function of the method of

fabrication used and is essentially indeterminate. Detectors which have been prepared by

the same procedure still exhibit spurious Si peaks which vary in size and this observation

was corroborated at this meeting (J. Russ, Edax Laboratories).

As a final note in this section, it seems worth pointing out that layers containing

silicon do not necessarily have to be connected with the detector itself. Vacuum oils and

greases and proprietary vacuum-sealant sprays often contain silicon, so spurious silicon

peaks and "dead layer" effects may be in part due to contamination of the entrance window.

1.4 Pulse detection efficiency

Because electrical noise is present, there is always a chance that a pulse may be

"missed" by the measurement electronics. This is easiest to understand and to quantify if we look at the behavior of an amplifier which produces unipolar output pulses which are detected if they exceed a threshold voltage [19]. If noise is present, there is a chance

that a pulse may arrive at the same time as a negative-going noise excursion so that the

resultant voltage waveform fails to exceed the threshold. The smaller the pulse, the more likely this is to occur so the pulse-detection efficiency decreases at low energies. The

effect on the overall spectrometer efficiency depends on the noise performance of the

channel which is effectively being used to detect the pulse. In many systems it is the

discriminator on the output of the main processing channel which open the ADC gate and

effectively decides whether a pulse is detected or not. In this case, the noise typically

corresponds to 100 eV f.w. h.m. so the efficiency would begin to decrease below an energy

of 250 eV if the threshold were set just into the noise distribution [19]. At this

meeting, Ron Musket (Kevex Corporation) pointed out that the carbon peak obtained with a "windowless" detector does not show the same low-energy tail as does the oxygen peak. It

is possible that the rapid fall-off in pulse-detection efficiency below 250 eV is responsible for this phenomenon since it would in effect attenuate the lower energy pulses

preferentially and tend to correct the asymmetry caused by incomplete charge collection.

The asymmetry would also be affected by non-linearity in the vicinity of the threshold

associated with "peak-detect" pulse measurement. With systems having a beryllium window,

the ADC threshold can be raised to exclude low-energy pulses. In this case, the pulse

detection efficiency goes through the transition from 100 percent to 0 percent in an energy

region extending from roughly 130 eV above to 130 eV below the effective threshold energy.

In some electronic processors, the detection of a pulse is determined by an auxiliary

inspection channel which can also be used for pile-up rejection. Since shorter time

constants are involved the noise is much greater than in the measurement channel so the

roll -off in efficiency occurs at a much higher energy. With a 1 us shaper, the efficiency

typically begins to fall below about 800 eV when the discriminator is set close to noise

[19]. However, if the discriminator thereshold is set well above noise, or if shorter time

constants are used, the roll-off will occur at higher energies and will affect the overall

efficiency in roughly the same way as would be observed with a thicker Be window.

1.5 Efficiency calculations

While engaged in my Ph.D. research [20], I had the opportunity to study two detector

systems because the first detector with a 1 mil Be window was replaced with an improved

version with a 1/3 mil window. Since the overall efficiency of both systems appeared to

be stable with time, the Ka peak intensities for all the electron microprobe standards were compared with the intensity for CoKa from a pure cobalt specimen and standard factors were stored on disk so that subsequently only the CoKa intensity was required to determine

all standard intensities. Now the intensity for a given Ka line is given by

132
Ka

Ka + Kp

R f(x) F fesc

e (1)

where C = element concentration by weight

N = Avogadros number

A = atomic weight

Q = ionization cross section

E Q = incident beam energy

E

c = critical excitation potential

S = stopping power

E = electron energy

fluorescence yield

Ka

Ka + Kp

R = backscatter correction

f(x) = absorption correction

F = fluorescence correction

*esc = correction for escape effect

e = spectrometer efficiency

so by calculating all the unknown terms, the ratio of efficiencies, e e

-| m-t

/'e cobal t ' cou ^ c' be determined from the individual standard factors, *elmt Ka^CoKa. Different calculations

were made using the data from one system . The same formulae were used for most of the

terms [20] with exception of the generation integral involving Q where two different methods were employed. In one, the integral was evaluated numerically using the Green/Cosslett [21]

expression for Q and the Bethe stopping power law [22] and the other used the approximate

formula (U-l) 1

' 67

, with U = E Q

/E

C

» which was suggested by Green and Cosslett [21] and has

also been used in a standardless approach by Russ [8]. Thus, from a single set of standard

factors, two sets of apparent relative efficiency values were calculated. The realtive

efficiencies, £

e i mt/

£

C obalt' were tnen fitted by a least squares technique to the formula

k exp / -1.82 \ Xb - 2.33 / H

, Be \ *si

p Be/ \ pSi/

J

2

where both the scaling constant k and the thickness parameters Xge and Xg,- were to be

determined. Heinrich's tables [23] were used for mass absorption coefficients, (u/p).

The purpose of the original study was to see whether a formula, as in equation (2), could

be used to parameterize the efficiency term in equation (1) so that standard factors could

be predicted from theory. As seen in Figures 5 and 6, equation (2) fits the derived

efficiency ratios to within about 5 percent relative error, irrespective of the formula

used for the generation term in equation (1). However, note that those window thickness

parameters which are necessary to ensure such a good fit are very different in the two

cases. In other words, differences in the theoretical formulae for x-ray generation are compensated by using different values for the Be and Si window thicknesses. It is not

surprising that the efficiency calculated by an indirect method should give results which

133
1.0

0.5

14pm Be

0.32pm Si

3 4 5

ENERGY keV

8

Figure 5. Efficiency relative to that for CoKa, calculated from intensity data from

system 2. The intensity integral was evaluated explicitly in equation (1)

and the solid curve shows the fit obtained for equation (2).

134
ENERGY keV

Figure 6. Efficiency relative to that for CoKa, calculated from intensity data from

system 2. The approximation (U-l) 1

' 67 was used in equation (1) and the

solid curve shows the fit obtained for equation (2).

135
depend on the accuracy of the theories involved and there is a danger of attaching too much significance to the predicted values without appreciating the sources of error. The

examples shown clearly demonstrate the folly of assuming that fitted window thickness

parameters accurately describe the detector structure; although the manufacturer's nominal

specifications for Xge and were 8 pm and 0.1 pm respectively, the fitting method

predicted 14 um, 0.3 pm and 23 pm, 0.6 pm as alternative descriptions derived from the same set of experimental data. The formalism described by equations (1) and (2) is

obviously useful in that it correctly describes the overall trends in generated x-ray

intensity and therefore can be used to interpolate between experimental values. However,

it is safer to treat the formulae as a convenient way to mathematically parameterize a system rather than as a route to determining fundamental constants.

Similar arguments apply to the fitting of the bremsstrahlung background. Although

one set of window thicknesses may produce a good fit when used in a theoretical expression

for the background spectrum obtained under a specific set of experimental conditions, there

is no guarantee that the same set of thickness parameters will result in an equally good

fit when the experimental conditions (such as kV, take-off angle, specimen composition) are changed. Again, this is because the window absorption term tends to compensate for

inaccuracies in the theory, which are likely to vary as conditions are changed. In general,

fitting to the background and fitting to characteristic line data yield different results

for the optimum set of window thickness parameters, both because of inaccuracies in

theoretical expressions and because of fundamental considerations such as the silicon

"dead" layer, which, as described above has a different effect on characteristic lines and

the continuum.

The gold coating was not considered in equation (2) because of its relatively small

effect and in any case, the scatter of the data suggest that a 5 percent step due to a gold layer could not be determined with any confidence. The effects of both the gold

layer and silicon dead layer should cancel to some extent when one considers the ratio of

efficiencies for two systems and the plot in Figure 7 shows the results of fitting just a beryllium window term to the ratio of intensities obtained for the two systems. The

quality of the fit suggests that assumption of a uniform thickness layer and use of fitted

absorption coefficients is satisfactory for Be windows approximately 1 mil (25 pm) in

thickness.

1.6 Advantages of poor efficiency

Although it is obvious that good efficiency is required at low energies if one is to observe the light elements C, 0, N, it is not often realized that this will compromise the

high-energy performance. As mentioned above, the detection efficiency for the electronics

is very poor below 1 keV when shaping times are shorter than about 1 ps so pile-up rejection is virtually non-existent for x-rays in this region. Therefore, it is desirable to absorb unwanted low-energy x-rays with a thicker window to avoid the peak shift and

broadening which results from undetected pile-up [24]. A further problem concerns the

transmission of electrons through the window. Transmitted electrons usually appear in the

spectrum as a broad hump superimposed on the background [17] and those of low-energy will

go undetected by the pile-up inspector and thereby induce peak shift and broadening.

Although an 8 pm Be window should stop 20 kV electrons effectively, experiments indicate

that enough 20 kV electrons can penetrate a nominal 8 pm window to generate a significant

silicon peak from dead zones of silicon and at 30 kV, roughly half the observed continuum

is due to transmitted backscattered electrons [18] (in a SEM). The ideal solution is to install an electron trap but this can involve the generation of fields which upset the

electron optics of the microscope and may be impossible because of space restrictions.

If this is the case, then again it is desirable to use a thicker window, at the expense

of low-energy x-ray efficiency, to absorb the troublesome electrons. For voltages up to

30 kV, a 4 pm mylar film provides an effective barrier to electrons when placed in front

of a nominal 8 pm Be window [18].

136
14.6jjm Be

012345678

ENERGY keV

gure 7. Ratio of efficiencies for system 1 and system 2. The solid line shows the

fit obtained for equation (2). System 1: 1 mil Be, System 2: 1/3 mil Be (nominal thicknesses).

137
2. Conclusions

The nominal thicknesses quoted by a manufacturer for the layers constituting the

detector entrance window are only a guide to overall performance and are subject to wide

tolerance even when some method of physical measurement is possible. (In the case of the

Be window, this tolerance typically corresponds to a variation of 15 percent in the

efficiency at 1 keV). Furthermore, the individual layers are non- uniform in thickness so conventional absorption calculations may be inappropriate. Indirect methods of measuring

efficiency are limited by the accuracy of fundamental parameters in theoretical expressions

and formulae such as equation (2) should really be treated merely as convenient functions

for parameterizing performance. The silicon "dead" layer is primarily a charge-collection

phenomenon so its effect on efficiency is really a question of definition and the overall

efficiency may vary with time because of contamination build-up on either the Be window or the detector surface.

This summary paints a bleak picture but is intended to emphasize why characterization

of spectrometer performance is non-trivial at low-energies. At present, probably the only

way to determine efficiency accurately is to use a flow proportional counter as a "transmission cell" and calibrate the spectrometer using the method described by Loomis

and Keith [25]. Such a calibrated spectrometer could in principle be used to categorize

a suitable portable secondary standard, such as a wel 1 -characterized material excited by an electron beam or x-rays of known energy. It is perhaps worth mentioning that the

action of comparing peak intensity from an unknown with that from a standard cancels out

the efficiency term, which is only of importance for fundamental measurements or standardless analysis. Although use of thinner entrance windows effectively removes some of the

uncertainty in the efficiency term, low energy x-rays and electrons have a detrimental

effect on spectrometer performance because of undetected pile- up so, in practice, it is

desirable to use as thick a window as the experiment will tolerate.

References

[1] Lifshin, E. , Ciccarelli, M. F. , and Bolon, R. B. , Proc. 12th Ann. M.A.S. Conf

.

, Boston,

Mass. , U.S.A. , (1977), paper 104 and Proc. 8th Int. Conf. X-Ray Optics and Microanalysis

(in press).

[2] Ware, N. G. and Reed, S. J. B. , vL Phys. E^: Sci . Instr. , 6, 286 (1973).

[3] Lifshin, E. , Proc. 9th Ann. Conf. Microbeam Analysis Society (M.A.S.

)

, Ottawa,

Canada, paper 53 (1974).

[4] Fiori, C. E. , Myklebust, R. L. , and Heinrich, K. F. J., Anal. Chem. , 48, 172-176 (1976).

[5] Small, J., Heinrich, K. , Fiori, C. , Newbury, D. , and Myklebust, R. , Proc. 13th Ann. Conf. Microbeam Analysis Society (M.A.S.

)

, Ann Arbor, MI , U. S.A. , paper 56, (1978).

[6] Statham, P. J., X-Ray Spectrom. , 5, 154-168 (1976).

[7] Goldstein, J. I., Costley, J. L. , Lorimer, G. W. , and Reed, S. J. B. , Proc. Scanning

Electron Microscopy , Vol, I, 315-323 (1977).

[8] Russ, J. C. , Proc. 9th Ann. Conf. Microbeam Analysis Society (M. A. S.

)

, Ottawa, Canada,

paper 22 (1974).

[9] Alfassi, Z. B. and Nothman, R. , Nucl. Instr. & Meth. , 143 , 57-60 (1977).

[10] Heinrich, K. F. J., in The Electron Microprobe (McKinley, T. D. , Heinrich, K. F. J.,

andWittry, D. B. , eds.) Wiley, N. Y. , 1966, pp. 296-377.

[11] Jaklevic, J. M. and Goulding, F. S. , IEEE Trans. Nucl. Sci. , NS-18 (1), 187-191 (1971).

[12] Echlin, P., Scanning Electron Microscopy , Vol. I, 109-132 (1978) (SEM Inc. AMF O'Hare,

II 60666, U.S.A.) (1978).

138
[13] Brombach, J. D. , X-Ray Spectrometry , 7, 81-91 (1978).

[14] Steele, W. J., Proc. 9th Ann. Conf

.

on Electron Microprobe Analysis , Microbeam

Analysis Society, Ottawa, Canada, paper 19 (1974).

[15] Reed, S. J. B. and Ware, N. G. , J. Phys. E. : Sci. Instr. , 5, 582-584 (1972).

[16] Llacer, J., Haller, E. E. , and Cordi , R. C. , IEEE Trans. Nucl. Sci. , NS-24 (1),

53-60 (1977).

[17] Fiori, C. E. and Newbury, D. E. , Scanning Electron Microscopy , Vol

.

I, 401-422

(1978) (SEMInc, AMFO'Hare, II 60666, U.S.A.) (1978).

[18] Statham, P. J., 14th Ann. Microbeam Analysis Society Conference , San Antonio, Texas,

U.S.A. , 247-253 (1979).

[19] Statham, P. J., X-Ray Spectrom . , 6, 94-103 (1977).

[20] Statham, P. J., Ph.D. Thesis, University of Cambridge, England (1975).

[21] Green, M. and Cosslett, V. E. , Proc. Phys. Soc. , 78, 1206 (1961).

[22] Bethe, H. A. and Ashkin, J., in Experimental Nuclear Physics , 1_, Wiley, N. Y. , 1953,

pp. 166-357.

[23] Yakowitz, H. , Myklebust, R. L. , and Heinrich, K. F. J., NBS Tech. Note 796 , (1973) on "FRAME" Program.

[24] Kandiah, K. , in Physical Aspects of Electron Microscopy and Microbeam Analysis , Chapter 24 (Siegel, B. and Beaman, D. R. , eds.) Wiley & Sons, 1975.

[25] Loomis, T. C. and Keith, H. D. , Appl. Spectroscopy , 29, 316-322 (1975).

139

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ELECTRONIC TECHNIQUES FOR PULSE-PROCESSING WITH SOLID-STATE X-RAY DETECTORS

P. J. Statham

Link Systems, Ltd.

Halifax Road, High Wycombe

Bucks. HP12 3SE

England

Abstract

This paper reviews the methods which have been proposed for

quantifying the output of a solid-state detector and summarise the

state-of-the-art with regard to noise and count rate performance for

x-ray systems. Ancillary techniques which improve the performance of

the basic pulse-processor such as pulsed excitation, pile-up rejection

and spectrum stabilisation will also be discussed and the fundamental

limitations will be analyzed.

1. Introduction

To a large extent, detailed knowledge of the operation of pulse processing electronics is no longer a pre-requisite for successful x-ray energy spectrometry because

sophisticated design techniques have gone a long way to reducing the apparent complexity

to that of a simple amplifier with just input, output and power supply connections.

Indeed, with the astonishing technological advancement in the field of digital electronics

and consequent trend towards automation, there has been a tendency to de-emphasize the

importance of analogue electronics and to concentrate on the more prestigious aspects of

a computer based system such as interactive colour graphic monitors with a variety of

display modes, multi-port data acquisition memories and automation packages for microscopes and XRF equipment. Certainly, the early problems of amplifier lock-up, drift,

excessive pile-up and count- rate-dependent peak shift have been solved to the extent that

if incident x-rays are limited to an energy range from 1 keV to 20 keV and to count rates

less than 10 kcps, then characteristic peaks will appear within one channel of their

expected position in a 1000-channel spectrum and qualitative analysis can be performed

without difficulty. However, comparatively new fields, such as light element analysis

with particle excitation and analytical electron microscopy, involve x-rays with energies

extending far outside the 1 keV to 20 keV range and an understanding of the limitations

of pulse processing electronics is essential to the correct interpretation of spectra.

Even with a restricted range of energies, second-order defects within the electronics

can have disastrous consequences for quantitative analysis when severely-overlapping

peaks have to be resolved and an appreciation of the fundamentals can help in deciding

whether or not a given pulse amplifier will be adequate for a particular application.

This paper will cover the salient features of various electronic techniques without paying

attention to the details of implementation (which are really the reserve of a design

engineer) and thus attempt to give the reader enough insight to predict situations where

performance of a given system is likely to suffer.

2. Head Amplifier

When a detector, appropriately biased, is exposed to a beam of x-rays, each detected

photon is converted to a packet of charge which is collected at the electrodes to give a voltage proportional to energy and inversely proportional to the effective capacitance of

the reverse-biased junction. Since this capacity changes with voltage, a charge-sensitive

amplifier configuration is normally employed as the first stage in the electronic chain

141
(figure 1). A field effect transistor (FET) is now exclusively used at the signal input

because of its low noise properties which are improved by cooling to about 100 °K. The

gate is connected directly to the detector to avoid the noise component and risk of damage

due to breakdown which would be present if a coupling capacitor were used. The detector

and FET leakage currents and the mean current due to arrival of photons at the detector

will cause the output to drift in such a dc-coupled system so some method of stabilisation

is required. The charge produced by a low-energy x-ray photon is of the same order as that

produced by a current of 0.1 nA flowing for 1 ps so the feedback current has to be very

small and could be achieved by connecting a very-high-valued resistor as shown in figure 1. However, such high-valued resistors have reactive components, may change resistance at low

temperatures, involve extra noise due to added capacitance and generate extraneous noise

[l] 1 so that resistive feedback offers a rather poor compromise between noise performance

and stabilisation for x-ray work. Several alternative feedback techniques have been sug- gested but the one that really revolutionised the field was opto-electronic charge neutralisation.

CRYOSTAT HOUSING

1

Cf

O

(DETECTOR)

7777777

Rf WW

6 ' EHT. -veL J

Figure 1. Schematic of head amplifier. The feedback capacitor, C

f

, is connected to make the amplifier charge-sensitive. In modern systems, the feedback

resistor, R^, is usually replaced by more effective electronic schemes

(see text). Components within the cryostat housing (dashed line) are kept cool by liquid nitrogen.

142
2.1 Opto-electronic charge neutralisation

In 1966, Kandiah [2] suggested the use of photodiodes for charge restoration at the

head amplifier input. Two GaAs lamps were placed near two small silicon photodiodes so that pulses from control logic could drive the input in either positive or negative directions (figure 2(a)). When the output of the head amplifier drifted outside limits defined

by threshold discriminators, suitable control signals were generated to restore the input

to zero volts thus providing "pulsed restoration". In 1968, this system was improved by

modifying the control to neutralise the head after every event. This technique of "pulseby-pulse restoration" avoided the massive restore pulses which were required to neutralise

the head after several events and by restricting the dynamic swing, also avoided the need

to maintain high linearity in the amplifier over a range many times greater than that

covered by individual pulses.

Unlike the high-valued resistor, the opto-electronic link behaves as an almost perfect

linear attenuator with good high frequency response and in 1969 Goulding et aj. , [4] decided

to use such a link in the continuous feedback mode (figure 2(b)). Although this was a significant improvement over resistive feedback, the feedback current still had to increase

with count rate to balance the mean detector current, with a consequent increase in noise,

and the LED/photodiode combination was still found to exhibit non-linearity which had a detrimental effect on high count rate performance of the system. Perhaps the most significant development described in the paper [4] was the use of the drain-gate junction of the

FET itself as the photodiode (figure 2(b)), thus avoiding the added capacity (and noise)

due to an extra discrete component. The advantages of this modification were only fully

realised when it was used in the pulsed restoration mode. In 1970, Landis et a]_. , [5]

described a system which neutralised the FET input after several events, as in Kandiah'

s

original scheme [2], and in 1975, Kandiah et a_L , [6] described a spectrometer using pulseby-pulse restoration, specifically designed for x-ray work, where they acknowledged the

improvement afforded by use of the FET as the restoring photodiode. Thus, systems in

current use are mostly based on the arrangement shown in figure 2(c).

(a)

Figure 2. Opto-electronic charge neu- tralisation schemes:

HT

(b)

(a) Pulsed feedback and

pulse-by-pulse restoration

using silicon photodiodes

(Kandiah 1966; Kandiah et aj. , 1968).

(b) Continuous feedback using

FET junction (Goulding et aj.

,

1969).

(c) Modern schemes using FET

junction. (Pulsed feedback,

Landis et aJL , 1970, pulse-bypulse restoration, Kandiah et

aj. , 1975).

143
The historical development is interesting because it shows the trend to remove non- essential components in order to reduce the noise component due to stray capacitance. In particular, note that only a single photodiode is shown in figure 2(c) because if the

detector leakage current is greater than the FET leakage current, then, with or without x- rays, the amplifier output will always drift in the same direction. Unfortunately, the

leakage current in modern detectors is often smaller than the FET leakage so, that in the

absence of x-rays, or even at very low count rates (< 100 cps), the amplifier may drift in

a direction where it cannot be restored by a single photodiode; so the amplifier "drops

out" or stops "ramping" and the spectrometer becomes inoperative. Although a burst of x- rays will bring the system back to life, this situation is obviously unsatisfactory. One

solution is to enhance the detector leakage by raising the EHT voltage, but this involves

the risk of breakdown and should not be done without consulting the manufacturer first.

However, it may transpire that the only realistic solution is to include a second photodiode, as in Kandiah's original system [2], and to accept the slight noise penalty.

2.2 Systems in current use

A comprehensive review of pulsed feedback techniques is given by Landis et aj. , [7]

but, for a variety of reasons, there are no serious competitors for opto-electronic restoration in the field of low-energy x-ray spectrometry. However, there is another continuous

feedback technique called "drain feedback" which is used in some commercial x-ray systems.

This method, introduced by El ad [8], involves regulating the FET drain voltage according to the input energy times count-rate product whereupon a component of gate current produced

by impact ionisation provides sufficient charge to neutralise the detector. Apart from

avoiding the complication of mounting extra components near the detector, drain feedback

apparently does not suffer from the potential problems caused by long-lived surface states

which are excited when light shines on the FET and there is no dead-time penalty associated

with resetting of the head amplifier. However, at high count rates, drain feedback, as with any continuous feedback approach, suffers the disadvantages of enhanced noise owing to the increase in gate current necessary to balance the x-ray- induced detector current.

Opto-electronic feedback is most commonly implemented in pulsed mode whereby the

charge due to several events is neutralised when the amplifier output exceeds a given

threshold level. The main amplifier has to be gated to inhibit the effects of the large

restore transient and some time is lost while the electronics recover but the dead time

penalty is normally only significant at count rates in excess of 10 kcps. A potentially

more serious problem is that of the after-effects of light pulsing. Charge stored in

surface states can sometimes take a long time to leak away and leads to degraded resolution

at high count rates. This effect has been noted in photodiodes [2] and FETs [5] but is

most severe in the detector [5] so it is very important to prevent stray light from reaching the detector surface. It should be emphasised that this is a potential problem and

with a good system, the high-count-rate performance is better than with drain feedback. As mentioned above, pulse-by-pulse opto-electronic restoration has the advantage that the head

amplifier output is kept within a small dynamic range and the restore transient is about

the same size as the original signal. In a time-variant processor, the reset recovery time

can be incorporated as part of the pulse processing interval so there is no dead time

penalty [6] and, because the restore signal is smaller, there may be less of a problem due

to after-effects of light pulsing, although this depends on the exact nature of the

problem.

To summarise, whereas there is no theoretical argument to suggest that drain feedback

or either of the pulsed opto-electronic techniques is substantially better than the others

for low-energy x-ray spectrometry at low count rates, the pulsed systems do perform better

when the energy times count-rate product gets large and the pulse-by-pulse opto-electronic

system, though difficult to implement, will handle the greatest dynamic range without any

compromise to the low-energy, low-count-rate performance.

144
3. Pulse Processing

The output of the head amplifier is a series of steps with heights proportional to the energy of the x-ray photons involved and the primary aim of the pulse processor is to

obtain an accurate measurement of the height of each step. Each measurement will be

affected by the history of the system before the event and electrical noise, and these

factors dictate the essential features of pulse shaping:

(1) High frequency noise must be suppressed. "Integration" is therfore involved which

smooths out the front edge of the step.

(2) The amplifier must be restored to its quiescent state as fast as possible after the

event so that succeeding events will be unaffected. This involves "differentiation"

which shapes the step into a pulse of a finite length.

(3) The height of the pulse must be measured and the result must be proportional to the

original step height. This is facilitated if the pulse has a fairly flat top.

Before considering the relative merits of various processing schemes, it is instructive to consider some of the fundamental problems which face the designer of such a system. In order to reduce the effect of dc drift and slow variations at the input, ac coupling can

be used, which also frees the designer from the problems of matching dc operating points

of the various stages in the amplifier. Thus, the pulse is subject to one or more "secondary" time-constants, due to interstage couplings and "decoupling" capacitors, which

are equivalent to CR-differentiator stages with long time-constants. In figure 3(a), the

effect of passing a unipolar pulse through a differentiating stage is demonstrated with

reference to an idealised rectangular pulse shape. After reaching a peak, the output

pulse decays exponentially according to the time-constant T

g = CRg and therefore "droops"

below the input waveform. When the negative-going edge arrives, the output undershoot

decays with time-constant, t

g

. Since the stage cannot transmit a dc level, the positive

and negative areas must be equal. If t

g is made bigger to reduce the "droop", the undershoot is smaller but lasts longer so that there is no easy way of avoiding the effect on a subsequent pulse which is shown as a baseline movement in figure 3(a). In figure 3(b),

the effect on a bipolar pulse is considered. In this case, the droop on the first lobe of

the pulse almost cancels the overshoot for the second lobe so the decaying residual

voltage is proportional to (T/T

g

) 2

. Thus, the residual effect on a subsequent pulse is

considerably smaller and bipolar shaping gives excellent baseline stability at high count

rates. Unfortunately, the poor signal-to-noise ratio precludes the use of bipolar shaping

for low-energy x-ray work, although it can still be useful for pile-up rejection and event

recognition for time-variant processors [3,6].

145
v "Hr

i

» vout

VCR,

^droopwl

baseline movement

overshoot ch I

\

Figure 3. Effect of secondary differentiating time constant on a pulse.

(a) Unipolar rectangular pulse of width T « t . (b) Bipolar rectangular pulse. A second pulse is shown to illustrate how residual voltage affects subsequent pulse

measurements.

3.1 Pole-zero cancellation

Although the undershoot problem for unipolar pulses cannot be cured by a simple

change of time-constant, the technique of pole-zero cancellation [9] can be used. If a step

waveform is passed through a single stage of differentiation, a unipolar pulse will be

produced. As shown in figure 3(a), if the unipolar pulse is passed through a secondary

stage of differentiation, an undershoot is produced. In the present context, the effect of

a secondary differentiating time-constant, x

g is easier to understand if we assume that the

input step has passed through such a stage and is decaying to zero with time constant x

g before it reaches the main shaping differentiator (see figure 4(a)). The output from the

shaper would decay to zero, were it presented with a flat-topped step, but undershoots

because of the decay introduced by x . The undershoot decays with time-constant x

g (x >> x) and the technique of pole-zero cancellation involves feeding some of the input

waveform through to the output to cancel the undershoot (figure 4(b)). By choosing the

shunt resistor, R

g

, so that CR

g = xg

, the output waveform no longer undershoots and decays

146
c

Figure 4. Pole-zero cancellation.

to zero with time-constant governed by the main shaper. One instance where pole-zero can- cellation is particularly relevant is with continuous feedback (e.g., drain feedback) where

the steps at the head amplifier output decay with a time-constant which is usually much

greater than that of the main pulse shaper. The technique can be applied more than once to cancel the effect of other secondary time-constants, if these are present, but note that

the modified stage in figure 4(b) involves dc coupling between input and output and this

may not be permissible in some instances.

The term "pole-zero" stems from the theory of linear circuits and cancellation of a "zero" by a "pole" is only totally effective if all circuits are linear. Non-linearity can appear in an amplifier which is driven close to saturation or in feedback components such

as high-valued resistors [4] and results in decays which are not purely exponential. In this case the pole-zero cancellation circuits have to be adjusted to provide the best

compromise in performance for a particular application. When pole-zero circuits have not

been adjusted properly, peaks exhibit distortion which may appear as low-energy tailing

[10]. Non-linearities and any pecularities in detector response combine to make the adjustment sensitive to both count rate and spectral content. This brings out the important

point that the performance of the pulse processor depends on the detector and head amplifier and cannot always be treated in isolation.

3.2 Baseline restorers

Although pole-zero cancellation goes some way to solving the problem, baseline fluctuations are invariably present because of residual pulse tails and undershoots due to uncancelled secondary time-constants. Therefore, some form of keeping the quiescent baseline level close to zero volts is required; the circuit used is called a baseline restorer

(BLR). Various schemes have been suggested and are reviewed by Fairstein [11] and Karlovac

[12]. Figure 5 summarises the basic features of each method and shows how the unit behaves

when presented with an idealised rectangular pulse and with a similar pulse with an undershoot represented by a negative lobe. The symmetric restorer [13] (figure 5(a)) affects

both positive and negative excursions equally. When presented with a positive pulse, the

left diode is switched off and the current, I, discharges the capacitor and produces a droop on the output pulse. When the back edge of the input pulse drives the output

negative, both diodes become conducting and a net current, 21- I = I, charges the capacitor

and rapidly restores the small undershoot to zero. When the input pulse has an undershoot,

or negative lobe, the restorer operates just as effectively and attempts to neutralise both

the main pulse and undershoot with equal force. At high count rates, there is greater

chance for a pulse to arrive before the effect of the previous pulse has been neutralised

and symmetric restorers cease to become effective at count rates >0.5/T, where T is the

pulse width at the baseline. If unipolar pulses are involved, an asymmetric restorer

147
(figure 5(b)) can permit operation at higher count rates. In this case, the rate of restoration for positive excursions is less than thatfor negative excursions; so there is less

droop in a positive pulse and the correspondingly-smaller undershoot is more rapidly

restored to zero. Unfortunately, if the input pulse has a negative undershoot, the capacitor stores charge during the undershoot period and an after-pulse is generated which being

positive, is only neutralised slowly (lower waveform, figure 5(b)). The resultant output

pulse has a greater positive dc component than the input pulse; this demonstrates the

rectifying property of an asymmetric restorer which will also rectify noise to give a baseline offset proportional to the noise level. These problems are removed if the restoring

currents are switched off during the pulse. Such a gated, or time-variant, BLR is shown in

figure 5(c). There is no droop on a pulse because the gate signal keeps the switch open and

any undershoot is restored symmetrically, without generating any long-lived after-pulses.

(c) —

I

•openGATE

SIGNAL

Figure 5. Baseline restorers,

(a) Symmetric.

(b) Asymmetric.

(c) "Time-variant" or "gated". Each circuit is an idealised

representation of one implementation of each type of

BLR. Tye top waveform in each case shows the response

to a unipolar rectangular pulse and the lower waveform

shows the response to a pulse with undershoot, approximated by a bipolar rectangular pulse.

148
The main function of the BLR is to remove baseline fluctuations which could be considered as low-frequency "noise" but this involves a compromise with respect to the response

to high-frequency noise. The noise "introduced" by a BLR has been calculated by Radeka

[14] but crudely speaking, the noise increases as the restoring currents are increased. In practice, this limits the effective restoring time-constant to be more than 10 x the pulse

duration [12]. On earlier amplifiers a switch was usually available to enable the amount

of restoration to be matched to the count rate but nowadays it is more common for x-ray

systems to be provided with a fixed restorer which is only optimised for low-count-rate

work with low-energy x-rays. This may mean that special modification will be necessary for

the system to be useful if it is to operate at high rates, as can occur with pulsed excitation, or with high-energy x-rays that are produced in a transmission microscope, for

example. If the BLR setting is not optimised, count-rate-dependent peak shift and broadening are apparent [10].

The gated BLR offers the best compromise between good baseline stability and noise

performance but, since the stabiliser is only switched on in the interval between pulses,

the characteristics are a function of count rate. Thus, suppression of low-frequency

fluctuations is less effective and noise is greater at high rates. Nevertheless, the highcount-rate performance is usually considerably better than with "passive" restorers [12].

At this stage, it is worth mentioning that figure 6 is only intended to point out the

essential features and practical circuits are considerably more sophisticated than is

suggested by the circuit diagrams.

4. The Kandiah/Harwel 1 Processor

So far, the pulse-processing techniques that have been described represent a "piece

meal" approach to the various problems and most systems are arranged as in figure 6 with a shaping unit, including pole-zero cancellation, followed by a baseline restorer and finally

the analogue-to-digital converter (ADC). However, Kandiah [2,3,6] took an entirely

different approach and redesigned the system to try and eliminate the source of many of the

problems. The operation of the whole system is co-ordinated by control logic which also

issues the signal to restore the head amplifier (figure 7). DC coupling is used throughout

which immediately improves high-count-rate performance and time-constants are switched by

the control logic to minimise the effective "memory" of previous events.

A simplistic view of the processing of a single pulse is shown in figure 8(a). The

main processing channel is effectively grounded until the arrival of a pulse is detected by

the fast recognition channel. At this time, the time constants are switched to change the

amplifier characteristics and the pulse is integrated for a precise interval. The charge

on the integrating capacitor is held long enough to allow the ADC to record and hence measure the voltage and then the time constants are switched again to rapidly restore the

system to its quiescent condition. In the pulse-by-pulse restoration mode, the head amplifier is also neutralised at this stage so that, in effect, all memory of the event is

removed. The "protect" interval which follows is necessary to allow the enhanced noise

[15], which results from the switching, to die away so that when the system is ready to accept another pulse; subsequent measurements will not be significantly affected by such

noise. The signal-to-noise ratio primarily depends on the effective time taken for the

measurement; so it is fair to compare the pulse duration with that for a typical "passive"

shaper which has a peaking time equal to the integration time for the Kandiah system

(figure 8(b)). Clearly, for similar noise performance, the dead time associated with the

time-variant filter is much less than for the passive shaper, despite the protect interval,

so that much higher throughput rates can be achieved.

149
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151
MEASUREMENT

READY TO ACCEPT

ANOTHER PULSE

PEAKING TIME

Figure 8. (a) Waveform showing various stages in the shaping of a pulse in the timevariant Kandiah scheme.

(b) Typical waveform for a non-time-variant or "passive" pulse shaper.

4. 1 Throughput rate

Deighton [16] has compared the theoretical performance of a time-variant processor

with that of an amplifier using "semi-Gaussian" shaping (one differentiating CR, four

integrating CR's). Besides the reduced residual voltage at the end of each pulse and

removal of baseline fluctuations, Deighton demonstrates the improved counting efficiency.

With the semi-Gaussian shaper, a time-to-peak of 20 us is typical for x-ray work and after

80 us, the residual voltage is 0.16 percent of the pulse height. If an amplifier is

designed to reject a pulse unless a preceding interval of 80 us and following interval of

20 us are free from other pulses, then the maximum throughput rate that can be achieved is

approximately 3.7 kcps. On the other hand, the switched time-constant, gated integrator

shaper can be designed with a time-to-peak of 20 us, measurement and restore time of 7 us and protect time of 13 us so that the maximum throughput rate is approximately 7.3 kcps.

4.2 Baseline stability

In practice, the Kandiah processor can still be used at input count rates of 100 kcps

with marginal peak shift but, of course, it is pointless to work under these conditions

when greater throughputs can be obtained at much lower rates. To put this into perspective, time-variant or gated BLR's can provide similar baseline stability up to 30 kcps,

which is adequate for x-ray work, but a conventional BLR becomes ineffective at relatively

low count rates and peak centroid shifts greater than 10 eV (0.29% at MnKa) are typically

observed at 10 kcps input rate. This comparison again assumes that a pulse peaking time of

about 20 us is involved (with four CR- integrators, peaking time = 4 x time constant).

152
4. 3 Zero stabil iser

With the problems of baseline fluctuation essentially solved, the added refinement in

the Kandiah system is a zero stabiliser. At regular intervals, and provided a pulse is not already being measured, the control logic is triggered to perform a pulse measurement

in the usual way and thus monitors the baseline level. The resulting measurements are added to the spectrum and constitute a "strobed zero peak" whose shape faithfully reflects

the electronic contribution to any peak in the spectrum. The zero stabiliser works by

comparing the result of each strobed-zero measurement with a fixed level and using the

result to modify the dc offset within the processor. The principles of "spectrum stabilisation" are described in detail by Dudley and Scarpatetti [17].

4.4 Overload recovery

Another feature of the system is the ability to restore the electronics at any time

to their quiescent condition, within a period not much greater than the "protect time"

(figure 8(a)). Thus, a "restore" can be issued to quickly remove a pulse rejected by

the pile-up inspector but, perhaps even more important, large pulses which would normally

overload the amplifier and cause peak broadening can be nullified as soon as they are detected, thus also saving much of the processing time and possibly paralysis time which

would otherwise be wasted. This rapid recovery from overload is very important for

spectrometers attached to high voltage electron microscopes because a 100 kV (or a 1 MeV!

)

photon can severely overload a system optimised for x-rays in the 1 to 10 keV energy

range. Walter [10] shows one example where the overload due to 122 keV y-rays from a Co[57] source degraded the resolution of an FeKa peak by 10 eV. Even in XRF systems

operating at much lower energies, the fast reject facility is useful in saving some of

the dead time normally spent in processing the dominant large-amplitude backscatter

pulses, and trace element x-rays can therefore be recorded at higher rates.

In reviewing the field of analogue signal processing in 1976, Goulding [18] suggested

that the unit described by Kandiah et aK , [6] designed as an integral processing package

with convenient operator controls, might well become the basis for the design of future

x-ray spectrometers. In fact, a prototype system was evaluated for use in microprobe

analysis in 1973 [19] and a commercial system has now been on the market for almost three

years [20]. Once the complexity of circuit design has been mastered, the advantages of

the time-variant, dc-coupled approach are readily apparent so it is indeed likely that

other systems based on these ideas will appear in the future.

5. Noise Performance

The electronic noise contribution to line width becomes progressively more important

at low energies and is the major factor in determining the resolution, and thus peakto-background ratio, at the NaKa position for example. The standard method of calculating

the signal-to-noise ratio involves integration of the noise power spectrum in the

frequency domain (see, for example, Tsukada [21]). However, Deighton [22] developed a time-domain method for noise calcuations which is much more suitable for time-variant

systems where the effect of a noise pulse depends upon its time of arrival relative to that of a signal pulse. Goulding [23] summarises this technique and introduces the

concepts of "step" noise and "delta" noise (previously termed "parallel" and "series"

noise): Charge due to electron flow in the input circuit is effectively in paral lei with

the signal current and is integrated by the input circuit capacitance to appear as tiny

voltage steps . Flow of current through the input and subsequent amplifying elements is

effectively added in series with the signal and the discrete nature of the charge can be

represented by a delta-function voltage generator in series with the input circuit. The

response of the processor to such signals involves straightforward calculations but there

are other noise sources, described by terms such as "generation-recombination",

"dielectric", "trapping", "surface", "excess resistor" and "flicker" or "1/f" noise, which

are not so easy to analyse. However, "1/f" noise has been discussed, for example by

Koeman [24], and Llacer [25] analyses the performance of various filters in the presence

of "step", "delta" and "1/f" noise.

153
Despite the apparent complexity of noise calculations, the general features can be

appreciated by the simplified representation shown in figure 9. If the (fwhm) 2 of

the noise distribution is plotted against the time constant, t, or alternatively the

peaking time for a given shaping circuit, the noise calculations can be separated as

fol lows:

C

2 1 noise (fwhm) 2 = k — - + k . I x + k, , s %

x P Vf (1)

where k

g

, kp and k^ are constants which depend on the design of the shaper and are associated with "series" ("delta"), "parallel" ("step") and "1/f" noise respectively. C is the

total effective capacitance at the input composed of the sum of detector, FET and feedback

capacities, I is the sum of detector and FET leakage currents and an equivalent feedback

current and gm

is the mutual conductance of the FET in mA/V. When the three components are

added, the total noise passes through a minimum at a particular time constant value. In figure 10, the optimum noise fwhm is shown for several different shaping schemes on the

assumption that there is no "1/f" noise. The exponential cusp (figure 10(a)) is the

theoretical optimum step response and all fwhm values are shown relative to that obtained

with such a filter. In practice, such a cusp can only be approximated and it is considerably easier to implement the shapers shown in figure 10(b) - 10(d). The semi-Gaussian

shaper (figure 10(c)) gives an optimum noise performance 16 percent worse than the cusp

but if more CR- integrators are included, or some other technique is used to make the pulse

shape more symmetrical [26], then the noise can be reduced by a further four percent

(figure 10(b)). The simple shaper with only a single stage of differentiation and integration (figure 10(d)) is rarely used, because of its poor noise performance and long tail,

and other shapers, such as the triangle or trapezoid, are presumably not used because of

expense of implementation, even though the noise performance is very close to that of the

cusp. The Kandiah-type shaper (figure 10(e)) with a switched integrating time-constant

and gated integrator gives a theoretical performance only two percent worse than the cusp

[22] which might suggest that the noise performance is 10 - 14 percent better than that of

the commonly-used passive shapers, 10(b) and 10(c). However, this comparison is unfair

because it only considers the minimum noise that can be achieved and this might occur at such a large value of time-constant to be physically impractical (figure 9). Furthermore,

"1/f" and other noise sources have been ignored in this treatment.

Time constant, X

Figure 9. Electronic noise as a function of time constant

used in a given pulse shaper. The various con- tributions are explained in the text and are given by equation (1).

154
(a) CUSP / \

/ V 1.00

1.12

1.16

1.36

1.02

Figure 10. Optimum noise performance for different pulseshaping schemes. The numbers on the right indicate

relative fwhm values. Calculations only account

for "series" and "parallel" noise contributions.

In spectrometers using pulsed opto-electronic restoration, there is effectively no noise- inducing feedback so that the "parallel" noise, which increases with time constant,

is relatively small. Llacer [25] suggests that, with the long measurement times used for

x-ray work, "1/f" noise is a dominant contribution so the filter should be optimised with

respect to this noise. His analysis suggests that a tailored cusp shape would improve the

noise fwhm by typically 8-10 eV over that for a 7th order Gaussian. Unfortunately, the

cusp is difficult to implement but Llacer [25] observed that the step response for

Kandiah's time-variant shaper (figure 9(d)) shows marked similarities to that for the cusp

(a point also noted by Deighton [22]) and in fact a complete analysis of a particular

example suggested that the Kandiah filter would perform even better than the cusp, giving a fwhm of 75 eV, as compared to 82.5 eV for the cusp and 90.5 eV for the 7th order Gaussian

(at a peaking time of 32 us). In a more recent paper, Llacer and Meier [27] followed up this theory by comparing the performance of a conventional 7th order pseudo-Gaussian filter

with that of a commercial processor using the Kandiah design [20]. They found that the

theoretical findings were borne out in practice and the improved performance of the Kandiah

design for both "series" and "1/f" noise resulted in as much as 15 eV reduction out of 90 eV fwhm at long time constants. In particular, at a peaking time of 20 us, the Kandiah

system gave about 95 eV fwhm as compared to 110 eV fwhm for the pseudo-Gaussian amplifier.

155
The "resolution-race" has traditionally been concerned with improvements in the fwhm

for the MnKa peak (at 5.9 keV) obtained with an Fe-55 radioactive source. Although this is a reasonable way of summarising overall improvements in both detector fabrication techniques and electronic noise, the graph with time [28] shows signs of an asymptotic approach

to a fwhm only slightly below 140 eV which might suggest that significant improvements in

spectrometer performance are unlikely. At 5.9 keV, the separation of Ka peaks is roughly

500 eV (figure 11) so indeed, there is little to be gained by improving the resolution

at 5.9 keV (unless, of course, it could be reduced sufficiently to allow KB/Ka overlaps

to be resolved). However, down at 1 keV, the Ka separation is only 200 eV and in addition

there is a significant problem with overlaps from L and M-series peaks so it is highly

desirable to improve the resolution. Two curves have been calculated to illustrate how a reduction in electronic noise from 100 eV to 50 eV would improve the overall resolution

for a spectrometer where the Fano- factor for the detector is 0.1 (figure 11). Although

halving the noise gives only about 25 eV improvement in the fwhm at 5.9 keV, the full

effect of the noise reduction is seen at 1 keV and below. Thus, reduction in electronic

noise is of vital significance in the field of x-ray spectrometry, and improvement is

most likely to arise from advances in FET design. Development of FET's with large gm and

small capacitance would obviously help to reduce the "series" noise, as indicted by

equation (1), but of equal importance is the reduction of "1/f" and other "extraneous"

noise contributions which dominate the behaviour at long time constants.

ENERGY keV

Figure 11. Resolution as a function of energy calculated for two systems, one with

a noise contribution of 100 eV fwhm and the other 50 eV fwhm, both with

a detector Fano-factor of 0.1. The thick curve shows the separation of

Ka peaks for adjacent atomic number elements (right hand scale) as a function of energy.

156
6. Pile-up Rejection

A full discussion on pile-up rejection appears elsewhere [29]; so here I will only

attempt to summarise the features which are of major interest in x-ray spectrometry.

The idealised diagram in figure 12 illustrates the most common method of pile-up rejection

which involves use of an auxiliary channel, with short time-constants, to separate pulses

which overlap in the main measurement channel because of the long time-constants involved.

Correct adjustment of the fast channel discriminator is essential if low-energy x-rays

(< 2 keV) are present in the spectrum; if the level is too high, small pulses due to these x-rays will be missed and pile-up rejection will be ineffective, while if the level

is too low, the amplifier will be saturated by the high rate of triggering of the discriminator by noise. A reasonable compromise is to set the discriminator to give a triggering rate of about 500 cps in the absence of x-rays. If the time constants are reduced to improve the pulse-pair resolution time, the noise is effectively increased and

the efficiency of detection is worse for low-energy x-rays, even with the discriminator

correctly adjusted. Although these considerations are of little consequence when the pileup is studied on a spectrum from Fe-55, they have serious implications for behaviour of

the spectrometer when presented for example, with a spectrum of aluminum excited by

electron bombardment. If the discriminator level is too high, or the pulse-pair resolution

is much less than 1 us, a continuum caused by unresolved pile-up, and excessive peak shift

will be observed even at count rates as low as 5 kcps.

6.1 Correcting the spectrum for pile-up

Even when the pile-up inspector is working properly, sum peaks can be troublesome

when they overlap with elements of interest (e.g., CaKa + SiKa ~ CrKa) or when they fall

in a region which is to be used for fitting a background curve. Automatic correction

for sum peaks would be a simple matter if the pulse-pair resolution time were constant

but unfortunately it usually depends on the relative heights of the pulses involved. A full correction of the spectrum could take longer than the original recording time [29]

and would therefore be no more effective than the solution of using a lower count rate with a larger recording interval. When high count rates are unavoidable, individual sum peaks can be corrected for by including suitably-shaped profiles to represent these peaks

in a least-squares fit to the spectrum but this is less satisfactory at low-energies

because sum peaks become distorted and merge into the pile-up continuum.

6.2 Electronic limitations

If no correction is to be applied to the spectrum it is advantageous to use two

inspection channels, one with longer time-constants, optimised for low-energy x-rays

and the other with much shorter time-constants to give a good resolving time for highenergy x-rays [6] (figure 7). The present state of the art in noise performance dictates

that 1 us is about the best resolving time that can be achieved for 1 keV pulses so in

microprobe analysis, where Na, Mg, Al , and Si peaks and bremsstrahlung continuum are present, the count rate has to be kept below 5 kcps in order to avoid significant pile-up.

Of course, with a windowless detector, the problems are much worse and shaping times of

the order of 10 us are required to detect x-rays down to 250 eV in energy which limits

the count rate to well below one kcps if pile-up is to be avoided.

6.3 Future developments

Pulse pile-up is perhaps the most serious limitation for low-energy x-ray spectrometry

if it limits the usable count rate to well below that which produces maximum throughput

rate. Reduction of electronic noise is therefore again of prime importance in the

development of improved spectrometers. One technique, which could, in principle, reduce

the effective resolving time significantly, involves "beam switching" [30]. In this case, since the source of x-rays is switched off as soon as a pulse is detected, the effective

resolving time is the time taken for a pulse to first exceed the threshold rather than

the time for the pulse to reach a peak and return below the threshold [29].

157
158
6.4 Guard-ring rejection

A refinement in detector design involves use of a double guard ring to define the

outer boundary of the sensitive volume of the detector [31]. The inner ring is connected

to a fast amplifier and discriminator so that if a charge cloud is collected via the

peripheral region, a signal will be detected from the ring. If such signals are used to reject subsequent processing of the main pulse, the system effectively provides electronic

collimation and thus eliminates many of the events that would otherwise suffer charge loss

in the peripheral region and would therefore add to the background shelf on the low-energy

side of a peak. The method is mentioned here because essentially the same principles

are involved as for pile-up rejection. The inspection channel connected to the inner

guard ring has a pulse detection efficiency governed by both electronic noise and the

discriminator level setting. If the ring only intercepts a small fraction of the main

charge cloud, the pulse will be correspondingly smaller and may be missed by the inspector. Therefore it is to be expected that the system would be ineffective at detecting

both pulses which have only been slightly degraded, and pulses which are severely degraded

where only a small fraction of the total charge is collected by central area and guard

ring alike. Thus, effects of poor charge collection would appear as counts just below

the main peak energy and also at very low energies. As the authors point out [31], it

is important to reduce noise so that small signals can be detected and the system does

indeed work better for higher energy x-rays, presumably because of the larger signals

involved. As in the case of pile-up rejection, the use of long-time-constant shaping

should improve the detection efficiency (unless of course, the guard ring channel is

subject to excessive low-frequency noise) and will give a greater signal if charge collection is slow due to weak fields at the periphery.

7. Pulsed Excitation

The pulsed excitation method operates in the following manner: as soon as an x-ray

photon is detected, the source of x-rays is switched off until the electronics has finished

processing the pulse. This ingenious idea was originally suggested in a paper by Jaklevic

et al . , [32]. These authors used a grid-modulated x-ray tube to pulse the exciting beam

in x-ray fluorescence equipment, effectively derandomising the arrival of pulses and hence

raising the efficiency and output rate that could be achieved. Basically, this a technique for avoiding pile-up and has already been mentioned above in the context of pile-up

rejection. By eliminating the time normally wasted on rejected pulses, the maximum

throughput rate can be increased by a factor of three or more over that attainable with

continuous excitation. In practice, this higher throughput rate cannot always be fully

exploited because of the time taken to switch off the source x-rays. When an x-ray

photon arrives at the detector, there is a delay before it is detected (which is dependent

on both photon energy and shaping time-constants used), a propogation delay while the

signal is transmitted to the grid control, and a decay while the field which deflects

the electrons is established. During this time, a second x-ray can arrive and "pile up"

with the first, so the count rate emitted when the x-ray source is on cannot be too high

or else significant sum peaks will appear in the spectrum.

Additional problems arise when pulsed excitation is implemented on an electron

microprobe [19,30] because x-rays are emitted from regions outside the probe spot while

the electron beam is being switched. The spectrum cannot be corrected for pile-up

involving such x-rays because they are emitted from areas of unknown composition. This

problem is minimised by switching the beam off very fast; with a voltage applied to parallel plates this can be achieved in 30 ns [30] and probably less with modern electronic components. Contamination of the spectrum with spurious x-rays can be avoided

by modifying the electronics to reject all pulses which arrive before the beam has

switched on and stabilised at the original probe position [30]. With these precautions,

quantitative analysis can be achieved at accummulation rates around 12 kcps [30].

Because of pile-up, the high-count-rate capability is not always easy to exploit

for low-energy x-ray work. However, there are still a number of advantages to be had

with pulsed excitation. Since the input rate is always less than with continuous

excitation for a given output rate, count-rate-induced effects such as peak shift and

159
broadening are reduced [33]. Furthermore, the same output rate can be obtained with a smaller average beam current so specimen damage due to electron bombardment may be less

with pulsed excitation.

7. 1 X-ray mapping

The technique has particular advantages to offer for x-ray mapping. Since the output

rate increases monotonically with input rate, there is no danger of contrast inversion;

this can occur with a continuous beam because the output rate on a region of high intensity

can actually be less than that from a region of low intensity. When mapping at TV rates,

the input count rate, and hence probability of pile-up rejection, varies with time so the resulting map is subject to intensity distortion. For example, the horizontal bars

on a copper grid appear darker than the vertical bars when a map is made using a conventional raster scan [33]. With a switched beam, vertical striations can appear near an edge due to the more periodic arrival of pulses at high rates. A theoretical analysis

of these anomalies [33] indicates that with a switched beam, higher rates can still be

used without incurring significant distortion and in a particular example, x-ray maps

could be obtained at 8 kcps accumulation rate as opposed to 3 kcps with a continuous

beam.

8. ADC

The operation of the Wilkinson run-down analogue-to-digital converter (ADC) is

discussed by Gedcke [34] and will not be described here. For low-energy x-ray spectrometry at relatively low count rates, such a converter is perfectly adequate; with a clock

rate of 100 MHz, 1000 channel precision can be obtained with a maximum conversion time

of 10 us which is still significantly less than a typical processing time for a pulse.

It is impractical to improve the speed much further if conventional circuitry is to be

used because the response time of logic elements and discriminators is of the order of

10 ns. In the future, with improved noise performance, smaller time constants and use of

pul sed-excitation techniques, the successive approximation ADC (as described, for example

by Kandiah et aj. , [3]) will come into its own because much shorter conversion times can

be achieved without the problems of a high speed clock.

8.1 Low-energy considerations

The sequence of operations in a run-down ADC is usually initiated by the pulse

exceeding a threshold level, so the efficiency of detection rolls off in the vicinity

of this threshold. With a noise fwhm of 100 eV, the efficiency would begin to decrease

for photon energies below about 250 eV if the threshold were set just into the noise

distribution [29]. Furthermore, the "peak detect and stretch" scheme, when used in

conjunction with a passive (non time-variant) pulse shaper, introduces non-linearities

in pulse height measurement for small pulses [35] basically because a positive-going

noise fluctuation can cause the maximum of the resultant waveform to occur either before

or after the signal pulse itself has reached a peak. These defects in behaviour are only significant for low-energy work with a windowless detector.

8.2 Differential linearity

With conventional spectrometers, it is differential linearity which is of prime

importance to the analyst. Differential linearity reflects the relative widths of adjacent

channels in the spectrum. The standard deviation of channel-to-channel fluctuations

1/2 due to Poisson counting statistics is N for a channel with N counts. If an electronexcited spectrum of, say, carbon is recorded for a long time, the bremsstrahlung

background exhibits channel-to-channel variations due to both differential non-linearities

and counting statistics. Thus, if each channel contains 10,000 counts, a differential

non-linearity of 1 percent or better will not be visible. However, if a channel contains

106 counts, differential non-linearities of the order 1 percent will account for most

of the observed channel-to-channel variation and will therefore affect the accuracy of

spectrum processing procedures.

160
9. Livetime Correction

ADC conversion time, computing overheads, pile-up rejection and head amplifier

restoration all affect the throughput rate and because the method of implementation

varies from system to system, it is desirable for the electronics to automatically correct

for dead-time losses since no mathematical formula is universally applicable. Briefly,

the purpose of livetime correction is to produce a clock so that if one accumulates a spectrum for "live" seconds on this clock and N counts are recorded within a given

energy window, then the true input counting rate for x-rays in this energy range is

N/T

L

.

In terms of pile-up rejection, there are two situations which have to be accounted

for. In "trailing edge" pile-up (figure 13(a)), the first pulse is measured success- fully but the second pulse is rejected. In the Covell [36] technique, the clock is

kept off until the processor is ready to accept another pulse (i.e., the dead time is

"extended") so that the clock is effectively only running when the processor is "live and

and waiting" (as would be the case in a perfect processor which took zero time to make

a measurement). In "leading edge" pile-up (figure 13(b)), the measurement of the first

pulse is corrupted by the second so the measurement is "lost". One way of compensating

for this "lost" measurement is to simply add a count to the spectrum and distribute such

counts statistically among the channels according to the form of the spectrum (figure

13(c)(ii)). This is basically the "Harms" method [37] which suffers from the drawback

that statistical errors cannot be calculated directly from the spectrum. An alternative

solution is to keep the clock off until the next successful measurement [38]. Thus, the

interval preceding the first pulse (figure 13(c)(i)), during which the clock was counting,

is not wasted and still is associated with a single measurement. This technique works

very well provided that the intensity of the x-ray source is not fluctuating on a time

scale comparable to the processing time. If this is the case, then the "Harms" method is to be preferred.

Livetime correction becomes important when the spectrometer is being used to measure

absolute x-ray yields, or when the peak intensities on a specimen and an unknown are being compared. Ideally, the correction should be tested by recording spectra of the

same material at known levels of excitation (e.g., beam current) for the same total

livetime and checking that the number of recorded counts are proportional to excitation.

161
Extend dead

Figure 13. Livetime correction techniques. The rectangular waveforms refer

to the livetime clock.

(a) Trail ing-edge pile-up.

(b) Leading-edge pile-up

(c) Methods for correcting for a "lost" measurement.

10. Summary

Pulsed opto-electronic feedback is still the most popular method for neutralising

charqe at the detector. Although "drain" feedback is also used on commercial spectrometers performance is about the same for low-count- rate, low-energy spectrometry while both

resolution and dynamic range are improved by pulsed opto-electronic techniques and in

particular pulse-by-pulse restoration. Whatever technique is employed, it is important to prevent light from reaching the detector; otherwise the electronic performance car. suffer.

This is just one example where the input to the pulse processor is modified and illustrates

the qeneral point that the detector and electronics cannot be treated in isolation. If the

head amplifier does not produce a perfect step for each x-ray photon then peak shift and

resolution degradation will be exaggerated at high count rates. Thus, non- linearity,

leakage and low-frequency noise sources such as ground loops and micropnony can all

seriously affect the performance of even a well-designed pulse-processor.

162
Over the last few years there has been an increasing emphasis towards the use of

time-variant pulse processing techniques. One example is the time-variant, or "gated"

baseline restorer but such methods have been most extensively used in the processor

designed by Kandiah. In Kandiah's system, count-rate-dependent baseline shift is virtually

eliminated and noise performance is in some cases better than what was originally thought

to be the theoretical optimum. Furthermore, a zero-stabiliser is incorporated and the

electronic noise peak is monitored so that each spectrum can be accurately corrected for

any resolution variations.

To date, developments have concentrated on optimising performance for low-energy

(1 - 20 keV), spectrometry but in the relatively new field of analytical electron microscopy, very-high-energy photons are generated which can overload and sometimes completely

paralyse a processor which has been set up to observe the spectrum below 20 kV. This

situation can be avoided by vetoing such high-energy events in a time-variant processor

such as that designed by Kandiah.

For low-energy spectrometry, particularly with windowless detectors, electronic

noise is the most important parameter affecting system performance. Resolution, and

thus peak-to-background ratio, is almost directly proportional to noise (fwhm) and pile-up

rejection is only effective for pulses which are significantly above the noise level.

In the future, improvements in FET design will therefore be of greater significance for

low-energy work than will the Fano-factor for detectors.

References

[1] Goulding, F. S. , Walton, J. T. , and Malone, D. F. , Nucl. Instr. and Methods, 7]_, 273 (1969).

[2] Kandiah, K. , Proc. Conf. on Radiation Measurements in Nuclear Power, Inst, of Physics

and the Physical Society, London, 420, 1966.

[3] Kandiah, K. , Stirling, A., Trotman, D. L. , and White, G. , International Symposium

on Nuclear Electronics, Versailles, 69-71, 1968; also A.E.R.E. Harwell Report

R5852, 1968.

[4] Goulding, F. S. , Walton, J. T. , and Malone, D. F. , Nucl. Instr. and Methods, J_[,

273 (1969).

[5] Landis, D. A., Goulding, F. S. , and Jaklevic, J. M. , Nucl. Instr. and Methods, 87,

211 (1970).

[6] Kandiah, K. , Smith, A. J. , and White, G. , Proc. 2nd ISPRA Nuclear Electronics

Symposium, Stresa, Italy, 153, 1975; also IEEE Trans. Nucl. Sci., Oct. 1975 (1975).

[7] Landis, D. A., Goulding, F. S. , Pehl , R. H. , and Walton, J. T. , IEEE Trans. Nucl.

Sci. , NS-18 , 115 (1971).

[8] Elad, E. , IEEE Trans. Nucl. Sci., NS-19 , 403 (1972).

[9] Nowlin, C. H. , Blankenship, C. L. , and Blalock, T. V., Rev. Sci. Instrum. , 38, 1063

(1965).

[10] Walter, F. J., "Characterisation of Semiconductor X-ray Energy Spectrometers", Energy

Dispersion X-ray Analysis, ASTM STP 485, American Society for Testing and Materials,

82-112, 1971.

[11] Fairstein, E. , IEEE Trans. Nucl. Sci., NS-22 , 463 (1975).

[12] Karlovac, N. , ibid. , 457 (1975).

[13] Robinson, L. B. , Rev. Sci. Inst., 32, 1057 (1961).

163
[14] Radeka, V., Rev. Sci. Inst., 38, 1397 (1967).

[15] Deighton, M. 0., Nucl. Instr. and Methods, 58, 201 (1968).

[16] Deighton, M. 0., Nucl. Instr. and Methods, 103, 1 (1972).

[17] Dudley, R. A. and Scarpatetti , R. , Nucl. Inst, and Methods, 25, 297 (1964).

[18] Goulding, F. S. , Lawrence Berkeley Laboratories, Report LBL-5368, presented at Int. Conf. Particle Induced X-ray Emission and Its Analytical Applications, Lund, Sweden,

1976.

[19] Long, J. V. P., Proc. Scanning Electron Microscopy: Systems and Applications,

Newcastle, U.K., 1973, Conf. Series 18, Institute of Physics, London and Bristol,

228 (1973).

[20] Link Systems, Ltd., Model 2010 pulse processor manufactured under license from

U.K.A.E.A. , Harwell, England.

[21] Tsukada, M. , Nucl. Instr. and Methods, 14, 241 (1961).

[22] Deighton, M. 0., Nucl. Instr. and Methods, 58, 201 (1968).

[23] Goulding, F. S. , Nucl. Instr. and Methods, 100, 493 (1972).

[24] Koeman, H. , Nucl. Instr. and Methods, 123, 161 (1975).

[25] Llacer, J., Nucl. Instr. and Methods, 130, 565 (1975).

[26] Mosher, C. H. , IEEE Trans. Nucl. Sci., NS-23 , 226 (1976).

[27] Llacer, J. and Meier, D. F. , Nucl. Instr. and Methods, 142, 597 (1977).

[28] Woldseth, R. , Proc. 11th Annual Conf. Microbeam Analysis Society, Miami, Florida, USA,

1976, paper 1 , 1976.

[29] Statham, P. J., X-ray Spectrometry, 6, 94 (1977).

[30] Statham, P. J., Long, J. V. P., White, G. , and Kandiah, K. , X-ray Spectrom. , 3, 153

(1974).

[31] Goulding, F. S. , Jaklevic, J. M. , Jarrett, B. V., and Landis, D. A., Adv. in X-ray

Analysis, 15, 470 (1971).

[32] Jaklevic, J. M. , Goulding, F. S. , and Landis, D. A., IEEE Trans. Nucl. Sci., 3, 392

(1972).

[33] Pawley, J., Statham, P. J., and Menzel , T. , Scanning Electron Microscopy/1977, !_> 297 (1977).

[34] Gedcke, D. A., X-ray Spectrom., 2, 129 (1972).

[35] De Lotto, I. and Gatti , E. , Int. Symposium on Nucl. Electronics, Versailles, 1968,

paper 64, 1968.

[36] Covell, D. F. , Sandomine, M. M. , and Eichen, M. S. , Anal. Chem. , 32, 1086 (1960).

[37] Harms, J., Nucl. Instr. and Methods, 53, 192 (1967).

[38] Bartosek, J., Masek, J., Adams, F. A., and Hoste, J., Nucl. Instr. and Methods, 104 , 221 (1972).

164
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

UNDERSTANDING MULTICHANNEL ANALYZERS

Eric Li f shin and Steven R. Hayashi

General Electric Company

Corporate Research and Development

Schenectady, New York 12301

Abstract

The multichannel analyzer (MCA) is a basic component of any solid

state detector system. It is responsible for the sorting of pulses from

the main amplifier, the storage and retrieval of spectra and, in most

modern computer-based systems, the various states of data reduction

required for both qualitative and quantitative analyses. Over the past

few years, the need for a detailed understanding of the inner workings of

MCA's has diminished through the development of more user oriented

systems where tedious tasks like counting dots, calculating energies

from channel numbers and looking up peak positions in tables have been

replaced by "push button controls". While on the surface this clearly

leads to improved operator productivity, it is important to periodically

examine the inner workings of such systems both from the point of view of

obtaining a general understanding of what's going on as well as to assure oneself that the various compromises which have been made are really

compatible with special purpose user needs. The intent of this paper is

to provide a general overview of what's behind the buttons.

1. Introduction

The multichannel analyzer (MCA) is a basic component of any solid state detector

system. It is responsible for the sorting of pulses from the main amplifier, the storage

and retrieval of spectra and, in most modern computer-based systems, the various stages of

data reduction required for both qualitative and quantitative analyses. Over the past few

years, the need for a detailed understanding of the inner workings of MCA's has diminished

through the development of more user oriented systems where tedious tasks like counting

dots, calculating energies from channel numbers and looking up peak positions in tables

have been replaced by "push button controls". While on the surface this clearly leads to improved operator productivity, it is important to periodically examine the inner workings

of such systems both from the point of view of obtaining general understanding of what's

going on, as well as to assure oneself that the various compromises which have been made are really compatible with special purpose user needs. To date the most detailed descriptions

of MCA operation are found in the manufacturer's operating manuals which often tend to be

highly specific. The number of published papers on the general use of such systems for

x-ray analysis is also quite limited, with the last review article published in 1971 [l] 1

. Therefore, the intent of this paper is to provide a general overview of what's behind the

buttons.

2. Basic Components

Figure 1 is a photograph of a modern multichannel analyzer, the principal components of

which are illustrated schematically in figure 2. They include an analog to digital con- verter (ADC), a memory unit and various output devices. Each voltage pulse from the main

x The figure in the bracket indicates the literature reference at the end of this paper.

165
Figure 1. Modern commercial multichannel analyzer.

A. CRT Display D. Analog to Digital Converter

B. Keyboard Entry E. Floppy Disc Storage

C. Detector Electronics F. Computer

8

- 6

- 4 - 2.5V 4.3V

2.1V

Time

Main

Amplifier and

Pulse

Pileup Rejector

Pulse —»

Busy

Busy

Analog

to

Digital Converter (ADC)

Dead Time

0

12 Memory j 3 Unit

4 *

or 5 : Computer

6 with

7

- Interface

Store

Clear Dead Time

5 a 4 §3

o 0

1

CRT Display

•

•

() 1 2 3 4 5 6 Channelr

i

Printer Channel

No. Counts

1

Figure 2. Simplified schematic of multichannel analyzer.

166
amplifier is transformed into a digital signal by the ADC. The output of the ADC then

becomes the address of a channel in the memory unit where one count is to be added. In effect, the memory unit acts as a set of independent scalers counting pulses in preselected

voltage increments. In the example shown, channel 0 corresponds to pulses from 0 to 1 volt,

channel 1 to pulses from 1 to 2 volts and so on up to channel 7 which counts pulses from 7 to 8 volts. In actual operation the memory is initially cleared, then the first pulse

(2.5V) is counted in channel 2, the second (4.3V) in channel 4, and the third (2.1V) is also

counted in channel 2. Upon completion of a preselected data collection time (clock or

live) the contents of the MCA memory can be printed out directly, displayed on a CRT or further processed by the computer. In the CRT mode the electron beam of the display tube

is scanned across the x-axis in such a way as to generate a series of dots through the use

of appropriate gating circuitry. The number of dots corresponds to the number of channels

selected and the horizontal displacement of each one is made directly proportional to the

channel number. Vertical deflection is then used to indicate the number of counts per

channel. This is accomplished by converting the digital signal stored in memory back to a proportional analog voltage. The proportionality constant and, therefore, the counts "full

scale" is set by the operator using a multiposition range switch on the instrument

panel. Since many instruments use 16 bit memories, the maximum number of counts per

channel is typically 65,535 although software or hardware options for greater storage

capacity are often available. Analog horizontal and vertical scale displacement and

expansion controls as well as a logarithmic display mode are also usually provided.

3. The ADC

A block diagram of a typical ADC is presented in figure 3 along with a corresponding

waveform diagram given in figure 4. Pulses from the main amplifier are first tracked by a voltage follower required to ensure compatabi 1 ity with the next stage of electronics. The

output (point A) is inspected by both upper and lower discriminators set by the operator.

Unacceptable pulses are blocked by a gate. Each accepted pulse, however, simultaneously

triggers an SCA output pulse, is examined by peak detection circuitry, and also charges the

peak stretcher capacitor to the maximum pulse voltage (point B). When the pulse voltage

exceeds the analog zero level value ZL (also set by the operator) an appropriate logic

signal (point D) combined with the peak detected pulse (point C) activates the pulse

stretcher busy (PSB) and ADC busy circuits.

Single Channel Analyzer

Upper and Lower

Level Discriminators

Main Amplifier

Voltage

Follower

Peak

Detector

Clock

Gate

Constant

Current Source

Peak StretcherL^. Busy

Zero

Level

.Stretcher — Capacitor

ADC

Busy

VF

Gate

i.G

Address

Scaler IT

Memory

Unit

Store Logic

Clear

Figure 3. Analog to digital converter block diagram.

167
A. Pulse

C. Peak

Detect

D. Zero

Level

E. PSB

F. Clock

G. Clock

Gate

Output

H. ADC

Busy

I. Store

J. Clear

_TL

AAAAAAA

_7VW\_

_J

Time

Figure 4. Waveforms from selected points in figure 3.

The PSB signal (point E) causes a gate to open which allows pulses from a clock to be

counted by a scaler. In addition it causes a constant current source to linearly discharge

the stretcher capacitor. When the zero level is again crossed the PSB signal is lifted

closing the gate to the clock (points F & G) thereby, leaving the address scaler containing

a count of clock pulses directly proportional to the original pulse voltage. Lifting of the

PSB signal combined with other test conditions not shown in figure 3 results in the output

of a store pulse (point I) which causes the address to be transferred to the memory unit

(see figure 2). The ADC busy signal (point H) which has been clocking all incoming pulses

is then reset by a clear pulse (point J) from the memory unit making it possible to accept

new pulses and the entire process can then be repeated until the predetermined data collection time is reached.

4. Definition of Terms

Examination of figures 3 and 4 should now make it possible to understand a number of

terms used in conjunction with MCA's. For example:

The conversion gain of an ADC refers to the total number of increments (addresses)

used to characterize a measured pulse distribution. Switch selectable values typically

range from 32 to 8192 in multiples of two. In effect the conversion gain determines the

168
ADC resolution (volts/channel) by controlling the discharge rate of the stretcher

capacitor. This determines the number of clock pulses to be counted for a main amplifier

pulse of a given size. In terms of figure 4, waveform B, the slope and therefore the time

required to reach zero level (ZL) are determined by the switch setting. For a maximum

acceptable main amplifier pulse of 8 volts, a conversion gain of 1024 means that at least

1024 pulses must be counted to achieve an ADC resolution of (8/1024) = .0078 volts. Use of

a 100 megahertz ADC implies that the clock frequency is 100 million cycles per second. For

the example just given the time required to convert an 8 volt pulse to a digital signal

will, therefore, be 1024 x 10~8 sec. =10 useconds. It is reasonable to ask if the conversion

time ever becomes the rate limiting factor in the overall spectrum collection process. This

question can be answered with the aid of figure 5, which shows three pulse shapes corre- sponding to 1, 6, and 10 usecond main amplifier time constants. As described elsewhere,

long time constants are required for high energy resolution at the cost of increased pulse

pileup rejection (high dead time) while short values are associated with lower energy

resolution and increased pulse throughput. In the upper part of the figure the ADC con- version time is plotted as a function of pulse height for a series of ADC speeds ranging

from 25 to 200 megahertz and conversion gains of 1024. Thus, for example, as shown with

line C an 8 volt pulse would take 20 useconds to convert following the sensing of the peak

maximum while a 4 volt pulse would take only 10 microseconds. Note also that although the

pulse amplitudes shown are for 4 volts, the horizontal scale for the pulse shape curves will

be independent of the pulse height. Therefore, in some cases the ADC sorting time may be

rate limiting while in others the main amplifier time to base line will dominate pulse

throughput. This, of course, assumes that a pileup rejector is used to block all incoming

pulses until the peak effectively returns to the baseline. For a 4 volt pulse and a 100

megahertz ADC, line A, it can be seen that pulse processing in the main amplifier is the

slow step for all of the pulses shown. It should also be mentioned that in addition to the

ADC conversion time which varies with pulse height, a fixed time increment is required for

the storage of each pulse in memory. This increment can be anywhere from a few useconds to more than 10 depending on whether the operation is done by hardware or software.

i I 1

!—r f 1 1 -20-10 O 10 20 30 40 ^ sec

TIME TO RETURN TO BASELINE

Figure 5. Relationship between pulse to baseline and analog to digital conversion times, conversion gain 1024.

169
Memory group size refers to how large a section of the memory is actually used for

spectral storage. In the simple example presented in figure 2, eight lines, each corre- sponding to a single address, were shown connected, one for one, to an eight channel

memory. In reality separate lines are not needed to each channel since the address can be

binary coded. Sixteen lines are sufficient to carry any address from 0 to 65,535 (2 16-1).

In the earliest MCA's coupled to EDS systems, hard wired magnetic memory units were used

which could be divided by the operator into specific groups. Consider, for example a system with a total of 2048 channels. Switch selectable memory groupings include:

Switch Setting Memory Size

1/1 full memory 2048 channels each

1/2, 2/2 first and second halves 1024 channels each

1/4, 2/4, 3/4, 4/4 four quarters 512 channels each

If the conversion gain is set equal to the memory size then, for the example given, either

one, two, or four complete spectra can be stored in memory depending on the resolution

required. Computer based systems allow for more dynamic allocation of memory, while the

newest systems have returned to dedicated memories which are readily accessible to other

parts of the system.

For practical purposes the user is interested in pulse energy rather than channel

number. This is often accomplished by specifying a number of eV/channel , typically ranging

from 5 to 40, and information is then displayed as energy. Consider first a calibrated

system in which the zero level has been adjusted to remove any DC offsets. Then, for

example, if a value of 20 eV/channel is used, and 512 is selected for both the memory group

size and the conversion gain, then the recorded spectra will cover x-ray energies of from

0 to 10.24 keV. Under these conditions copper Ka (8.048 keV) will give rise to a pulse

distribution centered around channel 402. Almost all of the peak will be contained within

a band defined by plus or minus one FWHM. Therefore, for a 160 eV detector measured at CuKa the entire peak will contain about 16 channels, which is a reasonable number for most

applications. Needless to say if one wishes to accurately measure detector resolution to better than 20 eV it is definitely desirable to use less eV/channel.

The analog zero offset controls the value of ZL as shown in waveform B of figure 4. By varying the point where the clock gate output pulse is lowered, it provides a way of

either adding or subtracting a constant number of clock pulses to the conversion of all

main amplifier pulses being processed. This, of course, shifts all of the addresses in

memory by a fixed amount. From the point of view of the MCA CRT display, varying ZL produces a linear displacement of the entire spectrum on the viewing screen (see figure

6a). In the example just mentioned, a ZL value of two volts would result in a displacement

of the spectrum by 25 percent. The display would cover a region of from 2.56 eV to 12.80

keV and the copper Ka would have the same width but appear centered around channel

402-256=146. In practice the analog zero adjust is usually restricted to removing unwanted

DC levels for calibration purposes. Shifts of the type just described are normally

performed by using digital offset switches (see figure 6b) which subtract fixed numbers from

the ADC scaler addresses. Their use makes it possible to use memory group sizes smaller than

the conversion gain. High ADC resolution can then be combined with limited memory capacity

for the examination of selected regions of a spectrum. This feature is parti culary useful

in accurately measuring detector resolution to a few eV by going, for example, to a com- bination of a conversion gain of 8192 with 512 channels of memory and then using digital

offset to bracket the peak position. Recognize, however, that digital offset leads to more

ADC time than the analog approach since the subtraction occurs after the stretcher capacitor

rundown.

170
ZERO SHIFT : : o. OFF. 128 6492

Figure 6a. Effect of analog offset. Figure 6b. Example of digital offset.

5. Calibration Procedures

The relationship between channel number and energy requires consideration of the role

played by the various components in the detection chain illustrated in figure 7. Each x-ray

photon absorbed in the active area of a lithium drifted silicon detector results in the

ejection of a photoelectron which in turn creates a number of electron- hole pairs. The

total charge is then converted to a voltage pulse by a charge sensitive preamplifier whose

output is essentially a step pulse. The first stage of the main amplifier then converts

this pulse to a near gaussian form by a combination of differentiations and integrations.

The corresponding waveform at this point (A) in the circuitry is shown in the figure. The

pulse amplitude is given by:

EeG„

(1)

where E is the pulse energy, e the charge per electron, the preamplifier gain, e the

average number of eV required to produce an electron hole pair, C the preamplifier feedback

capacitance and l x any DC offset that might be present (Z x and Z 2 are exaggerated for the

figure). The main amplifier output Point (B) is then given by

a a p 2 (2)

where G is the user controllable gain and Z2 is any DC offset at its output. Combining

a equations (1) and (2) gives

V

a = < Ga

G

p #E + G

a

Z

l + Z 2 where the total DC offset Z

Q

is given by

or

Z

o = G

a

Z

l + Z2

V = (G G -£)E + I

a a p eC J o

(3)

(4)

(5)

171
Input

X-ray

Energy E

> •

Detector,

Preamplifier

and first stage

main amplifier

Main Amplifier

Output

Channel Number

Cn = CQ (V.-ZL)

ADC

Conversion Gain C,

K

Figure 7. Schematic representation of factors determining the relationship

between pulse energy and channel number.

It is this pulse which is seen by the ADC (waveform A in figure 3) which will be

directed to channel C

n given by:

C n

C = 4 (Va

" ZL) (6)

n

where C is the conversion gain (number of channels used to span k volts (the maximum

y acceptable pulse size) and ZL and the user selected zero level setting (see figure 3 waveform B). In other words the conversion is done on a pulse whose effective height is V - ZL. Combining equations 3 and 6 gives the overall calibration equation:

for most calibration work ZL is adjusted such that:

ll = Z

o = Ga

Z

l + Z2

< 8)

and equation 7 transforms to: keC E(x-ray energy in eV) = ^ ^ ^

(eV per channel )X C

n (channel number) (9)

gap

The actual calibration process by which equation 8 is satisfied and G chosen to give a

a valid preselected eV/channel cannot be accomplished by the use of a single peak. The reason

is that any number of combinations of ZL and G, can be used to put a known peak in a prea determined channel; however, such a calibration will generally not apply to any other peak

since equation 8 may not be satisfied. Two peaks are therefore required with one usually

selected at low energy and the other at high energy to cover a practical energy range. If

the two peaks selected have energies E

1 and E2 then the peak centroids channel numbers can

be measured experimentally as C* and C

2

. Rewriting equation 7 for each case gives

172
C

n = k (Ga

G

p ef)

E ' + G

a

Z

l + Z2

" ZL < 10a >

C

n = k

(G

aG

p S§*

2 + G

a

Z

l + Z

2

" ZL < 10b >

Subtracting equation 10a from 10b gives:

C n" C

n = ^ ^p^^-^ <")

The significance of this equation (11) is that independently of Z and ZL the number

of eV/channel corresponding to the difference between two peaks can be set by varying G to

a give the desired value (example: 10 eV/channel) even though the absolute placement of the

spectrum with respect to zero may be wrong. The latter condition can be corrected by the

ZL adjustment to satisfy equation (4). This procedure avoids the need for iteratively

setting the controls to achieve calibration. As shown in figure 8 the process can be

performed in a semiautomated mode in a computer based system which signals the operator

how to make the needed adjustments. In those systems where the main amplifier gain and

zero level are under computer control the process can be fully automated.

Z£R0= 6

Figure 8. Visual output of automatic calibration program.

6. Multichannel Analyzer Evolution

The trend in advanced MCA development has been from completely hardwired to computer

based units, with the latest systems containing the best features of both using a combination of minicomputers and microprocessors. In the earliest MCA's, the user literally had

to count dots to determine channel numbers, which were then converted by hand to energy

values. Elements were identified by comparing the calculated peak positions with tables of

173
known x-ray line energies. The next generation included a dial controllable feature known

as a "bug" which made it possible to intensify the dot corresponding to the contents of

any specific channel on the CRT. At the same time its position, in either channel number

or energy, as well as the corresponding number of counts could be read directly from a numerical panel display. With the advent of low cost character generators, scale information and labels could also be displayed directly on the CRT. Later came a series of

special features including regions of interest (ROI) displays, line markers, and access to

a variety of auxiliary spectral storage and retrieval devices. The ROI approach allows

the user, often with the help of the bug, to define a series of energy bands in which a running count of the integrated intensities are recorded. In this manner integrated peak

intensities can be monitored either as a prerequisite for quantitative analysis, or pulses

corresponding to specific elements can be coupled to SEM displays for line scans or area

maps. Line markers are a series of vertical bars appearing at energies corresponding to the principal lines of any selected element. The line markers are introduced one element

at a time and are manipulated by the bug control or keyboard. They can be superimposed

over an unknown spectrum to facilitate rapid qualitative analysis. Figure 9 shows a composite of many of the types of information readily available to the user under keyboard

control in addition to line markers. The area within the box shows the actual information

that appears on the screen.

COUNTS IN CURSOR CHANNEL REGION OF INTEREST NUMBER 5 COVERING ENERGY RANGE

CURSOR POSITION

SHADING DENOTES

REGION OF INTEREST.

LOW ENERGY—!—.

CURSOR 6 430KEV =10-408 ROUCS] 6 090 6 740=257973

—3 860 FE 26 KA1=6 40KEV

100 \

\ CURSOR POSITION CORRESPONDS TO

Fff K„ LINE

COUNTS ABOVE BACKGROUND

Fe K.- AND Fe K/i PEAK POSITION MARKER

TOTAL COUNTS

FULL SCALE BACKGROUND

INTENSITY

U

Figure 9. Composite illustration of CRT display aids.

Data readout with the earliest systems generally consisted of unlabeled x-y recordings,

printed tables, or photographs of the CRT. Links to both large centralized computer systems

and dedicated minicomputers were generally slow and awkward. Although there was some effort

to interface the hardwired systems to auxiliary tape units for spectral storage and

retrieval this approach also proved limited. In the exclusively computer based MCA, information from the ADC is transmitted directly to the central processor unit of a dedicated

minicomputer which, through a combination of hardware and software, sees to it that pulse

distribution information is directed to specified memory locations. The operator normally

interacts with the system through an alphanumeric keyboard and/or various push button

switches. Desired actions can then be initiated under computer control. These include data

acquisition, spectral storage and retrieval on auxiliary devices and spectrum processing by

174
a variety of techniques. Central to this philosophy is the existence of an executive

program with responds to specific mnemonic commands issued by the user. Unfortunately,

unlike the more self-explanatory hardwired controls of early MCA's the software command

structures of the computer based system tend to be more unique to each different

manufacturer's product. The advantages are, however, much more flexibility and greater

speed in data interpretation.

With the computer based MCA the operator normally has the option of changing parameters

such as the memory group size, conversion gain, digital offset, the number of eV/channel

and a number of CRT display features under program control. Furthermore, since the memory

of the computer is used as the MCA memory, applications programs written in high level

languages can readily access the stored data for quantitation. Typical programs presently

available include ZAF, Bence-Albee, Cliff Lorimer and Rasberry-Heinrich calculations. The

storage and retrieval of both spectral data and processing or control programs is also a relatively straightforward matter since devices like tape drives and floppy discs are easily interfaced to almost all minicomputers. The major shortcoming of such systems has

been that simultaneous data collection and processing on an internal time shared mode tends

to slow them down since pulse sorting and storage can be almost a full time job. The

solution to this problem, as characterized by the latest generation of MCA's, has been to

use a combination of minicomputers and microprocessors to handle specific tasks. For

example, a dedicated microprocessor system with its own memory can be used for rapid pulse

collection, sorting and storage, its memory can be coupled directly to a minicomputer

which handles data processing. Other microprocessors can be used for controlling the CRT

display or performing specific computational or control tasks. Looking ahead, the increased

sophistication and reduced costs of semiconductor components will clearly lead to more integrated systems. Future microprobes, TEMs, SEMS, and x-ray fluorescence spectrometers

will be digitally controlled. In this manner it will be possible for an instrument operator

to supervise data collection, control, and processing through a single interface resulting

in better reproducibility, speed, and hopefully accuracy.

Reference

[1] Williams, Garry, "Role of Multichannel Analyzer in Data Handling" Energy Dispersive

X-ray Analysis: X-ray and Electron Probe Analysis, ASTM STP485, American Society for

Testing and Materials, 1971, pp. 125-139.

175

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

STATISTICAL CONSIDERATIONS OF DETECTABILITY LIMITS AND DEC0NV0LUTI0N IN ENERGY

DISPERSIVE X-RAY SPECTROMETRY

P. L. Ryder

Department of Physics

University of Bremen

West Germany

Abstract

The first part of the paper considers theoretically the experimental errors due to counting statistics in the determination of line

intensities by fitting Gaussian curves to the experimental values. The

line positions and the background are assumed to be known, and the background level is taken to be constant over the relevant energy range.

Expressions are derived for the standard deviations of the line

intensities a) for a single line and b) for two overlapping lines.

Numerical examples are given to illustrate the effect of line and

background intensities on these errors. In the case of two overlapping

lines, it is shown how the ultimate spectral resolution is limited by the

statistical errors, which increase as the line separation decreases.

The second part of the paper is concerned with the effect of

counting statistics on the deconvolution of unsmoothed ED spectra by

Fourier methods, when no assumptions are made concerning the line

positions or background spectrum. It is shown that a criterion for the

resolution limit of two overlapping lines is provided by the requirement

that the amplitude of the first maximum in Fourier spectrum of the line

pair must be equal to or greater than the mean amplitude of the noise

spectrum. In retransforming the Fourier spectrum the high-frequency

noise must be eliminated by a suitable frequency filter. It is suggested

that the filter shape should be that of the Fourier transform of the

spectrometer function (in this case a Gauss curve). The consequences for

computer evaluation of energy dispersion x-ray spectra are discussed.

1. Introduction

The accuracy with which the concentration C of a particular element may be determined

by x-ray analysis techniques such as electron probe microanalysis, x-ray fluorescence,

etc., is ultimately limited by the statistical error in the measured intensity. The

detectabil ity limit C

Q

is usually defined as the concentration at which the line intensity

is equal to some arbitrarily chosen multiple (usually taken to be 3) of the standard

deviation a of the background intensity [1-8] 1

. Assuming a linear relationship for small

concentrations:

the detectabi 1 ity limit is given by

I = YC (1)

C = 22 (2 ) o y figures in brackets indicate the literature references at the end of this paper.

177
In the case of energy dispersive x-ray spectrometry the situation is complicated by

the fact that, due to the limited resolution of the solid state detection, overlapping

lines are often encountered. The intensities of such lines must be determined by peak

fitting, peak stripping or general deconvolution methods [9-13]. It is the purpose of the

present paper to consider the effect of errors due to counting statistics in their

processes, and to show how such errors limit the effective resolution attainable, i.e., the

smallest line spacing for which the line intensities may be determined with a given required

accuracy.

The problem is first discussed for the special cases of a single line and two over- lapping lines on a constant background. Quantitative expressions are given for the relative

errors as a function of the line counts, line separation and background level, and the

results are illustrated by diagrams and numerical examples. The effect of statistical

noise on the general Fourier deconvolution method is then considered.

2. Line Fitting Methods

2.1 Single Line on a Constant Background

We suppose that the intensity of a single line on a constant, known background is to

be determined by fitting a curve of suitable shape to the experimental points. In energy

dispersive x-ray analysis the line shape is described with sufficient accuracy by a Gaussian

curve. The mean number of pulses collected in channel No. n is given by

P = B + If.

n hn (3)

B is the number of background counts per channel (assumed to be constant), I is the total

number of counts in the line (the parameter to be determined) and f. is given by

f hn = —— exp

sV2n

(h-n)'

L 2s

2 (4)

In this expression h denotes the position (channel No.) of the peak maximum and s is the

line width parameter, related to the half-width by

w = 2.35 s (5)

We denote the actual number of counts collected in channel n by Q

n

- The value of I is chosen so that the sum of the squares of the differences (P - Q n

) is a minimum. The condition for this is

HT E {

B + If hn-Qnf = ° (6)

n=l

(N = total number of channels). The leads to the following expression for I:

1 = & Vhn " BI fhn(7)

n n n

If we assume that the channel width is small compared with the line width (i.e., s»l),

which is indeed a condition for the validity of equation (3), then the sums I f. and

If? in equation (7) may be replaced by integrals:

178
' - I f. "C - 2 '^

dx= 1 (8) 1— /" exp (-xV2s£

)

00 2 f* i -L-2 /* exp (-x

2/s2

) dx = l/2sV7t (9)

n 2ns /

(10)

Inserting these results in equation (7) we obtain

I = 2sVn

| Z Q

n

f hn" bJ

( n- 1 J

The standard deviation o of the intensity I is given by

where a is the standard deviation of the channel count Q . Assuming Poisson statistics

n n and Q

n

>:> l» °" n is equal to the square root of the mean value:

a

n = VP; (12)

Substituting from (3), (10), and (12) in (11) gives

a

2 =4*s2 £ j Bf

2n+ Ifjji (13)

n ( )

Replacing the sums again by integrals, the following expression is finally obtained for

the standard deviation of the line intensity as a function of the line and background

intensities:

a

2 = 2sVn B + 2IA/3 (14)

The product sB, which will be denoted by U, is the number of background counts in a range s of the spectrum and is therefore independent of the choice of the channel width of

the multichannel analyzer. The relative standard deviation as a function of I and U is

given by

a\ 2 _ 2L^7t + _2_

, - (15)

1 j I 2 W3

Figure 1 shows the relative standard deviation as a function of the line count for various

values of the background parameter.

The horizontal dotted line at the value 0.33 indicates the level for which the

3a-error becomes equal to 100 percent of the line intensity.

179
10

o

i—<

>

LUQ

Q

CC<Q

Z

to

LU>

LU

1

0,33

0.1

0.01 10 10' 10J 10^ LINE COUNT

Figure 1. Relative standard deviation of the intensity of a single line as a function of the line count for

various values of the background parameter U = sB.

2.2 Two Lines on a Constant Background

This problem has been treated extensively in an earlier paper [13]. The mean count

in channel n is in this case

P n = B + I.f ln + I 9f n 1 In Z Zn9n (16)

I.J and 1^ ane the intensities of two lines centered on channel h^ and h^, respectively,

and f, f- are abbreviations for f. and f. . These parameters are chosen to give the

In Zn r^n hjn

best fit to the experimental points Q . The conditions for the minimum sum of the squares

of the deviations are:

0 (i=l,2) (17)

This given two simultaneous linear equations for 1^ and l^. If we again replace the sums containing f^

n

, and their products by integrals, we obtain the following solution:

h

=

2^$ - bs

2

- (1-b) b}

h

= ffz

{s

2 -bSl -(l-b)B}

(18)

180
with s. = Z Qn f in (1=1.2)

n

and b = exp |-(h

] -h2

) 2/4s

2 }

The standard deviations of the intensities 1^ and 1^ are given by

. 2

» 3Q

n \ n

From (18) and (20) we finally obtain

a

2 = XU + YI^ + ZI

2

a

2 = XU + YI

2 +

with

Z = 2(b4/3-2b7/3 * b

2

) V3 (1 - b

2

) 2

(19)

a

2 = E ]

• P

n (1=1,2) (20)

(21)

X = ^-2 1-b^

Y =

2 ^- 2b _%b

1 (22)

V3 (1 - bV

With the aid of equations (21) and (22) it is possible to calculate the errors as functions of the line intensities I-j and I^, the background count U (=Bs) and the line

separation. The following figures show numerical examples for various values of these

parameters. The line separation is denoted by the dimensionless quantity

h l"h

2

Figure 2 shows the relative error o^/I^ = °"2^*2 as a function of the line count

I-| = I 2 for two lines of equal intensity with a separation e = 0.5 (corresponding to about

1/5 half-width) and background counts of 0, 10, 102

, 103

, and 104 in the range s.

From such diagrams one may estimate how many counts are needed to obtain a given

desired accuracy under specified experimental conditions.

Figure 3 is analogous to figure 2, the only difference being that the line separation

e was taken to be 2.5, corresponding approximately to the half-width. Comparison with

figure 2 shows that the greater line separation leads, as one would expect, to lower

experimental errors in the line intensities.

181
>

LUQ

Q

or<Q

z

<

w 0.33

LU

or

10 10* 10

J TOTAL COUNT I 2

10

'

Figure 2. Relative standard deivation of the line intensities 1^ and I 2 as a function of I 2 calculated from equation (21)

for e = 0.5, I, = I £ and U = 0, 10, 102

, 103

, and 104

.

>

LUQ

Q

<

LO

LU

or

0,33 -

10 10

a 10

3 TOTAL COUNT l t

Figure 3. As for figure 2 with e = 2.5.

1<T

182
Figures 4 and 5 are corresponding diagrams for two lines with an intensity ratio of

1:10. The relative standard deviation of the weaker line (continuous curves) and the

stronger line (dashed curves) have been plotted against the total count of the stronger

line (^ for the same background levels as in figures 2 and 3. The curves illustrate in

particular the difficulty of detecting a weak line in the presence of a nearby strong

1 ine.

<

>

LUQ

Q

cr<Q

Z

<

in

LU>

LU

0,33

10* 10

3 TOTAL COUNT l 2

Figure 4. Relative standard deviation of the intensity I. (continuous

curves) and I 2 (dashed curves) as a function of for

e = 0.5, I

] = 0.1 I 2 and U = 0, 10, 102

, 103

, and 104 .

<

>

LUQ

Q

or<

Q

Z

<

LO

LU>

I-<

_lLU

or

TOTAL COUNT l z Figure 5. As for figure 4 with e = 2.5.

183
The horizontal dashed line in figures 2 to 4 representa a 3a limit of 100 percent. If

the error is greater than this, the line may be regarded as "undetectable" under the

given conditions. The points of intersection of curves with this horizontal line thus

indicate the line counts for which the given line separation may be regarded as the

"resolution limit", i.e., the smallest line separation for which the line concerned may

still be detected under the given experimental conditions. Values of the resolution

limit e Q

obtained graphically from diagrams similar to figures 2 to 5 are shown in

figures 6 and 7.

o

I—

r>

_i

o

CO

0.5

u=0 10 10

2 10

3 10

4

\

10 10*\\ \ \10 3 TOTAL COUNT I*

10*

Figure 6. Resolution limit e

Q

, determined graphically, as a function of I 2 for ^ = I £

and U = 0, 10, 102

, 103, and 104

.

Q 1 o

to

UJ

tr

0.5

u=0 10 10* 1 \

* A—i A103 0 10 10v10* 10* 10*

\ \ \

\—\—

\

u \ \

u \ \

\ \

\ \

\ \

\ x\

10 10

2 10

3 TOTAL COUNT Ia

10*

Figure 7. Resolution limit e

Q

, determined graphically for the

lines I.| (continuous curves) and I, (dashed curves)

with I

1 = 0.1 I £

and U = 0, 10, 102

, 103 and 104 .

184
Figure 6 shows, for example, that two lines with 100 counts each on a background of

100 counts may be detected separately in the sense defined above if their separation is

greater than about 5. As may be seen from figure 7, in the case of two lines of unequal

intensity, the value of the resolution limit depends upon whether one is trying to detect

the weak line or the strong line. The values for the stronger line (dashed curves) are accordingly lower than those for the weaker line (continuous curves).

For small line separation (e«l) we may use the approximation

b

n 5 1 - ne2/4 (24)

which gives

= a

2 = fan U + 8(I

1 + I 2 )/3V3^

1/2

/e (25)

The resolution limit e is defined as that value of e for which I. = 3a. (i = 1,2). This

o J J leads to the expression

e

0 = f: fan u + 8(I

i + V/3^}

1/2

(26)

3

with j = 1 for line 1 and 2 for line 2. For the special case I, = I- = I and U«I this

equation takes on the simple form

e

Q = 3.72/VT * 4/VT (27)

which may be used for estimating the ultimate resolution as a function of line intensity

for small backgrounds.

3. General Fourier Deconvolution

3.1 Principle of the Method

We first review briefly the Fourier method of deconvolution, which makes use of the

fact that the convolution of the spectrum is equivalent to multiplication of the Fourier

coefficients, so that the deconvolution may be effected by division.

The original intensity distribution P(E) in the x-ray spectrum is modified by the

spectrometer function G to give the measured spectrum P'(E)

00 P'(E) = / G(X) P(E - X) dX (28)00

-00

with

G(X) = —L exp — j :

V27t s

This integral transformation is sometimes written in the abbreviated form

P' = G * P (28a)

185
We assume that the spectrum is defined in the range 0< E <E and expand all functions as Fourier series. With a = 2n/E. we have

P(E) = P

k exp (-ika E) (30)

oo P'(E) = ^2 Pk exp ( " ika E) (31)

k=-oo

00 G(E) = ^ g

k exp ( " ik0' E) (32)

k=-oo

Replacing the functions in equation (28) by their Fourier series we obtain the following

relationship between the Fourier coefficients:

Pk

=

g

k Pk C33)

The Fourier coefficients of the spectrometer function are given by

E.

jU / -1— exp {ika E - ^-A dE (34)

o J sj2R ( 2s^j

g

k - r

For s<<E

Q we may take the limits of integration to be ±» and obtain

_ 1 /-a^k^s 2 \ ,~r\

g

k E~ exp \ 2 )

(35)

The coefficients p^ may be calculated from the measured spectrum and the coefficients p^

are then obtained by division

Pk

= P|/g

k (36)

The true spectrum is then reconstructed by Fourier synthesis from the coefficients p^.

Thus deconvol ution is equivalent to division of the Fourier coefficients.

The limits to this process are set, as we will see, by the noise. To deal with this

problem it is more convenient to handle a discrete spectrum. For a spectrum of N channels

the Fourier series equivalent to equations (30 and 31) are

P ^ \--^\ (37) / j k " ,! k=-oo

'H'-Ei"* h?*! <38)

k=-oo

186
Wl th t n n n \

Pk =

p

k exp

{ N

2

}

(39)

3.2 Noise

Statistical noise may be represented by a random variable R n added to the originally

smooth spectrum. The Fourier coefficients of this contribution are

n=l

In general, it is not possible to determine the amplitude and phase of the Fourier coefficients r^. One can, however, estimate the mean value of the amplitude |rv|. From (40) we

have

N

2 = Vk

* = ^ E E V>pjM^| (4D

For n ^ m, the mean value of R R is zero, since positive and negative values occur with

n m ' r

2 equal probability. For Poisson statistics the mean value of R

n is equal to P , the mean

number of counts in the channel. Equation (41) therefore reduces to

I I N

n N

where M denotes the total number of counts in the spectrum.

This calculation shows that the mean effect of the noise is a frequency- independent

contribution to the Fourier spectrum.

3.3 Background

The background spectrum is - apart from steps due to absorption edges - a slowly

varying function of the energy. As a simple extreme case we may take the number of background counts per channel (B ) to be constant (B = B). The Fourier coefficients are

N

/ \

b

k = NL Bn 6XP l~N~/

n=l

0 for k # 0

B for k = 0

(43)

Thus the background gives rise to a peak at the low frequency end of the Fourier spectrum.

3.4 Signal

The Fourier transform of the signal is a general function, which is modulated by the

transform of the spectrometer function according to equation (39). To illustrate the

principle we will calculate the Fourier spectrum of a simple spectrum consisting of two

lines of equal intensity separated by an energy interval a on a constant background.

187
The signal function is given in this case by

P = P A

= I r r+a

P =0 for n^r and n^r+a

n

(44)

The Fourier coefficients are

Pk

=

j /2nikr\

I exp {—^-)

N L

1 + exp nr-J (45)

The squares of the Fourier amplitudes are given by

2 _ 2T 1+ cos'2n_ka'

v N 7

(46)

and thus depend only on the separation of the two lines and not on their absolute position

in the spectrum. In the measured spectrum the Fourier coefficients are modified by the

spectrometer function:

2lf ( N

2

t

1 + COS exp

2 2 2 (47)

3.5 Deconvolution

Figure 8 is a schematic representation of the Fourier transform of the spectrum

considered above. In this figure the square of the Fourier amplitude has been plotted

against the frequency K. The background gives rise to the peak in the top left-hand corner

of the figure. The signal gives an oscillating function which is modified by the

spectrometer function, and the mean contribution of the noise to each Fourier amplitude

is constant.

The effect of noise on the deconvolution process is illustrated in figure 9. The

upper left-hand part of the figure shows the situation for an ideal spectrum without

noise. The deconvolution is effected, as explained above, by dividing the Fourier

coefficients of the measured spectrum by the Fourier coefficient of the spectrometer

function. In the presence of noise, however, (see right-hand and lower part of figure 9)

this leads to an exponential increase of the noise spectrum. This means that the deconvolution process magnifies the noise component. In the reverse Fourier transformation it is

therefore necessary to use a frequency filter to cut off the high frequencies.

188
Figure 8. Schematic representation of the square of the Fourier amplitude as a function

of the frequency for a spectrum consisting of two lines of intensity I and

separation a on a constant background B. N is the total number of channels

and M the total number of counts in the spectrum.

Figure 9. Schematic diagram showing the effect of noise on the Fourier deconvolution

of a spectrum. The original spectrum is assumed to consist of two lines on

a background (top left-hand figure). The next two figures to the right show

the spectrometer function and the folded spectrum, with the Fourier transform

(FT) below. The effect of Fourier deconvolution on a noisy spectrum (right)

is shown in the lower part of the figure.

189
The limiting frequency and hence the resolution limit of the spectrum may be estimated

from the criterion that the first maximum of the Fourier spectrum of the signal should be at least as great as the mean value of the noise level at the same frequency, i.e. ,

(48)

The corresponding line resolution limit is

s = f

=

2"

(49)

0 S / x V£n(4IVM)

For negligible background (M = 21) this reduces to

s o = —iS- (50)

^n(2I)

This does not agree with equation (27). The reason is that, in deriving equation (27),

it was assumed that the line positions and the background level were known, whereas the

general Fourier deconvolution method assumes nothing except a knowledge of the spectrometer

function.

3.6 Form of the Frequency Filter

It is well known that the shape of a frequency filter has a critical influence on the

quality of the reconstructed spectrum. A sudden cut-off may lead to spurious oscillation

which could be interpreted as lines. It is therefore suggested that the filter should

have the same shape as the Fourier transform of the spectrometer function - see equation

(35) - but with a different width. This may be regarded as a partial deconvolution,

effectively reducing the line width of the spectrometer. Instead of multiplying by

exp[2n2 k

2

s

2 /N2

] , as required by equation (39), we multiply by exp[2n2 k

2 (l-p2 )s 2 /N2

] giving

a

Pi! = Pi exp 2n

2

k

2 (l-p

2

)s 2/N

2 (51)

The relationship between p£ and the Fourier coefficients p

k of the true spectrum is

p£ = P|< exp I -2n

2

k2 (ps)

2/N2 (52)

Comparison with equation (39) shows that we have effectively reduced the spectrometer line

by a factor p. The choice of p (<1) will depend upon the particular noise level.

190
4. Summary and Conclusions

The object of the present paper was to show how the resolution ultimately attainable

by line fitting or Fourier transformation is affected by the counting statistics. The

results may be used to estimate the theoretical resolution limit under various experimental

conditions.

It was shown that, in the case of deconvolution by Fourier methods, a frequency filter

must be used to eliminate the high-frequency contribution from the noise. It is suggested

that the appropriate shape for the frequency filter is a Gauss curve corresponding to a reduced spectrometer line width. In a fast on-line deconvolution, the effective line width

could be made variable and adjusted to the prevailing experimental conditions.

Independently of the method of deconvolution employed, all computer programmes used

for evaluating energy-dispersive spectra should contain a test of the reliability of the

results.

References

[1] Kaiser, H. and Specker, H. , Anal. Chem. , 149, 46-66 (1956).

[2] Kaiser, H. , Anal. Chem., 209, 1-18 (1965).

[3] Kaiser, H. , Anal. Chem., 216, 80-94 (1966).

[4] Ziebold, T. 0., Anal. Chem., 39, 858-861 (1967).

[5] Svoboda, V. and Gerbatsch, R. , 2. Anal. Chem., 242, 1-13 (1968).

[6] Hubeaux, A. and Vos, G. , Anal. Chem., 42, 849-855 (1970).

[7] Kaiser, H. , "Methodicum Chimicum", I, Part 1, 1-20 (1973).

[8] Ryder, P. L. , Scanning Electron Microscopy/1975 (Part 1), 111-117.

[9] Ergun, S. , J. Appl . Cryst. , 1_, 19-23 (1968).

[10] Beevers, C. A. , Acta Cryst. , 5, 670 (1952).

[11] Stokes, A. R. , Proc. Phys. Soc. (London), A6J, 382-391 (1948).

[12] Cohen, J. B. , "Diffraction Methods in Materials Science", 80-105, MacMillian, New York

(1968).

[13] Ryder, P. L. , Scanning Electron Microscopy/1977, Vol. 1, 273-280.

191
Table 1. List of symbols.

I, I rI 2

'hn W

s

N

a

U

e

K

P(E)

P'(E)

G(E)

k

Pk' p

n' gk

a

R

n

M

channel number

mean number of pulses in channel n

actual number of pulses in channel n

background count per channel

total line counter

Gauss function, see equation (4)

line half-width

2.35 W

total number of channels in spectrum

standard deviation

background counts in range s (U = sB)

line separation parameter e = (h-j-f^Vs

(h.j = channel no. of maximum of line i)

Fourier frequency

original energy spectrum

measured energy spectrum

spectrometer function

Fourier frequency

Fourier coefficients of P (or P

n

), P' (or P^) and G, respectively

maximum energy of spectrum

2n/Eno statistical deviation of channel n ("noise")

Fourier coefficient of R

n total number of counts in the spectrum

192
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

CURVE FITTING TECHNIQUES AND THEIR APPLICATION TO THE ANALYSIS OF ENERGY DISPERSIVE SPECTRA

Frederick H. Schamber

Tracor Northern, Inc. 2551 W. Beltline Highway

Middleton, Wisconsin 53562

Abstract

Curve Fitting is a widely used procedure in many areas of analysis

and has become one of the dominant methods employed in the analysis of

energy dispersive spectra. Despite this popularity, curve fitting

methods are not universally understood by x-ray analysts and there is

thus the potential both for misapplication of the techniques and

disagreement as to the interpretation of curve-fitted results.

Curve-fitting analysis can frequently produce spectacular results

by extracting information which is not readily accessible by other

techniques. Since the basic principles of the method are generally

buried in the complexities of a computer program, the unfortunate

impression is sometimes given that curve-fitting is a form of statistical

alchemy which purports to extract unlimited information from minimal

data. Enthusiasts have been known to make extravagant claims for the

method, thereby confirming the worst fears of skeptics, who properly

mistrust any technique which seems to offer "something for nothing."

The primary area of misunderstanding seems to be one which is

addressed by the question: "What can curve fitting really do, and what

can it not do?" It is the intent of this paper to deal with this primary

question by discussing the basic theory and practice of curve fitting, at

a common sense level and as it specifically applies to problems in

extracting x-ray intensity information from energy dispersive spectra.

In so doing, the following basic questions will be addressed:

(1) What actually is meant when we speak of "curve fitting?" What

are the underlying principles?

(2) How does one actually implement a curve-fitting procedure?

What are the basic algorithms and how do they differ? What are

the practical considerations which determine which technique

is most appropriate?

(3) What kind of information does one have to supply to the procedure and what kind of information can one obtain? What is the

tradeoff between information supplied and information

extracted?

(4) What are the practical problems encountered when one applies

curve fitting methods to EDS spectra? How serious are these

problems and how might one propose to deal with them?

(5) What kind of confidence can one have in curve-fitted results?

How can one estimate statistical uncertainty and how reliable

are such estimates? Does the technique have the ability to generate grossly erroneous results, and if so, what

precautions must be exercised in interpreting fitted results?

193
It is clearly impossible to deal completely with each of these

topics - indeed, unqualified answers cannot be given to many of these

questions. It is instead the intent of this paper to clarify and briefly

explore these issues as they specifically relate to the suitability of

curve-fitting techniques for practical analysis of x-ray spectra.

1. Introduction

Curve fitting has become one of the major methods employed in the analysis of energy

dispersive x-ray spectra. Despite this popularity, curve fitting methods are not universally understood by x-ray analysts and there exists the potential both for misapplication

of the techniques and disagreement as to the interpretation of curve-fitted results.

It is the purpose of this paper to address two basic questions:

(1) What are the principles and practices of curve fitting?

(2) How reliable is curve fitting as an analysis technique?

2. Principles and Practices of Curve Fitting

This section of the paper will address itself to providing an overview of what curve fitting is and how it is performed. The primary intent will be to develop intuitive

concepts and to provide a working vocabulary for the non-expert.

As the name implies, "curve fitting" is a procedure whose goal is to match a mathematical representation of a curve (i.e., a fitting function) to a collection of data

points. Such procedures are applied to a wide variety of problems and for diverse

reasons. For example, if you pick up a book written by a statistician or a theoretician,

you will probably encounter curve-fitting used as a tool for functional analysis , that is,

the practitioner is attempting to determine the "true" representation of an observed

distribution of data by determining which function fits it best. (This is the kind of

analysis which is concerned with significance tests.)

Another kind of curve fitting, commonly used in numerical analysis and engineering, is

concerned with numerical approximation . Here one is interested only in obtaining a suitably

simple approximation for a collection of data or a transcendental function. One does not worry about the "true" functional representation or about the "meaning" of the parameters.

(This is the kind of application which makes frequent use of orthogonal polynomials and

"spline fitting".

)

In this paper we are concerned with a different type of curve-fitting application,

which we will refer to as unfolding . This type of problem is fundamentally different from

the first two types in that it is assumed that the functional representation of the curve

(spectrum) being fitted is already known; the goal is to determine the values of specific

parameters which are meaningful from an analytical viewpoint. Although the implementation

of an unfolding problem often involves aspects of other types of application (e.g., determining the best function to represent peak shapes or approximating the background with a polynomial) these are only preparatory steps to the main analysis problem.

The analysis of an x-ray spectrum is a classic example of an unfolding problem. The

spectrum consists of a "folding in" (summing) of a set of independent responses due to characteristic elemental emissions (peaks) and a relatively smooth underlying continuum

(background), upon which is superimposed a "noise" component of random statistical variations. The spectrum is to be analyzed by "unfolding" it into its constituents so that the

amplitudes of the relevant parameters (peak amplitudes) can be known. The "solution" to the problem is achieved when one determines the set of parameters which give the "best fit"

to the unknown spectrum.

194
NOTE: This type of curve fitting problem has sometimes been

referred to in the x-ray literature as "deconvolution" (a

practice which this author has also been party to). Howeve,

deconvol ution is the accepted name for a specific process

employed in Fourier Analysis (e.g. , refer to the paper in this

volume by P. L. Ryder). The use of this term to describe an entirely different type of analysis process should therefore

be avoided.

We can describe the major considerations involved in implementing a curve fitting

procedure as the following:

(1) Choosing the fitting function.

(2) Selecting a "best fit" criterion.

(3) Implementing an optimizing algorithm.

(4) Establishing the validity of the results.

These topics will now be discussed in some detail.

2.1 The Fitting Function

The first (and often the most difficult) step in implementing the analysis procedure

is to choose a "fitting function" which provides an appropriate representation of the data

to be fitted. We shall employ the following definitions:

y. = the measured amplitude of the jth data value

J (the "data points" to be fitted)

y. = the functional approximation of the jth data value

^ (the "fitting function" being employed)

In general, a complete fitting function is made up of the sum of several component

functions

y. = (function 1 ) .+(function 2) .+... +(function n)

.

J J J 3 For example, a spectrum consisting of two peaks and a "background" function might be

represented as:

y . = (peak 1 ) .+(peak 2) .+(background) . J J J J In turn, the peaks might be represented as Gaussian functions

(C.-x.)'

1 32 2S.

(peak i)^ = A^e

(1)

where A.. , C., and are respectively the peak amplitude, centroid, and width parameters

(S=FWHM/2.354; FWHM^Full Width of the peak at Half the Maximum amplitude).

195
A. Linear Functions

In many cases of practical interest, the fitting function can be expressed as a linear function of the parameters of interest. A fitting function is linear if it can be

expressed in the form

y

j

=Al F lj +A

2

F 2j + --- +A

n

F

n j «)

where the F. . are predetermined functions of the j index and the A^ are the linear coefficients to be determined from the fit. For example, the background function might be

chosen to be a quadratic function

(background) . = B,+B„x .+B,x . 2 (3)

where x^. is the energy-axis coordinate and the coefficients B-|,B 2

,B

3 are the free parameters

to be determined.

Similarly, if fitting with Gaussian peaks, one might be able to assume that the peak

centroids (C^ ) and widths (S..) are known so that only the amplitude coefficients (A^ ) need

to be determined. The complete fitting function is then linear and can be written as

y. = B 1+B

2x.+B3x

j

2+A

1 g

1 j+A

2

g

2

. (4)

where g^. and are Gaussian functions with fixed centroid and width parameters and

B-j^ B^.A-j^ are the free linear parameters to be determined.

Another example of a linear fitting function in x-ray analysis is the use of measured

single-element spectra as the fitting functions (a subsequent paper in this volume by

McCarthy and Schamber describes such a method).

B. Non-Linear Functions

Non- linear functions are those in which the parameters of interest appear in ways

other than as simple multiplicative coefficients. Thus, the centroid C| and peak width S. of the Gaussian function are non-linear parameters; if they are to be determined in the

fit, then the fitting function is non-linear.

Non-linear fitting functions must be used in x-ray spectrum analysis when the centroid

and/or width parameters are not predetermined. Such parameters would be variables if the

x-ray detector's calibration and/or resolution are uncertain. (The SIMPLEX program

described in a subsequent paper by Fiori, et al., employs non-linear Gaussian functions.)

In general, linear curve-fitting problems are much simpler to deal with than non-linear

problems. It will become obvious in the following that nearly all of the complications in

implementing a curve fitting program have to do with non- linear parameters.

2.2 The "Best Fit" Criterion

Another important consideration is to define what is meant by "best fit". For example,

one might propose to seek the solution which minimizes the sum of the absolute errors, which

is almost universally used in physical analysis problems is the "least-squares" criterion,

which is the solution (set of parameter values) which minimizes the quantity

i (Pj-y/wj (5)

where the w. are suitably chosen "weights". The least-squares criterion is intuitively

J \ attractive since it tends to concentrate on eliminating large errors and is relatively

insensitive to small errors.

196
If the weights Wj are all chosen to be one (or equi valently , all equal to the same

value) then all errors will be given equal importance and minimizing this function will

result in an unweighted fit. In practice, however, one knows that an error of 200 counts

is more important when y. is 10,000 than when it is 1,000,000 since the inherent statistical

uncertainty (standard deviation) in the former is a * 100 compared to a * 1000 for the

latter. Thus, the weights should account for the statistical precision of the data. It can be shown that when the weights are chosen such that

w

j = --| (6)

where o\j is the variance (standard deviation squared) of the jth data point, then the

choice of parameters which minimizes

X

2 - 1 ^/J J (7)

j °j

is the most probable solution [I] 1

. This is intuitively reasonable since if is the

uncertainty in the measured value y. and if the fitted value y. is the "true" value of this

J J point (e.g., equal to the average y. which would be obtained in an infinite number of

trials) then it follows that ^

( y.ry

.i )2(8) a

2

J

is the same (on the average) for all data points; thus all points receive "equal" weighting

(in the sense of proportionality to statistical precision). It can be shown that if the

measured points y. are normally distributed about y^ with standard deviation a then

X

2~(N-n) where N is the number of data points fitted, and n is the number of free

parameters.

It should be carefully noted that the quantity x

2 has specific statistical properties

and that strictly speaking, the quantity defined by Equation 7 is not x

2 unless it

possesses these properties [2]. In particular, Equation 7 is not the statistical x

2 unless the y.. are the true solution points. On the other hand, the solutions obtained by

minimizing Equation 1 are the most probable solutions only if this quantity is truly the

X

2 of statistical theory. Seldom (if ever) can we be certain of these facts. However, as long as we do not depart too far from these rigorous concepts, experience tells us that the

set of parameter values which minimize an approximate expression for x

2 will be reasonable

approximations of the "true" solutions.

A note on weighting seems appropriate at this point. It is often stated

that (assuming Gaussian statistics) the standard deviation of a count is

simply the square-root of the count. Therefore, it is common practice to use y. as the weighting factor in Equation 1. This is not strictly

J accurate since it is the true count value (average of all possible mea- ments) whose square-root is equal to the standard deviation. This

distinction can be safely ignored when the magnitude of y. is large, but

when it is small, the relative variation in yj is substantial and the use

Figures in brackets indicate the literature references at the end of this paper.

197
of w. = 1/y. is quite inaccurate; note particularly that when y. deviates

J J 3 to a small value the point is weighted more strongly in the fit than when

it deviates to a large value (e.g., for y. = 100±10, compare w. = 1/90

J 3 versus Wj = 1/110. This means that the fit will be consistently biased

towards low values. Fortunately, it is not hard to correct for this

problem; one needs only to smooth the weights in some fashion to minimize their statistical fluctuations.

Although other "best fit" criteria are frequently used in some fields (such as numerical analysis) the "least-squares" criterion is so commonly used that is nearly

synonymous with "curve fitting". Other designations which are also used to refer to the

minimization of Equation 7 include "error-weighted least-squares" and "chi-squared

minimization". In the remainder of this paper, it shall be assumed that the best-fit

criterion to be employed is the error-weighted least-squares procedure defined by Equation

7; the term x

2 will be used somewhat loosely, fully realizing that this may be only an approximation to the chi-squared function of statistics.

To reiterate, the "best" values for the parameters are assumed to be those which

minimize x

2

; the problem is to obtain those parameter values. As an aid to understanding

this problem it is useful to visualize the variation of x

2 versus the parameters as describing a surface. Figure 1 illustrates such a "chi-squared surface" in two dimensions

(parameters a and b). The least-squares solution is the point a . , b . which locates the

mi n mm

lowest point on the surface. As a simple means of representing x

2 surfaces, we often employ

a "contour plot", as in figure 1 where the contours of constant x

2 are projected onto the

(a,b) plane. As additional parameters are introduced, the x

2 surface must be represented

as an n-dimensional surface. Obviously, such higher-dimensioned surfaces are difficult to visualize, but the concepts are the same as for the simple two-dimensional case.

2.3 Optimization Algorithms

Having defined the optimization criterion (minimum x

2 )> it is now necessary to consider how this is to be accomplished. There are three main classes of methods which are commonly employed. These are:

(A) Pattern Search Methods

(B) Gradient Search Methods

(C) Analytic Solution Methods

A. Pattern Search Methods

Pattern-search methods are essentially "mechanical" methods of locating the minimum.

They do not make explicit use of the mathematical properties of the x

2 function.

Of the pattern search methods, the simplest conceptually is to divide the range of

possible parameter values into a grid of appropriate "mesh" size and to compute the value

of x

2 for each point on the grid. One then determines by inspection which point on the

grid has the smallest x

2 and this is taken to be the best solution. For example, in

figure 2, the value of x

2 for a hypothetical problem is tabulated for each of 11 possible

values of the parameters C x and C 2 respectively. The "best" solution is obtained in the

vicinity of C 1 =15, C 2 =17 where x

2=37. If we wish to locate the minimum with greater

precision, we can create a new grid of smaller mesh in the vicinity of this solution and

look for an even smaller value of x

2

-

198
CO
c2 >

16.95 16.96 16.97 16.98 16.99 17.00 17.01 17.02 17.03 17.04 17.05

14 .95 46762

j 38158 30664 24279 19002 14833 11770 9813 8959 9210 10562

14 .96

I 37490 29465 22911 17286 12769 9360 7056 5858 5765 6774 8886

14 .97 29475

I 22391 16417 11552 7795 5145 3601 3162 3828 5597 8468

14 .98

I 22718

I 16394 11180 7074 4077 2187 1403 1724 3149 5677 9307

14 .99

1 17217

I 11653 7198 3853 1615 485 460 1540 3725 7012 1 1402

15 00

l 12970

1 8166 4472 1886 408 -I -© 772 2612 5556 9602 14751

15 01

1 9977

1 5933 2998 1 172

I

I 454 843 2337 4937 8640 13446 19353

15 02

1 8236

1 4951 2777

I 1711

I 1753 2901 5155 8513 12976 18541 25208

15 03

1 7745-- 5221-- 3806-

I - 3500 4302 6209 9223 13341 18563 24887 32313

15 04 1 8504 6740 6085 6539 8100 10768 14541 19418 25399 32483 40668

15. 05 10512 9508 9613 10826 13147 16574 21107 26744 33484 41327 50271

Figure 2. Grid of x

2 values as a function of two parameters.

Such a "brute-force" procedure is infrequently used as a practical solution method

simply because it is so inefficient. However, it i_s a valid optimization technique and

illustrates how conceptually simple the problem really is. The more sophisticated methods

described in the following are superior only in the speed with which they can locate the

minimum.

Grid Search . An obvious improvement on the above crude method would be to inspect the

grid only for those points which lie in the direction of decreasing x

2 so as to eliminate

a great deal of unproductive computation. One such simple method is a "grid search" which

optimizes each of the parameters independently. This is illustrated in figure 2 for the same two-parameter example. Starting from the initial estimate C 1 =14.95, C2 =16.95 one first optimizes C 1

; the minimum in this column is found at C 1 =15.03. Then C 2 is optimized

while C x is held constant, and so forth along the path indicated in the figure. Again,

when a grid minimum has been found, the mesh size can be decreased and the search repeated

until the minimum has been located to the desired precision.

Grid Search Example . To illustrate the grid search procedure, a simple two-Gaussian

test case was devised (figure 3). The "unknown" spectrum to be fitted (dotted spectrum in

lower trace) consists of two Gaussians (top trace) of equal amplitudes (10,000) centered at channels 310 and 340 respectively, and of respective widths of 15 and 17 channels (FWHM).

The fit was started with the initial "guesses" shown (bars) and all six of the peak

parameters (two amplitudes, two centroids, two widths) were varied to obtain the best fit.

(The actual fitting procedure is modelled after one described by Bevington [3], including

the use of quadratic interpolation along an axis.) Figure 4 illustrates the six steps in

the first iteration: proceeding left-to-right and top-to-bottom the fit is first optimized

for the first peak amplitude (7277) and the second amplitude (12131), then for the first

peak center (305.4) and the second (335.29), finally for the width of the first peak

(14.70) and the second (19.73). In this first iteration, x

2 decreased from its initial

value of 6.8xl08 to a value of 1.38xl08

, but the fit to the curve is still very poor.

200
vFS-tsm 2 set

Figure 3. Two-Gaussian example problem:

Upper - the individual "true"

peaks. Lower — the composite

"true" spectrum (dots) and

the initial estimate (bars).

TRUE PAI PEAK1 PEAK2

AMPLITUDE: 10000 10000

CENTER: 310 340 FWHM: 15 17

IATES PEAK1 PEAK2

9000 12000

300 330

12 16 (FIT FROM CHANNEL 250 to 400)

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Figure 4. The six steps comprising the first iteration of a grid-search

fit of the two-Gaussian_ problem.

201
Figure 5 illustrates the progress of the fit in subsequent iterations. By iteration 24

the fit is starting to look quite reasonable, but x

2 is still 1.9xl0 5

. The fit will/

eventually converge to x

2 =6.2xl0 1 on the 55th iteration. (This is a synthetic example with'

no statistical error so x

2 is unreal istical ly small.)

Figure 5. Grid-search solutions for the two-Gaussian problem at the end of

2, 3, 4 (left), 8, 14, and 24 (right).

This example illustrates how slowly the grid-search method may converge in a complex

problem. Since each of the parameters is varied independently, the search for the x

2 minimum describes a tortuous path of many small incremental variations rather than a smooth trajectory.

Simplex Method . The sequential simplex method is a particularly elegant pattern-search

method. Like the grid-search, the simplex method is an intuitive geometric approach which

involves only simple computations. But unlike the grid-search, the simplex method varies

all parameters simultaneously. A simplex is defined as a geometrical figure of n+1 vertices, where n is the number of free parameters. For a single parameter the simplex is

a straight line connecting two points, for a two parameter problem the simplex is a triangle, for three parameters a tetrahedron, etc. Figure 6 depicts the simplex for a two dimensional problem (triangle ABC). The optimization technique is to move the simplex i across the parameter space in such a way that it always moves away from the least desirable j vertex and thus (presumably) moves towards the optimum parameters. This procedure is

governed by a set of simple rules (lucidly explained in reference 4) which will only be

illustrated here. Referring to figure 6, assume that x

2 is known for each of the three

?02
points, A, B, C and that A has the largest value of x

2

- Since the response at A is less

desirable than the response at either B or C a better response would presumably be found

if one moves away from point A; point A is discarded and a new vertex is chosen as the

reflection of point A across the line BC to obtain point D. The new simplex, BCD, is now evaluated, the least acceptable point is again reflected, and the process is repeated until

the simplex arrives in the vicinity of the minimum.

c

A

D

B

Figure 6. The basic two-dimensional simplex ABC, and its reflection to BCD.

An improved implementation of the basic simplex method employs "acceleration" which

permits the dimensions of the simplex to be expanded or contracted along the direction of

reflection. Figure 7 illustrates the moves in this modified simplex method (again a two- dimensional example). B. W, and N are respectively the Best, Worst, and Next-to-worst

responses. The first step, as before, is to compute the response at point R, obtained by

reflecting W across line BN. R is then compared with the other vertices according to precise rules [4] and expansion to point S, or contraction to points U or T can occur. Thus, the new simplex may be any of four possibilities: BNR - ordinary reflection, BNS -

expanded reflection, BNU - contracted reflection, or BNT - contraction on the same side.

These relatively simple modifications not only eliminate certain problems inherent in the

basic method ("straddling" and "circling"), but they also enable the simplex to automatically adapt to the optimization surface; the simplex expands to move rapidly when distant

and contracts as it nears the vicinity of the minimum.

203
Figure 7. Possible moves of the accelerated simplex modification.

B. Gradient Search Methods

Gradient methods represent an increased level of sophistication over the patternsearch methods. Instead of relying upon "trial-and-error" to direct the search, the

gradient methods attempt to predict the direction in which the minimum is to be found by

making use of the gradient (slope) of the n-dimensional x

2 surface. This is often referred

to as the "method of steepest descent" since the idea is that for any given set of parameter

estimates, the next set of estimates will be chosen along the direction of change which

causes x

2 to decrease most rapidly [3,5]. In two dimensions, this has a simple physical

analog: referring to figure 1, think of releasing an object at any point on the x

2 surface;

the direction in which it would initially move under the influence of gravity is the

direction of steepest descent (the negative of the gradient vector), and the path which it

would follow to the bottom (disregarding inertia) is the ideal route of the minimizing

procedure.

The mathematical basis of the method is illustrated in figure 8 for a two dimensional

X

2 surface. At an initial point, the negative components of the gradient are evaluated

with respect to the parameters a and b:

D = -§X? D = (9)

a 8a u

b 3b

and the "steepest descent" vector D is the vector sum of D

g and D

b

. (In practice, the

gradient may either be obtained exactly by evaluating the partial derivatives of the x

2 function, or approximated from the observed variation of x

2 for small incremental changes

in a and b.

)

204
Figure 8. The steepest descent vector D.

The steepest-descent vector is always perpendicular to the local contour of constant

chi-squared. For a "well-behaved" x

2 surface with elliptically shaped contours (c.f.

,

figure 22) the gradient vectors from all points on the surface will pass through the minimum. For an irregular surface (c.f., figure 19) the steepest descent vector at an arbitrary

point may aim far from the true minimum and it will be necessary to recompute the gradient

many times. In an "ideal" application of the method, the steepest descent trajectory would

be composed of a continuous series of infinitesimal gradient vectors. In practice, one must

use steps of finite size and therein lies a problem of the method; if one moves too far in

the direction indicated by the steepest descent vector, one can overshoot the minimum,

perhaps even to a point of higher x

2

- On the other hand, using very small increments and

continuously recomputing the gradient is time-consuming. In short, although the steepestdescent method tells us what direction to move, it does not tell us how far to move in that

direction. The usual procedure is to compute the steepest descent vector at an initial

point and then move in this direction in small increments until x

2 no longer decreases. It

is then customary practice to use quadratic interpolation to estimate the minimum point

along this vector and use this as the point from which to compute a new gradient vector.

Gradient methods would seem, in principle, to be very effective minimization procedures. This is not necessarily the case. First of all, gradient methods have difficulty

with narrow curving valleys ("ravines"); such situations are quite common in peak fitting

problems as will be seen in Part 2. Secondly, gradient methods converge slowly when the x surface is relatively flat, as tends to be the case in the vicinity of the minimum.

205
C. Analytic Solution Methods

The principle of the analytic method is to directly solve Equation 7 for the parameter

values which will minimize x

2

- By definition, the gradient ("slope") of x

2 with respect to each of the parameters must be zero at a minimum. To find the minimum, we solve for the

set of parameters for which

S-=0 (k=l,n) (10) da

k

If yj is a linear function of the parameters a.

then applying the condition of Equation 10 to Equation 7 results in the set of n coupled

equations

^i

F kin F ii

F ki I -Jy^ = I a.I -^L^ (12)

j o] i=l 'j

which can be stated more compactly in the form

G

k = j1 Vik < 13 >

where the following definitions have been employed

y.F. . G. = I -^2^ (14a)

F. .Z. . Z

ii =I_ 2 (14b)

j °j

Z is an nxn matrix ("curvature matrix") and G is a column vector of n elements ("normal

vector"). It should be noted that Equation 13 is nothing more than a set of n equations in

n unknowns (often called the "normal equations").

For illustration, we will write the appropriate equations for the simple

example of a quadratic fitting function.

y. = a,+a 9 x.+a~x. (fitting function) (15)

2 2

P (y.-a,-a?x.-a-x. ) X = I

-J— —^

(object function) (16)

j a

3

206
2 Using these definitions and approximating a. * y. we obtain for

Equations 14a and 14b: ^ ^

2 y. y.x. y.x.

G

] = I -1 G2 = 1 -^-1 G- = 1 -J-Jjyj

j y

j

j y

j (17)

Z

ll =

j y

12 i _ j y

x

.

Z = I

12 h

Z

21 Z 12 "22

Z 31 Z

ll Z 32

Z 23

2 2 x. x.

Z 33 = 1 -

J^JAfter computing the G and Z terms from these equations, one then obtains

the optimum solutions for a,, a

9

, and a

q by solving the normal equations:

G

1 = a

1 Z

11 +a

2Z21 +a3Z31

G

2 = al Z 12+a2Z 22+a3Z32

G

3 = al Z 13+a2

Z 23+a

3 Z33

Any reasonable method of solving the normal equations can be employed to obtain a solution. Most commonly, the solutions are obtained by first computing Z

-1

, the matrix

inverse of Z, and then solving for the a. via

nI

i=l

a. = G.ZT

1

. (19)

i ij

(For this reason, the analytic method is sometimes referred to as "the matrix inversion

method".)

One of the useful characteristics of this method is that it permits the uncertainty

(standard deviation) in the fitted parameters a. to be readily estimated from the relation

a

2 *z:!

a. = n (20)

Because of this property, Z

-1 is often referred to as the "error matrix".

Unlike the preceding methods which proceed to the minimum by a stepwise iteration

process, the analytic method obtains the optimum coefficients of a linear problem in a single computation. Not only is the analytic method invariably faster in such cases, but

it has the additional advantage that it requires no initial guesses, step sizes, or other

preparatory estimates. For this reason, linear least-squares problems are almost always

solved by the analytic method, leading sometimes to an unfortunate tendency to use the

207
terms as synonyms (which can result in confusing statements such as "an advantage of the

simplex method over the least-squares method is ...").

Analytic Solution of Non-1 inear Problems (Linearization) . The above remarks are all relevant for the case of a linear fitting function. In the case of a non-linear function,

application of Equation 10 to Equation 7 gives rise to a set of non-linear coupled equations

which are generally non-analytic and cannot be solved directly.

In certain special cases, one can convert a non-linear function to a linear function by

suitable manipulation of the problem. For example the case of fitting y". with the function

Bx. J yj = Ae J (21)

can be transformed to a linear problem by fitting to ln(y.), in which case the fitting

v function is of the linear form ln(A)+Bx. (note that the weighting function w. also needs to

J J be transformed appropriately). Such "tricks" are generally of quite limited utility and more general methods are needed for solution of routine problems.

The usual method used to solve non- linear problems by the analytic method is to "linearize" the problem by approximation in the vicinity of an estimate. The Taylor-series

expansion for a function f(x) about the point x=x° is given by:

f (x) = f (x°)+(x-x°) ax

, (x-x°)

2 32

f x=x° 2 3x2 (22) x=x

As a first-order approximation we disregard high order terms and use only the linear

expression (figure 9). For a function of n parameters

y(a). i y(a°).+ I Aa.y'(a0 ).. (23)

where a represents the complete set of parameters a-j . . ,a

n

; where a represents the set of initial estimates a^.a^.-.-.a^ of these same parameters; where y'(a0

).. is the first

partial derivative of y. with respect to a. and evaluated at a

0

; and where

o Aa. = a. -a. (24)

is a "small" variation of the ith parameter relative to the estimated value a.. Inserting

2 this approximation into Equation 1, one than has an approximation for x

where

(d. - 1 Aa.y'(a0 )..)

2

2 ~ v

J i=1

1 1J (25) x . 2

dj = y

j

-y(a°)

j

(26)

is the residual error in the jth fitted point for the current estimates a

0

. What we have

accomplished is to approximate a non-linear function of the a. parameters by a linear

function of the new parameters Aa. in the vicinity of the estimates a°. One now proceeds

208
by treating the Aa^ as the fitting parameters. By requiring the first partial derivatives

of x

2 to be zero with respect to these new parameters, one again obtains a set of linear

normal equations which can be solved for the increments Aa. (simply replace y. by d., F..

o ~ J J

1 J by y'(a ). ., and a. by Aa. in Equations 10 to 19). After solving for the Aa increments, one

o o again computes new estimates a by adding the increments to the old a estimates. The

process is then repeated until the fit converges (e.g., x

2 is minimized when all Aa. approach zero).

Linear approximation of the fitting function results in a parabolic approximation of

X

2 as shown in figure 10 for a one-dimensional case. The minimum of this approximation

parabola is closer to the true minimum than is the original estimate. Generating a new approximation parabola at this improved estimation point results in a still closer approximation, etc. It is clear that such parabolic approximation is potentially quite poor when

far from the minimum, but sufficiently close to the minimum any x

2 surface is parabolic

and the approximation is exact.

Equation 20 can still be employed for estimating the uncertainties in the computed

parameters (since a? are treated as constants all of the uncertainty is in the Aa.).

However, the validity of this equation relies upon the premise that the "true" solution

has been attained.

209
Figure 10. Parabolic approximation of the X

2 function at the point a .

Example of Li neari zed-Analytic Solution . To illustrate the application of the linearexpansion analytic solution method, we again consider the example of the two-Gaussian peak

fit with six free parameters (the same problem as for the earlier grid-search example).

For this problem, each Gaussian is linearized as follows:

Gaussian(A,C,S)j

where

= 9

0j AAg ACg

2j ASg3j

(27)

90j = A exp

•(C°-x.)

2

2(S

0

) 2 (28)

9t j = exp

2(S

0

) 2 (29)

J 2j -A exp

(C

0-x.)

2

2(S

0

) 2 (30)

9

3j = A (C°-x.)

2

3 J__

(S°)

3 exp

(C°-x.)

2

'

2(S

0

) 2 (31)

210
The fitting was thus performed for the six linear parameters AAj , AA2 , AC ls AC2 , AS X , AS2 (alternate ways of dealing with the linear A parameters could also have been employed).

Figure 11 illustrates the progress of the fit for the first five iterations: IT(0) (upper

left frame) is the fit for the initial estimates (with the same starting parameters as for

the grid-search fit); iterations 1 through 5 (IT(1) - IT(5)) then follow in a top-to-bottom

sequence. The example converges to a final values of x2= l-8xl0 1 on the 9th iteration.

Figure 11. First six iterations of the 1 i nearized-analytic fit of the two-Gaussian example.

Fai 1 ure of the Li neari zed-Analyti c Sol ution Method (Damping) . In view of the limited

validity of the linear expansion used in this method, it is not surprising that the procedure encounters difficulty when the initial parameter estimates are far from the minimum.

This is illustrated in figure 12 for a different example: the true solution in this case (dotted curve) is the same as for the earlier example (cf . , figure 3) except the peak

centers are now C 1 =310, C 2 —31 5 ; the starting estimates are as before (solid curve).

Initially, x

2 -2.1xl09 for the starting estimates, this decreases on the first iteration to 4.0xl08 and then i ncreases on the 3rd iteration to 5.6x10s

, to 9.7xl08 on the 4th, and

continues to increase on subsequent iterations! Instead of converging, the method has

diverged . This behavior can be understood as a break-down of the linear approximation; the

true x

2 surface is so poorly approximated that the solution obtained from this approximation

is farther from the true solution than was the original estimate. In short, the solution

subsequent iteration (being even further from the minimum) overshoots by a greater amount.

211
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A simple way of dealing with this oscillatory behavior is to introduce "damping" into

the procedure. The oscillations can be "damped" by changing the parameter estimates by

a fraction Q (Q<1) of the computed Aa^. Use of a Q factor of approximately 0.4 seems

to be a safe choice for the majority of problems. To expedite the procedure, Q can be

automatically increased by the program until QM as convergence is approached. Figure 13 illustrates the solution of the above problem after damping has been incorporated in the

program; the oscillations have been damped out and the fit will converge cleanly to the

correct solution on the 10th iteration.

It should be clearly stated that the sort of divergence problems which can be encoun- tered in the 1 inearized-analytic method do not make the method unreliable in the sense

of producing "wrong answers". When divergence is encountered, the problem is to obtain

any solution at all since the program tends to either "wander" indefinitely or "blow-up"

with attendant math overflow errors.

Comparison of the Algorithms . Four different solution method were programmed and

applied to the solution of the six parameter two-Gaussian problem already discussed (see

figure 3). The comparative performance data for these tests are summarized as follows:

Method Rel. Time Final x

2 Iterations

Linearized Analytic 1 18 9 Accelerating Simplex 5 22 246

Grid- Search 10 62 55

Gradient-Search 10 365 67

The "Final x

2 " value refers to the value of x2 which was achieved when the program was stopped; these values depend on the precise formulation of the stopping criterion in each

program and are thus somewhat arbitrary. The gradient-search method was stopped prematurely

since it was minimizing x

2 with excruciating slowness at this point; to obtain a x

2 value

as small as for the analytic method would surely have taken much longer than the factor of

10 indicated here!

212
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incorporation of damping.

"Relative Time" refers to the relative computation time for each program, arbitrarily

normalized to the fastest method (1 ineari zed-analytic). All of the fitting programs were

coded in the same higher-level language and run on the same computer. Actual computation

times for a specific implementation will depend strongly on the programming language, the

computation hardware used, and the details of program construction. The precise nature of

the fitting problem, (number and type of parameters, quality of initial estimates, number of

data points, stopping criterion, etc.) will influence both the absolute and relative speed

of each method.

For most problems, the analytic solution method is the best choice in terms of performance. Not only does it converge faster, but it also provides estimation of the statistical

uncertainties as a natural by-product of the computations.

A basic pattern-search procedure is probably the best choice for the user who is more interested in understanding curve-fitting technique than in absolute performance since the

"moves" of such a method are more readily conceptualized. The grid-search procedure is

varied. However, the simplex method is far more satisfactory in terms of performance.

Pattern-search methods are also a logical choice when optimizing is to be done via

criteria other than minimization of x

2 since these methods require only that some sort of

"desirability" criteria be established.

The gradient-search method would seem to be a poor overall method. Not only is its

convergence very sluggish in the region of the minimum, but its implementation is also

fairly complex. One point in its favor is that it is quite good at approaching the minimum

from afar (in the above comparison the initial rate of convergence of the gradient method

was significantly faster than that of the simplex or the grid-search). For this reason a gradient method is sometimes employed for the first rough-approximation, followed by the

1 inearized-analytic method for the final optimization.

213
2.4 Establishing the Validity of Fitted Results

It is an unfortunate fact that people tend to accept computed results uncritically

("If the number is printed to five decimal places, then the answer must be correct to five

decimal places"). For this reason, the program designer should make every reasonable

effort to provide the program user with relevant qualifying information.

It has already been noted the Equation 20 provides a useful estimate of statistical

uncertainty when the solution is obtained via the analytic method. Obtaining uncertainties

from the pattern and gradient-search methods is not so direct, but there do exist error estimation procedures for these methods also. Although statistical error estimates are just that, estimates which are not to be taken too literally, they do provide a useful

indication of precision.

Another useful indicator which should be reported for any least-squares program is X

2

. For convenience, x

2 is usually reported as "normalized chi-squared" , that is divided by

(N-n) where N is the number of data points fitted, and n is the number of free parameters.

When normalized, x

2~l indicates a "good" fit and x

2>> ! is indicative of a systematic

error, x

2 is not an ideal indicator since it is itself a statistical quantity which will

vary from analysis to analysis. Moreover, a small x

2 does not guarantee accurate results nor does a large x

2 guarantee inaccurate results since the sensitivity of x

2 to systematic error is directly related to the statistical precision of the data. This is because a systematic error of one percent in the measured content of a channel is not very significant

when the statistical uncertainty is ±10 percent but is very signficant when the uncertainty

is ±0.1 percent. As the statistics of a spectrum are improved (e.g., by longer counting

times) small systematic errors in the data or inadequacies of the fitting function will be

manifested in an increasing value of x

2

-

Of course, what a user would like is a fool-proof indicator of combined error (statistical and systematic), or at least a reliable way of knowing when the results are "bad".

Unfortunately, there is no completely satisfactory way of doing this, and the program user must bear some of the burden for qualifying the results (see the concluding Summary).

3. Reliability of Non-Linear Least-Squares: An Investigation

The first section of this paper developed the concepts of curve-fitting and detailed

specific methods of implementation. In this section, fundamental questions related to the

reliability of the method are explored in the context of investigating a particular

example.

Statement of the Problem

This investigation was motivated by a paper by Peter Statham in which he raised the

question of whether non- linear peak- fitting programs can be trusted to give correct

answers. As evidence, he cited a specific case in which non-linear fitting appears to fail

spectacularly [6].

The illustration is as follows: A synthetic two-peak problem is constructed from

Gaussian peaks of fixed widths (15 channels FWHM). The first peak has an amplitude of 5000

and is centered in channel 100, the second peak has an amplitude of 10000 and is centered

in channel 104 (simulating EDS spectral peaks of 150 eV resolution, centered at 1.00 and

1.04 KeV for an assumed calibration of 10 eV/channel). Simulated counting statistics are applied and this synthetic "unknown" spectrum is then fitted with two Gaussian peaks of

fixed width (150 eV) and unknown centers and amplitudes (4 free parameters).

Statham found that for one particular simulated spectrum the best fit was actually

obtained for the (keV, amplitude) parameters: Peak 1, (1.012,9119); Peak 2, (1.050,5903)

resulting in x2=0.925. In comparison, the "true" solution parameters Peak 1, (1.000,5000);

Peak 2, (1.040,10000) produce x

2=l - 173. That is, a "false" solution is found for this case in which the amplitudes of Peak 1 and Peak 2 are almost exactly reversed!

214
It is well known that false minima can exist in non-linear least-squares fitting

problems. Most commonly, however, false minima represent obviously absurd solutions, have

larger x

2 than the "true" minimum, and are encountered only when extremely poor starting

estimates are employed (e.g., figure 14). This is clearly not the case here since the

"false" solution is not only physically reasonable but has a lower x

2 than the "true" solution. This is a most disturbing example since it raises the prospect of getting wildly

wrong answers from a least-squares program and offers no apparent way to either prevent or sense such errors. The remainder of this paper describes an investigation of the important

questions raised by this example.

In figure 15, the upper left and upper right frames are the individual peaks for the "true" and "false" solutions respectively. Below, the composite solutions (dots) are overlayed on Statham's simulated spectrum data (bars). It is immediately clear that both

the "true" and the "false" solutions are very good fits to the data. Although there are subtle differences in the shapes of the two solution curves, they are effectively obscured

by statistical fluctuations in the data. There are several significant points to consider

in this example:

1. The false solution has a smaller x

2 in this instance simply because the statistical

fluctuations in the data happen to favor this solution. This implies that the

phenomenon is statistical in nature (rather than systematic) and should occur with a statistical probability distribution.

2. There is nothing particularly meaningful about the fact that the results are "reversed". If one peak amplitude is increased by 5000 then the other must naturally

be decreased by 5000 to maintain the correct number of total counts in the spectrum.

The important question is not why the amplitudes are reversed, but rather, whether

such a large error is reasonable.

3. The choice of a relatively narrow fitting window (channels 90 through 116) may be a contributing factor since there is a substantial amount of information which is

contained in the "skirts" of the Gaussians. Truncating the fitting region in this

way reduces the amount of information available to the fitting routine. This reduces

the precision of the analysis (i.e., increases a for each of the computed parameters)

and increases the probability of obtaining answers which deviate significantly from

the "true" solution.

215
cursor: i ee»pj =3000 Roi<ee> 0.900: 1 178=75528

pi (-5000. i.00J P2<ieeee, i.04>

cursor: i bbbkeu S414

piae&a, l ei4>

ROK00) 0.90B.' 1. 176=74432

P2<4889, 1.054)

Figure 15. Illustration of the "true" and "reversed" solutions of Statham's example case

(left and right): Upper - the individual solution peaks. Lower - comparison

of the composite solutions (dots) with the simulated data (bars).

Taken together, these observations suggest that if the fitting problem were reformulated so as to include the skirts of the Gaussians, the precision of the determination would

improve and the probability of obtaining the "false" solution would diminish (to "zero", one might hope). But there is a practical problem in broadening the fitting region; as one gets

far enough out into the skirts, the Gaussian function approaches zero and the statistical

distribution of the channel contents is governed by the Binomial distribution. Binomial

statistics are not only difficult to simulate, they also create unique problems in weighting

the least-squares fit properly. (It was for precisely this reason that Statham restricted

the fitting region as he did.) The approach used to circumvent this difficulty was to change the definition of the problem somewhat and to superimpose the Gaussian peaks upon a constant "background" of 100 counts per channel. The fitting region was then extended to include channels 70 to 135 (roughly the portion of the spectrum depicted in figure 15).

Test I

The test procedure was to generate a true spectrum using the following parameters:

Peak 1: amplitude = 5000, centroid = 1.00 keV, width = 150 eV

Peak 2: amplitude =10000, centroid =1.04 keV, width = 150 eV

Background = 100 (each channel)

(calibration: 10 eV/channel)

(Note that the above ampl itudes correspond to peak areas of approximately 80,000 and

160,000 counts respectively.)

216
This true spectrum was perturbed with random counting statistics (simulated via a com- monly used computer algorithm) and the resulting "unknown" spectrum was then fitted using

the lineari zed-analytic method with five free parameters (2 peak amplitudes, 2 peak centers,

1 background amplitude). In order to give the "reversed amplitude" solution plenty of

opportunity to occur, the starting estimates for the peak parameters were deliberately

chosen to be in the near vicinity of this false solution. This test was run repeatedly

(with different statistics each time) to simulate a series of independent physical measurements. Figure 16 depicts the distribution of the Peak 1 amplitude for a series of 1604 such

runs (the Peak 2 distribution is equivalent). The horizontal axis was divided into bins of

0.2 a width and each "+" represents an event which was counted in that bin. The horizontal

axis is labeled as follows: Bottom line - deviation from "true" in a units; Center Line -

number of results in each bin; Top Line - cumulative percentage of results which deviated

to this point from "true". The solid bar below the plot indicates +la. The open circles

depict an ideal Gaussian distribution.

FITTING LIMITS: 70-135

STARTING VALUES:

PEAK1: 10000 1.01

PEAK2: 5000 1.05

WIDTHS=.15 (FIXED)

BACKGROUND=100 (FITTED)

65

*•??*! ii iff i if if fff f fff f ff iifiifi?

1604 FITS

AVE=5046 +/- 21

PREDICTED SIGMA=787

DIST. CHISQD=1.12

WITHIN 1 SIGMA=67.77%

WITHIN 2 SIGMA=94.27%

WITHIN 3 SIGMA=99.13%

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t t I I I I I I I I t I t I t t I t I I I 1 I I t I HOOOOOO ooI I I I Figure 16. Distribution of 1604 solutions from Test 1.

217
"AVE=5046±21" is the observed average and observed standard deviation of the 1604

fitted values for the Peak 1 amplitude. "PREDICTED SIGMA=787" is the theoretical uncer- tainty (a) in the Peak 1 amplitude obtained via Equation 20 by fitting the "true" spectrum

(without simulated statistics) as if it were an unknown. "DIST. CHISQD" is a measure of

how well the observed values of the fitted amplitudes correspond to the expected result

of 5000±a (o=787).

m (A.-5000) 2 DIST. CHISQD = I —- =—

k=l (m-l)a^ (32)

where the A1 ,A2 ,...A

m are the 1604 independent measurements. Also tabulated on figure 16 are the percentage of results which fell within ±1, ±2, and ±3o (ideally 68.2%, 95.4%, and

99.7%).

(The reader may have noted that there is a single result which appears on the extreme right of the plot (+6. 5a). This point corresponds to a reversed- ampl itude fit, nearly identical to the one reported by Statham.

What is not apparent is the fact that this particular result occurred on the 26th trial! This incredible coincidence stimulated a great deal of

effort in attempting to understand what was unique about this particular

solution. It was only after hundreds of additional runs had been

performed that it began to seem plausible that this false solution might

only be a statistical anomaly. Eventually it was concluded that although

the theoretical probability of a deviation of this magnitude is less than

2xl0 -7 and although the odds are strongly against encountering the error

in only 1600 runs, it remains a valid statistical possibility.

Furthermore, we are here working with only an approximation to ideal

counting statistics.)

Test 2

Figure 17 is a plot of the distribution obtained using Statham 1 s original fitting

method; channels 90-116 for the fitting region, no background, and four free parameters (2

peak amplitudes, 2 peak centers). In this case, the ideal standard deviation of the Peak 1 amplitude is o=1486 (nearly twice that of Test 1). A "reversed" solution (Peak 1 amplitude

~10000) corresponds to an error of approximately 3.4a and we note that a number of fits

did produce results in this vicinity. In this case, the plot contains 1198 results. Not

included in the plot were four fits which were non-convergent (N.C.) and failed to produce

any solution.

It seems clear that the main features of figures 16 and 17 are consistent with the predicted statistical distribution of errors. Although large deviations, such as the "reversed

solution" case are improbable, they can occur and must be accepted as a normal hazard

imposed by the random statistics of the physical process. However, a careful formulation

will make maximum use of the available data and thereby minimize the probability of

obtaining such large absolute errors.

One of the hazards of a detailed investigation is that it tends to create new questions. Specifically, one might ask, why do both figures 16 and 17 tend to show more large

deviations on the positive than the negative side? Further, why is the average fitted

result signficantly greater than 5000 in both cases (5046±21 and 5161±41 respectively)?

One explanation for this skewing is that the initial estimates used to start the fit were chosen on this side of the true result. We can visualize the progress of the iterations as moving along the floor of a valley which gently slopes to the minimum. If one introduces statistical perturbations in the data, the floor of this valley will be distorted

and the gentle natural slope may be locally obscurred. If one adds to this picture the

presence of tiny "ripples" in the computed x

2 values due to computational round-off, etc. one can then imagine that the iteration procedure might well stumble into a local "pothole"

218
rather than proceeding to the "true" minimum. If this picture is correct, then the

probability of getting a result which deviates to a particular side of the true solution

is enhanced by starting the iteration from that side.

FITTING LIMITS: 90-116

STARTING VALUES:

PEAK1: 10000 1.01

PEAK2: 5000 1.05

WIDTHS=.15 (FIXED)

BACKGROUND=0 (FIXED)

oo

65

1198 FITS ( + 4 N.C. ) PEAK1 (5000 , 1. 01) : r\VE=5161 +/- 41

PREDICTED SIGMA=1486

DIST. CHISQD=.94

WITHIN 1 SIGMA=71.54%

WITHIN 2 SIGMA=95.83%

WITHIN 3 SIGMA=99.42%

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Figure 17. Distribution of 1198 solutions from Test 2.

Test 3

Test 3 was constructed in order to check this starting-point hypothesis. The procedure

was identical to Test 1 except the starting estimates were chosen to be the "true" solution

parameters. Figure 18 shows the resulting distribution for 711 trials. The computed

average of the distribution (5032±29) is now within approximately one standard deviation of

the true answer (compared to approximately two standard deviations for Test 1), tending to support the intuitive notion that the choice of starting estimates can be a factor in

determining precisely where the solution converges.

219
FITTING LIMITS: 70-135

STARTING VALUES:

PEAK1: 5000 1.00

PEAK 2 : 10000 1.04

WIDTHS=.15 (FIXED)

BACKGROUND=100 (FITTED)

oo 11

:: * O

711 FITS

AVE=5032 +/- 29

PREDICTED SIGMA =7 87

DIST. CHISQD=0.97

WITHIN 1 SIGMA=71.45%

WITHIN 2 SIGMA=95.78%

WITHIN 3 SIGMA=99.72%

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Figure 18. Distribution of 711 solutions from Test 3.

However, the distribution of figure 18 still seems somewhat skewed relative to a pure

Gaussian distribution. Note in particular that there seems to be more points than expected

above +2a and correspondingly less than expected below -2a, and in turn, there seems to be more results within -1 than within +la of the true. One explanation for this asymmetry is

simply that the distribution is really Poisson, rather than Gaussian. (This seems to be

borne out by figure 17 which has a decidedly Poisson-like distribution).

Interpretation of Uncertainty Estimates

In preparing distributions of fitted results such as figures 16, 17, and 18, one is

confronted with a subtlety of interpretation which has to do with the actual meaning of the

uncertainty estimates computed according to Equation 7. Consider, for example, the Peak 1 amplitude computed for the "anomalous" point in figure 16: 10135±445. A first inclination

is to consider the deviation to be (10135±5000)/445=1 1 . 5a! However, it so happens that

large positive errors in Peak 1 area are always accompanied by smaller than normal uncer- tainty estimates and vice versa. This is consistent with the fact that all of the measurements are drawn from the same parent population and must all have the same uncertainty

(since a is a property of the distribution, not the individual measurement). From

fundamental premises, the proper interpretation of a statement of the form "X±a^" is as follows: "Based upon available information, X is the best estimate of the "true" value.

If X is the true value , then the distribution of possible results has standard deviation

CTX

• 220
This conclusion is not confined to the interpretation of curve-fitted

results; it applies in ordinary counting statistics as well. For

example, assume that the "true" number of counts in a measurement is T,

but that because of a statistical fluctuation, one obtains R=(T+AT) as a measured count. The true value of the standard deviation of the

measurement is still given by a

2 =T, not a2 =R as one would infer on the

basis of one measurement. For a large relative value of AT this distinction is important.

This subtlety, though typically ignored, can potentially be important to the analyst. It

is an additional reason to distrust a result based upon one measurement since, if that

result should happen to be a statistical deviant, then its associated uncertainty estimate

is also invalid.

Factors Affecting Precision

An interesting question to consider is precisely how the uncertainties of the fitted

results are affected by specific conditions imposed upon the problem. Table 1 tabulates the

computed uncertainty (a) which would be obtained in fitting Peak 1, for various ways of

parameterizing the problem, and for several separations of the peaks. The "unknown" spectrum was always generated with the first peak fixed at 1.00 keV with amplitude 5000, the

second peak amplitude was always 10000, and both peaks had widths of 150 eV (calibration is

again 10 eV/channel). A fixed background of 100 counts/channel was used, and no statistical

perturbations were applied to the spectrum. This "unknown" spectrum was then fitted over a fixed broad range (channels 70 to 171) for 13 different positions of Peak 2 and nine different combinations of free parameters. The letters A, C, and S are used respectively to designate the amplitude, center, and width parameters for each peak. When circled, it means

that this is a free parameter in the fit; when the parameters of both peaks are circled as one, it means that both were fitted jointly (as a single parameter). Uncircled parameters

were held fixed at their "true" values. The "background" parameter was fitted in all cases.

Table 1. Uncertainty in Peak 1 area (5000) vs. fit method and Peak 2 position.

(h)

PEAK1

:

PEAK2:

(a)

®C S ACS

(b)

AlC S

C S

(c)

®C S ®C s

(d)

©(els

(e)

® cisj

(f)

®©s

(£)© s

1 .010 30 91 10 • 36 201 98 5884 94 202 49 47087 .47 69957 .66 ***** ** ***** **

1 .020 30 47 10 • 36 101 97 1466 42 102 99 5996 .48 8927 .70 24243 .80 ***** . **

1 .030 29 79 10 .36 69 .06 647 83 70 59 1812 .37 2675 59 5085 41 ***** . **

p

E

1 .040 28 94 10 .37 52 91 361 19 54 94 785 00 1147 .28 1743 19 37149 .46 AK

2

1 .050 28 00 10 • 37 43 45 228 49 45 98 414 .77 599 64 785 22 12304 88

1 .060 27 02 10 37 37 32 156 59 40 35 248 .77 355 47 420 91 4994 08

CE

1 .080 25 16 10 39 30 01 85 84 33 96 114 .00 158 63 167 75 1 198 21 N

T

1 .100 23 56 10 40 25 94 54 25 30 54 64 51 86 83 87 82 389 18 E

R

1 .125 21 98 10 42 22 98 35 58 27 64 38 86 49 79 49 79 121 95

1 .150 20 81 10 44 21 23 26 86 25 23 27 94 34 12 34 18 47 78

1 .175 19 98 10 46 20 15 22 62 23 32 22 97 26 92 27 00 27 67

1 .200 19 40 10 49 19 46 20 52 21 97 20 62 23 43 23 48 23 93

1 .300 18 54 10 63 18 54 18 56 19 88 18 56 19 90 19 91 23 04

221
The meaning of "jointly fitted" parameters requires some elaboration:

(A) Jointly fitted amplitudes were constrained to maintain a constant proportionality

(1:2 in this case). This can be a worthwhile strategy to use when fitting x-ray

peak multiplets (e.g., K-alpha/beta pairs).

(C) When fitted jointly, the peak centers were constrained to vary with constant separation. This would be appropriate when peak separations were known accurately but

there is uncertainty as to the "zero" of the energy scale.

(S) When fitted jointly, the widths were maintained at a constant ratio (1:1 in this

case). This might be useful if the spectrometer resolution was uncertain.

One factor which is immediately obvious from this table is that the greater the number

of free parameters the larger the uncertainty tends to be. This effect is especially

pronounced for small separations and tends to become less important for larger peak separations.

As the peak separation increases, the uncertainty of Peak 1 area typically diminishes

asymptotically (approaching the theoretical limit of o~17.6 in this case).

The special case (column b) where the amplitudes of the peaks are fitted jointly has

its minimum uncertainty when the peaks are exactly overlapped and gets worse as the peaks

are separated. This is logical because the two peaks are being treated as one structure;

the best precision is achieved when the counts are localized in the fewest number of

channels. The uncertainty of the joint-amplitude fit is always less than for any other case since the counts from both peaks contribute to its precision.

The ability to analyze a peak in the presence of another nearby peak will depend

strongly upon how much prior knowledge can be brought to bear on the problem. For example,

it is obvious that if every parameter except the unknown peak's amplitude is known (column

a), precise analysis is possible for a separation of as little as 10 eV (1 channel). At the other extreme, if all six peak parameters are unknown (column i), one would need to have a separation of more than 150 eV to attain comparable precision. Precision is a measure of how well-defined the x

2 minimum is. Large uncertainties therefore indicate

shallow ill-defined minima and convergence problems can be expected.

Induced False-Minima

From Table 1 one notes that for Peak 2 located at 1.04 keV, fitting with constrained

peak separation (column d) should result in a a for Peak 1 of less than half that for the case where the peaks are fitted independently (column f). Assuming that the same factorof-two relationship roughly holds for a narrower fitting region also (it does), then the

magnitude of a in Test 2 would be reduced by about half if the peak centers were constrained to a fixed separation in that fit. But Statham also reported observing a reversed- ampl itude solution when fitting with constrained peak separations [6]. Statistically, this is quite improbable and it seems suprising that such a solution was observed.

However, if one repeats Test 2 for the case of constrained peak separation (3 free

parameters) one finds that about half of the fits converge to the false "reversed" solution

and about half converge to the "true"! (When Test 1 was repeated with the same constraint

on peak separations, the "reversed" solution was not seen in a series of 130 trials and the

observed distribution of fitted Peak 1 values was completely consistent with the a predicted

by Table 1.) Clearly, some systematic property of the truncated fitting region causes the

constrained fit to consistently locate a false minimum.

In order to investigate this problem further, a simple computer program was constructed

to plot contour maps of x

2 surfaces for various conditions of interest. Figures 19 and 20 are such contour plots. They were obtained by fitting to the same statistics- less spectrum

as was used as the basis of Tests 1, 2, and 3. In this case the Peak 1 and Peak 2 centers

of the solution function were varied in fixed step sizes of 1 eV on a 31x31 grid; the

Peak 1 and Peak 2 amplitudes were then optimized to give the best fit for each grid point,

and the resulting value of x

2 was tabulated. The program then interpolated between these

222
31x31 points to obtain contours of constant X

2

- The contour levels were selected on an arbitrary power-of-two scale so that each contour line represents a value of which is

two times larger/smaller than its neighbor. The minimum point is shown as a small circle.

Innermost contour lines have been left out of the plot since they could not be resolved at this scale. (It must be noted that the "scalloped" appearance of the innermost contour

lines is due to the crude linear interpolation technique used to generate these plots - the

true x

2 contours are assuredly not so jagged!

)

PEAK2 CENTER ->

' / \ 'i I \ \ [ \ \\ \' \' ' I' ' ' J ' ' ' ' x '

' '

'

ccH

W

Du

1.00

-- 1.01

i i i i i i i i I i i i nTTrrh i t mrm

O

ino

FITTING REGION 0.90 TO 1.16 (NO BG)

Figure 19. Contour plot of x

2 surface obtained using a restricted

fitting region.

223
PEAK2 CENTER ->

FITTING REGION 0.70 TO 1.35 (BG=100)

Figure 20. Contour plot of x

2 surface obtained with an extended

fitting region.

Figure 19 employs the truncated fitting region used in Test 2 and Figure 20 employs

the broad fitting region used in Tests 1 and 3. If one visual lizes the solid surfaces

with these contours one gets a picture of the minima located in the bottom of deep curving

ravines. If one should initiate a least-squares fit with starting estimates of 1.02 and

1.06 respectively for the peak centers (lower-right-hand corner) one can imagine the

solutions moving down these long curving valleys to the minimum at 1.00, 1.04. Now imagine

the effect of constraining the peak separations to be fixed at 0.04 keV. The solution

must now move along the diagonal line drawn from the lower- right to the upper- left corner

of each plot. If one studies figure 19 carefully it is noted that this line intersects

one of the contours more than once. This means that along this path x

2 first decreases,

then increases, and again decreases on its way to the true minimum. Careful inspection of

figure 20 will show that the same diagonal path intersects each contour only once as it moves to the minimum.

The meaning of this difference is shown clearly in figure 21. Here the value of x

2 is plotted as a function of the combined Peak 1/Peak 2 center position (equivalent to taking a vertical cross section of the x

2 surface along the diagonal line of figures 19 and 20). One sees that for the truncated fitting region (solid line) the value of x

2 shows a pronounced local minimum in the vicinity of 1.014/1.054. For the broader fitting

region (dotted line) there is indication of an inflection in this region, but no minimum.

224
i i r I i i i i—i—

r

(O 10

r-. CO

T- 1— I I o o o o

Li. Li. o o o 9 o o

10 in oo o

c/)

LU

LUo

cn

s-

<*- o

CO

cuo

r—o

-Coos

s-o

M—

+-> c

•I—CO

$-cooco

4-> 03

Sn3Cl

CUini

03

CDQ-

-a

cux

4-O

cuco

4-O

cu

s-3

225
These plots clearly indicate why a false solution is consistently obtained when

attempting to fit Statham's test case with constrained peak separation. If the initial

starting estimate for the joint peak position is too large, there is a very high probability

of locating the "local" minimum in the vicinity of 1.014/1.054 rather than the "true" minimum at 1.000/1.040. The influence of statistics will be to further distort the x

2 surface and it can even happen that x

2 in the vicinity of 1.014/1.054 is smaller than in

the vicinity of 1.000/1.040.

Figure 22 is a contour plot similar to figures 19 and 20. In this case the true peak

centers are separated by 100 eV (10 channels) and the change in the shape of the contours

is dramatic. The contours are now effectively elliptical. This is a classic example of a "well-behaved" x

2 surface.

PEAK2 CENTER -»

o

FITTING REGION 0.74 TO 1.50 (BG=100 *)

Figure 22. Contour plot of x

2 surface obtained when peaks are separated by 100 eV.

226
Clearly, as one brings the peak centers closer together, the x

2 surface gets increasingly "wild". Figures 23 and 24 illustrate part of the reason for this. These plots are merely larger views of the x

2 surfaces depicted in figures 19 and 20 (the upper right hand corners of these plots correspond to the same portion of the surfaces depicted in 19 and

20). These plots clearly demonstrate that any two-peak problem has two valid minimum

points since one can simply switch the roles of the two peaks to obtain an indentical fit.

The broken diagonal line on these plots is a "singularity" line along which both peaks

have identical centers; the roles of Peak 1 and Peak 2 reverse as one crosses this line. As the peak separations decrease, the minima must move closer to this line and the contours

are increasingly distorted. When the fitting region is artificially restricted, this

seems to exaggerate the contour distortions and increasingly warp the minima valleys.

PEAK2 CENTER -»

o «— cm m zt o o o o

FITTING REGION 0.90 TO 1.16 (NO BG)

Figure 23. Expanded view of x

2 surface for the narrow fitting region.

227
PEAK2 CENTER ->

FITTING REGION 0.70 TO 1.35 (BG=100)

Figure 24. Expanded view of x

2 surface for the extended fitting region.

Figure 25 compares the contour maps for four different choices of the fitting region.

The top two plots of this figure are identical to figures 20 and 19. The lower plot is

restricted strongly on the low-energy side of the peaks but generous on the high-energy

side; the lower right plot is restricted strongly on both the low and high-energy sides of

the fitting region. Again, it can be seen that the effect of truncating the fitting region

is to stretch out and distort the valleys. As the valleys are elongated, the slope of

the bottom towards the minimum is reduced, and as a consequence, the precision of the

fitted parameters worsens and the probability of local statistical minima increases. This

exaggerated curvature also creates false minima when the peak-separation constraint is

imposed.

In summary then, one must recognize that there are a number of factors which are relevant to the successful solution of a least-squares problem. These include the uniqueness of potential solutions, the quantity and quality of available statistical information,

the presence of constraints, the quality of initial estimates, etc. When the overall

problem is basically wel 1 -determined (e.g., well separated peaks, good statistics, few free

parameters, etc.) the solutions tend to be "robust" and it is possible to take liberties

with some factors (e.g., poor starting estimates). However, when a problem is basically

ill-defined, the solutions are quite sensitive to minor sources of bias and careful

formulation is imperative.

228
PEAK2 CENTER ->

FITTING REGION 0.70 TO 1.35

II 1 1 II 1 1 II I Ml 1 1! 1 1 1 1 1 1 1 1 1 II 1

1

FITTING REGION 0.90 TO 1.16

FITTING REGION 0.95 TO 1.35 FITTING REGION 0.95 TO 1.09

Figure 25. Comparison of x

2 surface obtained with four choices of fitting region.

It is clearly possible to identify plausible circumstances under which non-linear

least-squares methods will produce unacceptably inaccurate results, but this investigation

has nonetheless given cause to feel reassured about the essential validity of the

principles. Although there are certainly "pitfalls" which one can encounter in their

application, these are not inexplicable hazards, but rather can be understood in terms of

fundamental statistical limitations and/or systematic factors related to implementation.

229
Summary

Least-squares curve fitting is a powerful analytic tool for the analysis of complex

spectra, yet inherently simple in concept. Linear fitting functions are particularly simple

to work with and have unique solutions. Non- linear functions are more difficult to deal

with and generally require an iterative optimization algorithm.

Assuming reasonable objectives and careful formulation, both linear and non-linear

least-squares problems can be solved reliably. This is not to say that unacceptably large errors (from the viewpoint of the analyst) cannot occur. Rather, it must be recognized

that the distribution of errors is governed by a statistical probability function and that

any error is possible in principle, although deviations which exceed a few standard deviations are quite improbable .

The precision of the fitted results is ultimately limited by the statistical content

of the data employed and the formulation of the problem. Precision is improved when:

(a) The statistics of the data are improved (more counts per channel).

(b) Redundant information is supplied (more channels).

(c) External information can be supplied to the procedure (e.g., use of "known"

parameters and judicious constraints between parameters).

Systematic biases can be introduced into a least-squares fit by improper program design

or careless formulation of the problem. Some factors which deserve consideration are:

(a) Care should be taken to ensure that the fitting region is sufficiently wide to include all of the information which is relevant to variations in the

parameters.

(b) The choice of starting estimates in an iterative procedure can affect the distribution of results obtained.

(c) Imposing parameter constraints can, under certain conditions, create systematic

false minima.

(d) Improper application of weighting can introduce biases into the solutions.

It is incumbent upon the user of any analysis program to exercise common sense and

reasonable caution. Some basic guidelines are:

(a) Employ estimates of statistical uncertainty as a guide, but do not rely solely on them for crucial determinations. For maximum safety, perform several measure- ments and compute the average and standard deviation of these independent

observations.

(b) Independently establish confidence in the performance of the analysis program by

running a few "known" specimens under typical conditions. If you cannot consistently achieve good results for the "known" samples, you cannot expect to do any

better with "unknowns" (regardless of what the program printout says).

(c) Use x

2 as an "early warning" of systematic problems. If x

2 begins to increase

significantly, you should make an effort to understand why, and to what degree

the accuracy of the results are being affected.

(d) Whenever possible, prevent systematic disturbances in the analysis rather than

correct for them. For example, a stable spectrometer is always preferable to the

introduction of non-linear parameters to accomodate calibration and/or resolution

changes. Even if such parameters cannot be absolutely controlled and non- linear

methods must be employed, the reliability of the correction is improved if such

parameters do not deviate greatly from their expected values.

(e) Be critical. Always ask "Is this reasonable?" No device or procedure is ever completely without flaw.

230
References

[1] Orear, J., Notes on Statistics for Physicists , UCRL-8417, University of California

Radiation Laboratory, Berkeley, Calif., (1958).

[2] Moore, R. H. , "On the Meaning and Use of "Chi-Square" in Curve Fitting", in Applica- tions of Computers to Nuclear and Radiochemistry , O'Kelley, 6. D. , Ed., NAS-NS 3107,

Office of Technical Services, Department of Commerce, Washington, D. C. , (1958),

p. 305.

[3] Bevington, P. R. , Data Reduction and Error Analysis for the Physical Sciences , mcGraw-Hill (1969), Chapters 6-12.

[4] Deming, S. N. and Morgan, S. L. , "Simplex Optimization of Variables in Analytic

Chemistry", Anal. Chem. , 45, (3), 379A (1973).

[5] Hamming, R. W. , Introduction to Appl ied Numerical Analysis , McGraw-Hill (1971),

Chapters 9 and 10.

[6] Statham, P. J., "Pitfalls in Linear and Non-Linear Profile-Fitting Procedures, X-Ray

Spectrom. 7, (3), 132 (1978).

231
I
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

SEQUENTIAL SIMPLEX: A PROCEDURE FOR RESOLVING SPECTRAL INTERFERENCE IN ENERGY DISPERSIVE X-RAY SPECTROMETRY

C. E. Fiori, R. L. Myklebust* and K. Gorlen**

Division of Research Services

National Institutes of Health

Bethesda, MD 20205

*Center for Analytical Chemistry

National Bureau of Standards

Gaithersburg, MD 20234

**Division of Computer Research and Technology

National Institutes of Health

Bethesda, MD 20205

Abstract

A method is described for the fitting of Gaussian profiles to the

pulse-height distributions of overlapping x-ray peaks measured with an energy-dispersive detector. The desired result is to know the areas under individual x-ray peaks in a spectrum in which the peaks are not resolved. A sequential simplex procedure for selection of the parameters

in a mathematical expression that describes a spectral peak is used.

Starting and stopping criteria and a method to reduce the number of

independent variables for the procedure are discussed, and a computer

program is included. Examples of the method applied to computer

generated peaks and to peaks obtained from a lithium-drifted silicon

Si (Li) detector are provided.

1. Introduction

The lithium-drifted silicon (Si(Li)) detector has become an established tool in x-ray

spectro-chemical techniques such as x-ray fluorescence spectrometry and electron beam x-ray

microanalysis. The popularity of the Si (Li) detector is due to a number of factors including high quantum efficiency and stability over the analytical energy range 1 - 20 keV, large

collection solid angle (typically 0.05 - 0.3 steradian), operational simplicity, low cost

and minimal maintenance.

The natural width of an x-ray line is in the order of several electron volts in the

energy range of interest measured at half the maximum of the peak intensity (the measured

quantity is designated "full width, half maximum" (FWHM)). However, the instrumental

response function of the Si C Li ) detector is closely approximated by a Gaussian distribution

having a width (FWHM) of approximately 150 eV. Consequently, a natural peak of 1.5 eV width having 1000 counts amplitude would be degraded to a peak having a width of 150 eV and

an amplitude of 10 counts. For some applications the available resolving power is sufficient. Often, however, peak overlap occurs in x-ray spectra observed using a Si(Li) detector. This paper will describe an application of an optimization procedure, the sequential

simplex [1.2.3] 1

, which is used by us to resolve spectral overlaps in the x-ray energy

range 1 - 20 keV.

Figures in brackets indicate the literature references at the end of this paper.

233
A sequential simplex procedure is a technique which can be used for selection of the

set of independent variables in a mathematical expression which cause the expression to be

a "best" fit to a set of data points. In this procedure, each of the n independent

variables in the function to be fitted is assigned an axis of an n-dimensional coordinate

system. A simplex, in this coordinate system, is defined to be a geometric figure consisting of n + 1 vectors (in this discussion we will use the purely mathematical definition of

a vector, i.e., an ordered n-tuple (X Xj X2 ...Xn). In one dimension a simplex is a line

segment; in two dimensions a triangle; and in three or more dimensions a polyhedron, the

vertices of which are the above-mentioned n + 1 vectors. The simplex is moved toward the

set of independent variables which optimize the fit according to a set of specific rules.

The function used to determine the quality of the fit for any set of independent variables

is called the "response function". The response function is a real valued function of n independent variables and is evaluated at the n + 1 vertices of a simplex. The value exhibiting the largest error (worst fit) is then replaced by another value, which is obtained by

reflecting the discarded vertex across the hyperplane of the remaining vertices. The methods

for expanding and contracting the simplex suggested by Nelder and Mead [2] have been

adopted.

The procedure uses a minimum of information at each step, and no record is kept of past

position of the simplex in the space being searched. Furthermore, only addition, subtraction, multiplication, and division are required in the simplex portion of the routine. Consequently the technique is well suited for use in a small computer.

We will use a simple example to demonstrate the essence of the method. Consider the

problem of finding the "best" straight line through a set of data points. A straight line

is defined by the two quantities m, the slope of the line and b, the y axis intercept of

the line. The functional form is:

The quantities we wish to optimize by a simplex search are m and b. We start the process by

estimating, as best we can, three lines each defined by its own slope m^ and y-axis intercept b.j , figure 1. Each of these three lines can be characterized by a vector (m- ,b.) on a m, b plane. The three vectors form a simplex - in this case a triangle - on the plane. If we now define a third orthogonal axis to represent the "response function" the simplex

search can be visualized (figure 2). The search proceeds by calculating the response above

each vertex of the simplex, discarding the vertex having the highest (worst) response, and

reflecting this point across the line connecting the remaining two vertices. This defines a new simplex and the process repeats subject to the rules specified in the flow sheet

(figure 3).

y = mx + b (1)

b

V

m

Figure 1. Linear data set with "best"

fit line and 3 initial assumption lines.

234
Figure 2. (m,b) plane and response surface showing

a single, arbitrary, simplex.

The problem of fitting a straight line to a set of data points is a linear problem.

However, the method of sequential simplex is an organized procedure to search for a minimum

on a response surface and is usable whether the function describing the data set is linear

in its coefficients or not. If the coefficients are linear, or can be made linear by a suitable transformation, the least squares determination can be accomplished analytically.

In this case a search procedure would not be used and is done so here only for the purpose

of explanation.

The function we fit to a given set of N overlapping spectral peaks is the additive

combination of N Gaussian profiles, i.e.,

where y. is the intensity at energy E., N is the number of x-ray peaks, A. is the amplitude

J J

' of each peak, E. is the peak energy, and a. is the width (standard deviation) of each peak.

The assumption of a Gaussian distribution is based on 1) the known distribution of noise in

the preamplifier of the Si (Li) detector and 2) the discrete nature of the counting process

of charge carriers created by capturing photons of a single energy in the active volume of

the Si (Li) detector. A small error is introduced by virtue of the fact that each channel

(data point) is in reality the integral of a Gaussian over a small energy range. This error

is very small if less than 10 eV/channel is utilized.

The fitting procedure involves the calculation of the function which is then compared

with the experimental data points. The purpose of the response function is to provide a measure of the goodness of fit y^ to the data points. The response function used is a normalized chi- squared function:

2. The Fitting Profile

N y . = I A. exp

i=l

1 (2)

235
Assume n+1 sets of sI

coefficients (Aj)

Compute Response

(Rj) for each set

; Order Rj from low

(R,) to high (Rh

) T

Compute Centroid (A) of all Aj except i=h _ ^

A=E(Aj/n). i*h

i=1 J.

Reflect Ah to A

A*=2A-An Compute R*

Expand A* to A** A"=2A*-A

Compute R

Contract An toA

A"=(A h+ A)/2

Contract A to A

A"=(A* + A)/2

Replace all

Aj by(A j+ A[)/2

Replace

A n by A*

Replace

A h by A"

£<End?>I»< STOP )

Figure 3. Flow diagram showing the logic necessary to minimize the response (R)

by moving a simplex.

?36
where M is the number of data points, Zj is the number of counts at each data point and is

consequently governed by Poisson counting statistics, is defined by equation (2), and f

is the number of fitted variables. The weighing factor, 1/y., normalizes the effect of all

data points used in the fit.

3. Reduction in the Number of Variables

From equation (2) we see that the independent variables (coefficients) A., and o. must be determined for each peak; therefore, the total number of coefficients, n, is:

n = 3N (4)

here, values for n + 1 different sets of coefficients are chosen to create the initial

simplex and the response, R, is determined for each set. The procedure outlined in figure 3 is then used to mimimize R to obtain the best set of coefficients.

We have introduced several simplifications which reduce the number of coefficients to

be fitted. Since the energies of x-ray lines are well known, they are entered as known

quantities and it is unnecessary to include all of the peak centroids (E.) as coefficients.

The energy of only the principal peak (Ep) is used as a coefficient to correct for small

shifts in energy due to mi seal ibration of the electronics of the Si (Li) spectrometer. The

energies of all other peaks are determined from the energy Ep, thereby reducing the number

of coefficients required in the fit. The stability and linearity of present-day amplifiers

permits this assumption. In addition, the width (a

p

) of only the principal peak must be

included since the widths of the other peaks (a.) are related to ap (for a Si (Li) detector

[3]) by:

/

o. = ([2500(E. - E p

) + (2355 a

p

) 2 ]) / 2355 E(keV) (5)

The number of coefficients required is, therefore, reduced from 3N to N + 2. Consequently,

only one more coefficient (amplitude) must be included for each additional peak used in a fit.

Since the energies and widths of small unresolved peaks are determined as functions of

the principal peak, opportunities for obtaining false minima are considerably reduced.

Significant savings in computation time are also realized.

4. Starting and Stopping Criteria

A difficulty common to all minimization methods is the possibility that a local mini- mum may be found before the global mimimum. As the number of peaks to be fitted increases

so does the possibility of finding false minima. Consequently, the simplex must be started

closer to the unique set of coefficients which determine the global minimum than to a set which determine a local (false) minimum.

The initial estimate of energy of a peak is considerably more important than estimates

for amplitude or width. Figure 4 demonstrates the relative significance of estimates of

these three parameters for a single Gaussian peak centered at 5 keV for three different

widths. Twenty channels symmetrically placed across the peak, 10 eV per channel, are used

for the fit. The amplitude is assumed to be 10,000 counts. Only one of the three parameters is varied for a given curve and the effect on the response function (equation 3) is

shown on the vertical axis.

237
a

^DIFFERENCE Figure 4. Relative change in response (R) for initial estimates of energy, amplitude, and standard deviation (a) to three different peaks centered at

5 keV with amplitudes of 10,000 counts and widths (fwhm) of (a) 100 ev,

(b) 150 eV, and (c) 200 eV, respectively. Twenty channels, 10 eV per

channel, are used for the fit. The abscissa is the difference, in

percent, between a given parameter of the theoretical Gaussian and the same parameter for the assumed Gaussian.

The results for the initial estimate of energy indicate that this parameter is at least

four orders of magnitude stronger in influencing the response than either the amplitude

estimate or width estimate. Fortunately, a good estimate of the energy of a peak is the

most easily supplied of the three parameters.

The first estimate of each coefficient establishes one vertex of the initial simplex.

The remaining n vertices are established by selectively perturbing several of the coefficients an appropriate distance. If the perturbations are too large the likelihood of

obtaining false minima increases. On the other hand, if the perturbations are too small,

computation time will be wasted as the simplex attempts to determine the direction in which

it must move. This occurs when the size of the simplex is of the same magnitude as the

statistical irregularity of the response surface. We have empirically determined that the

true minimum on the response surface is most often found with a smaller number of moves of

the simplex when the amplitude coefficients are initally perturbed by 10 percent; the width

coefficient of the largest peak perturbed by 1.0 percent; and the energy of the largest peak

perturbed by 0.1 percent.

238
A simplex search of a response surface can be terminated in at least two ways. A stopping criterion based on change of the response function for each new vertex of the

simplex as it moves across the response surface is one method. Another method is to examine

for changes in the coefficients as the simplex moves across the response surface. We use the second method. If all of the coefficients have changed by less than 1CT6 in three moves

of the simplex we assume convergence. This approach is less sensitive to changes in the

number of channels used in a fit and the number of peaks being fit.

5. Presmoothing of the Data

The largest amount of computation time in our procedure is involved in the calculation

of n Gaussian profiles at each data channel used in the fit. These calculations must be

made each time the simplex requires a new vertex. We have found that a significant reduction in computation time (greater than 50 percent) can be realized by first presmoothing the

data, we use an unweighted five point fit to a quadratic polynomial. The procedure then

uses every other data channel until convergence is close (all coefficients changing less

than 10 -3 in three moves of the simplex). At this point the procedure switches and uses

the original data from all channels until final convergence is reached.

6. Incomplete Charge Collection

The energy-dispersive x-ray peaks in our method are assumed to be Gaussian line

profiles. This assumption ignores the distortion on the low energy side of the observed

peak caused by incomplete charge collection in the detector [4,5,6]. The effect due to incomplete charge collection can be seen in figure 5 for chlorine Ka and potassium Ka peaks. The deviation from a Gaussian distribution (shown as a solid line) is a function of

energy. For example, the magnitude of the effect is significantly different for chlorine

and potassium, which are separated by only two atomic numbers. The distortion due to incomplete charge collection tends to diminish as the energy of the x-ray line increases.

In general, incomplete charge collection is a second order effect and can be neglected. For

those situations, however, in which it cannot (i.e., small peaks on the low energy side of

large peaks), it is possible to determine an expression of the form

-b(E. - E.)

Y. = aA. (E. - E.) e

1 J E. < E. (6)

J

i i 3 j i for a particular detector in a given experimental situation [7]. This equation can be functionally added to the Gaussian profile used to fit each peak in a spectrum. The terms in

equation 6 are as follows: E.. is the centroid of the particular Gaussian profile to be

modified and A^ the amplitude, y^ is the contribution due to incomplete charge collection at

energy E. , and "a" and "b" are coefficients which will, in general, be different for peaks

of different energy. The procedure to determine "a" and "b" for a given energy consists of

the following. A series of Ka lines in the vicinity of the energy in question are each

fitted with a Gaussian profile (K series lines are used because of their simplicity relative

to other x-ray lines). Only data from half the peak maximum on the low energy side to the

baseline on the high energy side are used for these fits. The fitted curve for a given peak

is subtracted from the real data and equation 6 fitted to the residuals. An example of such

residuals and fit for FeKa are shown in figure 6. The optimum "a" and "b" for any energy in

the characterized range can be interpolated. As long as significant changes are not made

in the experimental situation the values determined for "a" and "b" will remain valid.

239
Figure 5. Energy-dispersive spectrum of

KC1 illustrating distortion

caused by incomplete charge

collection (shaded area). The

solid line is a Gaussian fit to the data points.

Fe Koc INCOMPLETE

CHARGE RESIDUALS

Figure 6. The iron Ka incomplete charge residuals

remaining after subtracting a fitted

Gaussian curve from a real peak. The

smooth points are a fit to the residuals

using equation 6.

PEAK

POSITION

An example of a complete fit including the correction for incomplete charge collection

is shown in figure 7. The spectrum used in this figure is the same as used in figure 5.

240
CIKa • KK«.

Fit of Gaussian

incomplete charge * • function to original . £

C ' data ! Ec

= 15keV

Figure 7. Example of a complete fit including

the correction for incomplete charge

7. Constraints

A constraint is defined to be a limit to the values which a given coefficient may

assume as the simplex moves across the response surface. For example, it is known that

peaks may not have negative amplitudes and it would seem reasonable to forbid any peak from

assuming such a value. This can be accomplished by assigning a very high response to any

vertex which contains a negative amplitude. Similarly, one may place constraints on the

width and energy coefficients of the largest peak. The assumption made is that the simplex

will reflect off the constraint and the search will be confined to a defined, physically

meaningful, region. Constraints such as the above were suggested by Nelder and Mead [2]

and give significant positive benefits.

Figure 8 is a computer generated spectrum of two Gaussian peaks of similar amplitude

shown with their convolution. The peaks are separated by 50 eV and have the appropriate

widths for peaks at 5.0 and 5.05 keV arising from a detector having a resolution (FWHM) of

150 eV at MnKa (5.895 keV). Equation 5 was used to determine the widths. No background has

been superposed under the peaks nor have they been modulated by statistical uncertainty -

other than that due to computer round off error. To assess the utility of placing con- straints on amplitude, energy or width or a combination of all of these, the experiments

shown in figures 9, and 12 through 14 were performed..

Figure 8. Computer generated spectrum of two

Gaussian peaks of similar amplitude

with their convolution. Peaks are separated by 50 eV and have energies

of 5.0 and 5.05 keV arising from a detector saving a FWHM at MnKa of

150 eV.

241
In these experiments the convolution shown in figure 8 was subjected to a large number

of simplex searchs with various constraints and without constraint. The initial estimates

for width and energy were the correct values for the two peaks. Initial estimates for the

two amplitudes, however, were allowed to vary from zero to nearly twice the known values (in

increments of 10 counts) in both cases. The horizontal and vertical axis of the plots are respectively, the initial estimate for the amplitude of the low energy peak and the high

energy peak. A successful search was arbitrarily defined as all variables being within one percent of the known values. A success is denoted, at each increment, by an asterisk and a failure by a solid circle. In cases where a vertical line of adjacent failure is encoun- tered the solid circles are connected.

When the convergence test matrix for the no constraints situation is examined, as shown

in figure 9, failures are found principally near the axis, where the initial estimate of at least one of the peaks differs greatly from the true value. Somewhat surprisingly, isolated

failures are also observed in the region around the true value, with most of these failures

corresponding to initial estimates above the true value. Determination of the exact origin

of these failures will require additional work. It is felt that the extreme flatness of the

response surface, when peaks are separated by less than about 50 eV, is the principal cause

of the observed failures, figure 10.

Figure 9. Convergence test matrix, no constraints.

X marks correct solution.

242
a

Figure 10. Three dimensional slice through a six dimensional volume. The response

surface is shown above the two amplitude axes. The true peak amplitudes

are denoted by the solid circle, (a) 1000 counts each peak; (b) 200 counts

each peak. The response surface is essentially flat in the direction of

opposite amplitude variation.

The convergence test matrix was then recalculated with various individual constraints

placed upon the parameters. These constraints were (1) the peak energies were constrained

to ± 20 eV, figure 11; (2) the width of the lower energy (largest amplitude) peak was constrained to ± 5 eV from the given by equation (5), figure 12; and (3) negative amplitudes

were forbidden, figure 13.

Figure 11. Convergence test matrix, energy constrained.

243

As can be seen from figures 11 - 13, the various individual constraints produce a marginal improvement when both starting estimates are well below the true amplitudes

(denoted by a cross). For those failures where the starting estimates are both above the

true amplitudes there is no improvement. A curious result occurs when all the constraints

are included, figure 14. In this case there is a neglibibly small improvement over the case where no constraints are used at all. Clearly, more work needs to be done in this area as well. It must be noted, however, while only ten seconds is required for any one data point,

that 15 days of central processor time were required (PDP 11-60 computer with high speed

arithmetic capability) to produce the data used in figures 9 and 11 - 14.

0 COUNTS PEAK 1000

Figure 14. Convergence test matrix, all constraints included.

8. Fortran Program

A complete listing of our Simplex program is included in Appendix 1. This program is

written in Digital Equipment Corporation Fortran IV plus. Very little of the program,

however, is written in non-standard Fortran.

9. Conclusions

A sequential simplex procedure has been developed for the specific problem of unfolding

peak overlaps encountered in energy dispersive x-ray spectrometry. This procedure has been

tested to determine the importance of individual peak parameters, i.e., energy, width, and

amplitude, on the simplex response. The initial estimate of energy was found to be at least

four orders of magnitude more influential on the response than either the amplitude or width

estimates.

An exhaustive study was conducted to determine the capability of the simplex procedure

to find a correct solution for two computer generated overlapping Gaussian peaks separated

by 50 eV. A convergence test matrix was developed to test the sensitivity to convergence

on the initial amplitude estimates. Constraint of parameters, peak energy, peak width, and

restriction to positive peak amplitude, caused an improvement in the convergence. Anomalous

failures at individual points were observed in the region of the correct peak amplitudes.

Due to the extreme flatness of the response surface when peaks are separated by less than

about 50 eV it is recommended that great caution be exercised when attempting to resolve

such a spectral interference in real data modulated by statistical uncertainty and the

245
effects of background. The authors would like to thank Daniel Ferner for doing the graphics

utilized in figure 10.

References

[1] Spendley, W. , Hext, G. R. , and Himsworth, F. R. , Sequential Application of Simplex

Designs in Optimisation and Evolutionary Operation, Technometrics , 4, 441 (1962).

[2] Nelder, J. A. and Mead, R. , A Simplex Method for Function Minimization, Comput . J., 7,

308 (1965).

[3] Fiori, C. E. and Myklebust, R. L. , A Simplex Method for Fitting Gaussian Profiles, in

Computers in Activation Analysis and Gamma- ray Spectroscopy , Technical Information

Center/U. S. Dept. of Energy, 139-149 (1979).

[4] Fiori, C. E. and Newbury, D. E. , Artifacts Observed in Energy Dispersive Spectrometry in

the Scanning Electron Microscope, in Scanning Electron Microscopy/1978 , Los Angeles,

California, 0. Johari , ed. , Aug. 16-21, 1977, p. 401, S.E.M. , Inc., O'Hare, 111. (1978).

[5] Gedcke, D. A., The Si C Li ) X-ray Energy Analysis System: Operating Principles and

Performance, X-ray Spectrom . , 1^, 129 (1972).

[6] Elad, E. , Inskeep, C. N. , Sareen, R. A., and Nestor, P., Dead Layers in Charged Particle Detectors, IEEE Trans . Nucl . Sci . , 20, 534 (1973).

[7] Myklebust, R. L. , Fiori, C. E. , and Heinrich, K. F. J., FRAME C: A Compact Procedure

for Quantitative Energy-Dispersive Electron Probe X-ray Analysis, National Bureau of

Standards Technical Note 1106, U. S. Government Printing Office, Washington, D.C. (1979), available from authors.

246
C SIMPLX — X-RAY SPECTRUM DE-CONVOLUTION VIA SIMPLEX METHOD

CC

C AUTHOR:

CC KEITH E. GORLEN

C DIVISION OF COMPUTER RESEARCH AND TECHNOLOGY

C NATIONAL INSTITUTES OF HEALTH

C BETHESDA , MD. 20205

C JUNE 11, 1 979

C 0001 INCLUDE 'SMAT.COM'

* C

* C NAXES: NUMBER OF AXES (= NPEAK+2)

* C NVERT: NUMBER OF VERTICES ( = NPE AK+3

)

* C TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

* C IL: INDEX OF LOWEST RESPONSE VERTEX

* C IH: INDEX OF HIGHEST RESPONSE VERTEX

* C A: SIMPLEX MATRIX

* C A(*,I) = ITH VERTEX OF THE SIMPLEX

* C Ad, I) = SIGMA OF LARGEST PEAK (KEV)

* C A(2,I) = ENERGY OF LARGEST PEAK (KEV)

* C A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

* C R: R(I) IS THE RESPONSE OF THE ITH VERTEX

0002 * COMMON /SMAT/ NAXES , N VE R T , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7

)

0003 INCLUDE 'SPECTB.COM'

* C

* C NCHB: TOTAL NUMBER OF DATA CHANNELS IN BUFFER

* C ICHW: INDEX OF FIRST CHANNEL IN ANALYSIS WINDOW

* C NCHW: NUMBER OF CHANNELS IN ANALYSIS WINDOW

* C BKEVB: ENERGY OF FIRST CHANNEL IN KEV (BUFFER)

* C CHKEVB: ENERGY PER CHANNEL IN KEV (BUFFER)

* C CHB : ARRAY OF CHANNEL COUNTS (BUFFER)

0004 * COMMON /SPECTB/ NCHB , I CH W , NCH W , BK E VB , CHK E VB , CHB ( 1 0 28

)

0005 LOGICAL INPAR

C

C READ SPECTRUM

C 0006 1000 CALL RDSPEC

C

C READ PARAMETERS

C 0007 1100 IF (.NOT. INPAR(O)) GO TO 1000

C

C FIND PEAKS IN SPECTRUM

C 0008 CALL FPEAKS(CHB( ifiHW) ,NCHW, BKEVB, CHKEVB)

C

C PERFORM SIMPLEX FIT

C 0009 CALL SIMPLX(CHB( ICHW-2) ,NCHW+4, BKEVB, CHKEVB, ETIME)

C

C REPORT RESULTS

C 0010 CALL REPORT(A(1 ,IL),R(ID)

0011 WRITE (5,2000) ETIME

001 2 2000 FORMAT (/' ELAPSED T I ME = ',F7.1,' SECONDS*//)

C

C DISPLAY RESULT OF FIT ON KE VEX 7000 SCREEN

C 001 3 CALL DSPFIT(A(1 , IL)

)

0014 GO TO 1 1 00

0015 END

247
0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

001 2 0013

0014

001 5 0016

001 7 0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

0032

1 0 20

30

40

50

C

C

C

ENTER

1 00

200

SUBROUTINE AUTOPK ( L INLST

)

ESTIMATE PEAK ENERGIES AND AMPLITUDE FROM ATOMIC NUMBERS

LINLST = LIST OF LINE NUMBERS TO CONSIDER FOR FIT

LINLSTM): NUMBER OF LINES IN LIST

LINLST (2) - LINLSTC LINLSTC 1 ) ) : LINE NUMBERS

NZ: NUMBER OF ELEMENTS

IZ: ARRAY OF ATOMIC NUMBERS

PE: BEARDEN'S VALUES FOR PEAK ENERGIES OF AN ELEMENT

DIMENSION IZ(10),PE(17), LINLST (18)

INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN : FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE: PKE(I) IS ESTIMATED ENERGY CKEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE : DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NP E A K , F WH MMN , I BP , PK A ( 1 4 ) , PKE ( 1 4 ) , PKS ( 1 4 )

,

IZPKC14) , IDPKC14) , IXC 14) , DPKE (14)

INCLUDE 'SPECT.COM'

NCH : NUMBER OF DATA CHANNELS

BKEV : ENERGY OF FIRST CHANNEL IN KEV

CHKEV : ENERGY PER CHANNEL IN KEV

CH: ARRAY OF CHANNEL COUNTS

COMMON /SPECT/ NCH , BKEV , CHKEV , CH ( 1 024

)

DIMENSION T(14)

EQUIVALENCE (T(1),PE(1))

WRITE (5,20)

FORMAT (

' $ ENTE R NUMBER OF ELEMENTS (10 MAX): ')

READ (5,30) NZ

FORMAT (14)

WRITE (5,40)

FORMAT ( ' $ENTER ATOMIC NUMBERS : ')

READ (5,50) ( IZ( I ) , 1=1 ,NZ)

FORMAT (1015)

LOOK UP BEARDEN'S VALUES FOR PEAK ENERGIES

NPEAK = 0 DO 200 1=1 , NZ

IF (IZ(I) .LE. 0 .OR

CALL BEARDN( IZ( I ) , PE

)

DO 100 K = 1 , L I N LST ( 1 ) J=LINLST(K+1

)

IF (PE(J) .EQ. 0) GO TO 100

IF (PE(J) .LT. BKEV .OR. PE(J)

NPEAK = NPEAK+1

PKE(NPEAK) = PE(J)

IZPK(NPEAK) = IZ(I)

IDPK(NPEAK) = J CONTINUE

CONTINUE

IF (NPEAK . NE . 0) GO TO 400

WRITE (5,300)

IZ(I) .GT. 92) GO TO 900

GT. BKEV+CHKEV*NCH) GO TO 100

248
0033

0034

0035

0036

0037

0038

0039

0040

0041

0042

0043

0044

0045

0046

0047

0001

300

C

C

C400

C

C

C

C

C

C

600

700

800

C

C

C900

91 0

FORMAT ( GO TO 10NO PEAKS FOUND '

)

COMPUTE AMPLITUDE ESTIMATES

CALL ESTAMP

COMPUTE PERMUTATION TO SORT PEAKS BY INCREASING ENERGY

CALL SORTPK(NPEAK> PKE> IX)

OUTPUT ESTIMATES TO BE USED IN SIMPLEX FIT

WRITE (5,600)

FORMAT (/' PARAMETER ESTIMATES

' Z LINE AMPLITUDE ENERGY

DO 800 1 = 1 » NPE AK

WRITE (5,700) IZPK( IX( I ) ) , IDPK( IX( I )

)

PKE ( IX( I ) ) , PKS ( I X ( I ) ) FORMAT ( 1 X, 12 , 15, F9 . 0 , F9 . 3 , F9 .4)

CONTINUE

RETURN

ERROR PROCESSING

FOR SIMPLEX'/

SIGMA' )

PKA( IX( I ) )

WRITE (5

FORMAT (

' GO TO 1

0

END

SUBROUTINE

910)

ATOMIC NUMBER OUT OF RANGE ' )

BEARDN(IZ,PE)

BEARDN — GET PEAK ENERGIES FROM BEARDEN TABLE

ENTER

IZ ATOMIC NUMBER

c PE = 17 ELEMENT ARRAY

c

c PE(1 ) = KA

c PE(2) = KB

c PE(3) = L A

1

c PE(4) = LA2

c PE(5) = LB1

c PE(6) = LB2

c PE(7) = LB3

c PE(8) = LB4

c PE(9) = LG1

c PE(1 0) == LG3

c PE(1 1 )

== LN

c PE(1 2) == LL

c PE(13) == MA

c PE( 1 4) == MB

c PE(15) == MG

c PE( 1 6) == MZ

c PE(17) == M2N4

TO RECEIVE PEAK ENERGIES (KEV)

0002

0003

0004

DIMENSION PE(17) ,E(890)

REAL KALFA(48),KBETA(41),LALFA1(64),LALFA2(56),

+ LBETA1 (69),LBETA2(53),LBETA3(69),LBETA4(69),

+ LGAMA1 ( 53) , LGAMA3 ( 53 ) , L ETA ( 6 9 ) , L L AMD A ( 6 9 ) , + MALFA(36),MBETA(36) , MGAMA ( 52 ) ,MZETA(33),MIINIV(20)

EQUIVALENCE

+ .( KALFAC 1 ) , E ( 1 ) ) , + ( KBETA ( 1 ) , E ( 49 ) ) , + ( LALFA1 ( 1 ) , E ( 90 ) ) , 249
(LALFA2M ),E(154)),

( LBETA1 ( 1 ) , E ( 21 0 ) ) , (LBETA2M ) ,E( 279) ) ,

( LBETA3C 1 ) , E ( 332 ) )

,

( LBETA4( 1 ) ,E(401 ) )

,

( LGAMA1 ( 1 ) , E(470 ) )

,

( LGAM A 3 ( 1 ) ,E(523) )

,

( LETA ( 1 ) , E( 576) ) ,

( LLAMDAC 1 ) , E ( 645) ) , (MALFAC1 ) ,E(714)),

( MB E T A ( 1 ) ,E(750) ) ,

( MGAMA ( 1 ) , E ( 786 ) ) ,

( MZETA ( 1 ) , E(838) ) , (MIINIVC 1 ) , E(871 ) ) L0GICAL*1 IZL( 1 7) , IZU( 1 7)

DATA I ZL/O 3, 1 0, 2 9, 3 7, 2 4, 4 0,2 4, 2 4, 4 0, A 0,2 4, 2 4, 5 7, 57, 41 , 60,73/

DATA IZU/50, 50, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 92, 9 2/

DATA KALFA/.0543, .1085, .1833, .277, .3924, .5249, .6768, .8486,

1 .04098,1 .25360,1 .48656, 1 .739 78, 2. 01 341 ,2. 3 0 744, 2 .621 85,

2. 95701, 3. 31290, 3. 690 48, 4. 08910, 4. 50885, 4. 94968, 5. 41165,

5.89505,6.3991 ,6.92531 ,7. 47240, 8. 04112, 8. 63117, 9. 24277,

9. 87605, 10. 53181, 11 .20873,11 .90867, 12. 632, 13. 3754 7, 14. 14263,

14.93323,15.747,16.5837,17.4443,18.3283,19.2363,20.1686,

21. 1247, 22. 10 54, 23. 1104, 24. 1405, 25. 195 5/

DATA KBETA/. 8579 , 1.0711, 1.30222,1. 55 74 5, 1.83594, 2. 13908,

2. 46404, 2. 8156, 3. 1905, 3. 5896, 4. 0127, 4. 4605, 4. 93181, 5. 42729,

5. 94671 , 6. 49045, 7. 05798, 7. 64943, 8. 26466, 8. 90529, 9. 5720,

10.2642,10.9821,11.7262,12.4959,13.2914,14.112,14.9613,

15. 8357, 16. 73 78, 17. 6678, 18. 6225, 19. 6083, 20. 619, 21. 6568,

22.7236,23.8187,24.9424,26.0955,27.2759,28.486/

DATA LALFA1/. 9297, 1.0117, 1.09792, 1.1 88, 1. 282, 1. 3791, 1. 48043,

I .586, 1 .69413,1 .80656,1 .92256, 2. 04236, 2. 16 58 9, 2. 29316, 2. 424,

2. 55855, 2. 696 74, 2. 83861, 2. 984 31, 3. 13373, 3. 28694, 3. 44 398,

3. 60472, 3. 76933, 3. 93765, 4. 1099, 4. 2865 , 4 . 46626 , 4 . 65097 , 4 . 8402

,

5. 0337, 5. 2304, 5. 4325, 5. 6361 , 5. 84 57, 6. 0572, 6. 2728, 6. 4952,

6. 7198, 6. 9487, 7. 1799, 7. 4156, 7. 6555, 7. 899, 8. 1461, 8. 3976,

8. 6525, 8. 9117, 9. 1751, 9. 4423, 9. 7133, 9. 9888, 10. 2685, 10. 5515,

1 0 . 8 388, 1 1.1308, 11. 4268, 11. 727, 12. 0313, 12. 3397, 12. 652,

12.9687,13.2907,13.6147/

DATA LALFA2/1. 69256,1. 80474, 1.92047, 2. 0399, 2. 163, 2. 28985,

0., 2. 5 54 31, 2. 69205, 2. 83329, 2. 9 7821, 3. 12691, 3. 27929, 3. 43 54 2,

3. 59532, 3. 7 588, 3. 92604,0., 4. 2722, 4. 45090, 4. 6 34 23, 4. 823,

5. 0135, 5. 2077, 5. 4078, 5. 6090, 5. 8166, 6. 0250, 6. 2 38 0,6. 4577,

6. 6795, 6. 9050, 7. 1331, 7. 3673, 7. 6049, 7. 844 6, 8. 0879, 8. 3352,

8. 5862, 8. 8410, 9. 0995, 9. 3618, 9. 6280, 9. 8976, 10. 1728, 10. 4495,

10.73091 ,11 .01 58, 11 .30 48, 11 .5979,11 .895, 12. 1962, 12. 5008,

12.8096,13.1 222 , 1 3 .4388/

DATA LBETA1/. 5828,. 6488,. 7185,. 7914,. 86 88,. 9498, 1.0 347,

1.1248,1.2185, 1.317, 1.41923,1. 5259, 1. 6366, 1.75217,1.87172,

1. 99584, 2. 1244, 2. 25 74, 2. 39481 ,2.5368,2.68323,2.83441

,

2.990 2 2,3.150 94,3.31657,3.48721 ,3.66280,3.84357,4.02958,

4. 22072, 0., 4. 6198, 4. 82753, 5. 0421, 5. 2622, 5. 4889, 5. 7216, 5. 961,

6. 20 58, 6. 4564, 6. 7132, 6. 9 78, 7. 2477, 7. 5253, 7. 8109, 8. 101,

8. 4018, 8. 709, 9. 0227, 9. 3431, 9. 67235, 10. 0100, 10. 3553, 10. 7083,

II .0707,11 .44 23, 11 .8226, 12. 2133, 12. 6137, 13. 0235, 13. 447,

13. 876, 14. 316, 14. 770, 15. 23 58, 15. 713, 16. 2022, 16. 702, 17. 220 0/

DATA LBETA 2/2. 2194, 2. 3670, 2. 5183,0. ,2. 8360, 3. 0013, 3. 17179,

3.34 781,3.52812,3.71381,3.90 48 6,4.10078,4.3017,4.5075,0.,

4. 9359, 5. 1565, 5. 3835, 5. 61 34, 5. 850, 6. 0894, 6. 339, 6. 58 6, 6. 84 32,

7. 1028, 7. 3667, 7. 6357, 7. 911, 8. 1890, 8. 468, 8. 7588, 9. 0489, 9. 3473,

9. 6518, 9. 9615, 10. 2752, 10. 5985, 10. 9203, 11. 2505, 1 1. 584 7, 1 1 . 9241

12. 2715, 12. 6226, 12. 9799, 13. 34 04, 0.,0., 14. 45, 14. 84 14,0.,

15.6237,16 .024, 1 6 .4283/

250
DATA LBE TA 3/0. 6 38, 0.705,0. 790, 0.869,0.942,1 .0251 ,1 .1077,

1 .196,1 .2941 ,1 .3884, 1 .490,1 .569,1 .707,1 .82659,1 .94719,2.0722

2. 201, 2. 3348, 2. 473,0. ,2. 7634, 2. 9157, 3. 073, 3. 2344 6, 3. 40145,

3. 57311, 3. 75, 3. 9327, 4. 1204, 4. 31 34, 0., 4. 7167, 4. 9269, 5. 14 34,

5.3651,5.5918,5.8294,6.071,6.318,6.5713,6.8311,7.0959,7.3702

7. 6519, 7. 9392, 8. 231, 8. 5367, 8. 84 69, 9. 1634, 9. 4875, 9. 8188,

10. 1598, 10. 5108, 10. 8674, 11 .2308,11 .6103,11 .9953,12.3904,

12. 7933, 13. 2098, 13. 638, 14. 067, 14. 512, 14. 976, 15. 4449, 15. 931,

16. 4258, 16. 930, 17. 4550/

DATA LBE TA4/0. 638, 0.705,0. 790, 0.869, 0.942,1. 0251, 1.1077,

1 .196,1 .2861 ,1 .3884, 1 .490,1 .596,1 .697,1 .81771 ,1 .93643,2.06,

2. 1873, 2. 3194, 2. 4557, 0., 2. 7411, 2. 8908, 3. 04 54, 3. 20346, 3. 36719

3. 53528, 3. 7083, 3. 8864, 4. 0695, 4. 2575,0. ,4. 6494, 4. 8519, 5. 062,

5. 2765, 5. 4981, 5. 7216,0. ,6. 197, 6. 4389, 6. 6871, 6. 9403, 7. 2039,

7. 4708, 7. 7453, 8. 026, 8. 3132, 8. 6064, 8. 9054, 9. 2124, 9. 5252,

9. 8463, 10. 1754, 10. 5106, 10. 8 54 3, 11 .2047,11 .5630,11 .9306,

12. 306, 12. 6912, 13. 086, 0.,0.,0., 14. 7472, 0., 15. 6429, 16. 104,

1 6 . 5753/

DATA LGAMA 1/2. 302 7, 2. 46 18, 2. 6 2 35,0. ,2.9645,3.1438,3.3287,

3. 51959, 3. 71686, 3. 92081, 4. 13112, 4. 34779, 4. 5709, 4. 8009,

0. ,5.2804,5.5311 ,5. 7885, 6. 052, 6. 3221 ,6.6021 ,6.892,7.180,

7. 4803, 7. 78 58, 8. 102, 8. 4188, 8. 74 7, 9. 089, 9. 426, 9. 7801, 10. 1434,

10.5158,10.8952,11 .2859,11 .68 54, 12. 0953, 12. 5126, 12. 9420,

13. 3817, 13. 8301, 14. 2915, 14. 7644, 15. 2477, 15. 744, 16. 251,

16. 770, 17. 303, 17. 849, 18. 488, 18. 9825, 19. 568, 20. 1671/

DATA LGAMA 3/2 .5029, 2. 6638, 2. 8306,0. ,3. 1809, 3. 364, 3. 5533,

3. 74 98,0. ,4. 1605, 4. 3768, 4. 5999, 4. 829, 5. 0657,0. ,5. 5527,

5. 8092, 6. 0 74, 6. 34 09, 6. 616, 6. 902,0. ,7. 490, 7. 7961, 8. 105, 8. 423,

8.7532,9.087,9.4309,9.779,10.1431,10.511,10.8907,11.2776,

11. 6743, 12. 0824, 12. 4998, 12. 9240, 13. 3613, 13. 8090, 14. 265,

14.7368,15.218,15.7102,0.,0.,0.,0.,18.357,0.,

19.507,20.098,20.7127/

DATA L ETA/. 51 02 , 0.5675,0.628,0.694,0.762,0.832,0 . 9 06 , 0. 9842,

1 .068,1 .155,1 .2446, 1 .3396,0. ,1 .54177,1 .64933,1 .76095,

1 . 876 54,1 . 99620, 2. 1202,0., 2. 38 197, 2. 5191, 2. 6603, 2. 8061 , 2. 95675, 3. 11254, 3. 272 34, 3. 43661, 3. 60586, 3. 7801, 0., 4. 1421,

4. 3309, 4. 52 5, 4. 7315, 4. 935, 5. 1457,0., 5. 587, 5. 81 66, 6. 0495,

6. 2839, 6. 5 34 2, 6. 7883, 7. 0579, 7. 30 88, 7. 5802, 7. 8575, 8. 1393,

8. 428, 8. 7243, 9. 0272, 9. 3370, 9. 6522, 9. 975, 10. 3083, 10. 6512,

10.9 943,11 .349 3,11 .7122,0. ,0. ,0. ,0.

,

13.6630,0. ,14.5099,14.946,15.399 7/

DATA LLAMD A/. 5003, .5563, .6152, .67 78, .74 27, .8111, .884, .9572,

1. 0380, 1. 1198,1. 2044, 1. 2935, 0 . , 1 . 48238, 1.58215, 1.68536,

1.79201, 1.90225, 2. 01568,0. ,2. 2528, 2. 3765, 2. 5034, 2. 6337,

2. 76735, 2. 90 44 0,3. 04499, 3.1 88 60, 3. 33555, 3. 48502, 0.,

3. 7950, 3. 9 54 1,4. 124, 4. 2875, 4. 4532, 4. 6330,0. ,4. 9945,

5. 1772, 5. 3621, 5. 5467, 5. 7431, 5. 9434, 6. 152, 6. 3419, 6. 5455,

6. 7528, 6. 9596, 7. 1731, 7. 3878, 7. 6036, 7. 8222, 8. 0458, 8. 268,

8. 4939, 8. 7210, 8. 9532, 9.1 84 5, 9. 4204, 9. 664, 0.,0.,0.,

10.6222,0. ,11 .1186,11 .366,11 .618 3/

DATA MALFA/.833, .88 3, .9292, .9 78,0. ,1 .081 ,1 .131 ,1 .185,

1 .24,1 .293,1 .348,1 .406,1 .462,1 .5214,1 .5813,1 .6446,1 .7096,

1 .77 54,1 .8425, 1 .9102,1 .9799,2.0505,2.1229,2.1964,2.2706,

2.3455,2.4226,0.,0.,0.,0.,0.,0.,2.9961,3.0823,3.170 8/

DATA MB ETA/. 8 54, 0 . 902, 0.95,0. 977, 0.,1 .0998,1 .1533,

1.2091, 1.2661, 1.325, 1.383,1. 443, 1.503, 1.5675, 1.6312,

1 .6976,1 .7655,1 .8 34 9, 1 .9061 ,1 .9783, 2. 0535, 2. 1273, 2. 2046,

2.28 27,2.3621,2.44 27,2.5255,0.,0.,0.,0.,0.,0.,

3. 1 458, 3.2397, 3.3367/

251
0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

0032

0033

0001

0002

0003

*

*

x

x

xx

x

xx

x 0004 x 0005

x

x

x

0006 x

0007

0008

DATA MGAMA/. 356 , 0 . ,0.,. 462, .496, .531, .568, .606,

+ 0. ,. 691, . 733, . 788, 0. ,0. , 0. , . 973,1. 027, 1.0 749,

+ 1.1273, 1.180,0. ,1.291,1. 346, 1.402, 1.461, 1.522,

+ 1 .576,1 .643,0. ,1 .765,1 .832, 1 .895, 1 .964, 2. 035, 2. 1067,

+ 2. 182, 2. 254, 2. 331, 2. 410, 2. 490, 2. 571, 2. 6527, 2. 735, 0.,0.,0.,

+ 0 . , 0 . , 0 . , 3 . 370 , 3 . 4657 , 3 . 563/

DATA MZ ETA/. 753, 0 .,.831, .872, .914, .955, .998,1. 0450,

+ 1 .0901 ,0. ,1 .183,0. ,1 .28,1 .3303,1 .3835,1 .4368,1 .4919,

+ 1. 5458, 1.6022,1. 6605,0., 1 .7 78,1. 8395,1. 901 , + 0.,0.,0.,0.,0.,0.,2.364,2.435,2.50 7/

DATA MIINIV/2 .226, 2. 314,0. ,2. 502, 2. 594, 2. 695, 2. 797,

+ 0.,3.013,3.124,3.2 34,0.,0.,0.,0.,0.,0.,4.117,4.26,4.401/

LBASE = 1 DO 20 LN=1 , 1

7

PEC LN) = 0.0

IF ( ( IZ. LT. IZL(LN) ) .OR. ( IZ . GT . IZU ( LN ) ) ) GO TO 10

PE(LN) = E( LBASE+IZ-IZLCLN) ) 10 LBASE = LBASE+C IZIK LN)-IZL( LN ) ) +

1

20 CONTINUE

RETURN

END

LOGICAL FUNCTION CONVRG ( VERTEX

)

CONVRG — TEST FOR CONVERGENCE

ENTER : VERTEX

RETURNS

:

VERTEX WITH HIGHEST RESPONSE

.TRUE. IF TERMINATION CRITERIA SATISFIED

.FALSE. IF TERMINATION CRITERIA NOT SATISFIED

DIMENSION VERTEXC 1 6 ) , OLDVC 1 6 , 3)

INCLUDE 'SMAT.COM'

NAXES : NUMBER OF AXES (= NPEAK + 2)

NVERT: NUMBER OF VERTICES (= NPEAK+3)

TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

IL: INDEX OF LOWEST RESPONSE VERTEX

IH: INDEX OF HIGHEST RESPONSE VERTEX

A: SIMPLEX MATRIX

A(*,I) = ITH VERTEX OF THE SIMPLEX

A(1,I) = SIGMA OF LARGEST PEAK ( KEV

)

A(2,I) = ENERGY OF LARGEST PEAK (KEV)

A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

R: R(I) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SM AT/ N AX E S , N V E RT , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7

)

INCLUDE 'COUNTS.COM*

ITER8 : ITERATION COUNTER (PER PASS)

NRFLCT: REFLECTION COUNTER

NXP AND : EXPANSION COUNTER

NCONTR: CONTRACTION COUNTER

NSHIFT: SHIFT COUNTER

NTOTAL: TOTAL NUMBER OF ITERATIONS

COMMON /COUNTS/ I T E R 8 , N R F L CT , NX P AND , NCONT R , NSH I F T , NTOT A

L

EXIT IF MORE THAN 500 MOVES

IF (ITER8 .GE. 500) GO TO 20

CONVRG = .FALSE.

252
0009 I = 1 +MOD( ITER8, 3)

C REQUIRE THAT THE SIMPLEX MOVES AT LEAST THREE TIMES

0010 IF (ITER8 .LT. 3) GO TO 40

C

C TEST FOR CONVERGENCE OF COEFFICIENTS

C

0011 DO 1 0 J=1 ,NAXES

0012 D = 0 LDV ( J , I ) 0013 IF (D . EQ . 0) D = TOLER

C EXIT IF ALL THE COEFFICIENTS HAVE CHANGED BY LESS THAN

C TOLER IN THE LAST THREE MOVES.

0014 IF (ABS( VERTEXC J )-OLDV( J , I ) )/D . GE . TOLER) GO TO 40

0015 10 CONTINUE

0016 20 CONVRG = .TRUE.

0017 30 CONTINUE

0018 RETURN

C

C SAVE CURRENT VERTEX

C 0019 40 DO 50 J=1 , NAXES

0020 50 OLDV(J,I) = VERTEXCJ)

0021 RETURN

0022 END

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

001 1 0012

001 3 0014

0015

001 6 0017

1 0

20

SUBROUTINE DSPF I T ( V E RT E X

)

DSPFIT — DISPLAY SIMPLEX FIT ON KEVEX

ENTER

VERTEX VERTEX WITH LOWEST RESPONSE (I.E. A(I,ID)

1 00

DIMENSION VERTEXC16)

INCLUDE 'PEAKS.COM'

NPEAK : NUMBER OF PEAKS

FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE: PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPEAK , FWHMMN , IBP , PKA ( 1 4 ) , PKE C 1 4 ) , PKS ( 1 4 )

,

IZPK(14),IDPK(14),IX(14),DPKE(14)

WRITE (5,10)

FORMAT ( ' SQUADRANT NUMBER FOR SPECTRUM FIT DISPLAY:')

READ (5,20) KQUAD

FORMAT (110)

IF (KQUAD .LT. 1 .OR. KQUAD . GT . 4) RETURN

KQU AD= ( KQUAD-1 )*1 024

DO 200 J = 1 2 , 1 023

Y = 0 . E=J*. 01 DO 100 1=1 , NPEAK

Y=Y+VERTEX(I+2)*EXP(-0.5*( (E-(DPKE(I)+VERTEX(2)))/

PKSIG(VERTEX(1 ),DPKE(I)))**2)

CONTINUE

CALL KXWCH(J+KQUAD,Y)

253
CONTINUE

RETURN

END

SUBROUTINE ESTAMP

ESTAMP — COMPUTE AMPLITUDE ESTIMATES USING MULTIPLE LINEAR REGRESSION

DIMENSION AM 4, 14) ,B( 1 4) , C(1 4) ,T(14)

EXTERNAL GAUSPK

INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN : FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE: DPKECI) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPEAK>FWHMMN,IBP,PKA(14),PKE(14),PKS(14),

+ IZPK(14),IDPK(14),IX(14),DPKE(1<+)

INCLUDE 'SPECT.COM'

NCH: NUMBER OF DATA CHANNELS

BKEV: ENERGY OF FIRST CHANNEL IN KEV

CHKEV: ENERGY PER CHANNEL IN KEV

CH: ARRAY OF CHANNEL COUNTS

COMMON /SPECT/ NCH , BKEV , CHKEV , CH ( 1 024

)

COMPUTE PEAK SIGMAS

DO 10 J=1 , NPEAK

PKSCJ) = PKSIG(FWHMMN*4.2466E-4,PKE(J)-5.895)

CONTINUE

PERFORM MULTIPLE LINEAR REGRESSION

CALL R E GR ES ( NCH, CH, NPE AK, GAUSPK, A, B,T, PKA)

RETURN

END

SUBROUTINE FIT

FIT — PERFORM SIMPLEX FIT

DIMENSION ABAR ( 1 6 ) , ASTARM 6),ASTAR2(16)

LOGICAL CONVRG

INCLUDE 'SMAT.COM'

NAXES: NUMBER OF AXES (= NPEAK+2)

NVERT: NUMBER OF VERTICES (= NPEAK+3)

TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

IL: INDEX OF LOWEST RESPONSE VERTEX

IH: INDEX OF HIGHEST RESPONSE VERTEX

254
0005

0006

0007

0008

0009

0010

001 1

0012

001 3 001 4

001 5 001 6

0017

001 8 0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

00 30

0031

CC

C

1 000

1100

C

C

c

1 200

Cc

c

1 300

C

Cc

1 400

1 500

1 600

1 700

C

CC

1 800

A: SIMPLEX MATRIX

A(*,I) = ITH VERTEX OF THE SIMPLEX

A(1,I) = SIGMA OF LARGEST PEAK ( KEV

)

A(2,I) = ENERGY OF LARGEST PEAK (KEV)

A(J,I), 3 > = J < = NPEAK + 2 = AMPLITUDE OF JTH PEAK

R: R(I) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SMAT/ NAXES , NVERT , TOLER , I L , IH , A ( 1 6 , 1 7 ) , R ( 1 7

)

INCLUDE 'COUNTS.COM'

ITER8 : ITERATION COUNTER (PER PASS)

NRFLCT: REFLECTION COUNTER

NXPAND: EXPANSION COUNTER

NCONTR: CONTRACTION COUNTER

NSHIFT: SHIFT COUNTER

NTOTAL: TOTAL NUMBER OF ITERATIONS

COMMON /COUNTS/ I T E R8 , NR F LCT , NX PAND , NCONT R , NSH I FT , NTOT A

L

ALPHA: SIMPLEX REFLECTION COEFFICIENT

BETA: SIMPLEX CONTRACTION COEFFICIENT

GAMMA : SIMPLEX EXPANSION COEFFICIENT

DATA ALPHA/1 . 0/ , BETA/0 . 5/ , GAMMA/2 . 0/

INITIALIZATION

ITER8 = 0 IL = 1 IH = 1 COMPUTE RESPONSES FOR ALL VERTICES

DO 1100 1=1 , NVERT

R( I ) = RESP(A( 1 , I ) ) CONTINUE

TEST FOR CONVERGENCE

IF (CONVRG(A( 1 , IH) ) ) RETURN

ITER8 = ITER8+1

BEGIN NEXT ITERATION: FIND HIGHEST AND LOWEST RESPONSES

DO 1 300 1=1 , NVERT

IF (R(I) . GT . R(IH)) IH=I

IF (R(I) .LT. R(ID) I L = I CONTINUE

COMPUTE CENTROID (ABAR) OF ALL VERTICES EXCEPT A(,IH)

DO 1400 J=1 , NAXES

ABAR(J) = 0 DO 1600 1=1 , NVERT

IF (I . EQ . IH ) GO TO 1600

DO 1500 J=1 , NAXES

ABAR (J ) = ABAR (J )+A( J , I ) CONTINUE

DO 1700 J=1 , NAXES

ABAR (J ) = ABAR ( J )/( NVERT- 1 ) COMPUTE REFLECTION (ASTAR) OF ABAR

DO 1800 J=1 , NAXES

AST A R ( J ) = (1 .+ALPHA)*ABAR(J)-ALPHA*A(J,IH)

?55
0032 RSTAR = RESP ( ASTAR )

0033

0034

0035

0036

0037

0038

0039

0040

0041

0042

0043

0044

0045

0046

0047

0048

0049

0050

0051

0052

0053

0054

0055

0056

0057

0058

0059

0060

0061

0062

0063

0064

0065

1 900

C

C

c

2000

C

C

C21 00

2200

C

C

C

C

C

c

2300

C

C

C2400

CC

c2500

2600

C

C

C2700

C

C

C2800

TEST FOR EXPANSION

IF (RSTAR .LT. R(ID) GO TO 2500

TEST FOR REFLECTION

DO 1 900 1=1 , NVERT

IF (I . EQ. IH ) GO TO 1900

IF (RSTAR .LE. R(I)) GO TO 2700

CONTINUE

CONTRACT

IF (RSTAR . GT . R(IH)) GO TO 2100

DO 2000 J=1 , NAXES

A(J,IH) = ASTAR ( J

)

R(IH) = RSTAR

COMPUTE CONTRACTION ASTAR2

DO 2200 J=1 , NAXES

ASTAR2U) = BETA*A(J»IH) + (1 .-BETA)*ABAR(J)

RSTAR 2 = R ESP ( ASTAR 2

)

TEST FOR CONTRACTION FAILURE

IF (RSTAR2 .LE. R(IH)) GO TO 2400

CONTRACTION FAILED, DO SHIFT

DO 2300 1=1 , NVERT

DO 2300 J=1 , NAXES

A(J,I) = (A(J,I)+A(J,IL) )/2.

CONTINUE

NSHIFT = NSHIFT+1

GO TO 1000

CONTRACTION SUCCEEDED, REPLACE A(,IH) BY CONTRACTION ( ASTAR 2

)

NCONTR = NCONTR+1

CALL REPLA( ASTAR2 , RSTAR2

)

GO TO 1200

COMPUTE EXPANSION ( ASTAR 2 ) OF ASTAR

DO 2600 J=1 , NAXES

ASTAR2U) = GAMMA*ASTAR(J) + (1-GAMMA)*ABAR(J)

RSTAR 2 = RESP ( ASTAR2

)

IF (RSTAR2 .LT. R(ID) GO TO 2800

REPLACE A(,IH) BY REFLECTION (ASTAR)

NRFLCT = NRFLCT+1

CALL REPLA( ASTAR , RSTAR

)

GO TO 1200

REPLACE A(,IH) BY EXPANSION (ASTAR2)

NXPAND = NXPAND+1

CALL REPLA(ASTAR2,RSTAR2)

GO TO 1200

END

256
0001 SUBROUTINE F PEAKS ( CHB , NCHB , BKE VB , CH K EVB

)

C

C FPEAK — FIND PEAKS IN SPECTRUM

CC ENTER:

C CHB = DATA ARRAY TO BE FIT

C NCHB = NUMBER OF CHANNELS TO BE FIT

C BKEVB = ENERGY OF FIRST CHANNEL IN KEV

C CHKEVB = ENERGY PER CHANNEL IN KEV

C 0002 INCLUDE 'SPECT.COM'

* C

* C NCH: NUMBER OF DATA CHANNELS

* C BKE V : ENERGY OF FIRST CHANNEL IN KEV

* C CHKEV: ENERGY PER CHANNEL IN KEV

* C CH: ARRAY OF CHANNEL COUNTS

0003 * COMMON /SPECT/ NCH, BKEV, CHKEV, CH(1 024)

0004 DIMENSION CHB ( NCHB

)

C LINLSTC1): MAJOR PEAK ENERGIES, IE., KALPHA, KBETA, LALPHA1 , C LBETA1 , LGAMMA1 , MALPHA, MBETA, MGAMMA.

0005 DIMENSION LINLST(18,2)

C LINLST(,2) ALL PEAK ENERGIES.

0006 DATA LINLST/8,1 ,2,3,5,9,13,14,15,0,0,0,0,0,0,0,0,0,

+ 17,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17/

0007 WRITE (5,10)

0008 10 FORMAT ('CENTER 1 FOR MANUAL MODE, 2 FOR AUTO MODE: •)

0009 READ (5,20) MODE

0010 20 FORMAT (14)

0011 GO TO ( 1 00 , 200 ) , MODE

C

C MANUAL MODE

C 0012 100 CALL MANUPK

0013 RETURN

C

C AUTO MODE

C 0014 200 NCH = NCHB

0015 BKEV = BKEVB

0016 CHKEV = CHKEVB

001 7 DO 21 0 1 = 1 , NCHB

0018 CH ( I ) = CHB ( I

)

0019 210 CONTINUE

0020 WRITE (5,220)

0021 220 FORMAT ( ' $ENTER 1 FOR MAJOR PEAKS, 2 FOR ALL PEAKS: ')

0022 READ (5,230) MODE

0023 230 FORMAT (110)

0024 CALL AUTOPK( LINLSTd ,MODE) ) 0025 RETURN

0026 END

0001 SUBROUTINE GAUSPK ( M , T , I ) C

C GAUSPK — GAUSSIAN PEAK FUNCTION

C

C ENTER : C M = NUMBER OF PEAKS

C I = CHANNEL NUMBER

C

257
C RETURNS:

C T = TCJ) IS VALUE OF GAUSSIAN FOR PEAK J AT CHANNEL I C 0002 INCLUDE 'PEAKS.COM'

* C

x C NPEAK: NUMBER OF PEAKS

* C FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

* C IBP: INDEX OF BIGGEST PEAK

* C PK A : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

* C PKE : PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

* C PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

* C IZPK: ATOMIC NUMBER OF ITH PEAK

* C IDPK: LINE CODE OF ITH PEAK

* C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

* C DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

0003 * COMMON /PEAKS/ N P E AK , F WH MMN , I B P , P K A ( 1 4 ) , P K E ( 1 4 ) , PKS ( 1 4 )

,

* + IZPK(14),IDPK(14),IX(14),DPKE(14)

0004 INCLUDE 'SPECT.COM'

* C

* C NCH: NUMBER OF DATA CHANNELS

* C BKEV : ENERGY OF FIRST CHANNEL IN KEV

* C CHKEV: ENERGY PER CHANNEL IN KEV

* C CH: ARRAY OF CHANNEL COUNTS

0005 * COMMON /SPECT/ NCH , BKEV , CHKEV , CH ( 1 024

)

0006 DIMENSION T(M)

0007 ENERGY = BKE V+ ( I - 1 ) *CH K E

V

0008 DO 10 J = 1 ,M

0009 TCJ) = EXP(- . 5*( ( ENERGY-PKEC J ) )/PKS( J ) )**2 ) 0010 10 CONTINUE

0011 RETURN

0012 END

0001 LOGICAL FUNCTION INPAR

C

C INPAR -- INPUT PARAMETERS

CC RETURNS

:

C .TRUE. IF PARAMETERS RETURNED

C .FALSE. IF EOF ON INPUT REACHED

C 0002 INCLUDE 'LIMITS.COM'

* C

* C LIMSW: 1 = LIMITS ENABLED, 0 = LIMITS DISABLED

* C SIGMAX: MAXIMUM SIGMA OF BIGGEST PEAK (EV)

* C SIGMIN: MINIMUM SIGMA OF BIGGEST PEAK (EV)

* C EMAX: MAXIMUM ENERGY OF BIGGEST PEAK (KEV)

* C EMIN : MINIMUM ENERGY OF BIGGEST PEAK (KEV)

* C AMAX: MAXIMUM AMPLITUDE OF ANY PEAK

* C LCODE: LIMIT COUNTS

0003 * COMMON /LIMITS/ L I MSW , S I GMAX , S I GM I N , EMAX , EM I N , AMAX , LCODE ( 1 6

)

0004 INPAR = .FALSE.

0005 WRITE (5,10)

0006 10 FORMAT ( ' $ENABLE CONSTRAINTS? (1=YES, 0=NO): ')

0007 READ (5,20,END=99) LIMSW

0008 20 FORMAT (11)

0009 INPAR = .TRUE.

0010 99 RETURN

0011 END

258
SUBROUTINE INSIMP

CC INSIMP — INITIALIZE SIMPLEX MATRIX

C INCLUDE 'PEAKS.COM'

CC NPEAK: NUMBER OF PEAKS

C FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

C IBP: INDEX OF BIGGEST PEAK

C PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

C PKE: PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

C PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

C IZPK: ATOMIC NUMBER OF ITH PEAK

C IDPK: LINE CODE OF ITH PEAK

C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

C DPKE : DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPEAK , FWHMMN , I BP , PKA ( 1 4 ) , PKE ( 1 4 ) , PKS ( 1 4 ) , + IZPK(14),IDPK(14),IX(14),DPKE(14)

INCLUDE 'SMAT.COM'

C

C NAXES : NUMBER OF AXES (= NPEAK+2)

C NVERT: NUMBER OF VERTICES (= NPEAK+3)

C TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

C IL: INDEX OF LOWEST RESPONSE VERTEX

C IH: INDEX OF HIGHEST RESPONSE VERTEX

C A: SIMPLEX MATRIX

C A(*,I) = ITH VERTEX OF THE SIMPLEX

C A(1,I) = SIGMA OF LARGEST PEAK (KEV)

C A(2,I) = ENERGY OF LARGEST PEAK (KEV)

C A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

C R: RCI) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SMAT/ N AXES , N VE RT , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7 ) C

C SET SIGMA OF BIGGEST PEAK ESTIMATE

C AC 1 , 1 ) = PKSIG(FWHMMN*4.2466E-4,PKE( IBP)-5.895)

C

C SET ENERGY OF BIGGEST PEAK ESTIMATE

C A(2,1 ) = PKE(IBP)

CC SET PEAK AMPLITUDE ESTIMATES

C DO 10 J=1 , NPEAK

A( J + 2 , 1 ) = PKA ( J

)

10 CONTINUE

C

C COPY FIRST VERTEX TO ALL OTHERS

C DO 20 1=2, NVERT

DO 20 J=1 , NAXES

AC J , I ) = A(J,1)

20 CONTINUE

C

C PERTURB ONE PARAMETER ESTIMATE OF ALL BUT FIRST VERTEX TO

C FORM INITIAL SIMPLEX.

C

C PERTURB A SIGMA OF BIGGEST PEAK

C AM ,2) = A(1 ,2)*. 99

C

C PERTURB AN ENERGY OF BIGGEST PEAK

C 259
A(2,3) = A(2,3)*.999

C

C PERTURB PEAK AMPLITUDES

C DO 30 J=1 , NPEAK

A(J+2,J+3) = A( J+2, J+3)*.9

30 CONTINUE

RETURN

END

LOGICAL FUNCTION LIMIT(VERTEX)

C

C LIMIT -- TEST VERTEX AGAINST LIMITS

C

C ENTER:

C VERTEX = VERTEX TO BE CHECKED

C

C RETURNS:

C .TRUE. IF THE SPECIFIED VERTEX EXCEEDS A LIMIT

C .FALSE. IF THE SPECIFIED VERTEX IS WITHIN LIMITS

C DIMENSION VERTEXM6)

INCLUDE 'PEAKS.COM'

C

C NPEAK: NUMBER OF PEAKS

C FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

C IBP: INDEX OF BIGGEST PEAK

C PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

C PKE : PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

C PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

C IZPK: ATOMIC NUMBER OF ITH PEAK

C IDPK: LINE CODE OF ITH PEAK

C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

C DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPEAK , FWHMMN , I BP , PKA ( 1 <* ), PKE ( 1 4 ), PKS ( 1 4 ) , + IZPK(14),IDPK(14),IX(14),DPKE(14)

INCLUDE 'SMAT.COM'

C

C NAXES : NUMBER OF AXES (= NPEAK + 2

)

C NVERT: NUMBER OF VERTICES (= NPEAK+3)

C TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

C IL: INDEX OF LOWEST RESPONSE VERTEX

C IH: INDEX OF HIGHEST RESPONSE VERTEX

C A: SIMPLEX MATRIX

C A(*,I) = ITH VERTEX OF THE SIMPLEX

C A(1, I) = SIGMA OF LARGEST PEAK (KEV)

C A(2,I) = ENERGY OF LARGEST PEAK (KEV)

C A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

C R: R(I) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SMAT/ N AX ES , N V E R T , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7

)

INCLUDE 'LIMITS.COM'

CC LIMSW: 1 = LIMITS ENABLED, 0 = LIMITS DISABLED

C SIGMAX: MAXIMUM SIGMA OF BIGGEST PEAK (EV)

C SIGMIN: MINIMUM SIGMA OF BIGGEST PEAK (EV)

C EMAX: MAXIMUM ENERGY OF BIGGEST PEAK (KEV)

C EMIN: MINIMUM ENERGY OF BIGGEST PEAK (KEV)

C AMAX: MAXIMUM AMPLITUDE OF ANY PEAK

C LCODE : LIMIT COUNTS

COMMON /LIMITS/ L I MS W , S I GMAX , S I GM I N , EMAX , E M I N , AMAX , L COD E ( 1 6 ) 260
IF (LIMSW .EQ. 0) GO TO 100

CD L = 1 IF (VERTEXd) . GT . SIGMAX) GO TO 200

CD L = 2 IF (VERTEXd) .LT. SIGMIN) GO TO 200

CD L = 3 IF (VERTEX(2) .GT. EMAX) GO TO 200

CD L = 4 IF ( VERTEX ( 2 ) .LT. EMIN) GO TO 200

CD L = 5 C DO 10 1=1 .NPEAK

C IF (VERTEX(I+2) .LT. 0) GO TO 200

C IF (VERTEXCI+2) .GT. AMAX) GO TO 200

10 CONTINUE

C

C ALL PARAMETERS ARE WITHIN LIMITS

C100 LIMIT= . FALSE . RETURN

C

C PARAMETER OUTSIDE LIMITS

C200 CONTINUE

CD LCODE(L) = LCODE(L)+1

CD WRITE (5,*) L , ( VERTEX( I) , 1=1 ,NAXES)

LIMIT= . TRUE

.

RETURN

END

SUBROUTINE MANUPK

C

C MANUPK -- MANUAL ENTRY OF PEAK INFORMATION

C INCLUDE 'PEAKS.COM'

C

C NPEAK : NUMBER OF PEAKS

C FWHMMN : FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK ( C IBP: INDEX OF BIGGEST PEAK

C P K A : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

C PKE : PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

C PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

C -IZPK: ATOMIC NUMBER OF ITH PEAK

C IDPK: LINE CODE OF ITH PEAK

C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

C DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ N P E A K , F WH MMN , I B P , P K A ( 1 4 ) , P K E ( 1 4 ) , P K S ( 1 4

)

+ IZPK(14),IDPK(14),IX(14),DPKE(14)

5 WRITE (5,10)

10 FORMAT (

' $ ENTE R NUMBER OF PEAKS (14 MAX): ')

READ (5,20) NPEAK

20 FORMAT (12)

IF (NPEAK .GT. 16-2) GO TO 5 WRITE (5,30)

30 FORMAT (' ENTER AMPLITUDE AND ENERGY (KEV) OF EACH PEAK

DO 50 1=1 , NPEAK

READ (5,40) PKA(I),PKE(I)

40 FORMAT (2F15.0)

50 CONTINUE

CC SORT PEAKS INTO ORDER OF INCREASING ENERGY

C

261
CALL SORTPK(NPEAK,PKE> IX)

RETURN

END

FUNCTION PKSIG(SIGMAP.DELTAE)

C

C PKSIG -- COMPUTE SIGMA OF PEAK FROM SIGMA OF BIGGEST PEAK

C

C ENTER:

C SIGMAP = SIGMA OF BIGGEST PEAK IN KEV

C DELTAE = ENERGY DIFFERENCE FROM BIGGEST PEAK IN KEV

C

C RETURNS:

C SIGMA OF THE PEAK IN KEV

C PKSIG=SQRT( 2 50 0*DELTAE+(2 354.82 5*SIGMAP)**2)*4. 2 46 6E-4

RETURN

END

SUBROUTINE RDSPEC

RDSPEC -- READ SPECTRUM

THIS ROUTINE READS THE SPECTRUM DATA TO BE FIT (IN THIS CASE FROM

A KEVEX 7000 ANALYTICAL SPECTROMETER), DETERMINES THE ENERGY

RANGE TO BE PROCESSED, PADS THE DATA AT BEGINNING AND END FOR

POLYNOMIAL SMOOTHING, CHANGES ANY ZERO CHANNEL COUNTS TO ONE

TO AVOID DIVIDE OVERFLOW, AND SETS FWHM(MN).

INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PK A : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

P K E : PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPE AK , F WHMMN , I BP , PK A ( 1 4 ) , PK E ( 1 4 ) , PKS ( 1 4 )

,

+ IZPK(14),IDPK(14),IX(14),DPKE(1<+)

INCLUDE 'SPECTB.COM'

NCHB : TOTAL NUMBER OF DATA CHANNELS IN BUFFER

ICHW: INDEX OF FIRST CHANNEL IN ANALYSIS WINDOW

NCHW: NUMBER OF CHANNELS IN ANALYSIS WINDOW

BKEVB: ENERGY OF FIRST CHANNEL IN KEV (BUFFER)

CHKEVB : ENERGY PER CHANNEL IN KEV (BUFFER)

CHB : ARRAY OF CHANNEL COUNTS (BUFFER)

COMMON /SPECTB/ NCHB , I CH W , NCH W , B K E VB , CH K E VB , CHB ( 1 0 28

)

INPUT QUADRANT NUMBER

00 WRITE (5,110)

10 FORMAT ( ' $ENTER QUADRANT NUMBER: ' )

262
READ (5,120) KQUAD

FORMAT (110)

IF (KQUAD . G E . 1 .AND. KQUAD . LE. 4) GO TO 140

WRITE (5,130)

FORMAT (' INVALID QUADRANT')

GO TO 100

LOCATE "A" WINDOW

ICHNL= ( KQUAD-1 )*1 024

CALL KXWIN( ICHNL+1 2 , ICHNL+1 023, "001 ,ICHW,NCHW)

IF (NCHW .GT. 0) GO TO 160

WRITE (5,150)

FORMAT (' NO WINDOW SPECIFIED')

GO TO 100

READ CHANNEL DATA FROM KEVEX

CALL KXRCHS(ICHNL+12,1012,CHB(3))

SET PARAMETERS

CHKEVB = 0 .01

BKEVB=ICHW*CHKEVB

EKEVB=BKEVB+(NCHW-1 )*CHKEVB

ICHW=ICHW-1 2+3

NCHB=ICHW-1 +NCHW+2

ADD 2 ONES AT BEGINNING AND END FOR POLYNOMIAL SMOOTHING

CHB( 1 ) = 1 . CHB( 2 ) = 1 . CHB( 2 + 1 012 + 1) = 1 . CHB( 2 + 1 01 2 + 2 ) = 1 . CHANGE ALL CHANNELS WITH 0 COUNTS TO HAVE 1 COUNT

DO 165 I=ICHW, ICHW+NCHW-1

IF CCHB(I) .EQ.O. ) CHB(I) = 1 . CONTINUE

DISPLAY SPECTRUM INFORMATION

WRITE (5,170) NCHW,BKEVB,EKEVB

FORMAT MX,' PROCESSING ' , 14 ,

1 CHANNELS FR0M',F7.3,

' TO' ,F7.3, ' KEV )

GET FWHM(MN)

WRITE (5,180)

FORMAT ( ' $ENTER FWHM(Mn) (EV): ')

READ (5,190) FWHMMN

FORMAT (F10 .3)

RETURN

ERROR PROCESSING

WRITE (5,910)

FORMAT (' NO ANALYSIS WINDOW SPECIFIED')

STOP

END

263
0001 SUBROUTINE R EGR ES ( N , Y , M , F , A , B , T , C)

CC REGRES — PERFORM MULTIPLE LINEAR REGRESSION

C

C ENTER:

C N = NUMBER OF DATA POINTS

C Y = Y(N) ARRAY CONTAINING DATA POINTS

C M = NUMBER OF PARAMETERS

C F = EXTERNAL SUBROUTINE F(M,T,I) TO RETURN THE M FUNCTION VALUES

C AT DATA POINT I IN ARRAY T

C A = A(M,M) SCRATCH MATRIX

C B = B(M) SCRATCH ARRAY

C T = T(M) SCRATCH ARRAY

C

C RETURNS:

C C = C(M) — PARAMETER VALUES

C

C CALLS:

C MINV — MATRIX INVERSION

C GMPRD — GENERAL MATRIX PRODUCT

C (SEE THE SCIENTIFIC SUBROUTINES REFERENCE MANUAL,

C DIGITAL EQUIPMENT CORP. FOR DOCUMENTATION)

C 0002 DIMENSION Y ( N ) , A ( M , M ) , B ( M ) , T ( M ) , C ( M

)

C

C INITIALIZATION

C 0003 DO 1 00 J = 1 ,M

0004 B ( J ) = 0.0

0005 DO 100 K = 1 ,M

0006 A(J,K) = 0.0

0007 100 CONTINUE

C

C COMPUTE WEIGHTED SUMS OF CROSS PRODUCTS

C 0008 DO 200 1 = 1 ,N

0009 CALL F(M,T,I)

0010 DO 200 J = 1 ,M

0011 BCJ) = B(J)+T(J)

0012 DO 200 K=1 ,M

0013 A(J.K) = A(J>K)+T(J)XT(K)/Y(I)

0014 200 CONTINUE

C

C INVERT MATRIX A

C 0015 CALL MINV(A,M,D,C,T)

C

C COMPUTE PARAMETERS

C 0016 CALL GMPRD(A,B,C,M,M, 1 ) 0017 RETURN

0018 END

0001 SUBROUTINE REPLA ( VERTEX , RESP

)

CC REPLA — REPLACE A(,IH) BY SPECIFIED VERTEX

C

C ENTER:

C VERTEX = VERTEX TO REPLACE A(,IH)

C RESP = RESPONSE TO REPLACE R(IH)

C 264
0002

0003

DIMENSION VERTEXM6)

INCLUDE ' SMAT . COM

0004

0005

0006

0007

0008

0009

1 0

NAXES:

NVERT

:

TOLER

I L : IH : A:

COMMON

DO 1 0 J A( J , IH

)

R(IH) = RETURN

END

NUMBER OF AXES (= NPEAK+2)

NUMBER OF VERTICES (= NPEAK+3)

TOLERANCE FOR PARAMETER CONVERGENCE

INDEX OF LOWEST RESPONSE VERTEX

INDEX OF HIGHEST RESPONSE VERTEX

SIMPLEX MATRIX

AC*, I) = ITH VERTEX OF THE SIMPLEX

AM, I) = SIGMA OF LARGEST PEAK ( K E V

)

A(2,I) = ENERGY OF LARGEST PEAK (KEV)

A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

R(I) IS THE RESPONSE OF THE ITH VERTEX

/SMAT/ NAXES, NVERT, TO LE R , I L , IH , A ( 1 6 , 1 7 ) , R ( 1 7

)

=1 , NAXES

= VERTEX(J)

RESP

0001

0002

0003

0004

0005

0006

SUBROUTINE R E PORT ( VE R T EX , R ESP

)

REPORT — REPORT RESULTS

ENTER

VERTEX = VERTEX WITH LOWEST RESPONSE (I.E A ( 1 > I L ) ) RESP = RESPONSE OF SPECIFIED VERTEX

DIMENSION VERTEX M 6 ) INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE: PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPEAK, FWHMMN, IBP, PKAM4),PKEM4),PKS(14),

IZPKM4),IDPKM4),IXM4),DPKEM4)

INCLUDE 'SMAT.COM'

NAXES

NVERT

TOLER

IL : IH : A:

NUMBER OF AXES (= NPEAK+2)

NUMBER OF VERTICES (= NPEAK+3)

TOLERANCE FOR PARAMETER CONVERGENCE

INDEX OF LOWEST RESPONSE VERTEX

INDEX OF HIGHEST RESPONSE VERTEX

SIMPLEX MATRIX

A ( * , I A ( 1 , I A ( 2 , I A ( J , I R ( I )

= ITH VERTEX OF THE SIMPLEX

= SIGMA OF LARGEST PEAK (KEV)

= ENERGY OF LARGEST PEAK (KEV)

, 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH

S THE RESPONSE OF THE ITH VERTEX

PEAK

COMMON /SMAT/ N AXES , NVE RT , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7

)

265
0007 INCLUDE 'COUNTS.COM'

*

c

c ITER8

:

ITERATION COUNTER (PER PASS)

* c NRF LCT : REFLECTION COUNTER

* c NXPAND : EXPANSION COUNTER

X c NCONTR : CONTRACTION COUNTER

' x c NSHIFT : SHIFT COUNTER

c NTOTAL : TOTAL NUMBER OF ITERATIONS

0008 x COMMON /COUNTS/ ITER8,NRFLCT, NXPAND, NCONTR, NSHIFT, NTOTAL

c

c

c OUTPUT COUNTERS

0009 WRITE (5,10) NTOTAL , NRF LCT, NXPAND, NCONTR , NSHIFT

0010 1 0 FORMAT (///1X,I5,' ITERATIONS' , 15, ' R E F L ECT I ONS ' , 1 5

,

c + ' EXPANSIONS' , 15, ' CONTRACT I ONS *, I 5 ,

' SHIFTS')

c

c OUTPUT FIT DATA FOR BIGGEST PEAK

001 1 FWHM = 23 54.825xPKSIG(VERTEX(1),0)

0012 XFWHMM = 2354.82 5xPKSIG(VERTEX(1) , 5 . 8 9 5- VE R TE X ( 2 )

)

001 3 WRITE (5,20) RESP,VERTEX(2), FWHM, XFWHMM

0014 20 FORMAT (1X, 'RESPONSES ,G12.6,' ENERGY= ' , F7 . 3 ,

c + ' FWHM= ' , F7 . 2

,

' FWHM(MN)=' ,F7.2)

c OUTPUT FIT SUMMARY FOR ALL PEAKS

001 5 0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

30

40

50

WRITE (5,30)

FORMAT (/' AMPLITUDE ENERGY SIGMA

DO 50 1=1 , NPEAK

ENERGY = DPKE(IXd) )+VERTEX(2)

SIGMA = PKSIG( VERTEX( 1 ) ,DPKE ( IX( I ) )

)

ASIGMA = VERTEX(IX(I)+2)*SIGMA

WRITE (5,40) VERTEX( IX( I ) + 2 ), ENERGY, SIGMA, ASIGMA

FORMAT (1X,4G12.5)

CONTINUE

RETURN

END

A*S I GMA ' )

0001

0002

0003

0004

FUNCTION RESP(VERTEX)

RESP — COMPUTE RESPONSE OF SPECIFIED VERTEX

ENTER

VERTEX VERTEX TO CALCULATE RESPONSE FUNCTION FOR

RETURNS:

RESPONSE OF THE SPECIFIED VERTEX

DIMENSION VERTEXM6)

LOGICAL LIMIT

INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN: FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE: PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

266
x

X

X 0005 *

x 0006

x

x

x

x

0007 * 0008

x

0009 ^ 0010

0011

0012

0013

001 4 001 5 0016

0017

0018

0019

0020

0021

0022

0023

0024

0001

0002

xx

x

x*

x

C IDPK: LINE CODE OF ITH PEAK

C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

C DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ NPE AK , F WHMMN , I BP , P K A d 4 ) , PK E (1 4 ) , PKS ( 1 4 )

,

+ IZPKC1 4) , IDPKC1 4) , IX ( 1 4) , DPKEC1 4)

INCLUDE 'SMAT.COM'

C

C NAXES: NUMBER OF AXES (= NPEAK+2)

C NVERT : NUMBER OF VERTICES (= NPEAK+3)

C TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

C IL: INDEX OF LOWEST RESPONSE VERTEX

C IH: INDEX OF HIGHEST RESPONSE VERTEX

C A: SIMPLEX MATRIX

C A(*,I) = ITH VERTEX OF THE SIMPLEX

C A(1, I) = SIGMA OF LARGEST PEAK (KEV)

C A(2,I) = ENERGY OF LARGEST PEAK (KEV)

C A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

C R: R(I) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SMAT/ NAXES , N VE RT , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R ( 1 7 ) INCLUDE 'SPECT.COM*

C

C NCH : NUMBER OF DATA CHANNELS

C BKEV: ENERGY OF FIRST CHANNEL IN KEV

C CHKEV: ENERGY PER CHANNEL IN KEV

C CH: ARRAY OF CHANNEL COUNTS

COMMON /SPECT/ NCH , B K E V , CH K E V , CH (1 0 2 4

)

IF ( LIMITC VERTEX) ) GO TO 30

RESP=0

.

DO 20 J=1 , NCH

Y = 0 . E = BKEV + CHKEV*( J-1 ) DO 10 1=1 , NPEAK

Y=Y+VERTEX(I+2)*EXP(-0.5*((E-(DPKE(I)+VERTEX(2)))/

+ PKSIGCVERTEXd )>DPKE(I)))**2)

10 CONTINUE

RESP = RESP+(CH(J )-Y)**2/AMAX1 ( Y, 1 . 0 ) 20 CONTINUE

RESP=RESP/(NCH-NAXES)

RETURN

30 RESP=1E19

RETURN

END

SUBROUTINE SETL IM C CH , NCH

)

C

C SETLIM — SET PARAMETER LIMITS

C

C ENTER:

C CH = ARRAY OF CHANNEL COUNTS

C NCH = NUMBER OF DATA CHANNELS

C INCLUDE 'PEAKS.COM'

C

C NPEAK: NUMBER OF PEAKS

C FWHMMN : FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

C IBP: INDEX OF BIGGEST PEAK

C PKA : PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

C PKE : PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

C PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

C IZPK: ATOMIC NUMBER OF ITH PEAK

C IDPK: LINE CODE OF ITH PEAK

267
* C IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

* C DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

0003 * COMMON /PEAKS/ NPE AK , FWHMMN , I BP , PKA ( 1 4 ) , PKE ( 1 4 ) , PKS ( 1 4 )

,

* + IZPKC 1 4) , IDPKC 1 4) , IXC 1 4) ,DPKEC 1 4)

0004 INCLUDE 'SPECTB.COM'

* C

* C NCHB: TOTAL NUMBER OF DATA CHANNELS IN BUFFER

* C ICHW: INDEX OF FIRST CHANNEL IN ANALYSIS WINDOW

* C NCHW: NUMBER OF CHANNELS IN ANALYSIS WINDOW

* C BKEVB: ENERGY OF FIRST CHANNEL IN KEV (BUFFER)

* C CHKEVB: ENERGY PER CHANNEL IN KEV (BUFFER)

* C CHB: ARRAY OF CHANNEL COUNTS (BUFFER)

0005 * COMMON /SPECTB/ NCHB , ICHW>NCHW, BKEVB, CHKEVB>CHB(1 028)

0006 INCLUDE 'LIMITS.COM'

* C

* C LIMSW: 1 = LIMITS ENABLED, 0 = LIMITS DISABLED

* C SIGMAX: MAXIMUM SIGMA OF BIGGEST PEAK (EV)

* C SIGMIN: MINIMUM SIGMA OF BIGGEST PEAK (EV)

* C EMAX: MAXIMUM ENERGY OF BIGGEST PEAK (KEV)

* C EMIN: MINIMUM ENERGY OF BIGGEST PEAK (KEV)

* C AMAX: MAXIMUM AMPLITUDE OF ANY PEAK

* C LCODE: LIMIT COUNTS

0007 * COMMON /LIMITS/ L I MSW , S I GM AX , S I GM I N , E M AX , E M I N , AMAX , LCOD E ( 1 6 ) C

C DFWHMM: FWHM(MN) UNCERTAINTY (EV)

C DEBPK: ENERGY UNCERTAINTY OF BIGGEST PEAK CKEV)

0008 DATA DFWHMM/5./, DEBPK/. 02/

C

C SET MAXIMUM AND MINIMUM SIGMA OF BIGGEST PEAK

C 0009 SIGMAX = PKSIGCCFWHMMN+DFWHMM)*4.2466E-4>PKE(IBP)-5.895)

0010 SIGMIN = PKSIGC CFWHMMN-DFWHMM)*4.2466E-4,PKECIBP)-5.895)

C

C SET MAXIMUM AND MINIMUM ENERGY OF BIGGEST PEAK

C 0011 EMAX = PKE C IBP)+DEBPK

0012 EMIN = PKE ( IBP)-DEBPK

CC FIND MAXIMUM AMPLITUDE

C 001 3 AMAX = -1 . 0014 DO 10 I=3,NCHB-2

0015 IF (CHB(I) .GT. AMAX) AMAX = CHB(I)

0016 10 CONTINUE

C

C SET AMPLITUDE LIMIT

C 0017 AMAX = AMAX+3.2*SQRT(AMAX)

C

C RESET LIMIT COUNTERS

C 0018 DO 20 1 = 1 , 1 6 0019 LCODE(I)=0

0020 20 CONTINUE

0021 RETURN

0022 END

268
SUBROUTINE SIMPLX(CHB,NCHB,BKEVB,CHKEVB,ETIME)

5IMPLX — SIMPLEX FIT CONTROL ROUTINE

ENTER : CHB = DATA ARRAY TO BE FIT, INCLUDING 2 ADDITIONAL CHANNELS

AT BEGINNING AND END

NCHB = NUMBER OF CHANNELS TO BE FIT +4

BKEVB = ENERGY OF FIRST CHANNEL ( CHB ( 3 ) ) IN KEV

CHKEVB = ENERGY PER CHANNEL IN KEV

FOLLOWING COMMON VARIABLES MUST BE INITIALIZED:

NPEAK, FWHMMN, PKA, PKE

RETURNS:

ETIME = ELAPSED TIME TO PERFORM FIT IN SECONDS

DIMENSION CHB ( NCHB

)

INCLUDE 'PEAKS.COM'

NPEAK: NUMBER OF PEAKS

FWHMMN : FULL WIDTH AT HALF MAXIMUM FOR MANGANESE PEAK (EV)

IBP: INDEX OF BIGGEST PEAK

PKA: PKA(I) IS ESTIMATED AMPLITUDE OF ITH PEAK

PKE: PKE(I) IS ESTIMATED ENERGY (KEV) OF ITH PEAK

PKS: PKS(I) IS ESTIMATED SIGMA (KEV) OF ITH PEAK

IZPK: ATOMIC NUMBER OF ITH PEAK

IDPK: LINE CODE OF ITH PEAK

IX: ARRAY OF INDICES WHICH SORT PEAKS BY ENERGY

DPKE: DPKE(I) IS PKE(I) - PKE(IBP)

COMMON /PEAKS/ N PE AK , F WHMMN , I B P , PK A ( 1 4 ) , P K E ( 1 4 ) , PKS ( 1 4 )

,

+ IZPKM4),IDPK(14),IX(14),DPKE(14)

INCLUDE 'SMAT.COM'

NAXES : NUMBER OF AXES (= NPEAK+2)

NVERT: NUMBER OF VERTICES (= NPEAK+3)

TOLER: TOLERANCE FOR PARAMETER CONVERGENCE

IL: INDEX OF LOWEST RESPONSE VERTEX

IH: INDEX OF HIGHEST RESPONSE VERTEX

A: SIMPLEX MATRIX

A(*,I) = ITH VERTEX OF THE SIMPLEX

AM, I) = SIGMA OF LARGEST PEAK (KEV)

A(2,I) = ENERGY OF LARGEST PEAK (KEV)

A(J,I), 3 >= J <= NPEAK+2 = AMPLITUDE OF JTH PEAK

R: R(I) IS THE RESPONSE OF THE ITH VERTEX

COMMON /SMAT/ NAX E S , N V E R T , TO L E R , I L , I H , A ( 1 6 , 1 7 ) , R (1 7

)

INCLUDE 'SPECT.COM'

NCH: NUMBER OF DATA CHANNELS

BKEV: ENERGY OF FIRST CHANNEL IN KEV

CHKEV : ENERGY PER CHANNEL IN KEV

CH: ARRAY OF CHANNEL COUNTS

COMMON /SPECT/ NCH, BKEV, CHKEV, CHM 024)

INCLUDE 'C0UNTS.COM'

ITER8: ITERATION COUNTER (PER PASS)

NRFLCT: REFLECTION COUNTER

NXPAND : EXPANSION COUNTER

NCONTR : CONTRACTION COUNTER

NSHIFT: SHIFT COUNTER

NTOTAL: TOTAL NUMBER OF ITERATIONS

COMMON /COUNTS/ I T E R8 , NR F LCT , NXP AND , NCONT R , NSH I FT , NTOT A

L

269
0011

0012

001 3 001 4 0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

0032

0033

0034

0035

0036

0037

0038

1 00

C

Cc

200

Cc

c

c

c

c

300

C

C

C

INITIALIZATION

NAXES = NPEAK+2

NVERT = NAXES+1

LOCATE BIGGEST PEAK

IBP = 1 DO 100 I=2,NPEAK

IF (PKA(I) . GT . PKA(IBP)) IBP=I

CONTINUE

COMPUTE ENERGY DIFFERENCES FROM BIGGEST PEAK

DO 200 1=1 , NPEAK

DPKE(I) = PKE(I)-PKEdBP)

CONTINUE

SET SIMPLEX PARAMETER LIMITS

CALL SETLIM

SETUP INITIAL SIMPLEX MATRIX

ETIME = 0. TIME = SECNDSC 0 . ) CALL INSIMP

APPLY POLYNOMIAL SMOOTHING TO INPUT SPECTRUM

NCH = 0 DO 300 I=3,NCHB-2,2

NCH = NCH+1

CH(NCH) = (-3.0*CHB(I-2)+12.0*CHB(I-1)+17.0*CHB(I)

+12.0*CHB(I+1)-3.0*CHB(I+2))/35.0

CONTINUE

RESET COUNTERS

NRFLCT = 0 NXPAND = 0 NCONTR = 0 NSHIFT = 0

FIT ALTERNATE POINTS OF SMOOTHED SPECTRUM

BKEV = BKEVB

CHKEV = CHKEVB*2

TOLER = 0.001

CALL FIT

NTOTAL = ITER8

ETIME = ETIME+SECNDSCTIME)

OUTPUT RESULTS

CALL REPORTCAC1 ,IL),RCID)

FIT RAW SPECTRUM

270
0039 TIME = SECNDS( 0 . ) 0040 DO 400 1=1 ,NCHB-4

0041 CH(I) = CHB(I+2)

0042 400 CONTINUE

0043 NCH = NCHB-4

0044 CHKEV = CHKEVB

0045 TOLER = 0.00001

0046 CALL FIT

0047 NTOTAL = NT0TAL+ITER8

0048 ETIME = E T I M E + S ECNDS C T I ME

)

C

C OUTPUT RESULTS OF FINAL FIT

C

C CALL REPORT/CAM , IL.) ,R(IL) ) 0049 RETURN

0050 END

0001 SUBROUTINE SO R T P K ( N , A K E Y , I NDE X

)

C

C SORTPK — CALL SORTUP TO SORT SPECTRUM PEAKS

C

C ENTER:

C N = NUMBER OF KEYS TO SORT

C AKEY = KEY ARRAY TO COMPUTE PERMUTATION INDEX FOR

C

C RETURNS:

C INDEX = PERMUTATION WHICH SORTS KEY ARRAY INTO ASCENDING ORDER

C 0002 DIMENSION AK EY ( N ), INDEX ( N

)

0003 DIMENSION TEMP ( 1 4

)

0004 DO 10 I = 1 , N 0005 TEMP(I) = AKEY ( I ) 0006 INDEX(I) = I 0007 10 CONTINUE

0008 CALL SORTUPCN, TEMP, INDEX)

0009 RETURN

0010 END

0001 SUBROUTINE SORTU P ( N , AKE Y , I DATA

)

C

C SORTUP — SORT ARRAY INTO ASCENDING ORDER

CC ENTER:

C N = NUMBER OF ITEMS IN ARRAY

C AKEY = ARRAY CONTAINING SORT KEY

C IDATA = AUXILIARY DATA ARRAY TO BE SORTED

CC RETURNS:

C AKEY = INPUT ARRAY SORTED IN ASCENDING ORDER

C IDATA = INPUT ARRAY SORTED BY AKEY ARRAY

C

271
0002 DIMENSION AKE Y ( N ) , I DATA ( N

)

0003 IF (N .LT. 2) RETURN

C

C SELECTION SORT

C 0004 L = N-1

0005 DO 20 1 = 1 , L 0006 K = N 0007 DO 10 J = I » L 0008 IF (AKEY(J) .LT. AKEYCK)) K=J

0009 10 CONTINUE

0010 TEMP = AKEY(I)

0011 AKEY(I) = AKEY(K)

0012 AKEY(K) = TEMP

0013 TEMP = I DATA ( I ) 0014 I DATA ( I ) = I DATA ( K

)

0015 I DATA ( K ) = TEMP

0016 20 CONTINUE

0017 RETURN

0018 END

272
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

LEAST-SQUARES FIT WITH DIGITAL FILTER: A STATUS REPORT

Jon J. McCarthy and Frederick H. Schamber

Tracor Northern, Inc. 2551 W. Beltline Highway

Middleton, WI 53562

Abstract

The technique of incorporating digital filtering within a linear

least-squares fitting procedure was initially conceived and reported in

1973. At this time, it was one of the very few x-ray analysis procedures

which attempted to perform quantitative analysis on spectra containing

complex peak overlaps. Since then, the method has been used extensively

in a number of applications and continues to be a powerful analysis

technique.

The continued success of the filter-fit method is more than a little

surprising in view of its essential simplicity of concept. Unlike

continuum-modeling which achieves quantitative accuracy by careful

attention to the details of the x-ray production and measurement process,

the filter-fit method revolves about a single empirical fact: the

continuum (background) components in a spectrum are relatively "smooth"

structures compared to the characteristic x-ray peaks and thus can be

selectively suppressed by application of an appropriate digital filter

operator. In view of the considerable success which has been achieved

with this very simple concept, it seems clearly worthwhile to analyze why

the method works as well as it does.

It is the purpose of this paper to review the current state of knowledge

of the technique, based upon the past five years of experience in its

application. In addition to reviewing the basic concept of the technique

and its application to spectral analysis, the presentation will focus

attention on the inherent strengths and limitations of the technique and areas for future improvement.

1. Introduction

Quantitative analysis of the x-ray spectrum obtained with a solid-state silicon

detector requires a procedure that can accurately unfold overlapped peaks in the presence

of a relatively high background component. Linear least-squares fitting procedures may be

applied to the problem of overlapped peaks provided that the continuum background can be

eliminated by a suitable technique. A very simple, yet extremely powerful method for

continuum suppression is embodied in the concept of digital filtering. A simple modification of the usual least-squares fitting procedure allows continuum suppression via an appropriate digital filter. The application of this technique to x-ray analysis was first

described in 1973, and discussed in some detail in the following publications [1, 2,3,4]*.

This method has been implemented for use in an on-line minicomputer-based x-ray analyzer.

In this form, the technique has been applied with considerable success by a large number of users to problems in both electron and photon excited x-ray analysis [5,6,7,8].

1 Figures in brackets indicate the literature references at the end of this paper.

273
This paper will review the basic concepts of the technique and discuss its application

to analysis of x-ray spectra. Particular attention will be focused upon the effect of

absorption edges in the continuum and sources of error in peak unfolding. In the following

the technique will be referred to as the "filter-fit" method.

2. Basic Concept of the Filter-Fit Method

2.1 Linear least-squares fitting

The basic assumption of the linear least-squares method is that each channel of a measured spectrum can be represented by the sum of the corresponding channels of a set of

reference spectra in correct proportions. We can express this basic assumption in the

fol lowing manner:

Measured Spectrum = A x * (Reference 1) +A2 * (Reference 2) +A3 * (Reference 3)

+. .

.

+statistical noise

Analysis of the measured spectrum requires a procedure that will determine the set

of proportionality coefficients (A x , A2 , A3) ...) that provide the best fit to the

measured spectrum. The reference spectra may be any set of curves that provide an accurate model of the shape and amplitude of the particular x-ray lines of interest. Normally

such references are a set of measured pure element spectra.

If we denote channel j of the measured spectrum by Y. and channel j of the i^ refer- ence spectrum as R^., we may express the "word" equation given above in a conventional

mathematical form:

m

Y. = V A.R. . + E.

i=l (1)

where Ej is the statistical fluctuation in channel j.

Following the conventional least-squares procedure, we assume that the best set of

values for the A., coefficients are those that minimize the value of the quantity x

2 (chisquared), which is defined as follows:

m 99

j 1=1 (2)

a. is the standard deviation which represents the statistical uncertainty of Y.. When

J — J the measured values Y. are normally distributed with variance a. 2

, the set of A. coeffiJ J cients that minimize x

2 are the "maximum- likelihood" solutions in a rigorous mathematical

sense.

In the ideal case in which the reference spectra would provide a complete and exact

description of the constituents of the unknown spectrum, the solution coefficients of

Equation (2) would provide an accurate analysis of the unknown x-ray spectrum. Unfortunately, even when the reference spectra are obtained from pure elements and acquired under

274
identical conditions as the unknown spectrum, there is no assurance that the continuum in

the references will sum to duplicate the continuum of the unknown spectrum. To obtain

accurate results, any procedure which employs least- squares fitting must incorporate a treatment of these continuum differences.

Suppression of the Continuum

The filter-fit method approaches the problem of continuum removal by transforming the

actual spectra in question into a set of related spectra to which least-squares fitting

may be directly applied. This transformation is performed by the application of a digital

filter to both the unknown and reference spectra prior to the least squares analysis. An obvious distinction between the continuum and the peaks of a spectrum is the slow variation

of the former with energy compared with the much faster variation of the latter. A digital

filter is a simple correlating function which can be applied to a spectrum to suppress particular "frequencies" in a digital data array in much the same way that electrical filter

networks are used to suppress unwanted frequencies in an electrical signal. The effect of

the filter operator used here is to suppress both low frequencies (primarily continuum) and

higher frequencies (primarily statistical fluctuations) and pass the characteristic frequencies of the peak shapes. In this manner the continuum is effectively suppressed without

the need for a detailed model of continuum shape. In addition, no errors due to subtraction

of background are introduced into subsequent analysis procedures.

Digital Filtering

Mathematically, a digital filter is defined by a set of coefficients which are multiplied against corresponding channels of a spectrum. The sum of these products is the

"filtered" value for one point in the spectrum and can be expressed as

t FILT (Y.) v y

= Ljy fY.A s j+s

s=-t

where the f coefficients are the set of filter constants.

Figure 1 illustrates one of the most elementary such filters; the central lobe (upper

width) consists of UW positive coefficients (f = C/UW); the outer (lower width) lobes

contain LW negative coefficients (f = -C/2LW), where C is an arbitrary constant. With

C = 1, such a filter is nothing more than the average of UW central channels minus the

average of two regions of LW channels on either side. When this filter is applied to the

spectrum in figure 1 (consisting of a Gaussian shaped peak superimposed on a straight line)

it results in the filtered spectrum shown immediately below, there is a strong response to the peak and the straight line is suppressed entirely. It can be easily shown that all

filters of this type (symmetric and zero-area) will filter linear components to exactly

zero. Since a smooth curve can be approximated by a series of straight lines, and since

the continuum is assumed to be a slowly varying smooth curve, it follows that the continuum component of a spectrum can be suppressed by the application of a filter of appropriate dimensions. This is illustrated in figure 2 and figure 3. Several complex spectra

are shown before and after application of a digital filter of the type shown in figure 1. The spectrum of figure 2 was generated by electron excitation, while the spectrum of

figure 3 was produced by photon excitation. In each case, despite the fact that the

continuum shape is markedly different, the continuum has been effectively suppressed

and only the peak structures remain.

Although digital filtering introduces obvious distortions to peak shapes, this distortion is predictable and consistent, and does not alter the primary information content

of the peaks. Most importantly, digital filtering is a linear operation; a composite

spectrum, after filtering, is still the linear sum of its individually filtered peak

components. This is illustrated in figure 4. On the upper left of this figure is a composite spectrum which is a linear sum of the two Gaussian peaks and the smooth continuum traces shown below it. The right hand side of figure 4 shows the filtered composite

275
276
277
278
279
spectrum, and the two filtered Gaussian peaks. The contribution of the filtered continuum is zero and can be ignored. To a very good approximation the filtered composite

spectrum is the sum of the filtered peak spectra.

A detailed treatment of the effect of various filter dimensions in processing x-ray

spectra has been discussed by Statham [4]. The optimum filter dimensions reported by

Statham agree remarkably well with those first suggested by Schamber [2] from an empirical

viewpoint. The filter dimensions used for the illustrations in this paper are those of

Schamber.

The observations made in the preceding paragraphs form the basis for the filter-fit

method of spectral processing. Since digital filtering is a linear operation that effectively suppresses background, the peak structure in a filtered measured spectrum can be

represented by a linear sum of filtered reference spectra. If both the measured data

and the reference peaks are digitally filtered a conventional least squares fitting

procedure is justified to obtain the best fit of peak structure.

Filter-Fit Procedure

Consider the measured spectrum to be a linear sum of reference spectra, statistical

noise, and a slowly-varying continuum component. We can then rewrite Equation 1 in the

following manner:

To suppress the continuum, we apply a digital filter to both sides of this equation,

m

Y. = £ A iRij + contj + E.

i=l (3)

FILT (Y.) = £ A.[FILT(R. .)] + FILT(Ej) + FILTCcontj)

i=l

Since the continuum is assumed to be a slowly-varying component,

FILT(cont-) = 0

FILT(Yj) = £ A.[FILT(R. j)] + FILT(Ej)

i=l

with the definitions

= FILT(R

fJ

) and

(a 1 .) , the variance of Y'

.

m

We can define a new chi-squared variable by

2 _

X

- ECY'j " £ A^^.]2/^.)2

j i=l

280
Assuming that observed errors are due to counting statistics of the unknown spectrum only, we have

s=-t

From this point, the mathematical development employs the well-known Chi-Squared minimization procedure, except that the filtered quantities Y

1 . and R'.. are substituted for the

original channel variances a. 2

. J J

In this format, the least squares method has been previously discussed in detail [2,4]

and will not be treated further in this paper.

Estimate of Uncertainty

The results of any computerized data reduction procedure should not be accepted without

an adequate estimate of the errors to be expected. It is therefore important that the procedure calculate and output the statistical uncertainty of the results. A virtue of the

least-squares procedure is that it allows the uncertainties of the fitted coefficients to be

estimated by use of a simple approximation [2].

Another useful indicator of error is the quantity Chi-Squared (x

2

)- If X

2 is normalized (by dividing by the number of fitted points minus degrees of freedom) it should have a value of about one (on the average) for an ideal fit. Values of x

2 which are much greater

than one are indicative of some systematic error in the fit. x

2 can be quite useful in this

regard as a figure of merit for the entire procedure. A more detailed discussion of the

interpretation of Chi-Squared in the filter-fit procedure can be found in Reference 3.

3. Application to Spectral Analysis

The preceding development of the filter-fit method was based upon two key assumptions:

1) the continuum is adequately suppressed by a digital filter and 2) the measured refer- ence spectra provide accurate peak models. In view of the simplicity of these assumptions,

it is worthwhile to explore in some detail the performance of the method under various

conditions.

In order to demonstrate the performance of the filter-fit method, a simulation was devised which requires that both continuum suppression and peak unfolding be carried out

accurately. Six Gaussian peaks, each with a full width at half maximum (FWHM) of 150 eV, were constructed with their centroids separated by 150 eV. These Gaussians were summed

into another spectrum which contained a broad continuum profile. Statistical noise was introduced into the resultant composite spectrum with a radioactive source. This

composite spectrum is shown in figure 5. The final spectrum was then analyzed with the

filter-fit method using the original Gaussian peaks as reference profiles. This entire

process was repeated for a total of 54 trial cases. The results of the simulation are presented in Table 1. The high accuracy obtained illustrates the performance of the

filter-fit procedure for peak unfolding and continuum suppression.

Additional simulations have been devised to explore in more detail the strengths

and weaknesses of the filter-fit method. These simulations will be discussed briefly in

the following. Particular emphasis will be placed on the relationship of each simulation

to the key assumptions on which the filter-fit technique rests.

281
282
Table 1, Six Overlapping Gaussian Test Case.

r*E

c (kev; FWHM (evj Tni^ Pat i n Avpranp Ml Rat

i

nvci aye ill- i\au i u Standard Deviation3

1..000 150 0.1 0.100 ± 0.004 0.033

1.,150 150 1.0 1.002 ± 0.008 0.047

1..300 150 0.5 0.498 ± 0.005 0.042

1. 450 150 0.6 0.599 ± 0.006 0.047

1..600 150 5.0 4.997 ± 0.005 0.038

1..750 150 1.0 1.000 ± 0.004 0.033

Average of 54 independent noise-perturbed runs.

3.1 Discussion of continuum suppression

Two important considerations come to mind in respect to continuum suppression by

digital filtering. The first is apparent in the low energy region of the spectrum where

attenuation by the beryllium window of the detector causes an abrupt roll -off in the

spectrum. This effect is most pronounced in electron-excited spectra due to the larger

proportion of bremsstrahlung continuum at these energies. In order to perform well at these

energies, the filter-fit must be insensitive to this continuum curvature. The second con- sideration is related to the appearance of steps in the continuum at the energies of the

absorption edges of the various elements being excited. Despite the fact that these steps are smooth in appearance due to instrumental resolution, the abrupt appearance of an edge in

the continuum represents a severe departure from linearity over a region of the spectrum.

Sensitivity to Continuum Curvature

Figure 6 is a plot comparing the filter-fit method with two other methods of removing

continuum. To perform these comparisons, a Gaussian peak of fixed energy and width (REF) was superimposed upon a variable-width Gaussian peak (BGND) simulating a curving background. By varying the width of the BGND peak and plotting the error in measurement of

the REF peak by each of the three methods, the inaccuracies of the three methods can be

compared as a function of continuum curvature. The top trace was obtained by simple integration of the area above the straight line segment (A,B). This curve represents the error

of a linear background interpolation and direct peak integration. The middle curve was obtained by performing an ordinary least-squares fit (without filtering) across the region

(A,B), but introducing slope and intercept parameters into the fit to account for the

background. This curve represents the error obtained using ordinary least-squares fitting

with linear background. The bottom curve was obtained using the filter-fit method with

the REF peak as the sole fitting reference (filter dimensions as shown in figure 1).

The vertical axis of the plot was normalized to 100 percent error for the case when

the FWHM of the BGND curve was the same as the reference peak. As expected, the error of

all three methods approaches zero as the FWHM of the BGND peak is increased. What is

remarkable is the rapidity with which the error of the filter- fit method decreases with

decreasing continuum curvature. Note that the filter-fit method is subject to an error

of only M percent when the BGND width is four times the peak width. When the BGND width

is increased to eight times the REF peak width the analysis error has decreased to less than

0.05 percent. This demonstrates that digital filtering is highly effective in suppressing

smooth continuum.

This plot illustrates why the filter-fit method has been successful in the analysis

of low energy peaks which lie in the spectral region where detector window absorption

causes a strong curvature in the observed bremsstrahlung spectrum. The plot also indicates

that the filter-fit method is superior to linear-interpolation methods on the basis of

continuum rejection alone, confirming the results reported by Beaman and Solosky [6].

283
I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 FWHM RATIO : (BGND/REF)

Figure 6. Comparison of continuum removal performance by three methods.

284
Sensitivity to Absorption Edge Steps in the Continuum

Matrix absorption introduces steps into the continuum due to elemental absorption

edges. To first order, least-squares fitting with measured reference spectra compensates

for these edges since the edges also appear in the references. However, the amplitude of

an edge depends not only upon the concentration of the element, but also upon the amount

of bremsstrahlung, and is therefore not strictly proportional to peak amplitude.

In order to estimate the maximum error expected due to absorption steps, several

simulations were devised. In the first simulation, the continuum spectrum from a specimen

of pure iron was constructed using the method suggested by Fiori et al., [9]. A portion

of this spectrum which includes the step due to the iron K edge is shown in the lower

trace of figure 7. The upper trace is the result of filtering this spectrum. This trace

definitely shows a non-negative residual in the edge area after filtering. A single

Gaussian peak was generated and used as a peak reference for filter-fitting to this con- tinuum spectrum. The results of this study are shown in figure 8, where the area obtained

by a filter-fit of the continuum step is plotted versus the center channel of the fitted

peak. The maximum absolute value of the error yields a peak with area of about 45 counts,

centered about 5 channels (approximately 1/2 of 0.5 FWHM) on either side of the edge.

This represents an error of less than .01 percent of the Ka peak for pure iron. In a similar analysis [4], Statham concludes that the maximum error that can occur amounts to

a peak height of less than 1/3 of the height of the step. It is well to remember at this

point that these errors are the maximum expected because of the absence of an edge step

in the reference spectrum. In normal work, the reference spectrum would include the

edge step.

A second simulation was designed to explore the effects of edge steps in the region

below 2 keV where the continuum is large and sloping due to detector absorption. The magnitude of this effect was investigated for mathematically synthesized aluminum-silicon

spectra. This is a particularly interesting case since the absorption edges are large and

the aluminum edge (1.56 keV) falls close to the silicon K-alpha peak (1.74 keV). Table 2 displays the k-ratios obtained using synthesized aluminum and silicon reference spectra to fit "unknown" spectra synthesized in 1:9, 5:5, and 9:1 proportions of aluminum to silicon.

All spectra were generated using Gaussian peak shapes added to the bremsstrahlung continuum (including absorption edges) modeled by the method described by Fiori, et al., [9]

(excitation voltage = 15 keV, take-off angle = 57.5 degrees). These results strongly

suggest that absorption edge steps are not a significant source of error in the application

of the filter-fit method.

Table 2. Effect of Continuum Absorption on Fitted K-Ratios of Al and Si for Concentration Ratios of 1:9, 5:5, and 9:1.

Reference Expected Fitted Error

Al

Si

Al

Si

Al

Si

0. 10260

0.82781

0.49566

0.34311

0.89740

0.05395

0.10269

0.82882

0.49588

0.34353

0.89747

0.05397

-.09%

-.12%

-.04%

-.12%

-.01%

-.04%

285
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287
3.2 Discussion of peak shape models

The accuracy of a least-squares analysis is highly dependent upon how closely the peak

models used as fitting references represent the actual peak profiles of the measured spectrum. Shifts in peak position due to electronic drift, slight changes in detector resolution, or inaccurate reference peak shapes will produce inaccurate results. Since a measured

spectrum of good statistics provides the most accurate description of peak shapes and

positions possible, the use of measured reference spectra as fitting functions results in an analysis method which is simple to implement and is nearly optimum in terms of accuracy

and precision. Even certain spectral artifacts (such as escape peaks and peak tailing) are reproduced in measured reference spectra, and are thus compensated in the fit. For this reason, measured fitting references are normally employed in the filter-fit method. In cases where suitable pure-element standards are not available, appropriate compound stan- dards or even synthesized pure-element spectra may be employed.

Despite the inherent advantages of using measured fitting references, various situations can arise which limit the accuracy obtained by any least-squares procedure, including

the filter-fit method. Some examples of the effects of inaccurate peak models are discussed

in the fol lowing.

Differential Absorption

Spectrum peaks due to emissions of different line series (K,L,M) do not normally appear

in constant proportions in the spectra of different matrices due to the effect of matrix

absorption. For example, since softer x-rays will be more strongly absorbed, it is wise

to fit the K, L, and M lines of an element separately. Even if this precaution is observed,

a question remains as to what degree lines within the same series can be affected by the

appearance of an absorption edge between the lines. This problem will be most severe when

a strong absorption edge lies between two characteristic peaks. Figure 9 depicts four such

situations. Figure 9A illustrates the positions of the L-alpha and L-beta emissions of

lead (top trace) and the L absorption edges of tungsten. The LI and L2 edges of tungsten

will absorb the lead L-beta line more strongly than the L-alpha, thereby altering the ratios

of these lines. Similarly, figures 9B, C, and D illustrate differential absorption of the

zinc K-beta line by the copper K edge, mercury L-beta by the tungsten L2 and L3 edges, and

platinum L-beta by the zinc K-edge, respectively. Tests were conducted for these cases by

synthesizing reference spectra with Gaussians. Unknown spectra were constructed for various

concentrations of the absorbing and absorbed elements, with appropriate attenuation of the

beta peaks. The relative errors in determination of the concentration as a function of

absorbed element concentration are plotted in figure 10. As previously noted by Corlett

and McDonald [5], differential absorption is more severe due to the rough parity in size

of the L-alpha and L-beta lines. The most serious problem arises when a K absorption

edge falls between the L-alpha and L-beta lines (case D).

It is clear from these results that differential absorption can be a significant source

of error. A practical solution to this problem is to partition the fit when a strong

absorption edge is known to be present. For example, if fitting low concentrations of

platinum in the presence of large concentrations of zinc, it is advisable to prepare two reference spectra for platinum, one containing the L lines below, and the other containing

the lines above the zinc K edge. This technique has been successfully employed in

instances where such differential absorption was found to be a problem and is only a minor

i nconvenience.

288
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289
290
Unreferenced Peaks

Figure 11 illustrates the response to an unexpected peak of three different peak-area

measurement methods. This figure was constructed by determining the amplitude of a fixed

peak (analyzed peak) at various separations from a movable "interference" peak of identical

shape and size. The direct integration method simply used the area determined for a region,

extending one FWHM on either side of the centroid of the analyzed peak. The (unfiltered)

least-squares fit and the filter-fit curves were made with a single reference (for the

analyzed peak). At zero separation, the analyzed and interference peaks overlap exactly

and all three methods converge to 100 percent error. In the region of small peak separation (below about 1 FWHM) the filter-fit method has the least error of all methods tested.

This is gratifying since it is here that a minor overlapped peak is likely to be over- looked and not referenced in the fitting procedure. At peak separations of 1 to 3 FWHM,

the filter-fit performance is slightly worse than other methods, but the absolute errors

are much smaller in this region. Beyond 3 FWHM all methods show little interference. The

filter-fit method approaches the "no-interference" condition the most slowly since convolution with the filter function broadens the effective width of a peak shape. Because of

this extended response, it is important that either: (1) all peaks present in the spectrum

be fitted, or (2) the fitting region be carefully selected so that nonreferenced peaks

are excluded. We conclude that for small peak separations the filter-fit method exhibits

superior performance and is comparable to other methods at large peak separations.

An interesting characteristic of the filter-fit response is that it exhibits regions

of both positive and negative correlation. Thus, the presence of an unreferenced peak

might either increase or decrease the intensity determined for a nearby analyzed peak; there

are even two nodes of orthogonality where an interfering peak does not affect the fitted

amplitude of another peak at all. A large Chi-Squared value, of course will always be

associated with the presence of such unreferenced peaks.

Spectral Energy Shift

Since the least-squares analysis method is predicated upon the assumption that the

reference spectra provide an accurate peak-shape model, it is apparent that a difference

in energy calibration between the analyzed and reference spectra will introduce fitting

errors. Figure 12 illustrates the results of a study to determine the magnitude of this

effect. A Gaussian line shape (FWHM 160 eV) was used as the unknown spectrum and a peak

of identical width was used as the fitting reference. The position of the peak in the

unknown spectrum was shifted by small increments and the resulting relative error and

computed value of Chi-Squared were plotted.

A more realistic example of this type of error occurs in the case of analyzing a very

small peak which is overlapped by a large one. An example of this type of difficult problem

is the barium La and titanium Ka lines which are separated by only 44 eV. Spectra were

obtained from barium and titanium under identical conditions (25 kV, 80° take-off angle, 10

eV/channel ) , and used as fitting references. Both references were used to analyze a spectrum of pure titanium using the filter-fit procedure. The analysis was repeated as the

titanium spectrum was shifted in 10 eV steps with respect to the reference calibration. The

purpose of the test was to determine how much false barium would be reported due to the

energy shift. The results of these measurements are shown in figure 13. The value of the

maximum barium concentration reported is about four percent and occurs when the titanium has

been shifted approximately 0.5 FWHM.

291
(3na±/v3av onvo)=ouvd

292
293
WT I BA IN PURE TI

4

3 •

•

• • •

2

1

wm •

•

•

• • •

-8 -6 -4 -2 •

2 6 8 10 12 14

• -2 TI PEAK SHIFT (CHANNELS) •

•

-3

•

•

-4

Figure 13. Measured concentration of barium in a pure titanium spectrum as a function of the energy shift of the titanium K alpha peak.

It is clear from the examples above that the filter-fit method is (like all leastsquares techniques) quite sensitive to calibration shifts. Mathematical techniques for

restoring the calibration of spectra have been developed and found to work quite well . Gain

and zero-level shifts can be measured periodically by acquiring a calibration spectrum

containing two known peaks. Using this information, subsequently acquired spectra can be

restored to correct calibration by a quadratic interpolation procedure. A similar

procedure may be employed to examine each spectrum after acquisition and restore it to true calibration automatically. An example of the result of such an automatic calibration

is shown in figure 14. In this figure, the value of Chi-Squared obtained in an analysis

of each of twelve data sets is plotted versus the data file index. The data were acquired

on a spectrometer known to have substantial electronic drifts. The upper trace is the

result of analysis without automatic calibration. The lower trace is the result of

analysis with automatic calibration. The substantial reduction of Chi-Squared is an indication of the reduction of systematic error due to improper energy calibration.

294
AUTOMATIC TWO POINT ENERGY CALIBRATION
4. Summary

This paper has reviewed the development of the filter-fit method of spectral processing and its applications to the analysis of x-ray spectra obtained with the Si C Li

)

detector. The method has been used successfully for the last five years in a variety of

appl ications.

The filter-fit method relies on the simple postulates that the spectrum continuum is

essentially smooth and that measured reference spectra provide the best peak shape models.

The consequence of these assumptions is that no explicit knowledge of the continuum shape

or complex peak shape models are required for spectral analysis. The continued success

of the filter-fit method can be attributed to the fact that most x-ray spectra do indeed

satisfy these assumptions.

A number of practical concerns related to physical and instrumental effects has been

considered and shown to produce systematic errors. When the errors are small, as are absorption edge effects, they can safely be ignored. More serious errors arise from

incorrect energy calibration and differential absorption which require special procedures

to reduce their size. Development efforts focused on eliminating these systematic errors offer the best direction for improvement of the filter-fit method.

References

[1] Schamber, F. H. , "A New Technique for Deconvolution of Complex X-ray Energy Spectra,"

Proceedings of the Eighth National Conference on Electron Probe Analysis, New Orleans,

Paper 85, 1973.

[2] Schamber, F. H. , "A Modification of the Linear Least-Squares Fitting Method Which

Provides Continuum Suppression," in X-ray F1 uorescence Analysis of Environmental

Samples , T. G. Dzubay, Ed., Ann Arbor Science Publishers, Inc., 1977, p. 241.

[3] Schamber, F. H. , Wodke, N. F. , and McCarthy, J. J. , "Least-Squares Fit with Digital

Filter: The Method and Its Application to EDS Spectra." Proceedings of the Twelfth

Annual Conference of the Microbeam Analysis Society, Boston, Paper 98, 1977.

[4] Statham, P. J., "Deconvolution and Background Subtraction by Least-Squares Fitting

with Prefiltering of Spectra," Anal. Chem. , 49, No. 14, 2149-2154 (1977).

[5] Corlett, M. I., and McDonald, M. , "Quantitative Analysis of Sulphides and Sulfosalts

Using an Energy Dispersive Spectrometer," Proceedings of the Ninth Annual Conference

of the Microbeam Analysis Society, Ottawa, Canada, Paper 23, 1974.

[6] Beaman, D. R. , and Solosky, L. F. , "Advances in Quantitative Energy Dispersive

Spectrometry," Proceedings of the Ninth Annual Conference of the Microbeam Analysis

Society, Ottawa, Canada, Paper 26, 1974.

[7] Servant, J. M. , Meny, L. , and Champigny, M. , "Energy Dispersive Quantitative X-ray

Microanalysis on a Scanning Electron Microscope," X-ray Spectrom. , 4, 3, 99 (1975).

[8] Shuman, H. , and Somlyo, A. P., "Quantitative Electron Probe Analysis of Ultra-Thin

Biological Specimens," Proceedings of the Tenth Annual Conference of the Microbeam

Analysis Society, Las Vegas, Paper 41, 1975.

[9] Fiori, C. E. , Myklebust, R. L. , Heinrich, K. F. J., and Yakowitz, M. , "Prediction

of Continuum Intensity in Energy-Dispersive X-ray Microanalysis, Anal. Chem., 48, 1,

172 (1976).

296
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg , Maryland, April 23-25, 1979.

(Issued March 1981)

MULTIPLE LEAST SQUARES FITTING FOR SPECTRUM DEC0NV0LUTI0N

John C. Russ

EDAX Laboratories

P.O. Box 135

Prairie View, IL 60069

The multiple least squares fitting approach to energy-dispersive x-ray spectrum

deconvolution has been described by several authors [1-3] 1 and is used in some commercial

ED software [3]. It is also commonly used in nuclear spectroscopy [4]. The fundamental

assumption in this method is that the measured spectrum is a summation of spectra measured

on each of the pure element standards. For the peaks this is often a good first approximation as we shall see.

Since for x-ray spectra, and particularly for electron excited x-ray spectra, the

background is quite significant and not a linear summation of the individual pure elements'

spectral backgrounds, it is questionable whether the fitting should include the background. One approach [3] has been to fit not the measured spectrum, peaks and background,

but a filtered spectrum. By passing a simple digital filter over the spectrum (similar to but much simpler than the background-removing frequency filter [5], used by EDAX), the

spectrum is replaced by an approximation to its second derivative. This supresses the

background which varies more slowly than the peaks. A typical filter is shown diagrammatical ly in figure 1. Each channel is replaced by the summation of itself and its

neighbors multiplied by the function. Since all the coefficients are integers (in this

example 1 or 2) the filtering procedure, if written in machine language using simple

addition and subtraction, can be extremely rapid. Since the filtering process is linear,

it seems that linear least squares fitting of the filtered library of spectra to the

filtered unknown spectrum should also be possible, and this is done.

+ 1

0

Figure 1. Typical "Digital Filter" used to weight channels

to produce second derivative spectrum.

Another approach [6] has been to use weighted least squares fitting in which the

individual data points in the measured spectrum and each library spectrum are weighted in

importance according to their statistical precision. This is approximated as simply being

the square root of the number of counts in that channel, so the peak areas are more strongly

weighted than the background and major peaks more strongly than the minor ones. These

methods discriminate against the background but do not remove it. They are therefore

different from that used by the programs which calculate and/or fit background discretely

[7] and we believe inferior both in their practical consequence and philosophic approach.

Removal of the background by fitting or interpolating a line (straight or a simple function)

can also be used on the measured spectrum and the library spectra before fitting, and gives

good results.

Figures in brackets indicate the literature references at the end of this paper.

297
It is the purpose of this article to illustrate the basic method and evaluate potential

problem areas for it. Fitting of both generated and measured library spectra are con- sidered. A series of applications are shown using spectra having different characteristics

to test the performance of the various approaches. The particular features chosen are very severe overlaps of K and L lines where the peak separation is small, and families of lines

are involved.

First let us consider a simple example showing the combination of pure copper and pure

zinc spectra to match the spectrum measured on a copper zinc brass alloy. Figure 2 shows

the measured spectra each analyzed for 40 seconds, at 25 kV with the specimen inclined at

20° with a 36° takeoff angle. Tables 1-3 show the results of the different fitting

approaches with intensities and K ratios, that is, the ratios of the intensity of each

element measured on the brass alloy divided by the intensity measured using the same methods

on the pure element spectrum. Table 4 lists the resulting calculated concentrations. All

of the fitting methods give reduced chi-squared values very close to 1.0 (between .9 and

1.6) indicating excellent fits. In this example all of the results are roughly similar and

it is not obvious which method is to be preferred in general. Figure 3 shows typical

fitting results.

9 4 6 S E C

S : 1 0 K H S : 9 E V / C H

' mi i ll irrmwYf , >jBti,^.'^~iiiiJkk

|0 7 |0 8 [0 9 |10

CU 4 ZN_ EDflX

Figure 2a. Spectrum from 70 percent Cu - 30 Figure 2b. percent Zn Brass. Library spectra from pure Cu (dots) and Zn (bars).

Table 1. Intensities, ratios, and concentrations for 70 percent

Cu - 30 percent Zn spectrum.

Generated Peaks Library Spectra

Linear Weighted Linear Weighted Digital Filter

Cu 37710 39503

5264 5750 0.67311 0.68978 0.68754

Zn 13719 14538

1658 1745 0.30432 0.32954 0.28563

298
Table 2. Intensities on pure Cu and Zn.

Linear Weighted

Cu 53706

7497

53728

7537

Zn 42821

5886

42783

5843

Table 3. K- ratios from intensities in

Tables 1, 2.

Linear Weighted

Cu 0.7022 0.7390

Zn 0.3157 0.3349

Table 4. Normalized ZAF results from Tables 1, 3.

%Cu

%Zn

Generated Peaks

Linear Weighted

69.12 68.92

30.88 31.08

Library Spectra

Linear Weighted Digital Filter

69.03 67.81 70.80

30.97 32.19 29.20

0 0 S E C

S : 2 5 0 0 HS :

0 C / 5

2 0 E V / C H

0 7 |y 8 |0 9 |1 0

J / 2 N FITS EDHX _J

Figure 3a. Fit of individually generated

peaks (bars) to Brass spectrum

(dots).

Figure 3b. Fit of summation of library Cu and Zn spectra (bars) to measured Brass spectrum (dots).

Applying the same method to nominal 90 percent Cu - 10 percent Zn and 95 percent Cu - 5 percent Zn alloys (figure 4) produces the results summarized in Table 5. As there was some indication above that the weighting method introduces bias when mixtures of very large and

small peaks are present (this will become more apparent in a subsequent example), it was

not used for these calculations.

299
Table 5. Fitting results on 90-10 and 95-5 Brass alloys.

90-10

K ratio

CuZn

Cu

Zn

Generated Peaks

0.90874

.11086

89.19

10.81

Library Spectra

Linear Digital Filter

0.86402

.10054

89.65

10.35

0.87298

. 08893

90.82

9.18

95-5 Cu .96969 .91481 .91744

K ratio Zn .05942 .05112 .04772

Cu 94.26 94.10 95.09

% Zn 5.74 5.90 4.91

The next example is a sample containing lead and arsenic, as a mixture of arsenic

tri oxide and lead acetate (figure 5). This presents a major problem in the interference of

the AsKa and PbLa. Fitting of generated peaks was tried using one Ka and one La peak, at 10.53 and 10.55 keV respectively and also using separate Ka lf Ka2 , La x , and La2 peaks at

10.541, 10.505, 10.551, and 10.448 keV. The results listed in Table 6 are accompanied by

chi-squared values of 1.036 and 1.004 for the two fits, and excellent appearance of the

fitted result (figure 6). However, they are clearly wrong. In the one case the ratios of

Ka x to Ka2 and La x to La2 are wrong, and in the other, the ratio of As to Pb is about 3.5

while from all of the other peak ratios (and results of library fits to be shown below) the

ratio should be about 2.60. Furthermore, the values are extremely sensitive to changes in

peak position of as little as one eV, and even careful calibration cannot guarantee

adequate reliability.

300
79 10550EV L Z82 PB

VS: 25K HS: 28EV/CH

I

79 10530EV K Z33 AS

VS : 25 K HS: 20EV/CH

*

m ««JL JBL , - . , PB/flS OVERLAP- EDflX

Figure 5a. Spectrum measured on lead-arsenic Figure 5b. Spectrum measured on lead-arsenic

mixture, showing major Pb peaks. mixture, showing major As peaks.

Table 6. Intensities from generated peak fits to Pb-As overlap.

Case 1

Peak

AsKa

PbLa

keV

10.53

10.55

Total Counts

179720

52418

Case 2 AsKa x AsKa 2 PbLa!

PbLa 2

10.541

10.505

10.551

10.448

24567

77377

128945

1234

Figure 6a. Fit of two generated peaks (bars)

to AsKa - PbLa overlap (dots).

Figure 6b. Detail near baseline of

figure 6a.

301
Figure 6c. Fit of four generated peaks (bars) Figure 6d. Detail near baseline of

to AsKa - PbLa overlap (dots). figure 6c.

To judge the effect of mi seal ibration, the As Ka and Pb La peaks were fit at energies

varying over a narrow range, in one eV steps, keeping the separation of the peaks and their

widths constant. The results (Table 7, figure 7) shows huge variations in intensities, with

only slight variations in the reduced chi-squared value. Note that the minimum chi-squared

value occurs for nearly equal size As and Pb peaks, which again does not agree with the more probable ratio of about 2.6. Statham has shown another example of minimum chi-squared

occuring at the "wrong" answer [8].

Table 7. Variation of Pb La/As Ka with position.

(Energy in eV) As-Pb Mixture

Chi-squared *As

T Pb

-10 1.0167 -11317 242794

9 1.0159 -115 231590

8 1.0153 11081 220394

7 1.0147 22297 209177

6 1.0142 33492 197981

5 1.0137 44700 186772

4 1.0133 55906 175565

3 1.0130 67124 164345

2 1.0127 78341 153128

- 1 1.0125 89546 141922

0 1.0124 100761 130705

+ 1 1.0123 111979 119487

2 1.0123 123198 108266

3 1.0124 134415 97048

4 1.0125 145636 85827

5 1.0127 156853 74608

6 1.0130 168077 63383

7 1.0134 179297 52163

8 1.0138 190525 40934

9 1.0142 201745 29713

10 1.0148 212975 18482

11 1.0154 224206 7249

12 1.0160 235423 -3969

13 1.0168 246660 -15206

+14 1.0176 257878 -26425

302
t 1 r i -10 -5 0 +5 +10 +15

Calibration Shift (eV)

Figure 7. Effect of calibration shift on chi-squared and intensity

values for overlapped peak.

If we use the same series of energies for generated peaks of both elements to fit the

pure element spectra measured on a pressed pellet of arsenic tri oxide and a metallic lead

disc, respectively, we find results as shown in Table 8 and figure 8. Again note that the

absent element can have either positive or negative values, and that the reduced chi-squared

value varies only slightly, and is not at its minimum when the absent elements's intensity

passes through zero.

Table 8. Variation in intensities measured on pure elements.

Energy in eV Pb As_

Chi-squared Pb int. "As Int." Chi-squared As int. "Pb Int.

-5 1 0337 136516 -10052 1 0011 425197 202411

-4 1 0329 130549 -4086 1 0013 456201 171388

-3 1 0322 124566 1897 1 0017 487244 140359

-2 1 0315 118588 7874 1 0020 518196 109393

-1 1 0309 112611 13851 1 0025 549272 78324

0 1 0304 106633 19829 1 0031 580292 47302

+1 1 0299 100647 25805 1 0037 611359 16232

+2 1 0295 94655 31797 1 0044 642340 -14752

+3 1 .0292 88669 37783 1 0051 673368 -45779

+4 1 .0289 82679 423772 1 0059 704438 -76849

+5 1 .0288 76683 49769 1 0068 735405 -107816

303
800

- 1.010

Figure 8. Effect of calibration

shift on chi-squared

and intensity values

for individual element peaks.

1.025

10 12

Calibration Shift (eV)

If we choose the value of the pure intensities at the energy position where the absent

element's peak is zero, then we obtain ratios of intensities on the mixture (determined at minimum chi-squared) to pure intensities equal to 0.1874 for As and 0.8609 for Pb. These

are very different from the more likely values determined, for example, from a consideration

of the arsenic Kp and lead Lp lines (values typically about .25 for As and .65 for Pb). The

mixture of lead acetate and As 2 03 contained by weight 76.44 percent lead acetate and 23.56

percent As 2 03 , from which we might expect ratios of 0.236 for the arsenic spectrum and about

0.55 for the lead spectrum (taking into account the fraction of lead in lead acetate).

We would have to conclude from this that generated peaks cannot separate this overlap

by themselves, and either knowledge of peak ratios to the other a and p peaks, or the use

of those other peaks by themselves, would be required.

304
Using library spectra measured on the pure arsenic and lead compounds gives generally

more reliable results, but still with some variations. Since they were measured immediately

before and after the mixture, the calibration of peak positions has not changed, (although

it may be 1 or 2 eV off, even with careful adjustment used in this example). A major

variable left to the user's discretion is still the energy range to be used for making the

fit. Table 9 shows the change in results using linear fitting as the range is increased

from the major KaLa peak to the entire family of lines. Note that the As:Pb ratio increases

as more and more Pb lines are included in the fit. Figure 9 shows the library spectra.

These data indicate the ability of fitting of library spectra to give fairly consistent and

useful results if all of the available lines are used, but in this case equally useful and

less potentially questionable values could be obtained using only the As K0 and Pb Lp lines,

which are not overlapped.

Table 9. Variation of Pb-As fitting factors with energy range.

Limits As Pb

(keV)

Low High

10 11 0.2889 0.4238

10 12 .2535 .6011

10 13.5 .2432 .6533

11 13.5 .2395 .6673

8.1 14.0 .2444 .6584

8.1 15.9 .2451 .6623

Figure 9a. Library spectra measured for lead Figure 9b. Detail of figure 9a showing

(bars) and arsenic (dots). background and minor peaks.

Using the weighting method over the full 8.1-15.9 keV range, introduces a significant

error because of the assemblage of very large and small peaks. Table 10 shows the results,

and figure 10 illustrates the problem; the fit is strongly influenced by the larger a peaks

which have the major error because of the close overlap. This results in a too large As K0

and too small Pb Lp. The chi-squared results for /the weighted fit are correspondingly poor,

while for the linear and filtered fits, the values are 1.09 and 1.63 respectively. Figure

11 shows the filter fit results; note that the background is not well fit, while the major

peak areas are quite close, showing the general discrimination against the background

regions. The opportunity for biased results from the weighted fits argue against that

method, and in fact it has been dropped from subsequent examples.

305
Table 10. Variation of Pb - As fitting factors with method.

As

Pb

Li near

0.2451

.6623

Weighted

0.2746

.6791

Filter

0.2369

.6683

9 0 S E C

V S : 2 5 K H S :

0 C / S

2 6 E V / C H

'

Figure 10. Comparison of composite spectra

(bars) obtained by weighted

fit. AsKp (11.7 keV) is larger

and PbLa (12.6 keV) smaller

than measured spectrum (dots).

Figure 11a. Comparison of composite from

filtered fit (bars) with mea- sured spectrum (dots) showing

agreement on major peaks but

not background and minor.

Figure lib. Detail of figure 11a, showing

background and minor peaks.

A further useful test is to evaluate the purely statistical effects of fitting, figure

12. For the Pb-As mixture, and for each of the pure samples, duplicate spectra were

recorded. Each had a total of at least one-half million counts in the 8.2-15.6 keV fitting

region. Using all possible combinations of spectra gave the results shown in Table 11. The

variance is approximately one percent, while from a simple consideration of the number of

counts we would expect about half that value. Note also that there is a consistent difference between the linear and digital filter results.

306
6 S E C @ C / S 10 K H S : 2 0 E V / C H

i

|1 6 |1 1

P B RE S I D U fl

L

E D fl K

Figure 12a. Residual spectrum (bars) for As left after subtracting fitted

library standard for Pb, showing effect of changed peak

ratios.

Figure 12b. Residual spectrum (bars) Tor

Pb left after subtracting

fitted library standard for

As, showing effect of changed

peak ratios.

Table 11. Results of fitting of duplicate spectra.

Spectrum No. Fitting Factors

Linear Filter

As Pb As-Pb As Pb As Pb

1 1 1 0.2451 0.6623 0.2396 0.6758

1 1 2 .2470 .6622 .2423 .6733

1 2 1 .2454 .6669 .2389 .6830

1 2 2 .2473 .6668 .2416 .6807

2 1 1 .2422 .6513 .2380 .6616

2 1 2 .2440 .6512 .2401 .6607

2 2 1 .2424 .6559 .2373 .6686

2 2 2 .2443 .6558 .2394 .6677

Mean .2447 .6591 .2396 .6714

var. .0019 .0064 .0017 .0082

(0.8%) (1.0%) (0.7%) (1.2%)

As a third example, figure 13 shows library spectra of Ca K and Sn L peaks. The main

overlap here is between the Ca Ka and Sn Lp x measured on a sample composed of CaC03 and

Sn02 in H 3 B03. Since the Sn La, is relatively free from interference, we may expect it to determine the Sn contribution fairly well, but the Ca intensity will depend strongly on the

ratio of the Sn L0, to La peak heights. Table 12 shows the fitting ratios for varying

ranges, illustrated in figure 14; these are a) the Sn La, b) the Sn L0 - Ca Ka, c) the

Ca Ka and K0 with Sn Lp lines, d) the entire range. All of the fits give reduced chisquared values between .97 and 1.03, indicating excellent fits.

307
vs

4 6 S E C 5 0 K H S

W C / S

2 6 E V / C H

8 2

C fl

H!:'ii

S N

|0 4 FITS. E D fl %

Figure 13. Library spectra for Ca (dots) and Sn (bars).

Table 12. Ca-Sn fits to overlapped spectrum.

Range (keV) Linear Filtered

From To Ca Sn Ca Sn

a) 3.2 3.56 0.15827 0. 60213 0.15853 0.60158

b) 3.56 3.86 .16831 57355 .16247 .60073

c) 3.56 4.28 .16276 61625 .15607 .61721

d) 3.2 4.28 .16265 56756 .15558 . 59801

0SEC154180INT 0 . 0 S E C 0C/S

2 5 K H S : 2 0 E VV C H V S : 5 0 K H S : 2 0 E V / C H

|0 3 |0 4

fl / S N FITS. EDfi*

Figure 14a. Fit from 3.2 to 3.56. Figure 14b. Fit from 3.56 to 3.86.

308
Figure 14c. Fit from 3.56 to 4.28. Figure 14d. Fit from 3.2 to 4.28.

Figure 14. Ca:Sn library fits (bars) compared to measured spectrum (dots) over different

energy ranges. Note variation of ratios of minor peaks.

Note however, that the factors obtained vary substantially. Figure 13 illustrates the

fits, and shows the reason for the variation. The composite fitted spectrum does not match

the peak ratios of the measured one (note for instance the lower Sn La and higher Ca Kp -

Sn Lp2 at 4 keV); this is due to changes in the peak ratios of these elements in the

mixture, as compared to the pure elements. The same problem was also evident in the As-Pb

overlaps, since the As K absorption edge lies between the Pb La and Lp, and reduces the L0

in the composite.

This problem is increased if the excitation conditions are not the same for the mixture

and the pure library spectra. We analyzed the Ca-Sn mixture again using monoenergetic

nickel Ka excitation. The fitting coefficients (Table 13) show a different ratio of Ca to

Sn for fits performed over the entire overlap range 3.2-4.6 keV. Figure 15 shows that the

generated composite has greater discrepancies with the measured spectrum, because the

different L edges of Sn are differently excited in this case.

Table 13. Ca-Sn fits to monoenergetically excited spectrum.

Linear Filtered

Ca 0.17416 0.17512

Sn .57871 .58742

309
Figure 15. Fit of library spectra (bars) to spectrum measured on Ca:Sn mixture (dots)

using quasi-monoenergetic excitation which changes peak ratios.

The examples shown so far have all had quite large peaks and low backgrounds. The

fourth example concerns the overlap of Cr K0 - Mn Ka, complicated by low concentrations of

both unknowns and standards. This gives small peaks, and the samples have organic matrices

which scatter significant amounts of background. Figure 16 shows the standard spectra,

each measured for 400 seconds with quasi-monoenergetic excitation obtained by placing a gold

filter in front of a gold- target x-ray tube operated at 25 kV, 500 uA. The Mn standard

contains 100 ug Mn in a 2 gram sample, or 50 ppm by weight. The Cr standard contains 20

pg Cr, or 10 ppm, with a significant Fe contaminant peak.

The unknown spectrum (figure 17) was obtained from a sample containing 20 pg Mn,

100 pg, Cr. This spectrum is clearly not a simple linear sum of the two standards; figure

18 shows the results of fitting from 5 to 6.2 keV, using the standard spectra. The

coefficients are 2.1066 and 0.0466 for Cr and Mn respectively (the ratios of concentrations

are 5.0 and 0.2, respectively). Figure 19 shows the fit obtained using the digitally

filtered spectra; the coefficients are 0.9416 and 0.0909, for Cr and Mn. The filtering out

of the background changes the ratios of Cr to Mn dramatically, but the values are not "better". It also changes the reduced chi-squared from 0.88 to 0.47. (The low number of

counts in these spectra make the chi-squared values appear better than for spectra with

Figure 16. Standard (library) spectra for trace Mn (dots) and Cr (bars).

310
higher intensities.) However, it is obvious that the result does not match the measured

spectrum. It appears from these results that background suppression using filtering is

only useful when the background is in fact small relative to the peaks to begin with.

@ 408SEC 0C/S

VS:2500 HS: 28EV/CH

6 |9 7 |0 8 EDA X

Figure 17. Spectrum measured on sample

containing Mn and Cr.

Figure 18. Linear fit (bars) of library Cr

and Mn spectra to unknown com- posite (dots).

0 0SEC 0C/S

vs : 2 5 0 0 H S : 2 \i E V / C H

|@ 7 |e s #1 FILTER FIT. E D A X

Figure 19. Fit of library spectra (bars) to composite (dots), using digital

filter.

311
In cases like this, the use of generated modi f ied-Gaussian peaks with a separate

removal of background gives substantial improvement. Generating five peaks (Cr Ka and Kp, Mn Ka and Kp, and Fe Ka) above a linear background interpolated from 4.9 to 6.66 keV,

returned intensities as listed below and gave a reduced chi-squared of 0.98 (Table 14).

Figure 20 shows the overall fit, and the lines for each element singly.

Table 14. Intensities of simultaneously fit generated peaks.

El erne nt/ Li ne Intensity (Counts per 400 seconds)

Unknown Cr std. Mn std.

Cr Ka 11421 2311 -4 Cr Kp 1782 349 63 Mn Ka 3715 202 15148 Mn Kp 312 62 2221

Fe Ka 791 1018 665

0 0SEC 0C/S

VS : 2500 HS: 20EV/CH

0 0SEC 0C/S

VS : 2500 HS : 2 0 E vV C H

|0^|06

#1 GEN FIT.

7 |0 8 EDflX

Figure 20a. Agreement of individually

generated Cr, Mn, and Fe peaks

to measured composite spectrum.

0 0SEC 0C/S

VS:2 5 00 HS: 20EV/CH

#1 GEN CR_ ED fiX

Figure 20b. Detail of figure 20a showing

only generated Cr peak.

0 0SEC 0C/S

VS:2500 HS: 20EV/CH

#1 GEN MN

|0 7 |0 8 EDflX

Figure 20c. Detail of figure 20a showing

only generated Mn peak.

|05 fee |0 ? |0

»1 GEN FE. EDflX

Figure 20d.

312

Detail of figure 20a

only generated Fe peak.

showing
The 0/a ratios are reasonable, confirming the general success of the fit; the low value

obtained for the absent element in each standard is a measure of the error of fitting. The

ratios of intensities in unknown to standard are within the error expected from statistics

and the chemical preparation procedure. For such low concentration specimens, the intensity

ratio (using the sum of Kot and K0) should be close to the concentration ratio so we predict

97.5 ug of Cr and 23.1 ug of Mn (the amounts expected from preparation are 1-- and 20 ug, or

50 and 10 ppm, respectively).

Similar computation on a total of sixteen "unknown" samples confirmed the applicability

of fitting individual peaks with a separate background removal [9]. The same conclusion

would be expected for much larger concentrations of elements excited by an electron beam,

since the background is larger; this is particularly true at low energies (1-3 keV) where

the background shape is strongly influenced by concentration.

In summary, these examples show that multiple least squares fitting of library spectra

can be useful in cases where overlaps are too complete, or involve too many lines, for

practical peak generation. However, calibration is absolutely critical, as errors of even

1 or 2 eV can cause large changes in results. Fitting over all the lines involved is

perhaps the most consistent and defensible method, but introduces errors if peak ratios

change. This will happen even if the identical excitation conditions are used, and changes

in excitation conditions exacerbate the errors. Fitting of generated peaks would be

equivalent if all the major and minor peaks were used and their ratios known. If the overlap involves only a few lines and others are available without overlaps, it may be

preferable to ignore the overlapped region altogether. The digital filtering of the

spectrum changes the fitting ratios slightly, but does not appear to significantly increase

the overall error. It may improve results somewhat in cases of varying and/or large backgrounds. Although the filter and fit method may be used for this type of deconvolution, in

most cases of practical analysis methods involving peak generation after discrete background

removal are preferred because of their greater versatility and lesser dependence on availability of appropriate library spectra with identical peak positions and ratios.

References

[1] Applications of Computers to Nuclear and Radiochemistry, NAS-NS 3107 (U.S. Dept.

Commerce, Office of Technological Services, 1962).

[2] Gehrke, R. J. and Davies, R. C. , Anal. Chem. , 47, 1537-1541 (1975).

[3] Schamber, F. H. , in "X-Ray Fluorescence Analysis of Environmental Samples", T. Dzubay

(ed.) Ann Arbor Press, 1977, p. 241-257.

[4] Gunnink, R. and Niday, J. B. , Computerized Quantitative Analysis by Gamma-Ray

Spectrometry. Volume 1, Lawrence Livermore Laboratory, March 1, 1972.

[5] Russ, J. C. , "Processing of X-Ray Spectra", EDAX EDIT or Volume 6, Number 3, page

4-39.

[6] Arinc, F. , Gardner, R. P., Wielopolsi, L. , and Stiles, A. R. , Advances in X-Ray

Analysis, 19, 367 (1976).

[7] Russ, J. C. , A Fast Self-Contained No-Standards Quantitative Program for EDS, Proc. 9th Annual MAS Meeting, 1978, p. 46.

[8] Statham, P. J., X-Ray Spectrometry, 7, 132-137 (1978).

[9] Specimens courtesy of Dr. W. Wegscheider, U. Graz, Austria.

313

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ARTIFACTS OBSERVED IN ENERGY DISPERSIVE X-RAY SPECTROMETRY IN ELECTRON

BEAM INSTRUMENTS - A CAUTIONARY GUIDE

C. E. Fiori*, D. E. Newbury, and R. L. Myklebust

*Di vision of Research Services

National Institutes of Health

Bethesda, Maryland 20205

Center for Analytical Chemistry

National Bureau of Standards

Gaithersburg, Maryland 20234

Abstract

Microanalysis with the energy dispersive x-ray spectrometer on electron beam instruments is complicated by the existence of spectral

artifacts which are introduced during the x-ray detection, amplification

and display processes. The artifacts of the x-ray detection mechanism

include absorption in the window and silicon dead layer, peak broadening,

formation of parasitic peaks by escape of silicon x-rays, sum peaks due

to pulse coincidence, and effects due to system deadtime. Pulse

coincidence effects are exacerbated when high energy (> 30 keV) x-rays

strike the detector. In the transmission electron microscope high energy

x-rays cause pulse saturation to occur. Artifacts arising from

interactions of the spectrometer with its environment include

microphonics, electromagnetic interference, ground loops, and stray

radiation in the sample chamber. Spectral artifacts influence the

strategy for qualitative analysis since artifact peaks may be

misassigned to a chemical element not actually present in the specimen.

Key Words: Energy dispersive x-ray spectrometry, microanalysis, electron probe microanalysis, qualitative analysis, quantitative

analysis, scanning electron microscopy, spectral artifacts,

transmission electron microscopy.

1. Introduction

1.1 The x-ray spectrum

In carrying out microanalysis with the energy dispersive x-ray spectrometer attached

to the scanning electron microscope, electron probe microanalyzer, or transmission electron

microscope, the analyst is frequently confronted with artifacts introduced into the spectrum

by the measurement process [I] 1

. These artifacts can adversely affect both qualitative

analysis, leading, for example, to mis identification of peaks, and quantitative analysis,

causing improper measurement of x-ray intensities. To aid the analyst both in recognizing

these artifacts in spectra and properly accounting for them, we shall provide a catalogue

of typical artifacts, listing their characteristics and providing examples of them in real

spectra. It is beyond the scope of this paper to give details of the physical origin of

these artifacts; references to more complete descriptions of each artifact will be provided. Following the catalogue of artifacts, a systematic approach to qualitative analysis

will be described, with attention given to the recognition of artifacts. The treatment of

artifacts in measuring x-ray intensities for quantitative analysis will be considered.

1 Figures in brackets indicate the literature references at the end of this paper.

315
As a starting point, it is useful to consider the nature of the x-ray spectrum

generated in the sample by the incident beam. For a target such as copper bombarded with

electrons at an energy of 20 keV, the form of the x-ray spectrum generated in the specimen

is illustrated in figure 1. The spectrum consists of two constituents: (1) characteristic

peaks which result from electron transitions following ionization of an inner shell of an atom by a beam electron; (2) Bremsstrahlung ("braking radiation") or continuous x-radiation

which is formed when a beam electron undergoes deceleration in the coulombic field of an atom [2]. The continuous x-radiation forms a background extending from zero energy up to the energy of the incident beam. In the context of this paper, it must be pointed out that

the x-ray continuum is not an artifact since it is a product of the beam interaction with

the solid, and that it carries information about the average atomic number of the sample.

Artifacts are the deviations from the natural spectrum introduced into the observed spectrum

by the measurement process. The generated continuum of energy E can be reasonably well

described by the equation of Kramers [3]:

I « Z(E

0

- E)/E (1)

where I is the continuum intensity in an energy interval AE, Z is the average atomic number

of the target, E

Q

is the incident beam energy, and E is any energy less than EQ. At very

low energies, the continuum intensity predicted by equation (1) increases toward infinity.

Figure 1. Characteristic and continuum

x-ray spectrum generated in a bulk copper target by a 20 keV

electron beam as calculated by

Monte Carlo electron trajectory

simulation. Note that only the

alpha member of the characteristic x-ray family is plotted.

°0 1 2 3 I 5 6 7 5 5 ?0 E (keV)

The x-rays are generated over a range of depth in the sample, and during propagation

out of the sample, absorption occurs. Absorption significantly alters the shape of the

emitted spectrum, which is shown in figure 2(a), as calculated by a Monte Carlo electron

trajectory simulation [4], Low energy x-rays (<300 eV) are very strongly absorbed, and

only those low energy x-rays generated near the surface escape. High energy x-rays are

less likely to be absorbed. The mass absorption coefficient for x-rays decreases with

increasing x-ray energy, except near the energy of an absorption edge. An absorption edge

corresponds to the ionization energy of an inner shell; x-rays at an energy just above the

absorption energy are very strongly absorbed relative to energies just below the edge.

316
f

N >10 6 photons/e" ster

6-

Figure 2(a). Characteristic and continuum

x-ray spectrum emitted from a bulk copper target excited

with a 20 keV electron beam

as calculated by a Monte

Carlo electron trajectory

simulation.

The x-ray spectrum which is finally observed with the energy dispersive x-ray spectrometer is further modified by the measurement process. The artifacts introduced by the

measurement process can be broadly classified into two main categories: (1) those which

arise during the detection process and (2) those associated with the signal manipulation

following the detector. In addition, artifacts may arise because of certain unfortunate

interactions between the energy dispersive x-ray spectrometer and the instrument environment

in which it operates.

2. Artifacts of the Detection Process

2.1 Absorption effects

The structure of a typical energy dispersive x-ray spectrometer containing a lithiumdrifted silicon detector is illustrated in figure 3. In order to enter the active portion

of the detector, x-rays must pass through a beryllium window with a nominal thickness of

7.6 urn (0.3 mil), which maintains the vacuum integrity of the spectrometer, a gold surface

electrode about 20 nm thick, and an inactive or dead layer of silicon, 20 - 200 nm thick.

The absorption which occurs during passage through those layers eliminates virtually all

x-rays with an energy below 500 eV. Above an energy of about 2 keV the absorption becomes

relatively insignificant so that nearly all x-rays pass into the active detector region.

For intermediate energies the transmission is about 45 percent at 1 keV and 70 percent at

1.5 keV. The modifications to the spectrum which result from absorption in the detector

components can be seen in figure 2(b) for the calculated copper spectrum and in an experimentally measured spectrum, figure 2(c). These effects are further illustrated in the

experimentally measured spectrum from high purity carbon, figure 4. The decrease in the

measured continuum below 1 keV is evident, and discontinuities are observed in the continuum

at the absorption edges of silicon and gold, where the abrupt change in mass absorption

coefficient is observed as a discontinuity in the continuum. In the windowless variety of

the energy dispersive x-ray spectrometer, the measurable x-ray range extends as low as carbon Ka (0.282 keV), although absorption still occurs in the silicon dead layer and gold

electrode [6].

317
01 23456789 10 E IkeVI

Figure 2b. Spectrum of 2a corrected for

absorption in the beryllium

window and silicon dead layer.

Figure 2c. Experimental spectrum of copper; 20 keV beam. Chlorine,

titanium, and iron are trace

contaminants in the specimen or mounting medium. Note that the

CuKa escape peak at 6.30 keV

is obscured by the FeKa peak at 6.40 keV. In the peak intensities "K" equals 1000.

318
200 A* Au

X-Rays

electrons

holes

1000 V oi ( lithium drifted

region

)

O'lftm p( inactive)

8 fitn Be

Figure 3. Schematic diagram of a typical lithium-drifted

detector of an energy dispersive x-ray spectrometer.

to

Amplifier

Carbon E^IOkeV Figure 4. EDS spectrum from carbon excited

by a 10 keV electron beam. The

silicon and gold absorption

edges can be observed as well as

a silicon peak resulting from

internal fluorescence of the

silicon dead layer. The solid

line is a theoretical fit to the

continuum; the shaded region is

the distortion caused by incomplete charge collection of the

silicon internal fluorescence

peak.

E (keV)

The absorption of x-rays in the inactive silicon layer leads to fluorescent emission of

silicon K x-rays which are detected in the active region of the spectrometer. This causes

the production of a small artifact silicon peak in the spectrum which is not representative

of a silicon constituent in the specimen. In the analysis of a bulk specimen, this artifact

silicon peak is typically equivalent to an apparent concentration of about 0.2 percent or

less by weight of silicon in the sample.

2.2 Peak broadening

In the Si C Li ) detector the process of detection consists of converting the energy of

the incident photon into charge carriers (electron-hole pairs), proportional in number to the photon energy, within the detector material [7]. The charge carriers are collected by

applying a suitable bias to the faces of the detector. All subsequent operations involve

the measurement of this charge.

The natural width of an x-ray peak is in the order of 1 eV in the energy range of

interest, measured at half the maximum peak intensity (the measurement is designated full

width, half maximum, "FWHM"). The detection process in the energy dispersive x-ray detector

results in statistical uncertainty being introduced which distributes the x-ray information

over a range of energies such that the measured peak has a FWHM about two orders of

319
magnitude greater than the natural width [8]. For example, a typical detector will produce

a peak for manganese Ka with a FWHM of 150 eV. The natural width of Mn Ka is 2.3 eV. Consequently, a natural peak of 2.3 eV width having an amplitude of 1000 counts will be

degraded to a peak of 150 eV width having 15 counts in the peak channel (assumed to be 10 eV

in width). The natural x-ray line would be totally contained in one, or at most two, such

channels. The peak width varies as a function of peak energy, and a convenient measure of

the FWHM for any peak is given by [1]:

where FWHMX and E x are the peak width and energy (eV) of the peak of interest, and FWHM2 and

E2 are the known values for a reference peak, usually manganese Ka, recorded with the same detector under similar conditions.

The peak produced by the detector is reasonably well described by a Gaussian function,

figure 5. Deviations from the ideal Gaussian shape occur on the low energy side of the peak

due to the phenomenon of incomplete charge collection [9,10]. Incomplete charge collection

produces a noticeable shoulder on the low energy side of the peak, figure 6. The relative

magnitude of this shoulder is dependent on the energy of the peak. A second consequence of

the incomplete charge collection phenomenon is the formation of a "background shelf" of

counts lost from the peak extending from the peak energy down to zero energy. The background shelf is normally obscured by the continuous x-ray background present in electronexcited spectra. The effect is illustrated in the spectrum of figure 7, where the manganese

radiation is obtained with virtually no x-ray continuum from a radioactive 55 Fe source. The

background shelf at half the peak energy has an intensity of about 0.1 percent of the peak

intensity. The total intensity distributed in the background shelf is about 1 percent of

the integrated peak intensity.

(2)

2.3 Peak distortions

Figure 5. Theoretical Gaussian distribution which describes an EDS

peak, a is the standard deviation of the distribution, A. is the peak amplitude, and y is the

amplitude at any energy, E.

320
30K20KKCl E =15keV Figure 6. EDS spectrum of KC1 illustrating (1) peak overlap and

(2) the incomplete charge

collection distortion. The CI Ka and Kp peaks are not

resolved while the K Ka and

Kp are nearly resolved. The

solid line is a Gaussian fit to the data points. The shaded

areas are the distortions

introduced by incomplete charge

col lection.

E IkeV)

2K- MnK«

310 K

Mn Kotlesc)

AIK

CIK« CqK«

MnK/9

Figure 7. EDS spectrum derived from 55 Fe radioactive source. The background shelf is easily observed

at energies below Mn Ka to the

threshold energy, 300 eV. The

silicon escape peaks from the

Mn Ka and Kp are noted. Extra- neous characteristic peaks from

the source holder are observed

(aluminum, chlorine, and

calcium signals).

E (keV)

2.4 Silicon escape peaks

The process of capturing an x-ray in the detector involves the ionization of a silicon

atom. De-excitation of the ionized atom involves the emission of a silicon K x-ray in about

4 percent of the ionizations. Because an element has a relatively low absorption for its own characteristic x-rays, these silicon x-rays have a significant probability of escaping

the active detector area, robbing the x-ray pulse being measured by an amount corresponding

to the silicon K energy, 1.74 keV for Si Ka and 1.83 keV for Si Kp. An artifact peak is

formed, the so-called silicon escape peak, at an energy of E x - 1.74 keV, where Ej is the

energy of the parent peak. Due to the low detector resolution and the low probability for

silicon Kp formation, only one escape peak per parent is observed. The escape peaks for

manganese Ka and Kp radiation can be seen in the radioactive source spectrum in figure 7 and

the electron-excited spectrum of manganese in figure 8. The relative magnitude of the

silicon-escape peak varies with the energy of the parent peak, ranging from about 1 percent

of the integrated parent peak intensity for phosphorus Ka to 0.01 percent for zinc Ka x-rays. Below the excitation energy for silicon, 1.840 keV, the silicon K escape peak

cannot be generated.

321
Nx104

fr Manganese

20 keV beam

MnK«1012K

Mn K* esc

MnKf 136 K

Figure 8. EDS spectrum of manganese

excited by a 20 keV electron

beam. As compared to figure 7,

the continuum spectrum makes

the observation of the distortion due to incomplete charge

collection difficult except

near the peak. The background

shelf is obscured in this

spectrum but it is still

present.

2 3 4 5 6 7 E IkeVI

3. Artifacts Introduced During Signal Processing

The task of amplifying the small charge collected during the capture of a photon by the

detector to yield a signal which is useful for classification by a multichannel analyzer

(MCA) requires extensive signal processing [11]. Moreover, in order to minimize the effects

of noise to produce an amplified signal capable of yielding a high resolution spectrum, long

pulse widths must be employed, in the range 10 - 100 us. Since the amplification circuits

are occupied for such a relatively long time with the pulse developed from a single photon,

there is a high probability that a second photon may enter the detector during this occupied

period (or deadtime) and lead to a distorted signal. Artifacts of the signal processing

chain arise principally from pulse coincidence effects.

3. 1 Pulse pi leup

If x-rays of a peak energy E x are being measured, pulse coincidence can lead to a con- tinuous band of overlapping pulses in the range E 1 to 2E X

, the so-called pulse pileup

continuum, figure 9 [12]. A modern system contains an efficient pulse pileup rejector which

functions to eliminate overlapping pulses. However, there is a limitation in the capability

of the rejector to separate two pulses closely spaced in time. Therefore, even an optimally

adjusted rejector will allow a pile-up sum peak, which is composed of pulses in nearly exact

coincidence, to appear in the spectrum, figure 10. Moreover, for low energy pulses, the

pileup rejector may fail. For example, a properly adjusted pileup rejector may function

adequately for silicon, figure 10, and fail badly for magnesium, figure 9, due to the

differences in the energies of the pulses. If a spectrum contains major peaks of energy

Ej, E 2 , and E 3 , sum peaks may be found at 2E ls 2E2 , 2E3 , (E x + E2 ), (E2 + E3 ),

and (Ei + E 3 ).

322
MgKa./S

61-9 K

Magnesium

E0

= 15 keV

Figure 9. EDS spectrum of magnesium

excited by a 15 keV electron beam. Pile-up peaks

and pile-up continuum can

be observed.

2t 3 E IkeV)

4K- SiK«./5

71-8 K

Silicon E =15keV Figure 10. EDS spectrum of silicon

excited by a 15 keV electron beam. Only a small sum peak can be observed,

and the pile-up continuum

is not significant.

* E IkeV) 3

3.2 Deadtime Correction

The system deadtime which results from the long pulse widths is normally compensated

automatically by the signal processing system through the function known as "live time

correction". If the desired live time is T seconds, the clock time is allowed to run to T + Xn.x., where is the pulse width appropriate to each different energy pulse and n^

is the number of each type of pulse. The circuitry of the deadtime correction section of a signal processing system is capable of making a sufficiently accurate correction to satisfy

most practical analysis situations. The analyst must not, however, presume that he can use an arbitrarily high count rate with the assumption that the deadtime correction will automatically accommodate that rate. Indeed, at high count rates, sum peak loss becomes

significant. Every pulse in a double energy peak represents the loss of two pulses from the

parent peak. In general, pulses lost from a peak due to pile-up are not correctly accounted

by the deadtime circuit. However by using a count rate where the sum peaks are not in

evidence in the spectrum, this effect is reduced to an acceptable level.

For those applications where the output pulses of the amplifier are used directly,

such as intensity modulation of a cathode ray tube for x-ray area mapping and x-ray line

profiling, which are applications where a high count rate is especially useful, the analyst

323
should be aware of the following insidious artifact. In figure 11, the output count rate from a processing system is plotted versus input count rate for several different pulse

widths. For the longest pulse widths (which would yield the highest spectral resolution),

the output count rate actually begins to decrease above an input count rate of about 10,000

cps. Such a system should not be operated above a count rate of about 15,000 cps. Above

the maximum in response, a situation could be encountered where two widely differing input

count rates, which could correspond, for example, to the signals obtained for an element

from regions of radially different composition, yield the same output count rate, and

hence, the same apparent composition in an area or line scan.

Figure 11. Relationship of output and

input count rates for a generalized EDS system for

three different values of

pulse width.

A second important point about figure 11 is the fact that the input count rate considered is the total spectrum count rate , from the minimum cutoff energy of the discriminator,

about 500 eV, to the incident beam energy. All x-rays entering the detector, characteristic

as well as continuum x-rays, contribute to the limitation on input signal rate.

3.3 High energy x-ray effects

The extension of energy dispersive x-ray spectrometry to the transmission electron

microscope operating at 100 keV or more raises the problem of processing pulses derived from

the capture of x-rays of very high energy, as well as x-rays in the practical analytical

range of 1 - 20 keV. Under normal operation, the main amplifier is adjusted so that the

useful x-ray energy range (1 - 20 keV) spans a specified group of channels (typically

quarter, half or full memory) in the multichannel analyzer. Under bombardment with high

energy electrons, the specimen will emit a continuum spectrum with x-ray energies up to the

beam energy, and possibly high energy characteristic x-rays as well (Ka - Kp). Significant

numbers of high energy x-rays will strike the detector which will produce voltage pulses

outside the allowed range as specified above. These high energy pulses will produce

electronic saturation, figure 12, and can lead to two undesirable effects, (1) loss of

resolution and (2) incorrect operation of the deadtime circuit. The loss of resolution,

which results from electronic artifacts caused by the sharp transitions associated with

pulse cut-off, can be of the order of 5 eV at MnKa. The high energy pulses are proportionally wider than the low energy pulses of interest. For an acceptable deadtime, the

amplifier will spend a significant amount of its time processing the wide high energy pulses

leaving less time for the low energy pulses of interest.

324
4. Artifacts of the Detector-Microscope System

Several artifacts may be observed in spectra which result from certain interactions

of the EDS detector and/or amplifier with the microscope environment. These artifacts can,

in principle, be removed by employing suitable precautions. In order to aid the analyst

in recognizing these artifacts, examples will be given.

4.1 Microphonics

The extraordinary sensitivity of the detector and amplifier of the EDS system produces

a marked susceptibility to interference from mechanical and acoustic vibration and stray

electromagnetic radiation. One form of artifact introduced into the spectrum from these

sources is shown in figure 13. Figure 13a is the spectrum of a chromium- iron alloy obtained

in the absence of interference. In figure 13b, the same spectrum recorded with an acoustic

noise source (a telephone) operating contains an enormous noise background extending from

zero energy up to about 2 keV, which eliminates all useful spectral information in this

region. Further, the peaks in the high energy region of the spectrum of figure 13b are broadened, i.e., the resolution has degraded in the presence of the acoustic noise source, and consequently the peak amplitudes have been diminished.

CrK«

Cr-Fe Alloy E„=15 keV Figure 13(a). EDS spectrum of a chromiumiron alloy excited by a 15 keV electron beam. For Al and Si , the Ka - Kp pair is unresolved.

FeK*

AIK SiK '. ,i '. »*. /

' ;

: •' • FeK/i

E (keV)

325
Noise 2«104

Cr -Fe Alloy E =15keV

CrKoc Figure 13(b).

Fe K«

E IkeV)

EDS spectrum obtained

under identical conditions

to 13a, except with a source of acoustic interference operating (nearby

telephone). Note the

noise spectrum exciting to about 2 keV and the degradation in resolution as compared to 13a.

4.2 Electromagnetic interference

One does not normally think of the electron beam instrument as a source of electromagnetic interference. However, several intrinsic components of electron beam systems can be sources of electromagnetic radiation which can couple with the silicon detector system and

introduce artifacts. If the silicon detector-cryostat assembly, main amplifier, or interconnecting cable between these components is placed in proximity to a source of

electromagnetic radiation, difficulties can arise. Typical instrumental sources for this

radiation include power supply transformers, scan generators, scan coils, computers and

logic circuits, and motors. For example, a poor location for the main amplifier of the

EDS system would be in the same electronic bin as the SEM scan generator or placed near a computer. An example of a particularly difficult electromagnetic interference problem is

that arising from interaction of the beam scan coils with the silicon detector. While the

problem is not usually manifest in low beam energy SEM's the problem can be observed on high energy (200 keV) STEM-TEM instruments where the scan coil excitation is necessarily

much higher and where the instrument configuration places the silicon detector adjacent to the scan coils, figure 14. The electromagnetic interference can introduce a spurious peak,

figure 15, whose apparent energy is a function of the STEM magnification, since the scan

coil excitation varies with magnification. Reversal of the flow of current in the scan coils would eliminate the obvious manifestation of the artifact, namely the spurious peak,

but close examination of the spectrum would reveal the resolution still degraded and the

deadtime correction adversely affected.

Figure 14. Schematic explanation of anomalous peak in

figure 15 arising from electromagnetic

coupling of scan coils.

326
ML Z=82 PB

PR " S 45SEC 0 INT

U-4896 H«29KEU 1 1Q AQ*28KEU IQ

Figure 15. Anomalous peak observed in 0 - 2 keV region in an analytical electron microscope; the peak

position varied with the magnification setting

(magnification 9500X, 7200X, and 5000X, left

to right peaks, frame time 1/s).

This type of artifact is a member of a class of problems which result from the marriage

of two independently designed systems, the x-ray spectrometer and the electron microscope.

Generally, the user must be prepared to recognize and correct such problems, since it is

unlikely that the manufacturers of the individual systems have had the opportunity to test

the composite instrument.

4.3 Ground loops

We might normally assume that the metal components of the spectrometer/microscope

system are all at ground potential with no current flowing between them. In fact, small

differences in potential of the order of millivolts to volts, can exist between the components. The potential differences cause currents to flow, ranging from microamperes of dc

current to many amperes of high frequency ac current. These extraneous currents are commonly referred to as "ground currents" or "ground loops" since they are flowing in

components of the system which are supposedly at ground potential, such as the chassis or outer shields of co-axial cable. Since high frequency ground currents have associated

electromagnetic radiation, such currents flowing in co-axial cable shielding can modulate

low level signals passing through the center conductor. In EDS systems, the signals being

processed are at extremely low levels, hence ground loops must be carefully avoided. The

interference from a ground loop can manifest itself as degraded spectrometer resolution,

peak shape distortion, background shape distortion, and/or deadtime correction malfunction. An example of peak shape distortion is shown in figure 16, where a normal MnKa - K{5

pair appears to be split into subsidiary peaks in the presence of a ground loop.

327
328
Ground loop artifacts are best dealt with by prevention rather than cure. At the time

of installation of the detector, care should be taken to insure that ground loops are not

inadvertently introduced. For example, as shown in figure 17, the grounding path should

be created in a logical fashion avoiding cross-connections between microscope components

and spectrometer components. An important ground loop path to avoid is that between the

cryostat housing (preamp) and the microscope. The resistance between the cryostat assembly

(disconnected from its cabling) and the microscope column should typically exceed 5 x 106Q.

Cross connections can be inadvertently introduced when the EDS amplifier and/or multichannel

analyzer units share common racks with microscope components such as scan generators, video

amplifiers, power supplies, etc. Ideally, it is best to keep the two systems electrically

isolated through the use of separate racks. A logical ground path should be established

in the sub-systems to a high quality ground. Note that a high quality ground is not

typically available at the wall power plug. A high quality ground might consist of a 1 cm diameter, or greater, copper wire leading through the shortest distance possible to an external assembly consisting of three separate 5 m or longer copper rods driven vertically

into the ground and reaching into the water table. The EDS user should note that modifications to establish a high quality ground necessarily involve altering the electrical

distribution network of the microscope/EDS system, and as a result, such modifications must

be carried out under supervision of a qualified electrician.

5i(Ll)

o

oo oo

oo o oo

SEM

ELECTRONICS

m

oo o oo

Si(Lih

ELECTRONICS

MCA

Figure 17. Schematic illustration of ideal

ground paths in a SEM instrument

equipped with an x-ray detector.

=HIGH

- QUALITY

GROUND

4.4 Stray radiation

The EDS detector accepts any x-rays emitted in a large solid angle, illustrated

schematically in figure 18. This fact makes the spectrometer sensitive to stray x-rays

generated over a wide portion of the sample as well as parts of the specimen stage and

sample chamber [13,1]. Several potential sources of stray radiation which can excite areas remote from the beam impact point are shown in figure 19. Electrons may scatter out of the

beam after interacting with the final aperture. Backscattered electrons can strike the

chamber walls or stage, and upon scattering a second time, they can excite the specimen at large distances from the focused beam. Characteristi-c and continuous x-rays generated in

the aperture can cause x-ray fluorescence effects over the entire sample. Collimation of

the detector can serve to minimize these effects, but the analyst should be aware that even

a well collimated detector may have an effective acceptance area several millimeters in

diameter at the position of the sample. The key to avoiding difficulties with artifacts

introduced into the spectrum by stray radiation is recognition of the problem. Two speci- mens serve as suitable tests for stray radiation: (1) It is useful to obtain the spectrum

329
Polepiece

Figure 18. Solid angle of x-ray

collection from the point

of view of the specimen

(dark shading) and the

detector (light shading).

Si (Li) Detect<

Figure 19. Possible sources of remote

x-ray excitation in an electron beam/EDS system.

Solid lines - electron

paths; dashed lines -

x-ray paths.

of spectrographical ly pure carbon. The appearance of characteristic x-ray peaks in the

carbon spectrum is an indication of stray radiation striking the surroundings of the specimen. (2) For assessing the magnitude of stray radiation near the beam, a special Faraday

specimen cup is useful, figure 20. The Faraday cup consists of a small aperture (10 - 100 um diameter) pressed into a blind hole drilled in a block of material dissimilar to that of

the aperture. Spectra are recorded for a fixed live time with the beam striking on the

block, figure 21(a), on the aperture material, figure 21(b), and in the hole, figure 21(c).

If the "in-hole" spectrum contains characteristic x-rays for the block and/or aperture

materials, a stray radiation problem exists. In the example of figure 21, the stray radiation represents about 0.6 percent of the production of the direct beam. Elimination of

such problems is entirely dependent on the nature of the individual instrument.

330
Pt Aperture

1

lOOyum Hole

LJVJ

Ti Block

Carbon Backing

Figure 20. Schematic illustration of a cross-section through a Faraday

cup suitable for detecting

remote sources of x-ray excitation in an electron beam

instrument.

3CHK

20

10

3.o *

20

10

20 keV beam

Pt M

Pt M

Ti K«

Titanium block

' '• Ti K/3 J \J\

Platinum disc

jt %m "In-hole"spectrum

PtL« 23456789 10 E (keV)

Figure 21. Spectra obtained in an electron probe microanalyzer

from the components of a Faraday cup: (a) titanium

block, directly excited; (b) platinum aperture (c)

"In-hole" spectrum. The spectra indicate a source

of stray radiation at a long range from the incident

beam since only titanium x-rays are observed in the

"in-hole" spectrum. The stray radiation represents

about 0.6 percent of that excited by the direct

beam.

Electrons backscattered from the specimen may penetrate the beryllium window of the

energy dispersive spectrometer and be measured by the detector. Just as in the case of an x-ray, the detector will produce a signal pulse proportional to the energy of the electron

and add this to the x-ray spectrum being measured. Because electrons are scattered from a specimen with a continuum of energies, measurement of stray electrons from the specimen has

the effect of increasing the background above that expected from the normal x-ray continuum. A spectrum containing a significant electron contribution is shown in figure 22. The

background under the peaks of interest is mainly due to the electron contribution. The

predicted x-ray continuum, extrapolated by the method of Fiori et al., [14] from the high

energy region above 25 keV, which is unaffected by the electrons, is shown for comparison.

The scattered electron problem is most significant for high atomic number specimens. For

such targets, a larger fraction of the incident electrons are backscattered and these electrons retain a significant portion of their incident energy. The problem of direct electron

entry into the detector becomes insignificant for beam energies less than about 25 keV,

since the beryllium window on the detector is effective at stopping electrons up to this

energy. For windowless detectors, a magnet is used to deflect the electrons away from the

detector.

331
AsL« CuK« *sK« 16 4 K

18-1 K

" AsKf

FeK« .

. 3 0 K

Arsenic E =40keV Figure 22. EDS spectrum of arsenic

excited by a 40 keV electron beam showing an anomalous background

below 25 keV due to direct

entry of electrons scat- tered from the specimen

into the detector. The

x-ray continuum was fitted

from the high energy

end of the spectrum.

5. Artifacts in Qualitative Analysis

A correct qualitative analysis forms the basis for an accurate quantitative analysis.

Artifacts principally influence qualitative analysis through misidentification of peaks

which may be taken to represent elements not actually present in the specimen. With the

important exception of artifact peaks generated by stray electrons which excite remote portions of the sample and/or surroundings and which may produce major peaks in the spectrum,

all of the artifacts described in previous sections will be present at a level of the order

of one percent or less of the parent peak. Thus, artifact peaks primarily cause difficulty

in the identification of low intensity peaks which usually represent minor constituents of

a sample. In order to properly account for artifact peaks, the following general scheme for

qualitative analysis is useful [1].

5.1 General guidelines for qualitative analysis

(1) An adequate spectrum must be accumulated in order to give statistically valid

peaks. This condition is especially important for the low intensity peaks which are close

to the background intensity. If it is difficult to determine whether a peak exists above

the statistical fluctuations, the analyst should either ignore it or else accumulate more counts in the spectrum to "develop" the peak against the background.

(2) In order to comply with condition (1), the tendency of the analyst is to operate

the system at a high count rate. However, the energy dispersive spectrometer becomes

increasingly prone to pulse coincidence-related artifacts as the input count rate increases. For a typical system, the input count rate should be kept below 5,000 cps for

the full spectrum. Further, it must be noted that the "full" spectrum includes all x-rays

up to the incident beam energy. Thus, if a 100 keV beam is employed but the analyst

monitors only the region 0-10 keV in the multichannel analyzer, x-rays in the range 10 - 100 keV still contribute to system deadtime.

(3) The most important principle of qualitative analysis with EDS is to identify the

fami 1y of lines produced by an element. With the exception of light elements with atomic

numbers less than 16 (sulfur), each element produces at least two x-ray lines which are sufficiently separated in energy to be resolved with an ordinary EDS system (150 eV FWHM at manganese Ka). When the principal lines of an element are noted, all minor lines of the

family, which can be quite extensive for a heavy element, should also be marked. This makes

the task of identifying the low intensity peaks belonging to minor elements easier and

reduces the chance of mis identifying a low intensity peak. The degree of confidence which

can be placed in assigning an element as present in a specimen goes up rapidly as the number

of lines identified increases. In the following specific guidelines, families of x-ray

lines which can be observed under practical analysis conditions (high quality spectra with

5 x 106 counts in the full spectrum) are noted along with the approximate weights (relative

transition probabilities) of the lines. These weights are only approximate, since they vary

332
with the atomic number of the element. When the weights observed in an analysis differ

radically from these values, an interference or misidentification should be suspected.

(4) The energies of x-ray lines for qualitative analysis are usually obtained from

charts, "slide rules", tables, or "KLM" markers available on some multichannel analyzers.

Generally, all of these sources have deficiencies, either in terms of incomplete data or

lack of simultaneous display. The analyst generally must use several sources of information. The ultimate reference is a definitive compilation of x-ray energies, such as Bearden's table [15], but the detail in such a table is too great for quick reference; for

example, twenty-five L series lines are listed for heavy elements whereas at most nine are observed in practical analysis. To augment the existing sources, we present a plot of x-ray

line energies for all lines observed in spectra obtained under practical conditions, including the silicon x-ray escape peak energy of the principal line for each series, figure 23. The utility of this plot is that the analyst can look at a particular energy and note all

possible peaks which can occur. The FWHM produced by a 155 eV (Mn Ka) detector is shown as

a bar for each keV segment at the top of the plot. It should be noted, however, that for a heavy atomic number specimen, such as uranium, and sufficient counts in the full spectrum

(5.106 counts) a quite complex spectrum results, shown in figure 24, which requires the use

of a table such as that of Bearden [15].

333
B re

=3

S- l/> 4J S- O (0

1) -Q

Cl

to TO

i— o

3 V)

CO -S=

•rt * ?

HaawnN oiwoiv o_ I— +->

CO

CM

s-3

CD

334
MS

99.999 % URANIUM 20keV

Hp

Figure 24. Spectrum of high purity

uranium containing 5x1

0

6 counts in full spectrum.

5 keV

5.2 Specific guidelines - major peaks

(1) The analyst should begin with the peaks of high intensity, working from the high

energy end of the spectrum toward the low energy end. The reason for this approach is that

the members of a family of lines are separated by the largest differences in energy at high

energy and are likely to be resolved by the energy dispersive spectrometer.

(2) The energy at which a large amplitude peak occurs should be determined. If it is

suspected to be a Ka peak, look immediately for a Kp peak at the proper energy and at approximately 10 percent of the Ka peak height. For K lines of elements with atomic number

16 (sulfur) and above, the Ka - Kp peaks will be resolved with an ordinary 150 eV detector. If a peak is identified as a K peak of an element, try to find the L series of lines

for this element; L lines become visible for atomic number 26 (iron). Always locate the

complete family of lines for each element.

(3) If a Ka - Kp pair doesn't fit, L lines should be examined. If an La peak is sus- pected, look for the series La lJ2 (1), Lp x (0.7), L02 (0.2), Lv x (0.08), Ly3 (0.03), U

(0.04), and Ln, (0.01). If a high energy L family is located, look for M series lines which

may occur. M series lines become visible for atomic number 58 (cerium).

(4) If neither K series or L series lines fit the observed lines, M series lines should

be examined. The observed lines are Ma (1), M0 (0.6), M£ (0.06), V\y (0.05), and the line

obtained from the transition MjjNjy (0.01).

(5) At energies below 3 keV, the L and M series lines are not well resolved by an EDS

detector. Because several of the peaks of the family have similar weights La 1>2 (1), LPi

(0.7); Ma (1), Mp (0.6)) the resulting composite L or M peaks are frequently assymmetrical

.

This asymmetry aids in distinguishing L or M peaks from K peaks, figure 25 (a), (b), (c).

At low energies, the Kp weight decreases toward one percent of the Ka intensity and the

energy separation becomes less, so that the low energy composite Ka - Kp peak is nearly

symmetrical, figure 25(a). Near 1 keV, nearly all K, L, and M peaks appear symmetrical

and distinguishing the different types becomes difficult.

335
Figure 25. Low energy x-ray peaks in

the region of 2.3 keV.

The K family in this

region appears as a single, nearly symmetric

peak, while the L and M families appear distinctly

asymmetric.

5.3 Specific guidelines - minor peaks

Only after all of the high intensity peaks have been satisfactorily identified, and the

complete family of lines for each element so identified has been noted, is the analyst ready

to proceed with the remaining low intensity peaks.

(1) Spectral artifacts associated primarily with high intensity peaks should be

systematically identified. Each peak with an energy above the silicon K-absorption energy

(1.84 keV) has an associated silicon escape peak at 0.1 - 1 percent of the main peak intensity. The silicon escape peak is likely to be seen only for high intensity peaks. The

energies of silicon escape peaks associated with principal lines are plotted in figure 23.

(2) If the spectrum was obtained at a high counting rate, pulse coincidence effects

will lead to the formation of sum peaks. Only high intensity peaks will have a sufficient

count rate to give significant pulse coincidence. If major peaks exist at energies E^, Eg,

and E^, low level coincidence peaks can be expected at 2E^, 2Eg, 2E^, (E^ + Eg), (Eg + E^,), and (EA

+ Ec

).

(3) A low level silicon peak will be found as an inevitable result of the fluorescence

of the silicon dead layer following absorption of x-rays. This silicon signal represents

an apparent silicon constituent in a bulk sample at a level of 0.2 weight percent or less,

in a good detector.

(4) Artifact peaks due to stray radiation, previously determined on known samples,

should be noted as extraneous to the sample being investigated.

(5) Any low intensity peaks remaining after the above procedures have been carried out

have a reasonable chance of arising from an elemental constituent of the sample. The same procedures are repeated to identify these peaks as described above for major elements. How- ever, the level of confidence with which the identification of a minor element can be made

is generally poorer than for major elements, since the peak intensities are usually so low

that only the main peak of a family is visible.

As an example of the application of the above guidelines to the qualitative analysis of

a complicated spectrum, we shall analyze the spectrum of NBS glass K-252 [16].

The spectrum is shown in figure 26.

5.4 Example

336
15

NBS Glass K252

10

4 5 6 E (keVI

Figure 26(a,b).

11 _

Spectrum of NBS glass

K-252 at two different

scale expansions. The

qualitative analysis identifying the numbered peaks

is described in the text.

16* 17

if

NBS Glass K252

2 b

i r

E (k.VI

(1) Starting at the high energy end of the spectrum, figure 26(a), peak (1) has an energy which corresponds to zinc Ka (8.64 keV) and the appearance of peak (2) at the proper

energy for zinc KB (9.57 keV) and at about 10 percent of the intensity of peak (1) confirms

the assignment. Next we look for a zinc L line, and find peak (3) at the energy for zinc

La (1.01 keV).

(2) Peak (4) is found to correspond to copper Ka (8.04 keV); the corresponding Kp peak

at 8.91 keV, peak (5), appears unresolved from the zinc Ka peak and is only seen as a distortion on the high energy side of that peak. Copper La (0.93 keV) is not resolved from

zinc La (1.01 keV).

(3) Peak (6) is found to be cobalt Ka (6.93 keV) with peak (7) as cobalt KB (7.65

keV). The peaks are found to be in the correct ratio which confirms the assignment. Similarly, peaks (8) and (9) are found to correspond to manganese Ka (5.89 keV) and KB (6.49

keV).

(4) At first appearance, we seem to have two additional Ka - KB pairs, peaks (10) - (11) and (12) - (13), since the peak ratios are approximately correct. However, the energies of the nearest candidate elements, vanadium and titanium, are sufficiently different

from the measured energies to throw out these assignments. When L energies are examined, we find that barium La (4.47 keV) fits peak (12), LB X (4.83 keV): peak (10); LB? (5.15

keV): peak (13); and Ly x (5.53 keV): peak (11). To complete the barium L series, we

locate barium LI (3.95 keV) at peak (14). Barium Ln. is unresolved.

337
(5) The last major peak, (15), is identified as silicon Ka (1.74 keV); the detector

will not resolve silicon KB (1.83 keV) and so only one peak is available for identification.

Peak (15) might be confused with tantalum M, except for the difference in energy (1.71 keV

compared to 1.74 keV), the symmetry of the observed peak, which implies a K peak, and the

absence of a tantalum L series at 8.15 keV.

(6) Expansion of the vertical scale, figure 26(b), reveals only two additional peaks,

(16) and (17). Searching first for artifacts of the major peaks, we identify peak (16) as the silicon escape peak of barium La and peak (17) as the escape peak of barium LB^ No other peaks remain unidentified, so the qualitative analysis is complete. Due to the large

number of major elements in the specimen, the available regions of the spectrum where minor

elements might be observed are very limited. This example points out that a consequence of

the poor resolution of the detector is the difficulty in recognizing minor elements in

complicated spectra.

6. Conclusions

Successful x-ray microanalysis with the energy dispersive x-ray spectrometer requires

knowledge of the types of artifacts which can occur. Artifacts which originate in the

physical process of x-ray detection cannot be eliminated from the spectrum. Artifacts

arising from the specimen environment can frequently be reduced to an insignificant level.

Accurate analysis can be carried out with a logical method for proper peak assignments and

artifact recognition.

References

[1] Fiori, C. E. and Newbury, D. E. , "Artifacts Observed in Energy-Dispersive Spectrometry

in the Scanning Electron Microscope", SEJM/78, Om Johari , ed. , 1978, pp. 401-422.

[2] Reed, S. J. B. , Electron Microprobe Analysis , Cambridge University Press, Cambridge,

England, 1975.

[3] Kramers, H. A., "On the Theory of X-ray Absorption and of the Continuous X-ray Spectrum", Philosophical Magazine , 46, 1923, p. 836.

[4] Heinrich, K. F. J. , Newbury, D. E. , and Yakowitz, H. , Use of Monte Carlo Calculations

in Electron Probe Microanalysis and Scanning Electron Microscopy , NBS Special Publication 460, U.S. Government Printing Office, Washington, D.C. 20402, 1975.

[5] Woldseth, R. , X-ray Energy Spectrometry , Kevex Corporation, 1101 Chess Dr., Foster

City, California 94494, 1973.

[6] Freund, H. U. , Hansen, J. S. , Karttunen, E. , and Fink, R. W. , "Photon Spectrometry in

the Low Energy Region of 500 eV to 100 keV", Proceedings Inter. Conference on Radioactivity and Nucl. Spect. , 1969, p. 623.

[7] Gedke, D. , "The Si(Li) X-ray Energy Analysis System: Operating Principles and Performance", X-ray Spect. , 1(4), 129 (1972).

[8] Heath, R. L. , "The Applications of High-Resolution Solid State Detectors to X-ray

Spectrometry - A Review", Advan. X-ray Anal

.

, 15(1), 1 (1972).

[9] Elad, E. , Inskeep, C. N. , Sareen, R. A., and Nestor, P., "Dead Layers in Charged Particle Detectors", IEEE Trans, on Nucl. Sci. , 20(1), 534 (1973).

[10] Jaklevic, J. M and Goulding, F. S. , "Detection of Low Energy X-rays with Si(Li) Detectors", IEEE Trans, on Nucl. Sci. , NS-18 (1), 187 (1971).

[11] Fitzgerald, R. and Gantzel , P., "X-ray Energy Spectrometry in the .1 to .10 A Range",

Energy Dispersion X-ray Analysis : X-ray and Electron Probe Analysis , ASTM-STP-485,

American Society for Testing and Materials, 1971, p. 3.

338
[12] Statham, P. J., "Pile-up Rejection: Limitations and Corrections for Residual Errors in

Energy-Dispersive Spectrometers", X-ray Spect. , 6(2), 94 (1977).

[13] Bolon, R. B. and McConnell , M. D. , "Evaluation of Electron Beam Tails and X-ray Special

Resolution in the SEM" in Scanning Electron Microscopy , 0. Johari , ed. , 163 (1976).

[14] Fiori, C. E. , Myklebust, R. L. , Heinrich, K. F. J., and Yakowitz, H. , "Prediction of

Continuum Intensity in X-ray Dispersive Microanalysis", Anal . Chem. , 48(1) (1976).

[15] Bearden, J. A., "X-ray Wavelengths and X-ray Atomic Energy Levels", Rev. Mod. Phys. , 3JO) (1967).

[16] Fiori, C. E. , Heinrich, K. F. J., Marinenko, R. B. , Darr, M. M. , Blackburn, D. H.

,

Newbury, D. E. , and Small, J. A., "An Overview of the Glass Standards Program for

Microanalysis at the National Bureau of Standards", Proc. 11 th Annual Conference of

MAS , Miami Beach, Florida, 1976.

[17] Henoc, J. , Heinrich, K. F. J. , Myklebust, R. L. , "A Rigorous Correction Procedure for

Quantitative Electron Probe Microanalysis (COR 2)", NBS Technical Note 769 , U.S. Government Printing Office, Washington, D.C., 1973.

[18] Yakowitz, H. , Myklebust, R. L. , and Heinrich, K. F. J., "FRAME: An On-Line Correction

Procedure for Quantitative Electron Probe Microanalysis", NBS Technical Note 796 , U.S. Government Printing Office, Washington, D.C., 1973.

[19] Schamber, F. H. , "A Modification of the Linear Least-Squares Fitting Method which

Provides Continuum Suppression" in X-ray Fluor. Analysis of Environmental Samples , T. G. Dzubay, ed. , Ann Arbor Science Pub. , Ann Arbor, Mich. , 1977, p. 241.

[20] Smith, D. G. W. , Gold, C. M. , and Tomlinson, D. A., "The Atomic Number Dependence of

the X-ray Continuum Intensity and the Practical Calculation of Background in Energy

Dispersive Electron Microprobe Analysis", X-ray Spect. , 4, 149 (1975).

[21] Fiori, C. E. , Myklebust, R. L, and Heinrich, K. F. J., "A Method for Resolving Overlapping Energy-Dispersive Peaks of an X-ray Spectrum; Application to the Correction

Procedure FRAME B" , Proc. 11th Annual Conf. MAS , Miami Beach, Florida, 1976, paper 12.

[22] Heinrich, K. F. J. and Yakowitz, H. , "Absorption of Primary X-rays in Electron Probe

Microanalysis", Analyt. Chem. , 47(14), 2408 (Dec. 1975).

[23] Reed, S. J. B. and Ware, N. G. , "Escape Peaks and Internal Fluorescence in X-ray

Spectra Recorded with Lithium Drifted Silicon Detectors", Phys. E: Sci . Instru. , 5, 582 (1972).

[24] Myklebust, R. L. , Fiori, C. E. , and Heinrich, K. F. J., "FRAME C: A Compact Procedure

for Quantitative Energy-Dispersive Electron Probe X-ray Analysis", Proc. 12th Annual

Conf. MAS , Boston, Mass., 1977, p. 96A.

339

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ARTIFACTS ENCOUNTERED IN ENERGY DISPERSIVE X-RAY SPECTROMETRY

IN THE ANALYTICAL ELECTRON MICROSCOPE

D. B. Williams and J. I. Goldstein

Metallurgy and Materials Engineering Department

Lehigh University,

Bethlehem, PA 18015

Abstract

The analytical electron microscope (AEM), consisting of an energy

dispersive spectrometer interfaced to a scanning transmission electron

microscope, produces its own particular spectral artifacts. These

artifacts are due specifically to the use of high kV electrons and thin

foil specimens. The detection of these artifacts, the identification of

their source, and the minimization of their effects are discussed. It is

concluded that an ideal microanalytical configuration for an AEM exists

and that now it is possible to reduce the effect of spectral artifacts to the level where they do not limit quantitation of the x-ray data.

1. Introduction

The interfacing of an energy dispersive x-ray spectrometer (EDS) to a scanning transmission electron microscope (STEM) is one of the principal configurations for analytical

electron microscopy (AEM). Ideally in the AEM the measured characteristic x-ray intensity

above background is related to the specimen composition directly through a known constant

of proportionality. For this to occur the source of x-rays measured by the EDS must be

the region of the specimen interacting with the primary electron beam. This is not

routinely the case in AEM where x-rays generated from other sources are often detected.

This undesired signal is the major artifact in the EDS and is sometimes called the spurious

x-ray signal. Spurious x-rays are insidious because they often seem to be part of the

desired spectral information and are thus not immediately obvious. Unless very specific

precautions are taken, AEM x-ray microanalysis is not quantitative under these conditions

and therefore the applications of AEM are limited.

The problem of spurious x-rays has received much attention recently [1.2] 1

. It appears that the spurious x-ray signal is caused by two major effects, 1) stray radiation

in the AEM illumination system, and 2) specimen interaction with the primary electron

beam. The resultant spectral artifacts are specific to AEM because the system uses high

kV electrons and the specimen is transparent to the high kV beam. These points contrast

with other instruments (e.g., scanning electron microscope and electron microprobe) in

which EDS is used, where bulk specimens are typically irradiated by electrons with energy

< 30 keV.

This paper summarizes the present state of knowledge and describes how the instrument-specific sources can be identified and eliminated or minimized. In the ideal

configuration that is described in the conclusion of the paper, it is considered that AEM

x-ray microanalysis using EDS can be free of quantitation-1 imiting spectral artifacts and

that quantitation is then a straightforward process.

1 Figures in brackets indicate the literature references at the end of this paper.

341
2. Sources of Spurious X-rays

2.1 Artifacts due to stray radiation in the AEM illumination system

The existence of this type of spurious x-rays is easy to determine. Figure la shows a typical EDS spectrum obtained in ~60 seconds from a thin Ag disc specimen in a Phillips 300

TEM-STEM operated at 100 kV. Figure lb shows a similar spectrum obtained when the primary

beam was placed down a hole in the foil. Since it can be assumed that the primary beam

continued uninterrupted down the microscope, it is clear that something other than the

primary beam is exciting specimen-characteristic x-rays. This and similar phenomena have

been reported by many investigators and the effect has been ascribed to both uncoil i mated

electrons [3-5] and stray x-rays [6-8]. However it is possible to distinguish between the

two sources because the Ag spectrum contains high and low energy x-ray lines (Ka at 22.2 keV

and La at 3.0 keV) and the relative height of the lines is a useful indicator of the source

of the spurious x-rays [2]. By performing this type of experiment and observing the

'hole-count' spectral artifact, it is possible to identify unambiguously the sources of the

spurious x-rays from the illumination system as described below.

2048-FS

mJ'^- iWL

O E(kV)

Figure la. Typical spectrum from the thin region

of a Ag disc (FS = full scale).

512 -FS

K

_y i- ./w/.

O E(kV) 40

Figure lb. Spectrum obtained when the primary beam

is placed down a hole in the same Ag

specimen as figure la.

342
The use of 100 kV electrons in STEM gives rise to x-rays detected from areas other

than the point of interest on the specimen, principally in the following two ways shown

schematically in figure 2:

1. The STEM electron optical system is not always designed to colli mate the electron

beam completely. In normal TEM and STEM imaging and diffraction operation, it is of little

consequency if stray electrons hit the specimen and holder at distances well away from the

point of interest. However this is clearly undesirable in an AEM when the EDS can detect

x-rays from a wide area of the specimen chamber.

2. 100 kV electrons generate continuum (bremsstrahlung) and characteristic x-rays from

various apertures as they are collimated by the illumination system. If these x-rays reach

the specimen chamber, they can act as efficient sources of fluorescent x-rays from large areas of the specimen and its surroundings.

PRIMARY / ELECTRONS

UNCOLLIMATED / ELECTRONS-^

X-RAYS / FROM C2

' APERTURE

SPURIOUS

X-RAYS

C2 FIXED

-C2 VARIABLE

\ FOCUSED

1 ELECTRONS

i .UPPER

\ /OBJECTIVE

/ POLE

/ PIECE

SPECIMEN

Figure 2. Principal sources of spurious x-rays

in the illumination system of an

AEM.

AXIS

The two problems outlined above are instrument dependent, since illumination systems

differ. Therefore solutions to eliminate 'hole count' spectra are instrument dependent.

The first problem is to determine the existence of the spurious x-rays and identify their

source(s). After that one can attempt to minimize the effect.

In the silver spectrum (figure la) obtained with the primary beam incident on a thin

region of the disc, it is assumed that the La to Ka peak height ratio is primarily characteristic of an electron-induced spectrum. Observing the hole spectrum (figure lb), a similar conclusion might be drawn. Thus in this instrument (a Phillips EM 300) some of the

uncoil imated electrons are not stopped by the C 2 aperture and are focussed onto the specimen. This effect is particularly acute in that series of Phillips instruments and was first

determined in 1976 [4]. A simple solution to stop uncollimated electrons (which are in fact

scattered around the beam-defining apertures) is the insertion of a spray aperture or plug

beneath the variable C 2 aperture. If this is done, the hole count spectrum is as shown in

figure 3. The reduction in scale (x4) compared with figure lb indicates that a substantial

proportion of the spurious contribution to the specimen spectrum has been eliminated.

343
K

128-FS

O E (kV) 40

Figure 3. The hole-count spectrum obtained after the contribution

of uncoil i mated electrons has been removed by a spray

aperture. Compare the F.S. with figure lb.

There is still a detectable hole count but, as shown in figure 3, the Ka peak is now substantially higher then the La (compare with figures la and lb). The remaining source of

stray radiation is the hard x-radiation generated in the illumination system. A simple

calculation [1] shows that even conventionally thick platinum apertures (^25 urn in the

vicinity of the hole) transmit substantial fractions of the continuum x-rays generated.

Figure 4 shows the calculated spectrum of continuum x-rays which would travel down the col- umn through increasing thicknessess of a platinum aperture. Since the efficiency of x-ray

fluorescence depends on the energy of the exciting x-rays, it is clear that the continuum

x-ray intensity from the aperture will excite the high energy Ag Ka line more efficiently

than the low energy Ag La line. This explains the hole spectrum shown in figure 3.

Figure 4. Calculated spectrum of bremsstrahlung x-rays

generated by 100 kV electrons at the final

beam-defining C 2 aperture. The reduction in

intensity due to absorption by increasing

thickness of Pt is shown as well as the relative position of the AgKa and La x-ray lines.

344
This problem of fluorescence due to continuum x-rays is inherent in all AEMs, but will

be reduced if thin film specimens are used, since x-ray absorption is minimized. However,

in metallurgical and ceramic specimens, which are usually thick discs with a central electron transparent zone, the fluorescence problem will always be significant.

Two ways to combat the fluorescence problem have been proposed. First the source may

be removed by using extra thick [9] or cylindrical (no taper) [10] apertures. Figure 5 shows an SEM image of a 200 urn thick, 20 urn wide aperture designed for this application and

produced by Microengineering2

. In this case the hole count is further reduced using both a spray aperture and an extra thick C 2 aperture by a factor of four as shown in figure 6

(compare scale with figure 4). Extra thick apertures are now available for Phillips EM 300

and 400 series AEMs. The second approach to the hard x-ray problem is to insert non beamdefining apertures further down the column [7]. This has the effect of cutting down

substantially on the area fluoresced. This solution is incorporated into all JE0L 100CX

and 200CX 'hard x-ray kits'.

Figure 5. SEM image of extra-thick Pt C 2 variable aperture. The aperture is cylindrical, no regions

being <200 urn thick.

32 -FS

K

L

0 E (kV) 40

Figure 6. The hole-count spectrum obtained with both a spray aperture and extra thick C 2 aperture.

There is still a small amount of AgKa and La due to residual stray radiation.

2 Microengineering, Inc., R.D. #3, Box 501, Williston, VT 05495.

345
With all the modifications to the illumination system discussed above, the problems

of spurious x-rays from electron and x-ray sources can be reduced to acceptable levels. Two

criteria of acceptability have been defined [2,11] but they have the same aim - i.e., to ensure that these spurious x-rays are not the limiting factor in the accuracy of x-ray

quantitation. Given a reasonable microanalysis situation, if the spurious hole count for a given element is less than the continuum background generated when the beam is on the

specimen, then at least the spurious contribution to the spectrum is of the order of the

detection limit of the characteristic line of interest. For example, selecting a thin

region of a specimen such that 10,000 counts (statistically significant) can be accumulated

in ~60 seconds (reasonable time) with a typical probe size (70 - 100 8) used for microanalysis, then the characteristic x-ray hole count, using the same probe for the same time,

should be less than the background for all peaks of interest in the specimen spectrum.

The modifications described above are currently being incorporated in the latest AEM

instruments. However, all earlier first generation AEMs suffered from the problem of

spurious x-rays induced by stray radiation and the individual user should check his

instrument in the way outlined above and correct the problem accordingly.

2.2 Artifacts due to beam interaction with the specimen

Assuming that the illumination system problems are removed, it is still not certain

that all the detected characteristic x-rays are generated solely from the point of interest. The primary reason for this is that the thin specimens allow transmission of high kV

electrons, and also scatter electrons in both the forward and backward directions.

The sources of x-rays that can reach the EDS due to this effect are summarized in

figures 7a and 7b. These problems can be identified as follows.

INCIDENT INCIDENT f —- ELECTRONS wv DESIRED X RAYS

Wl/ SPURIOUS XRAYS

ANTI-CONTAMINATOR

BACK SCATTEREO f

DISC SPECIMEN

ANTI-CONTAMINATOR

OLE

PIECE TRANSMITTED e" CURRENT SITUATION -DISC SPECIMEN

ELECTRONS ww DESIRED X RAYS VW SPURIOUS X RAYS

THIN FILM SPECIMEN

SPECIMEN GENERATED CONTINUUM

BACK SCATTERED

SCATTERED t"

TRANSMITTED «"

CURRENT SITUATION -THIN FILM SPECIMEN

Figure 7a. Sources of spectral artifacts

due to electron beam- thin disc

specimen interaction in an

AEM.

Figure 7b. Sources of spectral artifacts

due to electron beam interaction with a thin foil specimen

mounted on a support grid.

346
1. Incident high energy electrons are backscattered into the microscope specimen

chamber and generate x-rays from this region (e.g., cold trap, upper pole piece, EDS

collimator). The possibility of direct backscatter into the EDS itself also cannot be

discounted, although the strong prefield of upper objective polepiece should prevent this.

2. Incident high energy electrons are transmitted, scattered or diffracted into the

specimen chamber (or specimen support grid if it is very close to the region of interest),

and generate a) characteristic and continuum x-rays, b) backscattered electrons from, e.g. lower objective polepiece, objective aperture drive.

3. Characteristic and high energy bremsstrahl ung from the point of interest on the

specimen fluoresce the specimen environment. This is probably the most important problem.

If the specimen is a disc and is tilted to some non-horizontal orientation to permit

detection of the characteristic x-rays, then the possibility of self-fluorescence (i.e.,

fluorescence induced by the continuum radiation generated in the sample) exists (figure

7a). This of course depends on the specimen shape, thickness, and microstructure, and the

actual energy and intensity distribution of the bremsstrahl ung x-rays. If a specimen

support grid is used (figure 7b), then this radiation will fluoresce the grid. This

probably accounts for the observation of copper peaks in otherwise 'clean' STEM systems,

when a copper support grid is used. Figure 8 which is a spectrum from a 200 nm particle

of Fe-Ni sulphide in a silicate matrix illustrates this problem. A Cu Ka peak is still

clearly visible although the primary beam is confined totally within the particle and the

grid bar is > 200 urn away from the focussed beam. The use of beryllium support grids will remove the Cu Ka peak in the spectrum since a) there will not be efficient fluorescence of

beryllium, and b) beryllium characteristic x-rays cannot be detected by EDS. However, even using beryllium support grids, there is still a danger of backscatter of scattered (diffracted) 100 kV electrons from this grid into the bulk of the specimen, if the grid is very

close to the region of interest.

Fek

Sk, Cuk

Nik

'•J v '"J LT» 71 CT« 73 4IKFS 11.24

Figure 8. Spectrum from an analysis situation as in

figure 7b. The Cu peak due to the Cu grid bars is clealy bisible.

347
The minimization of problems 1 and 2 above, will again be somewhat instrument dependent. For example, the JEOL 100C produced system counts from pole pieces, anticontaminator

and the guard tube in the stigmator coils [12]. All of these contributions to the spectrum

may be minimized by suitable light element inserts which either reduce the backscatter, or produce x-rays which are not detectable. Conversely, in some instruments, no such systems

peaks have been detected.

The use of thin foil specimens rather than discs may cut down the possibility of

specimen self-fluorescence. Furthermore, the development of high take-off angle detectors

(standard on some STEM/AEMs and being developed on others) is an essential step. Specimen

interaction with its own bremsstrahl ung is thus minimized. There is, however, the possibility of increased backscatter directly into the detector in this configuration, but

to-date, no conclusive evidence exists for this effect. The use of beryllium grids, when

grids are necessary, is of paramount importance, but even in this case, microanalysis

should be performed at least 0.1 mm away from any grid bars to minimize any backscatter of

diffracted or scattered electrons. Electron backscatter and x-ray production from the lower

objective polepiece can be minimized by covering it with an element of low atomic number

such as carbon, boron, or beryllium. The same could be done to minimize x-ray production

from the upper objective polepiece and/or anticontaminator. Taking all these points into

account, an ideal microanalysis situation from the specimen chamber point of view is

shown in figure 9.

INCIDENT e"

Figure 9. Ideal microanalytical configuration

which minimizes spectral artifacts

in an AEM. The configuration shown

is attainable in current AEMs, and assumes that any stray radiation

from the illumination system has

been effectively suppressed.

PIECE TRANSMITTED •

"

IDEAL SITUATION

3. Summary

To the unwary operator, spectral artifacts due to spurious x-rays in EDS in the AEM can contribute significantly to errors in quantitation. The spurious x-rays unique to AEM are insidious in their effect because more often than not, the signals they generate are similar

to that expected from the point of interest, although often originating from far away on the

specimen. Careful operation and ideal microanalytical configurations have now been identified and it is possible to reduce spurious contributions to the specimen to levels at which

they do not limit the accuracy of quantitation. It is considered that at this stage of

development EDS in the AEM is fully quantitative and may be confidently applied to a whole

range of materials problems awaiting the advent of the superior spatial resolution of this

technique.

This work was supported by NASA Grant NGR 39-007-043.

348
References

[1] Goldstein, J. I. and Williams, D. B. , SEM (IITRI), 1, 651 (1977).

[2] Bentley, J. , Kenik, E. A. , Zaluzec, N. J. , and Carpenter, R. , SEM, ]_ (1979).

[3] Geiss, R. H. and Huang, T. C. , X-ray Spectrometry, 4, 196 (1975).

[4] Shuman, H. , Somlyo, A. V., and Somlyo, A. P., Ultramicroscopy , 317 (1976).

[5] Russ, J. C. , SEM (IITRI), 1_, 335 (1977).

[6] Hren, J. J., Ong, P. S. , Johnson, P. F. , and Jenkins, E. J., Proc. 34th EMSA Meeting,

Claitor's Publishing Division, 418 (1976).

[7] Zaluzec, N. J. and Fraser, H. L. , Proc. 34th EMSA Meeting, Claitor's Publishing

Division, 420 (1976).

[8] Kenik, E. A. and Bentley, J., Proc. 8th ICX0M/12th Annual MAS Conference, Boston, 114A

(1977).

[9] Joy, D. C. and Maher, D. M. , Proc. Specialist Workshop on AEM, Cornell University, 111 (1976).

[10] Williams, D. B. and Goldstein, J. I., Proc. Specialist Workshop on AEM, Cornell

University, 174 (1978).

[11] Williams, D. B. , Goldstein, J. I., Proc. 8th ICX0M/12th Annual MAS Conference, Boston,

113A (1977).

[12] Nicholson, W. A. P., Robertson, B. W. , and Chapman, J. N. , Developments in Electron

Microscopy and Analysis, The Institute of Physics, Bristol and London, 373 (1977).

349

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

UNCOLLIMATED FLUORESCING RADIATION IN AN ANALYTICAL ELECTRON MICROSCOPE

(100-1000 KEV): SOURCES AND SOLUTIONS 1

Nestor J. Zaluzec

Oak Ridge National Laboratory

Metals and Ceramics Division

Radiation Effects and Microstructural Analysis Group

Oak Ridge, Tennessee 37830

Abstract

The interfacing of a solid state energy dispersive spectrometer

(EDS) to the column of a modern transmission or scanning transmission

electron microscope (TEM/STEM) can potentially provide an extremely

powerful analytical tool for the microcharacterization of materials.

Unfortunately, there are sometimes serious instrumental limitations

involved in this seemingly straightforward modification of a TEM/STEM

instrument into an analytical electron microscope (AEM) of the simplest

kind. The limitations arise due to the presence of uncoil imated

radiation, consisting of both hard x-rays and electron tails, which is

generated within the probe forming system of the microscope. In an unmodified instrument this radiation can reach the specimen area and thus

produce characteristic x-ray emission by fluorescence of the specimen

and its environment. Clearly, in any quantitative analysis scheme such

an effect is undesirable. Fortunately, these complications can be

minimized by judicious modification to the electron-optical column of

the AEM system. The correction procedures employed in instruments with

incident electron beam energies from 100 to 1000 keV will be outlined and

their effects on quantitative analysis discussed.

1. Introduction

The interfacing of a solid-state energy-dispersive x-ray spectrometer (EDS) to the

column of a modern transmission or scanning transmission electron microscope (TEM/STEM) can potentially provide an extremely powerful analytical tool for the microcharacterization of

materials. Such an instrument is complementary to a conventional electron microprobe in

that it can provide high spatial resolution (< 1 00 nm) x-ray analysis of microvolumes of

material. This increased spatial resolution is derived from the use of minute electron

probes (<20 nm), high incident beam energies (100-1000 keV), and thin electron transparent

specimens (<200 nm). Furthermore, the quantitative interpretation of data produced under

these conditions is fundamentally simpler than bulk analysis since the specimen thicknesses

employed are such that absorption and fluorescence corrections are usually negligible.

Unfortunately, there are sometimes serious instrumental limitations involved in this seem- ingly straightforward modification of a TEM/STEM instrument into an analytical electron

microscope (AEM) of the simplest kind. The limitations arise due to the presence of uncollimated radiation, consisting of both hard x-rays and electrons, which is generated within

the probe-forming system of the microscope. In an unmodified instrument this radiation can reach the specimen area and produce characteristic x-ray emission by fluorescence of the

specimen and its environment. In any quantitative analysis scheme such an effect is

undesirable. Fortunately, these complications can be minimized by judicious modifications

to the electron-optical column of the AEM system. The correction procedures employed in

Research sponsored by the Division of Materials Sciences, U. S. Department of Energy, under

contract No. W-7405-eng-26 with the Union Carbide Corporation and the E. P. Wigner Fellowship at 0RNL.

351
2. Uncoil imated Fluorescing Radiation: Sources

In an ideal AEM all of the information recorded by the x-ray detector system is produced solely by the interaction of the incident electron probe and the region of interest of

the specimen. However, the current generation of AEMs have evolved from the addition of

analytical attachments to TEM/STEM instruments, and as such are not necessarily optimized

for x-ray work. The most detrimental aspect of this is the nonlocalized fluorescence of the

specimen and its immediate surroundings by uncollimated radiation (i.e., electrons and/or

x-rays not confined to the immediate diameter of the incident probe). The presence of this

radiation is most easily detected by allowing the incident electron probe to pass through a hole in an appropriate test specimen. In this situation one would expect all x-ray emission

from the specimen to cease. This, unfortunately, is usually not observed. The x-ray emission detected during such an "in-hole" experiment is produced by fluorescence of the

specimen by remotely produced radiation. The source, amount, and nature (i.e., electrons

versus x-rays) of this radiation vary from instrument to instrument. Nevertheless, the net result is the same - a degradation of the high spatial resolution and accuracy of the

quantitative analysis ultimately obtainable in the AEM.

In order to produce the minute electron probes used in an AEM, it is necessary to intercept a relatively large electron flux by a series of beam-defining apertures. If the

geometry and/or operating conditions of the microscope are appropriate, these apertures act

as the principal sources of uncollimated radiation. Two types of radiation have been identified as originating from this source: electrons (not confined to the focused incident

probe) and x-rays (both characteristic and continuum).

Depending on the specific design of the condenser aperture system of the prospective

AEM, it may be possible for electrons (at a significant fraction of the incident beam

energy) to scatter around the periphery of the fixed and variable aperture support mechanisms. These electrons together with any that are scattered from the bore of contaminated

or thick apertures can result in the formation of significant electron tails and can become

an important factor in the production of characteristic x-ray emission from regions of the

specimen other than that defined by the electron probe. These beam-defining apertures can also act as a source of a second type of fluorescing radiation. In this case, one can consider the apertures not as electron collimators but rather as a source of hard x-ray

(both characteristic and bremsstrahlung) emission. Typical condenser apertures in modern

microscopes are relatively thin, high-atomic-number materials of sufficient mass thickness

to completely stop the incident electron probe. Conditions such as this are nearly perfect

for these apertures to act as thin target x-ray sources. The situation is further complicated by the fact that the emission of bremsstrahlung (continuum) radiation is highly

anisotropic and at the electron energies typical of AEM investigations (>1 00 keV) this

emission is peaked in the forward scattering direction. Thus, in an AEM one can find the

conditions and geometry nearly ideal for x-ray fluorescence of the specimen and its environment. Figure 1(a) schematically illustrates these sources of uncollimated radiation.

(a) Schematic illustration of remote sources of uncollimated fluorescing

radiation from Fixed (A) and Variable

(B) beam-defining apertures.

(b) Modifications to reduce the

intensity of uncollimated fluorescing radiation using: Thick Fixed

apertures (A) above the Variable

beam-defining aperture (B) and Thick

Spray apertures (C) below, as well

as non-beam defining apertures (D)

just above the Objective Lens Pole

Piece.

352
3. Uncoil imated Fluorescing Radiation: Solutions

In order to assess the amount, nature, and detrimental effects of uncoil imated radiation on quantitative analysis, the choice of test specimen used to observe the phenomenon

is an important consideration. Regardless of the nature of the radiation, the type of

specimen which most dramatically illustrates these effects if of the self-supporting thinned

disc variety. In this case, the periphery of the specimen is essentially a thick (M).l mm)

rim of material representing a bulk sample. The thin electron transparent region of these

specimens is confined to the edge of a small hole near the central area of the disc and is

usually produced by preferentially electrochemically thinning using a fine jet of electrolyte [l] 2

. An additional factor influencing the magnitude of fluorescence effect is the

specimen composition. The most pronounced effects are observed in specimens composed of

medium-to-high atomic number elements (or suitably homogeneous alloys) having both a lowenergy (<3 keV) as well as a medium-energy (>7 keV) characteristic x-ray emission line.

In selecting a suitable specimen, it is advantageous to choose a material Which has

characteristic x-ray lines in the 0-18 keV range (above ~20 keV, the transmission of

high-energy x-rays by the solid state detector becomes significant and thus their efficiency

of detection decreases). Some suggested systems are molybdenum and its alloys (MoKa ~17.4

keV, MoLa ~2.3 keV), or some of the homogeneous aluminum nickel alloys such as BNiAl

(NiKa ^7.5 keV, AlKa ~1.5 keV).

There are four steps in the elimination of remotely produced uncollimated radiation.

They are: (1) identification of the fluorescing species; (2) consultation with the instrument manufacturer as to their solutions to the problem; (3) modifications to the electron

optical column; and (4) corrections to measured spectra for quantitative analysis. It is

beyond the scope of this review to discuss at length the details of such modifications, and

the prospective investigator is referred to the recent tutorial specifically devoted to that

subject; however, the most important points are discussed below [2].

The identification of the fluorescing species is probably the most difficult task of

the four steps outlined above, because the procedures are more qualitative than quantitative

in their interpretation. One first examines the design of the probe- forming system of the

microscope to ascertain if it is physically possible for electrons to reach the specimen by

scattering around the periphery of the various aperture mechanisms. Next, with the variable

second condenser (C2 ) aperture in its normal position and using a thick solid disc in place

of the normal aperture, one should attempt to detect electron scattering around the aperture

supports using a suitable Faraday cup and sensitive electrometer (the currents involved may

be less than 10" 13 A). Third, using a suitable test specimen as previously described,

acquire a representative in-hole spectrum and carefully note its characteristics. If x-rays

are the principal fluorescing species, then one would expect the intensity of the low energy

characteristic line to be extremely small relative to the higher energy line. This result

is to be expected because the x-ray production cross sections for fluorescence by x-ray

photons favors excitation of the higher energy lines, while electrons favor the lower energy

lines. This can be seen by reference to Tables 1 and 2 which document the relative cross- section ratio of a high to low energy x-ray lines for the elements molybdenum and nickel due

to excitation by x-ray photons and electrons, respectively. In addition, the relatively

weak interaction of x-rays with solids (relative to electrons) results in the majority of

characteristic x-ray production of x-ray fluorescence occurring deep within the sample.

Thus, the lower energy lines will suffer significant absorption effects (relative to the

higher energy line), making their detection more difficult. In comparison, the cross sections for electron excitation favor the lower energy lines and, although absorption

effects will reduce their intensity, they will also be of greater relative intensity since

generation will occur nearer the sample surface. The final test in the qualitative evaluation is to irradiate the bulk regions of the test specimen with the incident beam and record

an x-ray spectrum resulting from electron excitation of this area. The overall characteristics of this (bulk specimen) electron-excited spectrum can then be compared qualitatively

with the "in-hole" spectrum. The combination of the results of all these tests is usually

sufficient to identify the dominant source.

2 Figures in brackets indicate literature references at the end of this paper.

353
Table 1. Cross section ratios °"Kc/°~La for molybdenum and nickel

Characteristic x-rays by x-ray excitation

X-ray Photon / x-ray / x-ray \ / x-ray / x-ray

Energy Mo {o^ /a

la )

NT [o

La j

25 205 625

50 230 610

75 235 590

Table 2. Cross Section Ratios o^/a^ f° r molybdenum and nickel.

Characteristic x-rays by electron excitation

Incident Electron

Energy (keV)

/ electron , Mo Wa 1

1 electron \

°La }

1 electron /

Ni Wa 1

' electron

CT Lo

25 0.002 0.010

50 0.011 0.012

75 0.014 0.013

100 0.016 0.015

The various AEM manufacturers have begun to offer modifications or accessory attachments to their respective AEM instruments in order to reduce the magnitude of uncoil imated

radiation. These modifications are based on the various solutions proposed in the literature to reduce this problem; these modifications include: the insertion of spray apertures

above and below the variable (C 2 ) drive assembly to reduce electron tails [3], the use of

thick apertures in the probe-forming system to reduce the emitted hard x-ray flux [4] and

finally the addition of non-beam defining apertures between the condenser aperture system

and the top of the objective lens pole piece [5]. Figure 1(b) illustrates the use of these

solutions to reduce the intensity of the uncoil imated radiation. The details of these

modifications can be found in the paper by Bentley et al . , (1979) [2].

It is pertinent to note the fluorescence effects by uncollimated radiation have been

reported in every transmission or scanning transmission electron microscope in which the

interfacing of an EDS system for quantitative analysis has been reported. The experimental

evidence includes the following instruments:

Philips EM-400 FEG (Carpenter and Bentley, 1979) [6]

Philips EM-400 (Clarke, 1978) [7]

Philips EM-301 (Geiss and Huang, 1975) [8]

Philips EM-300 (Shuman et al . , 1976) [3]

JE0L 100-B (Joy and Maher, 1976) [4]

JEOL 100-C (Kenik and Bentley, 1977) [9]

JE0L 100-CX (Headley and Hren, 1978) [10]

354
JEOL JSEM-200 (Zaluzec and Fraser, 1976) [11]

VG HB-5 (Fraser, 1979) [12]

Siemens ST-100F (Oppolzer and Knauer, 1979) [13]

Hitachi H-700 (Fiori and Joy, 1979) [14]

Hitachi HU-1000 (Zaluzec et al . , 1978) [15]

Although this discussion may be taken as discouraging for x-ray microanalysis in an AEM, it is important to point out that the situation is correctable. In the worst case reported for a TEM/STEM instrument it was possible to reduce the "in-hole" spectrum to £2

percent of the measured spectrum from a 100 nm thick region of an appropriate test speci- men [16]. Initial studies on an HVEM indicate that the situation in this case is also

correctable [15].

Figures 2 and 3 show an example of the reduction of the in-hole spectrum to an acceptable level (the term "acceptable level" will be defined shortly) in a modified AEM. The

data presented to illustrate this effect and its solution was chosen from an extreme case.

It should not be taken as representative of all microscopes or even for newer versions of

the instrument used herein. The results presented here were obtained from a JEOL JSEM 200

TEM/STEM instrument ( circa 1973) capable of operating up to incident beam energies of 200

keV. The test specimen used was a 3 mm diameter single crystal of p-NiAl (50/50 at. %) of

the self-supporting disc variety and was mounted in a modified stage designed for x-ray

analysis [11]. All measurements were performed in the STEM mode at 200 keV with a probe

diameter of ^10 nm, and exciting an area of approximately 100 nm thick. The spectra shown

in figure 2 were obtained from the instrument in its "as-received" condition. Spectrum 2(a) was measured with the electron probe positioned on a 100 nm thick region of the specimen,

while 2(b) is the corresponding in-hole signal (both spectra were acquired for a detector

live time of 200 s). The fluorescing species in this instrument was identified as hard x- rays and reduction of the uncollimated radiation was accomplished using a non-beam defining

aperture (0.5 mm in diameter, 2 cm long) positioned just above the upper objective lens pole

piece (see figure 1(b)) [11]. Figure 3 shows the resulting changes in the measured specimen

(3(a)) spectra after this modification. By comparing these data with figure 2 one can see that the in-hole signal was reduced by nearly two orders of magnitude. For example, in

figure 2(b) the NiKa in-hole intensity makes up over 95 percent of that signal measured when

the probe is interacting with the specimen, while in figure 3(b) the in-hole NiKa intensity

is ~3 percent. Furthermore, a substantial change in the relative NiKa/AlKa intensity ratio

for the two cases is also apparent, indicating that in the initial experimental conditions

quantitative analysis were seriously compromised.

200 kev

am 2.00 energV (KEV)

6.00 8.00 Figure 2. Specimen (a) and in-hole (b) spectra

obtained in an "as-received" AEM (see

text for details).

0.00 2.00 (KEV) 355
Ea = 200 kev

Figure 3. Specimen (a) and in-hole (b) spectra obtained

in an AEM modified to reduce uncollimated

fluorescing radiation. Note the changes relative to figure 2.

The effect of this large in-hole spectrum on quantitative analysis is shown in greater

detail in figure 4. The measured intensity ratio of th NiKa/AlKa lines is shown as a function of specimen thickness for the unmodified instrument (curve A). Also plotted on this

figure is a theoretically calculated variation of the NiKa/AlKa intensity ratio (curve C).

These calculations were based purely on electron excitation of the sample and, as one would

expect, initially begins with the ratio being nearly constant and then slowly increasing as absorption effects become important [17]. In the as-received state of the microscope,

quantitative analysis using the thin-film electron excitation model is invalid. The initial

decrease of the Ni/Al ratio in curve A results from the increasing contribution of the

electron probe excited signal with specimen thickness relative to the nearly constant inhole signal. As this signal begins to dominate and finally exceed that produced by the

uncollimated radiation, the measured ratio will approach but not necessarily reach that

corresponding to electron excitation. The amount of the discrepancy will depend on the

magnitude of the "in-hole" signal and, as seen in this case, may be in error by a substantial amount. The ratio eventually goes through a relative minimum and then begins to increase as absorption effects become important. Although this particular example was a result of x-ray fluorescence, identical trends will be observed if electron tails are the

problem. The results of the reduction of the uncollimated radiation on quantitative

analysis are also shown in figure 4 (curve B) and one can see that the agreement is now reasonable. The discrepancy in the thinner regions of the foil is related to the detrimental aspects of contamination [18]. The deviations at the last few points investigated

can be attributed to errors in the thickness measurement as well as a neglect of beam

broadening effects in the calculations.

356
1 -I 1 1 1 1—i—i—i—j~r 1 1 1 1—i—i—

m

0-1 .2 A -6 -8 1.0 2- 4. 6. 8.10. THICKNESS 1pm)

Figure 4. The effects of an in-hole signal on the measured

intensity ratio in a NiAl alloy as a function of

specimen thickness. A - as-received instrument;

B - modified instrument; C - calculated variation

based on electron excitation model.

Ideally, one would like to completely eliminate the in-hole signal in order to perform

quantitative analysis. Realistically, this may be extremely difficult. The pertinent

question to then ask is: At what level does one reach an acceptable in-hole signal for

which the errors in quantitative analysis will be small and hence correctable or negligible? There is obviously no general answer to this question, principally due to the wide

range of materials as well as AEM instruments used in the scientific community. The following general criteria is a reasonable definition of "acceptable" in-hole signal; however

individual investigators may wish to propose more stringent requirements. When the

integrated intensity (above background) of each characteristic x-ray line recorded during an "in-hole" measurement is less than ~2 percent of the intensity of its corresponding line

with the incident electron probe positioned on the thinnest area of the specimen to be

studied, then the in-hole signal is acceptable. It is assumed that both spectra are measured under identical operating conditions which include: constant geometry, incident

beam energy, emission current, and detector "live time". For example, typical metallurgical

specimens usually involve the analysis of regions 100 - 500 nm thick; thus, if the in-hole

signal is <2 percent of the signal measured from the specimen where the probe is exciting

a region <1 00 nm thick, then one can consider this as an acceptable "in-hole" spectrum for

all measurements when the specimen thickness is > 1 00 nm.

An alternative definition of an acceptable in-hole signal has also been suggested

[19]. In this definition, when the peak intensity of each characteristic x-ray line in the

in-hole spectrum is less than the average continuum intensity generated by the incident

probe exciting the region of interest of the specimen, then the in-hole signal is deemed

neglibible. Under an ideal set of experimental conditions, this average specimen bremsstrahlung criterion and the previous definition based on a fraction of the integrated peak

intensity are in fact related. This is due to the fundamental relationship between the

generation of characteristic and continuum x-rays in a thin specimen by electron excitation

[17]. The definition of acceptability based on integrated peak intensities in this author's

opinion is a more suitable choice for several reasons. First, by using the integrated peak

intensity criterion one is comparing the in-hole signal to the actual quantity of interest

for analysis rather than an indirect measure using the continuum. Furthermore, the specific

experimental conditions used - namely, incident beam energy and detector-specimen geometry - have more pronounced effects on the measured continuum intensity [17] than on characteristic x-ray emission. Finally, the detected continuum intensity is extremely sensitive to specimen contamination effects [18,3]. It was shown that the accumulation of foreign

material on the specimen surface results in a net increase in the generated continuum intensity with neglibible effects to characteristic emission. Based on these considerations,

these two criteria can easily yield different results. For example, consider the spectra

shown in figure 3. The net peak intensity of the in-hole signal for the NiKa line (^7.5

keV) in figure 3(b) is 28 counts, while the corresponding average specimen bremsstrahlung

under this peak (figure 3(a)) is 35 counts. This result implies (using the average

bremsstrahlung criterion) that the in-hole signal has reached an acceptable level and can be

357
neglected. On the other hand, the ratio of the integrated peak intensities of NiKa for the

in-hole/specimen spectra yields a value of 0.035, thus exceeding the integrated peak

intensity criterion. The errors involved are not extremely large; however, it is ultimately

up to the investigator to propagate the errors through the quantitative analysis schemes

and clearly the lower the in-hole signal the better.

It should also be re-emphasized that the thinner the region of interest the more pronounced is the detrimental effect of the in-hole signal. This can be simply understood

by recalling that the in-hole signal is, to a first approximation, independent of the

position of the incident electron probe. The specimen-related signal is, on the other hand,

directly related to specimen thickness. Thus, an in-hole signal which may be unacceptably

high for a region of the specimen ~50 nm thick may be acceptable at a thickness of MOO

nm. There are, however, compromises in analyzing thicker regions of a specimen since

bream-broadening effects will ultimately decrease the attainable spatial resolution, as well

as the need to include absorption effects in quantitative analysis schemes [20].

Even though we have defined an acceptable in-hole signal it is still important to correct all subsequent specimen measurements for the small but still detectable in-hole

spectrum. This is particularly relevant during analysis of heterogeneous specimens since

any residual in-hole peaks (which reflect the average specimen composition) may be errone- ously interpreted as trace elements in the inhomogeneous specimen being studied. To a good

approximation the specimen measurement may be corrected by the simple subtraction of the

reduced in-hole spectrum for each successive specimen measurement [18]. An important

assumption in this case is that the in-hole signal essentially remains constant over the same period of machine time as the sample measurement. Furthermore, it is tacitly assumed

that the locally produced uncoil imated radiation resulting from the interaction of the

electron probe with the sample itself (i.e., secondary and backscattered electrons, specimen

generated characteristic and continuum x-rays, etc.) are important only to second order.

The first of these assumptions is usually the most important, and it is assumed that the

remotely produced uncoil imated radiation is directly related to the emission/probe current

which is taken to be constant during both sample and in-hole measurements. This is a good

approximation for stable thermionic guns; however, for AEM instruments using field emission

sources, particularly cold field emission, this may be quite inaccurate. In such

instruments, the emission current can vary substantially and the data acquistion time

should be controlled by a measurement of the integrated electron dose rather than using

the conventional live-time corrections [12].

Four points are important to remember when applying this subtraction correction for

quantitative analysis. First, the in-hole spectra will change from one specimen to another;

thus, it must be monitored for each sample. Second, for each specimen measurement, one should acquire a corresponding in-hole spectrum to ensure that long-term changes in the

emission/probe current do not invalidate the subtraction assumptions. Third, for the case

of trace element analysis, extreme care must be taken to ensure the statistical validity of

minor peaks which may be detected both in specimen and in-hole data. Fourth, it is essential to stress that the in-hole signal must be first reduced before subtraction; otherwise,

additional errors may be introduced into the data.

Figure 5 is an example of the improper subtraction of the in-hole signal to correct for

uncollimated fluorescence effects. The curve labeled (A) in this figure is a measurement of

the NiKa/AlKa intensity ratio as a function of specimen thickness in the unmodified instrument described previously. Curve (C) is the result of a simple subtraction of the in-hole

data from the data of curve (A) without first reducing the in-hole intensity. The middle

curve (B) is the result of first reducing the in-hole signal to an acceptable level and then

subtracting it from each successive sample measurement. The error bars indicated on each curve were taken from the specimen data after subtraction of the in-hole signal.

358
E - 200 kev

1 . 1—i

1

1

1

1

I

' ' 1—'—i

' i 0 .1 .2 .4 .6 .8 1.0 2 4 6 8 THIC K NESS (ym )

5. Subtraction of the in-hole signal to correct for uncoil imated fluorescence.

A - as-received instrument; B - modified

instrument with reduced in-hole signal

subtracted; C - erroneous subtraction of

in-hole signal without reduction of inhole signal

.

Up to this point we have considered the problem of uncollimated fluorescing radiation

in commercially available AEMs with incident beam energies <200 keV. There is considerable

interest in employing High Voltage Transmission Electron Microscopes (HVEMs) for x-ray

microanalysis where the incident beam energies are >1000 keV. This interest stems from the

potential advantages in thin-film x-ray analysis using the highest energy electron probes

available. One of these advantages is the decrease in the detrimental beam broadening

effects with increasing electron energies [26,22], which serves as the ultimate limit to the

spatial resolution of thin-film analysis. Secondly, theoretical calculations [17,23] as well as experimental measurements [15,25] have shown that the characteristic peak-tobackground (P/B) ratio in thin specimens increases with incident beam energy. This result

becomes important when considering the mass sensitivity of x-ray analysis which is directly

related to the measured P/B ratio. Figure 6 illustrates the variation in the calculated P/B

ratio for Ka x-rays from thin specimens of nickel, calcium, and aluminum as a function of

the overvoltage (U

Q

) for the geometry found in most AEMs (i.e., the detector axis perpendicular to the electron-optical axis). Here, the overvoltage has its conventional definition,

U

Q = E0/Ec

. that is, the ratio of the incident probe energy (E

Q

) to the critical excitation

energy (E

c

) of the x-ray line in question. From this figure one can see that the calculated

P/B ratio for nickel at 1000 keV is ~2.5 times the value at 100 keV. Combining these

factors with the relatively large pole piece gaps and the availability of a wide variety of

in situ experimental stages (heating, cooling, straining, etc.) makes an HVEM a potential

candidate for real-time microanalytical measurements during controlled phase transformation

studies.

Figure 6. Calculated Ka peak-tobackground for thin speci- mens of nickel, calcium,

and aluminum as a function

of overvoltage of incident

electron beam energy.

359
As in the case of conventional AEMs the specific details of the probe- forming lens

system of the HVEM will have a great influence on the magnitude of any uncoil i mated radiation present in the instrument. Unfortunately, there have been very few systematic studies

made on the optimization of an HVEM for x-ray analysis [15,24]. As such at the present time

it is difficult to generalize this discussion to cover all HVEMs; therefore, the remainder

of this text will concentrate on the author's own experience with a Hitachi HU-1000 HVEM at

ORNL.

The source of uncoil imated radiation in this instrument is fundamentally the same as in

the lower voltage AEMs; however, the problems it produces are different. In addition to fluorescing the specimen and producing characteristic x-ray emission, there is also a more serious problem in an HVEM based system - namely, the detection of an anomalously high

continuum intensity in all measured spectra. This continuum is most likely a result of a large high energy x-ray flux produced by the interaction of the 1000 keV electron beam and

the various beam defining apertures in the probe-forming lens system of the microscope.

This hypothesis is substantiated by the detection of the continuum x-ray spectrum shown in

figure 7 which was obtained by operating the instrument in the TEM mode using a probe

diameter of 0.75 urn and a probe current of 2.5 x 10 -9A, and positioning the x-ray detector

physically outside of the microscope at the nominal height of its insertion. These results

demonstrate quite clearly that there is a substantial x-ray flux at this level, which

undoubtedly is greater inside the electron-optical column. Figure 8(a) is a typical x-ray

spectrum from a self-supporting TEM disc specimen of 0-NiAl acquired after the detector has

been appropriately interfaced to the microscope while figure 8(b) is the corresponding inhole spectrum. These and all subsequent spectra were acquired at an incident electron

energy of 1000 keV, a probe current of 2.5 x 10-9

A, a probe diameter of ^0.75 urn, and a detector live-time of 400 s. One can see that the continuum distribution obtained during

the excitation of the p-NiAl test specimen (8(a)) is essentially identical to that obtained

during the in-hole measurement. Furthermore, characteristic fluorescence of the specimen

by the uncollimated radiation is low (<10% for the NiKa line) and exhibits preferential

excitation for higher energy x-rays indicating the presence of a hard x-ray flux rather than

electron tails.

128-1

Figure 7. Measured continuum intensity external to microscope column for a Hitachi HU-1000

HVEM instrument operating at 1000 keV

(see text for details).

Energy-keV

Since most Si (Li) detectors are enclosed in relatively thin stainless steel casings,

the first step toward reducing the magnitude of the detected bremsstrahlung flux consists of

increasing the shielding around the detector. This can be accomplished, for example, by

wrapping thin lead sheets around the detector head. Figures 8(c) and (d) show the resulting

changes in the specimen and in-hole spectra obtained by the addition of 3 mm lead to this

part of the detector. The overall continuum intensity drops by nearly a factor of two relative to figures 8(a) and (b). After this modification, it was no longer possible to increase shielding at this position and therefore a remote non-beam-defining collimator was added between the condenser and objective lenses. Modifications to the beam-defining

aperture at this time were not considered feasible due to their inaccessibility. The last

set of spectra (8(e) and (f)) were obtained after the insertion of lead non-beam-defining

collimators (1/2 mm bore diameter, 2 cm long) above the upper objective pole piece and the

addition of 2 cm of lead sheet over the remaining cross section; furthermore, the detector

collimator thickness was also increased to 1.5 cm from the 0.4 cm previously used. These

modifications, which are shown schematically in figure 9, successfully reduced the original

in-hole signal by ^90 percent; however, the continuum distribution between the specimen

360
and in-hole signal is still essentially indistinguishable, indicating that additional

shielding is still required. As well as reducing the anomalous continuum intensity, one can

also observe in spectra 8(a), (b) and (c) an apparent improvement in the energy resolution

of the system as each modification was implemented. This is also consistent with a large

high energy x-ray flux degrading the detector system performance.

1K 1

K

i i

1 1 r— 0 5

i i 1-

(B)

Oj r1 0

-i 1 1 1 1 r 5 10

1K -, (C) (D)

1 0

1K (E) 1K (F)

1 0 1 0

Energ y- keV Energy - keV

Figure 8. Variation in specimen and in-hole spectra recorded in an HVEM as a function

of modifications to the electron-optical column to reduce uncoil imated

radiation, (a), (b) unmodified instrument, (c), (d) after installation of

shielding around detector head, (e), (f) after installation of non-beamdefining collimator and additional shielding between the condenser and

objective lens (see text for details).

361
Figure 9. Schematic diagram of modifications to electron-optical column

of an HVEM to reduce the intensity of uncoil i mated radiation

reaching the specimen area (see text for details).

In order to continue it becomes necessary to decrease the incident probe current from

2.5 x 10"9 A to 2.5 x 10" 11 A which reduces the magnitude of the uncoil imated radiation to nearly manageable levels as shown in figure 10. Here in in-hole continuum signal (figure

10(b)) is <25 percent of the corresponding specimen continuum (figure 10(a)) and it was possible to qualitatively assess the proposed advantages of HVEM-based analysis on the P/B

ratio. By subtracting this reduced in-hole signal from the corresponding specimen measurement, an increase relative to 100 keV measurements of 2.4 and 1.6 times was experimentally

measured for the NiKa and AlKa lines, confirming qualitatively the expected increase. The

100 keV data used for comparison was obtained from the same specimen in a JE0L JSEM 100 C TEM/STEM instrument operating at nearly the same geometry. The characteristic fluorescence

problem in the HVEM is also reduced after these modifications as the in-hole signal for the

NiKa line is now ~5 percent and is essentially zero for the AlKa line relative to the

specimen measurement. The feasibility of replacing the remote lead shielding by appropriately machined depleted uranium is being investigated as well as possible modifications to the condenser aperture assembly to further reduce the magnitude of the hard x-ray flux

closer to its most probable source.

0RNL-DWG 79-11916

512 (A) 12 8. (B)

O

O 5 Energy - keV

1 O O 5 Energy-keV

1 O

Figure 10 Specimen (a) and in-hole spectra (b) measured after

reduction of bremsstrahl ung production by decreasing

incident probe current and addition of non-beam-defining

collimators (note scale changes between the two spectra).

362
4. Summary

The presence of uncol 1 imated fluorescing radiation in an Analytical Electron Microscope

is the most serious problem in the application of x-ray microanalysis to thin electron

transparent specimens. The magnitude of this radiation varies from one instrument to another, and its detrimental effects to a particular analysis can be judged easily by

performing an "in-hole" experiment on the specimen under investigation. Should the intensity of any characteristic x-ray peak in this in-hole spectrum be greater than 2 percent of

the intensity of the same line when performing a specimen measurement, then steps should be

taken to reduce the magnitude of the uncoil imated radiation by modifications to the

electron-optical column. Once the uncollimated radiation is reduced to an "acceptable

level", each subsequent specimen measurement can be corrected to a good first approximation

for any residual, yet still detectable, in-hole signal by the simple subtraction of the two

spectra.

The author would like to acknowledge the invaluable assistance and discussions with

Drs. E. A. Kenik, J. Bentley, and R. W. Carpenter, as well as the support of the

E. P. Wigner Fellowship at ORNL under contract no. W-7405-eng-26, U. S. Department of

Energy, with the Union Carbide Corporation.

References

[1] Hirsch, P., Howie, A., Nicholson, R. , Pashley, D. , Whelan, M. , Electron Microscopy of

Thin Crystals, Krieger Pub. Co. , New York (1979).

[2] Bentley, J., Zaluzec, N. J., Kenik, E. A., Carpenter, R. W. , SEM/1979/II, Washington,

D.C., 0. Johari, ed. , Chicago Press (1979), p. 581.

[3] Shuman, H. , Somlyo, A. V., Somlyo, A. P., Ultramicroscopy , 1_> 317-339 (1976).

[4] Joy, D. C. , Maher, D. M. , Analytical Electron Microscopy: Report of a Specialist

Workshop, Cornell University (August 1976), Materials Science Center, Clark Hall,

Cornell University, Ithaca, New York 14850.

[5] Zaluzec, N. J. and Fraser, H. L. , Phys . E. , 9, 1051-1052 (1976).

[6] Carpenter, R. W. and Bentley, J., SEM/1979/I, Washington, D.C., 0. Johari, ed. , Chicago

Press (1979), p. 153.

[7] Clarke, D. R. , SEM/1972/I, 0. Johari, ed. , Los Angeles, Chicago Press (1978).

[8] Geiss, R. H. and Huang, T. C. , X-ray Spectr . , 4, 196-201 (1975).

[9] Kenik, E. A. and Bentley, J., 35th Annual Proceedings of EMSA, G. W. Bailey, ed.

,

Claitor's Publishing Company, Baton Rouge (1977), p. 328-329.

[10] Headley, T. J. and Hren, J. J. , Ninth Inter. Conf. on Electron Microscopy,

J. M. Sturgess, ed. , Microscopical Society of Canada, Toronto (1978), Vol. 1, pp. 504-

505.

[11] Zaluzec, N. J. and Fraser, H. L. , Analytical Electron Microscopy: Report of a Specialist Workshop, Cornell University (August 1976), pp. 118-120, Materials Science Center,

Clark Hall, Cornell University, Ithaca, New York 14850.

[12] Fraser, H. L. and Woodhouse, J. B. , Electron Microscopy: Proc. of 2nd Workshop,

Cornell University, Ithaca, New York (1978), p. 191.

[13] Oppolzer, H. and Knauer, U. , SEM/1979/I, Washington, D.C., 0. Johari, ed. , Chicago

Press (1979), p. 111.

[14] Fiori, C. and Joy, D. C. , National Institute of Health, Bethesda, Maryland, private

communication (1979).

363
[15] Zaluzec, N. J., Kenik, E. A., Bentley, J., Analytical Electron Microscopy: Proc. of

2nd Workshop, Cornell University (July 1978), p. 179.

[16] Zaluzec, N. J., Ninth Inter. Conf. on Electron Microscopy, J. M. Sturgess, ed. , Microscopical Society of Canada, Toronto (1978), Vol. 1, p. 548-549.

[17] Zaluzec, N. J., An Analytical Electron Microscopy Study of the Omega Phase Transformation in a Zirconium-Niobium Alloy, PhD Thesis, University of Illinois, 1978; also

published as Oak Ridge National Laboratory Report ORNL/TM-6705. Copies available from

NTIS, U. S. Department of Commerce, Springfield, Virginia 22161.

[18] Zaluzec, N. J. and Fraser, H. L. , 8th Inter. Conf. on X-ray Optics and Microanalysis,

Boston (1977), Beaman, Ogilivie, Wittry, eds. , Science Press, Princeton, New York

(in press).

[19] Goldstein, J. E. and Williams, D. B. , SEM/1978/I, 0. Johari , ed. , SEM, Inc., AMF O'Hare,

Illinois (1978), pp. 427-434.

[20] Zaluzec, N. J., Introduction to Analytical Electron Microscopy, Hren, Joy and Goldstein,

eds. , Plenum Press, New York (1979), chapter 4.

[21] Goldstein, J. I., Costley, J. L , Lorimer, G. W. , Reed, S. J. B. , SEM 1 (1977), 315.

[22] Hutchings, R. , Loretto, M. J., Jones, L. P., Smallman, R. E. , Analytical Electron

Microscopy, Proc. of 2nd Workshop, Cornell University, Ithaca, New York (1978) , p. 166.

[23] Joy, D. C. , Maher, D. M. , SEM/1977, 0. Johari, ed. , IITRI, Chicago, Illinois (1977),

p. 325.

[24] Cliff, G. , Nasir, M. J., Lorimer, G. W. , Ridley, N. , 9th Inter. Conf. on Electron

Microscopy, J. M. Sturgess, ed. , Microscopical Society of Canada, Toronto (1978), pp. 540-541.

[25] Zaluzec, N. J. and Fraser, H. L. , Analytical Electron Microscopy: Proc. of the 2nd

Workshop, Cornell University, Ithaca, New York (July 1978), p. 122.

[26] Goldstein, J. I. and Williams, D. B. , SEM/1977/I, 0. Johari, ed. , Chicago Press, IIT Research Institute, Chicago, Illinois (1977), pp. 651-662.

36/1
National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

SPECTRAL PROCESSING TECHNIQUES IN A QUANTITATIVE ENERGY DISPERSIVE

X-RAY MICROANALYSIS PROCEDURE (FRAME C)

R. L. Myklebust, C. E. Fiori* and K. F. J. Heinrich

National Measurement Laboratory

National Bureau of Standards

Gaithersburg, MD 20234

^Division of Research Services

National Institutes of Health

Bethesda, MD 20205

Abstract

The correction procedure FRAME C was developed for quantitative

electron probe microanalysis with an energy dispersive detector.

Algorithms incorporated to compute the background due to the x-ray

continuum and to correct for overlaps due to x-ray peaks will be

described in detail. The method employed to determine the detector

efficiency for the x-ray continuum calculation will be discussed as well

as the computation of the continuum.

1. Introduction

The application of quantitative analysis procedures to x-ray data obtained with an energy-dispersive x-ray spectrometer (EDS) requires two computations not normally needed for

wavelength-dispersive spectrometer (WDS) data. First, since the peak-to-background ratio in

EDS is approximately an order of magnitude less than the peak-to-background ratio in WDS,

the background subtraction in EDS is of much greater importance than in WDS [l] 1

. Second,

there must be a method of resolving overlapping peaks in the EDS since the resolution of the

EDS system is much poorer than in a WDS system and thus, peak overlaps frequently occur in

practical analytical situations [2].

We will describe the methods used in the NBS theoretical matrix correction procedure,

FRAME C, for handling these two problems [2].

1.1 Background prediction

The problems associated with calculation of the background in the EDS spectrum can be

divided into three parts: (1) the continuum x-rays generated within the specimen, (2) the

absorption of the continuum x-rays by the sample, and (3) the losses of x-rays within the

Si(Li) detector (detector efficiency).

1.2 Generation of continuous radiation

Kramers proposed a relation describing the intensity distribution of the continuum

which, as a function of the energy of the emitting photons, is

I EAE = k

EZ(E

Q

- E)AE

(

1 Figures in brackets indicate the literature references at the end of this paper.

365
In this equation, I^AE is the average energy of the continuous radiation, produced by one electron, in the energy range from E to E + AE. The constant k

£ is Kramers' constant [3]

transformed for the energy scale, Z is the average atomic number of the target, and E Q

is

incident electron energy in keV. As shown by Green [4], varies moderately with Z, but

this effect is not germane to the present communication. The value of the photon energy,

E (keV), ranges from zero to E Q

. Kulenkampff [5] obtained measurements which, except for a small term involving Z

2

, agreed with equation 1. The probability, IAE, that one electron

will produce one photon of energy between E and E + AE is equal to

IAE = k

EZ AE

(

The number of photons, N(E), within the energy range E to E + AE, produced in time t with a beam current i (electrons per second) and observed by a detector of efficiency P

p

, subtending a solid angle L, is equal to

N(E) = ^ itf

E PE lAE

h UfE

P

E kE! AE

(3)

Here, the term f

E (absorption factor) denotes the probability of absorption of a photon of

energy E within the target. In equation 3, the term Q/4n is equal to the fraction of

photons emitted towards the detector only if the generation of the continuum is isotropic

in thick targets. This assumption is a reasonable approximation, although some degree of

anisotropy does exist [5-7]. If we define a = (Q/4n)itk

EAE, then we obtain

N(E) = af

E

P £Z (4)

Li fshin, Rao-Sahib and Wittry, and Hehenkamp and Bocker [7-10] have shown that the

energy dependence of N(E), expressed by equation 4, is only approximate. For a detailed

discussion of the subject, the reader is referred to reference [11] by Rao-Sahib and Wittry

who also show that the atomic-number dependence of N(E) is more complex than indicated by

Equation 4.

The residual correction has been carried out in at least two ways: (1) Applying more refined theoretical calculations, [11], and modifying the exponents, from unity, on the

energy term and atomic number Z in Kramers' equation (7,8); or (2) Using an additive

energy-dependent correction term in equation 4 as proposed by Ware and Reed [12] as follows:

N(E) = a f

E

P

EZ

E

o

" F(E)

(5)

where F(E) is taken from an empirically determined table. The authors indicated that for

continuum photon energies above 3 keV, the additive energy term was unnecessary. Li f shin

[10] empirically developed and tested a more general relation for N(E):

N(E) = f

E PEZ + b (E„ " E)'

(6)

in which b is a fitting factor, and the remaining terms are as defined for equation 4.

366
We have found that it is possible to determine a and b for any target by measuring N(E)

at two separate photon energies, and solving the resulting two equations in two unknowns.

This procedure presumes that both f^ and in equation 6 are known. Since the determination of N(E) is made from measurements on the target of interest only, the inaccuracies of

Kramers' equation with respect to atomic number do not affect the proposed method.

The points chosen for measurement of N(E) must be free of peak interference and detector artifacts (incomplete charge collection, pulse pile-up, double-energy peaks, and escape

peaks) and they should be the same for standard and specimen. With moderate count rates

and reasonable counting statistics, these limitations are not very restrictive,

1.3 The absorption factor, f^

For characteristic primary x-ray photons, the absorption within the target is taken

into account by a factor called fp or f(x). This factor has been expressed by [13] as

1/fp = (1 + ayx)

2 (7)

1 1 where v is E - E„ , E„ is the excitation potential in keV for the shell of interest,

o q q and x = u'csc i|> (u is the mass attenuation coefficient of the target for the line of

interest, and ^ is the x-ray emergence angle with respect to the specimen surface). The

presently used value for the coefficient a is:

a = 1.2 x 10" 6 (g/cm2 )/kV

] - 65

(8)

If it is assumed that the generation of the continuum has the same depth distribution

as that of characteristic radiation, which Monte Carlo calculation suggests is a reasonable

assumption [14] and if E = E , then equation 7 can be modified to l/f^O'a^-E1 - 65

*,.)

2

(9)

where x^, = esc is the mass attenuation coefficient of the target for continuum

photons of energy E and f^ is the absorption factor for these photons. Similar assumptions

and expressions have been used by other authors [8,10,12].

The criterion for assessing the usefulness of equation 9 is that it should remove the

effects of all absorption edges over the energy range of interest.

As pointed out by Ware and Reed [12], the absorption term f^ is dependent upon the

composition of the target. Consequently, when the composition of the target is unknown,

it is necessary to include equation 6 in an iteration loop, as shown in figure 1.

367
Assume Initial

Concentrations

Subtract

Background

Clear Overlaps

from Measured

Lines

Output

Results

Figure 1. Short block diagram showing the

principal operations within the FRAME C iteration loop.

368
1.4 The detector efficiency,

The continuous radiation emitted from the target towards the detector traverses a beryllium window, typically on the order of 8 urn thick, a surface-barrier contact (< 20 nm

of Au), and an inactive layer of silicon extending 200 nm or less into the detector. Then,

the radiation enters the active (intrinsic) region of the detector which has a thickness

typically between 2 and 5 mm. The absorption losses in the beryllium window, the gold, the

silicon dead layer, and the transmission through the active silicon zone can therefore be

calculated as follows:

P

E = exp - [(j(Be,E)tBe

+ u(Si,E)t°. + u(Au,E)t

Au

][l - exp - (uSi,E)t

$i )] (](

In this equation, tg. , t<-.j are the thicknesses (g/cm2

) of the silicon dead layer, and of

the active detector region, respectively. tg

g and t^ are the thicknesses (g/cm2

) of the

beryllium window and the gold contact layer. The mass attenuation coefficients of beryllium, silicon and gold at the energy E, u(Be,E), u(Si,E) and u(Au,E) are calculated as described in reference 15.

Since sufficiently accurate values were not available from the manufacturer, estimates

of the thicknesses of the beryllium window and the dead layer of the detector were adjusted

to optimize the fit between the calculated and the experimental spectra at energies below

2 keV.

The detection efficiency and, hence, the shape of the recorded background are also

affected by incomplete or slow charge collection, by Compton scattering and by the escape

mechanism for Si K photons [16]. However, for radiation below 25 keV, the Compton effect

is negligible. Incomplete charge collection can produce minor increases in the background

at energies less than major peaks and, thus, affect the accuracy of trace analysis. The

effects of the escape mechanism are negligible for our purposes. All of these effects

should be considered, however, where the detector efficiency must be determined for absolute

measurements. It should also be noted that the theoretical curves shown in figures 2 and

3 do not take into account the effects of limited detector resolution.

369
ENERGY (keV)

Figure 2. Spectrum of Kakanui hornblende with the fitted x-ray continuum background. The

large open triangles are the fit points.

370
40k

30k

20k

10k

No

T»

Co

15K

Po

Co A

4 6

ENERGY (keV)

8 10

Figure 3. Spectrum of Kakanui hornblende shown in figure 2 but

with continuum removed.

1.5 Line interferences

The original EDS version of the FRAME program, FRAME B, [17] required that all analytical x-ray peaks be clearly resolved from other x-ray peaks in the spectrum. This condition

often cannot be met in energy-dispersive spectra. We have developed a simple method [18]

to overcome this limitation which involves integrating peaks between the energy limits of

the regions of interest for each analytical x-ray line. One x-ray line from each element

is used for the analytical procedure; however, all lines are potential interferences which

must be accounted for if they fall within the region of interest of an analytical line,

i.e. , one that has been assigned a region of interest.

371
Given an x-ray peak near a region of interest, the fraction of this peak which falls

within the region depends only on the relative positions of peak and region of interest,

and the width resolution (a) of the Gaussian peak but not on the composition of the

specimen. Therefore, this factor can be computed once early in the procedure and stored

for all subsequent measurements. For simplicity we will develop the argument for two unresolved peaks, A and B (see figure 4). is the center of peak A and Eg is the center

of peak B.

E l EA E2 E3 EB E4

channel Dumber

Figure 4. Schematic representation of two overlapping Gaussian shaped peaks. Shaded areas

are the overlaps of each peak into the region of interest of the other peak.

We can calculate the width resolutions and a

g as a function of E

ft and Eg [18] if the

detector resolution has been measured for a peak at any energy (e.g., Mn Ka). Since peak B contributes counts to the region of interest for peak A and vice versa, the correction

procedure must be iterative.

372
The number of counts due to peak A at any channel whose energy is E. is:

N

i = A

A exp 1/2

(11)

Consequently, the integrated counts N

ft between the limits E-j and which define the

region of interest of element A, are:

E

l

1/2

EA

" E

i dE.

(12)

and the integrated counts between the limits E 3 and E^ of the region of interest of element

B are:

NAB = rexp -1/2

EA

" E

i dE. (13)

The interference factor is defined as:

H

N ab " n:AB

(14)

This factor is multiplied by the number of counts measured in peak A between the limits Ej

and E 2 to obtain the number of counts contributed to the region of interest of peak B by

peak A.

A different type of interference occurs when a peak from element A which is not used

for data reduction falls within the region of interest selected for element B (figure 5). In this case, the intensity of the interfering peak from element A must be obtained from

the count rate within the region of interest of the analytical peak for element A. If the

analytical line and the interfering line are both generated from ionization of the same subshell, their generated intensities are related by their respective relative transition

probabilities. A transition probability is the intensity of the line of interest divided

by the sum of the intensities of all other lines arising from the ionization of the same

subshel 1

,

* = T AKB Kp

/

(

X AKp

+ I AKa (15)

373
374
The measured intensities are also affected by the absorption of the x-rays within the speci- men and the efficiency of the detector. For example, if the interfering line is a Kp peak

of element A, then to relate the interference of AKp upon another peak BKor to the measured

intensity of AKa, the overlap factor of AKp into the region of interest of BKa is obtained

by:

H _ ^AKg m f P(AKa)

P E(AKa) m ^JaB AB ! AKa

f P(AKp)

P E(AKp) NA (16)

where ^Kp/I^Ka1S derived from the transition probabilities, fp is the absorption within

the specimen for each line energy, and is the detector efficiency at each energy. The

transition probabilities for the Kp and Ka x-ray lines of elements of atomic number 15

through 30 have been determined by us (see Table 1) as well as by others [19,20] and are better known than those of the L or M x-ray lines.

Table 1. Relative transition probabilities for Kp x-ray lines [19].

'c Number Kp/(Ka + Kp)

15 .050

16 .063

17 .085

18 .097

a

19 .104

20 .113

21 .114

a 22 .114

23 .116

24 .116

25 .119

26 .119

27 .120

28 .121

29 .123

30 .128

interpolated value.

We have estimated the transition probabilities for the L and M series of x-ray lines up to

10 keV. These estimates were made on five pure elements measured at three different beam

voltages. Since the L-lines (and the M-lines) were all related to the Lo^ line (or the Ma

line), these are not the true transition probabilities. However, the transition probability is incorporated within this factor.

To compute overlap factors for all the L x-ray lines, we relate the intensity of each

L line to the La x line (the measured line). We must, however, take into account that the

ratios of line intensities of the L spectra vary with excitation potential. This is due

to the fact that these lines arise from three subshells each of which has a different

critical excitation potential. To relate the intensity of any line to the measured La x line, we compute an intensity factor which is the intensity of the line "i" divided by the

intensity of the La x line (I. /I. ). To introduce an energy-dependent term into this

L

i Lal factor, we have integrated the expression:

(17)

375
where s is the mass thickness along the electron path (g/cm2 ), in the same manner as Philibert and Tixier [21] to obtain the number of ionizations, Y , , in the level nl.

nl

|u

o

- 1 - J

lnW[li(U

Q Y = * W) - li(W)]|nl AM ) "o " W

, ""t ' ,V V -vyj/ (i 8)

M and N/A are constant and can be eliminated since we are calculating the ratios of the

x-ray lines for the same element. U is the overvoltage E /E , where E , is the ionization * o

3 o nl nl energy of the edge nl , W is 1166 E nl /J where J is the mean ionization potential, U

Q

W is

1166 EQ/J and li is the logarithmic integral. We have fitted an expression which approximates the logarithmic integral (li(x) ~ 2.905 - 3.063) over the range of interest. The

number of electrons in each subshell , , is determined from basic quantum theory, and the

pertinent values for the subshells of interest are as follows:

Subshel

1

Number of 1

K 2 h

2

L2

2

L3 4 M

l 2 M

2 2 M

3 4

M

4

4 M

5 6

The ratios of lines arising from different absorption edges are then corrected for differences in overvoltage by multiplying each line by Y , (Equation 18). The total L line

intensity factor, for any L line, thus becomes:

L I. Y.

1 Lal (19)

The first step in computing the L overlaps is to determine the area of the Laj line

within its region of interest by subtracting the areas of all other L lines of the same element from the region of interest. This is done by calculating all the overlap factors

for these lines and stripping them out in the same manner as is used for overlaps from other

elements. Since the computed overlap factor for each L line is relative to the La 1 line

alone, this step is necessary. Overlap factors for the measured lines of other elements are then computed in the same manner as for K lines except that the overlapping area of an interfering peak is multiplied by that peak's intensity factor, S^. In the case of the

ratio Lofj/Ka, the ratio must be multiplied by the ratio of fluorescence yields, u^/u^, and

an atomic number dependent factor which is at least partly due to the Coster-Kronig transitions.

yL = 0.095Z - 1.69

(20)

376
M lines are treated in a similar manner. There are also provisions for determining

the intensity of an M line from an L line or vice versa. Since we were unable to find an appropriate cross section for the M-series of x-ray lines, we fitted an empirical expression to the generated Ma/Lor x ratio for three elements at five different operating voltages.

"34.17 30.03

Y

n1 yM

= exp

ln(U

p

) + 8.527 (21)

U is the product of the L overvoltage, E Q

/E

L

, and the M overvoltage, E Q

/EM

. Approximate

" 3 5 results can be obtained by this technique; however, the factors used are not well known at

this time.

In the previous discussion, we have assumed that the peaks have simple Gaussian

shapes. In reality, however, the peaks are distorted on the low-energy side by the effect

of incomplete charge collection. Consequently, the x-ray peaks observed by a Si(Li) detector are the convolutions of several contributions. Some peaks consist of several x-ray

lines of the same series (such as Ka x and Ka2 ). In most cases of multiple peaks (except

Ka 1)2 and KPi^.g), we compute the peaks separately for each x-ray line in the same manner

as the overlapping peaks from other elements. For the incomplete charge collection at the

low-energy shoulder of each peak, we have developed an empirical expression:

I c

i

= A

Aa(EA

" E i>

e *P[- b ( E A

' E i>]

(22)

in which a and b are energy-dependent coefficients fitted for a particular detector and the

other terms are as previously defined,

intensity from Equation 13.

A. was previously obtained as a measure of line

The remaining case of overlap that is computed is the interference of the escape peak

of a characteristic x-ray line with an analytical line. The expression used to predict the

number of counts in an escape peak based on the number of counts in the parent peak is that

due to Reed and Ware [22]:

k = 0.038e/(l - 0.038e)

esc (23)

where kgsc

is the integrated intensity ratio of the escape peak to the parent peak and

e = 1/2 1 - H(Si,Si)

u(Si ,EA

)

1 + M(Si,E

A

) u(Si ,Si)

(24)

u(Si,Si) is the mass absorption coefficient for SiKa in silicon and p(Si,E^) is the mass absorption coefficient for the parent peak in silicon. An overlap coefficient is then

computed in the same manner as previously described for a peak that has no region of interest. The standard deviation of the escape peak is that corresponding to its energy rather

than of the parent energy; k is the intensity ratio of escape peak to parent peak.

2. Conclusions

The effectiveness of the corrections is shown in Table 2 where several pure materials

have been analyzed for elements that are not present in the material. The last two columns

are the amounts of the absent element found with and without the overlap correction. The

results are satisfactory except when two measured peaks overlap almost completely (e.g.,

Ta + Cu in Table 2).

377
Table 2. Overlap corrections.

Specimen

0keVAnalyzed

for

Overlap

with

Mass fraction

found before

correction

Mass fracti<

found afte

correctio

MgO 15 Al MgKB .0026 . 0016

A1 20

3 15 Mg AlKa .0132 .0058

Si0

2 15 Al SiKa .0072 -.0010

Cr 15 Mn CrKB .1311 -.0024

Mn 15 Fe MnKB . 1045 . 0019

Fe 15 Co FeKB .0653 .0062

on<iU Zn LuKp . O^Oo r\ /i o Ci

. 0039

Zr 15 Bi ZrLB

2

, .0214 -.0113

Bi 15 Zr BiMa,MB, .0213 -.0009

Zn 20 W ZnKa .1329 .0010

Ta 20 Cu TaLa^ .1968 .0700

The regions of interest for Cu Ka and Ta La overlap. These lines are too close to be separated well by this method.

The electron beam incidence angle is 90° and the x-ray emergence

angle is 52.5°.

References

[1] Fiori, C. E. , Myklebust, R. L. , and Heinrich, K. F. J., Analytical Chemistry , 48 (1),

172-176 (Jan. 1976).

[2] Myklebust, R. L. , Fiori, C. E. , and Heinrich, K. F. J., FRAME C: A Compact Procedure

for Quantitative Energy-Dispersive Electron Probe X-ray Analysis, NBS Tech. Note 1106,

Sept. 1979.

[3] Kramers, H. A. , Phil . Mag . , 46, 6th series (July-Dec. 1923).

[4] Green, M. , PhD Thesis, University of Cambridge (1962).

[5] Kuhlenkampff , H. , Ann Phys . , 69, 548 (1922).

[6] Dyson, N. A. , Proc. Phy_s. Soc. , 73, 924 (1959).

[7] Rao-Sahib, T. S. and Wittry, D. B. , Proc. 6th Int. Conf. on X-ray Optics and Microanalysis, Tokyo, 1972.

[8] Rao-Sahib, T. S. and Wittry, D. B. , J. Appl . Phys . , 45, 5060, 5068 (1975).

[9] Hehenkamp, Th. and Bocker, J. , Mikrochim Acta Suppl . , 5, 29 (1974).

[10] Lifshin, E. , Proc. Ninth Annual Conference of Microbeam Analysis Society, Ottawa,

Canada, paper 53 (1974).

378
[11] Duncumb, P. and Reed, S. J. B. , Quantitative Electron Probe Microanalysis, K. F. J. Heinrich, ed. , U. S. National Bureau of Standards Special Publication 298, U. S. Government Printing Office, Washington, D.C. 20402, 1967, p. 133.

[12] Ware, H. G. and Reed, J. B. , J. Phys . E. , 6, 286-288 (1973).

[13] Heinrich, K. F. J. and Yakowitz, H. , Analytical Chemistry , 47 (14), 2408-2411 (1975).

[14] Reed, S. J. B. , X-ray Spectrometry , 4, 14 (1975).

[15] Yakowitz, H. , Myklebust, R. L. , and Heinrich, K. F. J., U. S. National Bureau of

Standards Technical Note 796, U. S. Government Printing Office, Washington, D.C. 20402 (1973).

[16] Freund, H. U. , Hansen, J. S. , Karttunen, E. , and Fink, R. W. , Proc. Int. Conf. on Radioactivity and Nuclear Spectroscopy (1969).

[17] Fiori, C. E. , Myklebust, R. L. , and Heinrich, K. F. J., Proc. 11th Annual Conf. of

Microbeam Analysis Society, Miami Beach, Florida (August 9-13, 1976) paper 12.

[18] Fiori, C. E. , Myklebust, R. L. , and Heinrich, K. F. J., Proc. 10th Annual Conf. of

Microbeam Analysis Society, Las Vegas, Nevada (August 11-15, 1975), paper 20.

[19] Heinrich, K. F. J., Fiori, C. E. , and Myklebust, R. L. , J. Appl . Phys . , 50 (9), 5589-

5591 (Sept. 1979).

[20] Smith, D. G. W. , Reed. S. J. B. , and Ware, N. G. , X-ray Spectrometry , 3, 149 (1974).

[21] Philibert, J. and Tixier, R. , in Quantitative Electron Probe Microanalysis , NBS Special

Publication 298, K. F. J. Heinrich, ed. , 13-33 (1968).

[22] Reed, S. J. B. and Ware, N. G. , J. Phys. E: Sci. Inst. , 5, 582-584 (1972).

379

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

THE EFFECT OF ELECTRON INCIDENCE ANGLE ON QUANTITATIVE ELECTRON PROBE MICROANALYSIS

J. D. Brown

Faculty of Engineering Science and

The Centre for Interdispl inary Studies in Chemical Physics

The University of Western Ontario

London, Canada

Abstract

<|) (pz) curves have been measured at 25 and 30 keV electron energies

and electron incidence angles from 60 to 90°. These data can be used to establish the effect of changing incidence angle on quantitative correction procedures and will be invaluable for comparison with Monte Carlo

predictions. None of the factors currently used to correct for inclined

electron incidence adequately compensates for this effect.

1 . Introduction

The usual ZAF procedure for converting x-ray intensities to composition involves separate correction for differences between unknown and standard in average atomic number, in

x-ray absorption and in fluorescence due to absorption of characteristic or continuum

x-rays. When measurements are made with non-normal electron incidence, electron backscattering and diffusion are modified. This results in a decrease in depth and a change in

shape and magnitude in the production of x-rays as a function of depth. To account for

these changes, several geometric factors have been proposed to modify the absorption

parameter x in the absorption correction [1-3] 1 and the backscatter factor R in the atomic

number correction [4,5]. Several authors [6,7] have commented on the inaccuracy of these

factors. Originally Castaing [8] proposed that quantitative analysis could be based on knowledge of the production of x-rays as a function of depth in the specimen. To measure

x-ray production, Castaing and Descamps [9] proposed a sandwich sample technique. Characteristic x-rays generated by electrons in a thin tracer layer of one element buried at

different depths in a second (matrix) element were measured at a fixed take-off angle i|> outside the specimen. From this angle, the overlayer thickness, and the mass absorption

coefficient, the measured intensity could be corrected for absorption to yield the x-ray

intensity generated in the thin layer. For normalization, the intensity from the identical

thickness of tracer layer isolated in space was measured. The ratio of intensity in the

thin layer in the specimen relative to the isolated layer plotted as a function of depth in

the specimen was defined as the <|> (pz) curve. If the <j> (pz) curves are known for both the

unknown and pure element standard, then the measured relative x-ray intensity K« can be

calculated from the equation:

/•OO Jo

K A = W

> (pz) exp (-u pz cosec i|i) dpz. (1)

A A r»

J o <|> A

(pz) exp (-pA pz cosec t|>) dpz.

where subscript S refers to the unknown specimen and subscript A refers to the pure

element A.

Figures in brackets indicate the literature references at the end of this paper.

381
is the weight fraction of A in the unknown and u is the mass absorption coefficient

for the characteristic x-ray line of element a used in the analysis.

In the absence of fluorescence effects, equation (1) is exact and includes both the

atomic number and absorption corrections. Fluorescence due to characteristic lines can be

simply calculated if the (pz) curves are known [6].

If the <|> (pz) curves are measured with an instrument in which the electron beam is

inclined with respect to the normal to the specimen surface, then the effects on the absorption and atomic number corrections can be evaluated. This paper will discuss measurements

of <J> (pz) curves for both normal and inclined electron incidence angle and the effects of

changing electron incidence angle on the absorption and atomic number corrections.

2. The Measurement of <J> (pz) Curves

Several sets of sandwich samples have been prepared for the measurement of <)> (pz) curves in different electron energy ranges. The tracer layer must be thick enough for

sufficient x-ray intensity but thin enough that the resolution in the depth distribution is not degraded at the lowest electron energy.

The specimens were prepared by polishing carefully to a final stage of 0.25 urn diamond

paste sets of 2 cm diameter pieces of pure matrix elements. The tracer layer was evaporated

simultanously on all matrix elements from a thermal source in a vacuum apparatus in which

the polished matrix pieces rotated during the evaporation about their own axis and about an axis through the evaporation source. The distance from source to specimen was great enough

that the expected variation in tracer layer thickness over the specimen surface and from

specimen to specimen due to the geometry of the evaporation apparatus was less than two percent in the absence of specimen rotation. The rotation during evaporation should have

ensured better uniformity. Tracer layer uniformity was maintained for the measurement of

<]> (pz) curves for the same tracer layer in different matrices. A tracer layer thickness of

10-15 ug/cm2 was used for the measurements at electron energies from 6-15 keV while a thickness of 30-40 ug/cm2 was used from 15-30 keV.

The matrix overlayers were similarly evaporated. The specimens were appropriately

masked and the thickness of each successive evaporation doubled so the 2

n different overlayer thicknesses were obtained. The thickness was measured from duplicate glass slides

masked to a standard area and exposed to the evaporation source at the same distance as the

sandwich specimens during the overlayer evaporation. The mass per unit area evaporated

onto the glass slides was determined by weighing on a microbalance, by x-ray fluorescence

analysis using an appropriate characteristic x-ray line of the matrix element and by atomic

absorption using the proper hollow cathode lamp and spectral line. The three methods of

measurement agreed to about two percent of the mass thickness. Table 1 lists the specimens

prepared for the measurement of $ (pz) curves.

Table 1. Sandwich samples used in <j> (pz) measurements.

Tracer Line Tracer Thickness Matrices Electron En< (ug/cm2

) KeV

Zn Ka •x, 40 Al ,Cu,Ag,Au 15-30

Si Ka -v 40 Al ,Cu,Ag,Au 15-30

Cd La <\, 40 Al ,Cu,Ag,Au 15-30

Bi La! ,Maj ~ 40 Al ,Cu,Ag,Au 15-30

Si Ka •v 15 Al ,Ni ,Ag,Au 6-15

Cu Ka •v- 15 Al.Ni.Ag 12-15

Cd La ~ 15 Al ,Ni ,Ag,Au 6-15

Bi Laj jMaj ^ 15 Al ,Ni ,Ag,Au 6-15

382
The x-ray intensity from the tracer layer for each overlayer was measured. The pure

tracer element was used as standard which was measured frequently to serve as a correction

for any beam current changes. The background was measured at the peak position of the

characteristic line for the tracer element on the pure matrix element. The data were corrected for dead time, background, drift and absorption in the overlaying matrix layers

to obtain the x-ray generation curves.

Four different electron microprobes were used in measuring the (pz) curves. In two

of the instruments, an Applied Research Laboratories electron microprobe and a Cambridge

Microscan-5, the electron beam is incident normal to the specimen surface while in the other

two, an Associated Electrical Industries SEM2A and a Materials Analysis Company electron

microprobe, the sample is inclined to the electron beam. The geometry of the four instruments is shown in figure 1. In addition a special specimen holder was constructed for the

AEI SEM2A so that the electron incidence angle could be accurately changed in the range

from about 50° to 80°. Table 1 indicates the energy ranges for which <|> (pz) curves have

been measured during the past few years. Some of these have been published in the

literature [10-13].

e" /

/"

'75° / / / / / /

a

e"

c

e" /x

/52.5°

Figure 1. Geometry of <J> (pz) measurements,

(a) Applied Research Laboratories (b) Cambridge Microscan- 5 (c) Materials Analysis Co. (d) Associated Electrical Industries

SEM-2A

The curves of the ratio of intensity from the thin tracer layer to the intensity of

the pure element are correct on a relative scale when compared from matrix to matrix. They

contain all the information about the atomic number and absorption corrections. However, to put them on the same basis as the <|> (pz) curves as defined by Castaing, it is necessary to normalize to the proper $ (0) value, i.e., relative to a thin isolated layer. The $ (0)

values chosen for normalization are those of Hutchins [14] and Reuter [15] - at least in

the case for normal incidence. For non-normal electron incidence the situation is not as satisfactory. The <|> (0) value will be affected by changes in backscattering as electron

incidence angle changes but it is not clear exactly how the <|> (0) value will be affected.

Love et al . [2] have suggested an equation for calculating <)> (0) as a function of incidence

angle and their equation has been used for normalization. A careful study of the resultant

<|> (pz) curves would suggest that a better function for <|) (0) needs to be found.

383
3. Experimental Results

Figure 2 is a plot of the <|> (pz) curve for the ZnKa line from a zinc tracer in a silver matrix measured in the two instruments with normal electron incidence. The agreement of the measured curves is indeed striking. The agreement proves that the tracer and

matrix layer thicknesses are accurately known since many of the points on the curve are

the sum of several individual layer thicknesses and that the electron energies were the same for both instruments. The agreement in the tail of the curve would suggest that the

background is being measured correctly and that the absorption correction applied is also

accurate.

Figure 2. Comparison of measured <)> (pz) in ARL and Cambridge Microprobes.

ZnKa from a zinc tracer in

silver at 25 KeV electron

energy. + Cambridge, A ARL.

0 .3 .6 .9

DEPTH, MG/SQ. CM

Figure 3 shows the effect of the matrix element on the <(> (pz) curve, in this case for

the CdLa x line from a cadmium tracer in the four matrix elements whose atomic number varies

from 13 to 79, measured in the Cambridge Microscan 5. These curves show clearly the effect

of atomic number on the <(> (pz) curves: a shift to greater depths of the maximum of the curve

and a broadening of the distribution as matrix atomic number decreases. An integration of

the area under the curves gives the atomic number effect since the differences in these

<t> (pz) curves result from differences in electron backscattering and stopping power as a function of atomic number.

384
3 3

2-

DEPTH, MG/SQ. CM

3

0 .5 I 1.5 2

DEPTH, MG/SQ. CM

DEPTH, MG/SQ CM

b

3

DEPTH, MG/SQ. CM

d

Figure 3. Effects of matrix atomic number on <J> (pz) curves. CdLa from cadmium tracer in

(a) aluminum (b) copper (c) silver (d) gold.

Figure 4 illustrates the effect of electron energies on the <j> (pz) curves, in this case at much lower electron energies. These curves were measured using the specimens with

thin tracer and overlayer thicknesses in the Cambridge Microscan 5. They show clearly, even

down to 6 keV for the SiKa line, the characteristic shape of an increase in <(> (pz) from the

surface value, through a maximum, then a decay to zero intensity. The same shape is

apparent for the CuKa line at 12 keV even though the overvoltage ratio is less than 1.5.

385
3

The data for the many 0 (pz) curves measured at normal electron incidence as a function of electron energy, exitation potential and matrix atomic number, have been fitted to the empirical equation

$ (R) = D-k-n (kR)

n_1

exp (-(kR)

n

)

where R = pz + pz

Q

.

The parameters pz, D, k, and n were obtained for each (pz) curve by computer optimization of the parameters. The dependence of these parameters on Eo, Ec, Z, and A was established by appropriate plots and the constants for the dependence established by least

squares linear regression [16]. One set of constants is required for low electron energies

(below 15 keV) and a slightly different set for the higher electron energies [17]. The curves plotted in figures 2-4 are those for the $ (R) equation. Quantitative analysis

using this equation yields excellent results when compared with the more conventional ZAF method with a significant improvement at low electron energies [16].

386
4. The Effect of Electron Incidence Angle

When the electron beam is inclined relative to the normal to the specimen surface, the

41 (pz) curve changes and x-rays are produced somewhat closer to the specimen surface. This

can be seen in figure 5 where the electron incidence angle has changed by 27.5° for the two

curves. Note that no simple geometric factor can reconcile the differences in the curves. The <1> (pz) curves can be converted to absorption correction curves by calculating the

fraction of x-rays which will escape from the specimen as a function of the absorption

parameter x- An example of the dependence of f(x) on electron incidence angle <t> is shown

in figure 6. Geometric factors such as 1 + 1/2 cos 2* or sin* have been suggested to change

the x scale to convert the f(x') curves for non-normal incidence. Although these factors

may be satisfactory for specific cases, in general they cannot be expected to compensate

for the changes in shape of the (pz) curves, with electron incidence angle which are different for different matrix atomic number, electron energy and excitation potential.

4

V

0 .3 .6 .9 1.2 DEPTH, MG/SQ. CM

387
0

I

0 2000

X

4000

Figure 6. f(x) curves from $ (pz) curves. Effect of electron

incidence angle. ZnKa in silver at 25 KeV electron

energy. (a) * = 90° (b) * = 67.5°.

The situation for the atomic number effect is even less satisfactory. The (pz) curves can be measured for different matrix elements with the same thin tracer layer to establish the magnitude of the atomic number effect at a specific *. To place the 4» (pz) curves on the basis of Castaing's definition relative to the thin isolated layer, the $ (0)

value must be known for the tracer layer in the same matrix element, (0) as a function

of electron incidence angle is not well known. Love et al. [4] have proposed a formula for

$ (0) which takes into account changes in backscattering and up to this time, we have used

<j> (0) values predicted by this equation. The shape of the curves and in particular the

crossing of the tails of the curves as incidence angle increases suggests that the (|> (0)

values are not accurate. More work needs to be done either experimentally or perhaps by

Monte Carlo calculations to better establish the dependence of (0) on electron incidence

angle. In practical analysis, errors fortunately tend to compensate provided all measurements are made at the same electron incidence angle as is normally the case in an electron

microprobe. In scanning electron microscopes fitted with energy dispersive x-ray detectors,

there is real danger since analyses are frequently performed on fracture surfaces. Even though there are methods to determine electron incidence and x-ray take-off angles from

stereo measurements, calculation of quantitative analysis will give less accurate results

when the unknown and standard are measured at different electron incidence angles since the

compensation of errors will not occur.

Changing the electron incidence angle affects the shape and magnitude of x-ray generation as a function of depth. These changes differ for different electron energies, average

atomic number matrices and x-ray energies. Simple geometric factors cannot be expected to exactly compensate for non-normal incidence. Fortunately for the usual analysis conditions

in an electron microprobe where both the unknown and standard are measured in the same geometry, errors in the correction equations tend to cancel. In the SEM with energy

dispersive analysis where this is frequently not the case, accuracy can be expected to suffer. Additional work needs to be done to establish more accurately the effect of

electron incidence angle on x-ray generation and on $ (0). This work should lead to more reliable analysis in the variable geometries used in scanning electron microscopes.

5. Concl usions

388
References

[1] Bishop, H. E. , Proc . Phys . Soc . (London) 85, 855 (1965).

[2] Moll, S. H. , Baumgarton, N. , Donnelly, W. , Proc . 8th Int . Conf . X-Ray Optics and

Microanalysis , Boston, Paper 33 (1977).

[3] Colby, J. W. , Proc . 6th Nat . Conf . on Electron Probe Microanalysis , Paper 17 (1971).

[4] Love, G. , Cox, M. G. C. and Scott, V. D. , J. Phys . D 1_1_. 7 (1978).

[5] Russ, J. C. , in Scanning Electron Microscopy , IITRI, Chicago, 1973, p. 114.

[6] Brown, J. D. , in Electron Probe Microanalysis , (Tousimis, A. J. and Marton, L. L. , eds.) Academic Press, N. Y. , 1969, p. 73.

[7] Bishop, H. E., J. Phys. D. 1_, 673 (1968).

[8] Castaing, R. , in Advances in Electronics and Electron Phys . , (Marton, L. L. and

Marton, C. , eds.) Academic Press, N. Y. , 13, 317 (1960).

[9] Castaing, R. and Descamps, J., J. Phys . Radium , 16, 304 (1955).

[10] Brown, J. D. and Parobek, L. , in Proc . 6th Int Conf X-Ray Optics and Microanalysis , (Shinoda, G. , Kohra, K. , and Ichinokawa, T. , eds.) U. of Tokyo Press, 1972, p. 163.

[11] Brown, J. D. and Parobek, L. , Adv. X-Ray Anal . , 16, 198 (1973).

[12] Parobek, L. and Brown, J. D. , Adv. X-Ray Anal . , 17, 479 (1974).

[13] Brown, J. D. and Parobek, L. , X-Ray Spect . , 5, 36 (1976).

[14] Hutchin, G. A. , 2nd Nat. Conf on Electron Probe Microanalysis , Boston, Mass. (1967).

[15] Reuter, W. , in Proc . 6th Int . Conf . X-Ray Optics and Microanalysis , (Shinoda, G. , Kohra, K. , and Ichinokawa, T. , eds.) U. of Tokyo Press, 1972, p. 121.

[16] Parobek, L. and Brown, J. D. , X-Ray Spect . , 7, 26 (1978) errata 8 V (1979).

[17] Brown, J. D. and Robinson, W. H. , Proc . 14th Nat . Conf . Microbeam Anal . Soc . , San Francisco Press (1979).

389

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ENERGY AND WAVELENGTH DISPERSIVE X-RAY FLUORESCENCE ANALYSIS OF HEAVY ELEMENTS USING

RADIOACTIVE, SECONDARY FLUORESCENCE AND X-RAY TUBE SOURCES, AND WITH

GERMANIUM, SILICON, AND SCINTILLATION DETECTORS

M. A. Short and S. Bonfiglio

Engineering and Research Staff

Ford Motor Company

Dearborn, Michigan 48121

Abstract

Comparative analytical results are presented for the analysis of

barium/lanthanum/cerium and osmium/iridium/platinum using both K and L series characteristic lines for four different excitation/detection

systems. These systems include a radioactive source with an intrinsic

germanium detector, a radioactive source fluorescing a secondary target

with an intrinsic germanium detector, an x-ray tube fluorescing a secon- dary target with a lithium drifted silicon detector, and an x-ray tube

used in a conventional wavelength dispersive spectrometer. The two

principal advantages of K lines of heavy elements are discussed: the

relative simplicity of the K spectra and the relatively great penetrating

power of high energy K radiation. It is concluded that better resolution

is obtained for L series lines using wavelength dispersive spectrometry.

An example of the high penetrating power is shown in the in- laboratory

analysis for platinum and lead on an automobile catalyst contained within

a stainless steel can.

Key Words: Catalysts automotive; energy dispersive analysis; heavy element fluorescence analysis; intrinsic germanium detectors;

secondary target fluorescence analysis; source excited

fluorescence analysis; x-ray fluorescence analysis.

1. Introduction

In the automobile industry, a considerable effort has been invested in recent years in

the elemental analysis of automotive catalytic converters. Artz [I] 1 has discussed the

quantitative determination of 17 elements ranging from phosphorus to lead in such catalysts

using a wavelength dispersive x-ray fluorescence technique. His method, which may be

described as "semi-destructive", that is the catalyst was pulverized but the powder itself

was not consumed in the analysis, utilized the L series characteristic lines for the

heavier elements ranging from barium to lead. Artz used these L lines because his x-ray

tube and generator, like those in the majority of wavelength dispersive x-ray fluorescence

analysis systems, had an upper limit of 60 kV (although systems are available going up to 80 kV and 100 kV). A tube voltage of 60 kV corresponds to a bremsstrahlung maximum

intensity in the region of 40 kV. Consequently, not only will the characteristic K lines

of elements such as barium, lanthanum, and cerium be inefficiently excited, the K lines of

elements such as iridium, platinum and lead will not be excited at all. In addition, the

angular dispersion of available analyzing crystals for the K lines of barium to lead is

poor.

Recently, we have been investigating the development of alternative x-ray fluorescence

techniques which would excite, detect, and measure the intensities of heavy element K lines

and examining the advantages which would result from the use of such lines. The techniques

The number in brackets refers to the literature reference at the end of this paper.

391
include the use of radioactive and secondary fluorescence sources and the use of lithium

drifted silicon and intrinsic germanium detectors. There are two clear advantages. First,

the K spectra of these heavy elements are relatively simple and free from interference with

L series lines, whereas the heavy element L lines are complex, overlap with other L lines

and overlap with the K lines of elements scandium (Z=21) through selenium (Z=34). Second,

the higher penetrating power of the K lines, and in particular of platinum (E~67 keV) and

lead (E^-74 keV) is so great that, by suitably collimating the excitation and fluorescence

x-rays, the platinum and lead contents of geometrically isolated volumes within a solid

material, particularly where the material is of a low average atomic number and/or density

such as is the case for a ceramic based monolithic catalyst, can be examined.

Additionally, because the absorption of platinum and lead K x-rays in steel is relatively low, the platinum and lead contents of automotive converters could perhaps be exam- ined in the laboratory, non-destructi vely , while the catalyst itself is encased within the

steel container which holds it in the automobile exhaust system. Such an analysis might be

useful in both catalyst development and testing. The ability to undertake this analysis

non-destructi vely means that the same catalyst could be examined before and after use, thus

obviating any possible variations in precious metal content from one catalyst to another.

2. Experimental

Four different experimental techniques have been investigated:

(1) A cobalt-57 radioactive source emitting 122 keV gamma rays in conjunction

with an intrinsic germanium detector of nominal resolution 189 eV (at 5.9

keV) used in an energy dispersive mode. Tin or uranium apertures were

utilized to limit the count rate and to reduce the dead time of the system

to 60 percent to 70 percent. Tungsten or tungsten-monel (monel is a nickel-copper alloy) shielding was employed to screen the detector from

the source.

(2) A cobalt-57 source fluorescing a uranium secondary target and an americium-241 source fluorescing a dysprosium secondary target, both used

with an intrinsic germanium detector (189 ev) in the energy dispersive

mode. Tin or uranium apertures were again utilized to limit the count

rate; tungsten was employed to screen the detector from the source.

(3) X-ray tubes fluorescing secondary targets, specifically tungsten fluorescing gadolinium, tungsten fluorescing nickel, and silver fluorescing

zirconium, were used in conjunction with a lithium drifted silicon

detector of nominal resolution 175 eV (at 5.9 keV), in the energy

dispersive mode.

And for comparative purposes:

(4) A chromium or molybdenum x-ray tube used in a conventional wavelength

dispersive spectrometer. For the longer wavelengths a PET crystal/flow

detector combination was utilized and for the shorter wavelengths a LIF (220) crystal /scintillation detector combination was used. A fine collimator was moved into the primary fluorescence beam.

Synthetic samples were prepared specifically to demonstrate differences in wavelength/

energy resolution shown by the four techniques described above and to show the effect of a light element matrix. Two sets of samples were prepared: one related to a comparison of

K and L spectra for elements in the region barium/lanthanum/cerium, and another for elements

in the region of osmium/iridium/platinum. The samples were not "infinitely thick" with

respect to the K series x-rays.

Not all combinations of techniques, samples, and x-ray lines were examined. Table 1 shows those combinations that were examined. The combinations not examined were those that

either preliminary experiments or intuition predicted to be of lesser interest. Peak and

background measurements to determine the optimum technique for any particular analysis were not made.

392
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3. Results - Energy Resolution

The relative simplicity of heavy element K series lines as compared with L series

lines is illustrated in figures 1 and 2. Figure 1 shows an energy dispersive (ED) platinum

K spectrum, figure 2 shows a wavelength dispersive (WD) platinum L spectrum. Note .that the

spectra contain 4 and 13 resolved peaks respectively. Note also the evident loss in

resolution shown by WD platinum K series and ED platinum L series spectra shown in figures

3 (right) and 4 respectively as compared with figures 1 and 2. Figures 1 and 3 (right)

illustrate graphically the superior resolution of energy dispersive spectroscopy over wavelength dispersive spectroscopy for short wavelength/high energy characteristic lines.

The WD spectrum of platinum in a light element matrix (lithium tetraborate), as shown in

figure 3 (left), shows the high background introduced by the bremsstrahl ung from the x-ray

tube, a discouragement for good quantitative analysis.

PT metal

EOS:

C057 SOURCE

W SHIELD

SN APERTURE

IG DETECTOR

(189 EV)

PTK£2

60 70

KEV

80 90 100

Figure 1. ED spectrum of Pt K lines: Co-57 source, intrinsic Ge detector.

394
395
Figure 3. WD spectra of Pt K lines: Cr tube at 95 kV, LiF(220) crystal.

396
PTL a l,2

—-M—

PT

1

1 — AG TUBE 45 IV : : ZR SECONDARY

Si (LI) DETECTOR

(175 EV)

:

-

_U_-f—i—

u

i.j. .... , . ; i ,

::vi:

!

'

i

; —•

,

1

i 1. s :,

q .

.

. ::.r .

_ _ PTL/84

X

:: PTLy,

HI

pi L/2,3 7 PTU ptlW

A

:

1 Iff J \

!

; 111 r 1 I

1 I i

: \ 1 I

' 1 4 6 8 REV 12 14

Figure 4. ED spectrum of Pt L lines: Ag tube, Zr secondary, Si (Li) detector.

Resolution becomes important in the analysis of elements with close or overlapping

characteristic lines such as might be exhibited by elements adjacent in the periodic

table. Osmium, iridium, and platinum, for example, in admixture, give a complex wavelength

dispersive L spectrum (figure 5). Nevertheless, as may be seen there is sufficient

resolution between the three La ± lines of these elements that a good quantitative analysis

could be made. The positions of the osmium, iridium, and platinum Lj , Ljj, and Ljjj

absorption edges are also indicated on this figure. Figure 5 may be contrasted with an energy dispersive L series spectrum of the same admixture, shown in figure 6, where there

is little resolution between the same lines. No quantitative analysis of osmium/iridium/

platinum could be made using energy dispersive analysis of the L series lines without

utilizing a least squares or stripping technique.

397
398
0S + IR + PT

4 6 8 KEV 12 14

Figure 6. ED spectrum of Os/Ir/Pt L lines: Ag tube, Zr secondary,

Si (Li) detector.

In the analysis of platinum on catalytic converters, we are concerned with small

amounts of platinum (of the order of 0.2%) on light element ceramic substrates. Figure 7 shows a much higher concentration of platinum (2%) in a light element matrix (lithium

tetraborate). This is an ED spectrum obtained using a cobalt-57 source and an intrinsic

germanium detector. The superiority of this ED spectrum over the WD spectrum of the same sample shown in figure 3 (left) with regard to both resolution and peak-to-background is

readily apparent. It can be seen, however, that there is a substantial tungsten peak

arising from the tungsten shield and also a small amount of lead from the general

shielding. A pellet containing a mixture of tungsten, platinum, and lead was made up to show the interference between these elements in ED spectroscopy. The spectrum obtained is

shown in figure 8. The principal interferences are between tungsten Kp x 3 and Platinum Ka x , and between platinum Kp x 3 and lead Ka 1

. If these interferences cannot be removed a good

quantitative analysis can still be made for both platinum and lead using the Ka2 lines.

It was found that if the tungsten shield surrounding the cobalt-57 source when used in the

primary excitation mode was replaced with the tungsten-monel shield, then the tungsten

399
2%PT IN Ll 2 B4 07

EDS:

C057 SOURCE

W SHIELD

Sn APERTURE

IG DETECTOR

(189 EV)

j PTKa

|

!; WK/3,,3

,1

h

1

1

: i

•\

1

i 1

1 !

; \

\ !

; \

1

i C057 COMPTON

1

PTKa 2

i

s

II

[i

l!

j

/ 1

1 PB Ka| ^

1

1

ii /

WKa, j WKa 2 }

1

j

1

i

1 1

; \

i

1

1

! !

1

j

!

i PBKa 2

1

jPTKft/

; ,-'

; jv

J 1 1 1

I

I I L_

60 70 KE v 80 90 100

Figure 7. ED spectrum of Pt K lines: Co-57 source, intrinsic Ge detector.

400
PT Ka | "01.3

WKa.

WKa,

PTKa 2

/ I A /

PBKa2

WK£2

Ka| 2%W + 2%PT + 2%PB IN Ll 2 B4 07 EDS:

CO 57 SOURCE, W-MONEL SHIELD

U APERTURE, IG DETECTOR

(I89EV)

PBKft

CO 57 COMPTON

,UKa 2 UKa

60 70

key

80 90 100

Figure 8. ED spectrum of W/Pt/Pb lines: Co-57 source, intrinsic Ge detector.

lines were reduced to a very low intensity, as can be seen in figure 9. Recently, a better

alternative has been found - to construct a shield of depleted uranium; no tungsten peaks were observed and the uranium K and L series lines at 94/115 keV and 13/18 keV respectively

do not interfere with most heavy element spectra. Unfortunately, neither the tungsten-monel

nor the uranium shields would fit in the radioactive source holder in the secondary excitation configuration, and even with a sample of pure platinum the tungsten shield gave a substantial tungsten peak (figure 10). A simple subtraction procedure was therefore

employed to remove the tungsten lines from the spectrum. The success of this procedure can

be seen in figure 1 1

.

401
2% PT IN Ll 2 B 4 07

EDS:

CO 57 SOURCE

W-MONEL SHIELD

onCu nrapfrtiirf l i\ i u i»l 16 DETECTOR

(189 EV)

-

PT Ka

,

'

\

PTKa 2 i

1

':

i Pft PflllPTflM

i ij

|i

l!

i

i

!

1

|

;! i

i

!

PTK^|, 3 ,PBKa,

\

. ^

ii

i

, PT Ky9 2

| n '

i

i

ii!

W Ka,

i

f 1

1

PB Ka 2

'.

'

' v'

i i

\<

i i J I I

I L_

60 70 KEV 80 90 100

Figure 9. ED spectrum of Pt K lines: Co-57 source, intrinsic Re detector.

402
PTKa J WK/9,

t3

PTKa'

WKa.

WKa

PT METAL

EDS:

CO 57 SOURCE

U SECONDARY

W SHIELD

SN aperture

16 DETECTOR

(189 EV)

I

, WK^2 —

/

60 70 KEV 80 90 100

Figure 10. ED spectrum of Pt K lines: Co-57 source, U secondary,

intrinsic GE detector.

*03
PTKa,

PT METAL (-W)

EOS :

i! CO 57 SOURCE

U SECONDARY

til A 1 1 IP 1 f\ W SHIELD

ONCm ArLl) apfrtiirf 1 unc 16 DETECTOR

(189 EV)

PTKa 2

i

i

|

i

i

1

1

1

! PTK/3. ,

i

! ji

i WKa 2 WKa, !

w .•••(

j

i

! /

- i

i

i

i ! a ptk/3

2

i

> j ~

;

;

i

• ^ - Ji i i i ... ^ L_ L

60 70

KeV

80 90

Figure 11. ED spectrum of Pt less W K lines: Co-57 source, U secondary,

intrinsic Ge detector.

Very large tungsten Ka peaks were obtained for a mixture of two percent platinum and

two percent iridium in a lithium tetraborate matrix, see figure 12. The tungsten peaks were again substantially removed by subtraction, as is shown in figure 13. The diminution

in height of the platinum Ka x peak resulting from this subtraction may be noted. Figure 14 shows a three element mixture of osmium, iridium, and platinum using the alternative

approach of a primary cobalt-57 source and the tungsten-model shield. It may be deduced

that energy dispersive analysis using K lines would only permit analysis of the end-members

osmium (using Os Ka2 ) and platinum (using Pt Kc^) unless a least squares or stripping

procedure is used.

404
LRKa g, PTRaj

Ma).!!*,,,

UKa CONPTON

2% PT ^ 2% IR

IN Ll 2 B4O7

EDS:

C057 SOURCE

U SECONDARY

W SHIELD

SN APERTURE

16 DETECTOR

(189 EY)

UK>8 CONPTON

60 70 KEV 80 90 100

Figure 12. ED spectrum of Pt/Ir K lines: Co-57 source, U secondary,

intrinsic Ge detector.

405
iRKai, PTRa 2

lRKa2

WKa2 WKai

I

PTKai

UKa COMPTON

2% PT + 2% IR (-W)

IN U2B4O7

EOS:

CO 57 SOURCE

U SECONDARY

W SHIELD

SN APERTURE

IG DETECTOR

(I89EV)

/ ,PTKi9| t 3,lRKi82

PT K>3 2

h \/wn

x

UKj8 COMPTON

60 70 80 90 100

Figure 13. ED spectrum of Pt/Ir less U K lines: Co-57 source,

U secondary, intrinsic GE detector.

406
OsKa g lRKa 2

0sKa2

WKa,

Ma|

PTKa 2

2% PT + 2%lR + 2% OS IN Ll 2 B 4 0 1 EDS^

C057 SOURCE, W-MONEL SHIELD

SM APERTURE, IG DETECTOR

(189 EV)

PTKa,

0SK/3 I>3

0sK£2 M£l,3

PTK£|,3 /

PTK/02

1

C057 COMPTON

60 70 KEV 80 90 100

Figure 14. ED spectrum of Pt/Ir/Os K lines: Co-57 source, intrinsic Ge detector.

WDS is superior with respect to resolution for L series lines in the region of barium/

lanthanum/cerium. Figures 15 and 16 show WD and Ed spectra, respectively, for cerium. The

superiority of WDS is especially marked for a mixture of barium, cerium, and lanthanum.

Figure 17 shows a WD spectrum, figure 18 shows an ED spectrum. It can be readily seen that

a quantitative analysis for these three elements could be made by WDS using a Ba La 1 2 > La La t 2 , and Ce L0J 4

. Also indicated in figure 17 are the positions of the barium,

lanthanum, and cerium Lj, Ljj, and Ljjj absorption edges. The ED spectrum would require a very sophisticated least squares stripping program to obtain quantitative information.

Excellent resolution of the K series lines of barium, lanthanum, and cerum can be achieved

with EDS. Figures 19 and 20 show spectra obtained with a tungsten tube, a gadolinium

secondary target, and a lithium drifted silicon detector, and with an americium-241 source,

a dysprosium secondary target, and an intrinsic germanium detector, respectively. The

resolution given by the two systems is very similar.

407
408
CE

EDS:

W TUBE, 45KV

Nl SECONDARY

Si (LI) DETECTOR

(I75EV)

1/1 CELr, I

Oil I I S\J

\ Aa

L/2,3

/

2 3 4 5 KEV 6 7 8 9

Figure 16. ED spectrum of Ce L lines: W tube at 45 kV, N1 secondary

Si (Li) detector.

409
410
BA + LA + CE

EDS:

W TUBE, 45KV

Nl SECONDARY

Si (Ll) DETECTOR

(I75EV)

LALa|,2, L77

j(CELa|

f 2,L«7

BAL/9, |3| 4

5 KEV

Figure 18. ED spectrum of Ba/La/Ce L lines: W tube at 45 kV, Nl secondary

Si (Li) detector.

411
412
Ba+LA + CE

EDS:

AM 241 SOURCE

DY SECONDARY

W SHIELD

U APERTURE

IG DETECTOR

(189 EV)

BAKai

LAKai

CEKai

h-+-

BAKa2

LAKa2 CEKa2

Ka ESCAPE PEAKS

BA LA CE

BaK/3

2 CEK>3

lt3

LAK/32 BAK/3,,3

LAK)S,

f 3

CEK£2 DYKa COMPTON

DYK£ COMPTON

DYKa 2,|

20 30 KEV 40 50

Figure 20. ED spectrum of Ba/La/Ce K lines: Am- 241 source, Dy secondary,

intrinsic Ge detector.

4. Results - High Energy Penetration

To demonstrate the high penetration power of platinum and lead K series x-rays and to illustrate the possible usefulness of the technique, two automobile converters were examined

under laboratory conditions using the cobalt-57 source (122 keV gamma rays) to excite the

platinum and lead, and the intrinsic germanium detector to detect the platinum and lead

fluorescence K radiation. One catalyst was fresh and unused, the second had, for test

purposes, been run in a car on leaded gasoline. Both contained in the region of 0.2 percent

by weight of platinum and both were contained within a standard stainless steel can in which

they are normally connected to the engine exhaust. The x-ray examinations were made through

the can. The resulting spectra, shown in figure 21, each took approximately 50 minutes to collect (1000 seconds "live time" at a 60% to 70% "dead time"). The experimental conditions

were not optimized to reduce the counting time to a minimum. It can be seen that platinum

is readily detected at this low level (note the change in full scale for the lead containing

catalyst) and that there is a substantial amount of lead present. No attempts have been

made to analyze converters i_n situ . Possible complications of i_n situ analysis include:

413
the additional attenuation of the x-rays by the converter heat shield, radiant heat damage

to the detector, and unacceptably long analysis times. Even a successful vn situ analysis

for lead would provide no immediate information as to the timing or quantity of leaded

gasoline which had been consumed by the engine.

AUTOMOBILE CATALYST - FRESH

EDS: Co 5T SOURCE

W-MONEt SHIELD

Sn APERTURE

16 DETECTOR

<l69«v)

FULL SCALE:

18000 COUNTS

I

H

PTKot,

PTK42 n f

- h /

/

60 80

KEV

AUTOMOBILE CATALYST

+ LEADED GASOLINE

EDS:

Co 57 SOURCE

W-M0NEL SHIELD

Sn APERTURE

IG DETECTOR

(I89ev)

FULL SCALE:

64000 COUNTS

PB K<1,

PTKot,

PTKot,

PBKol,

60

KEV

80

Figure 21. ED spectra of Automobile Catalysts: Co-57 source, intrinsic Ge detector.

414
5. Discussion

Wavelength dispersive and energy dispersive spectra have shown that better resolution

for barium/lanthanum/cerium and osmium/iridi urn/platinum L series lines is provided by

wavelength dispersive spectroscopy, while better resolution for the K series lines is given

by energy dispersive spectroscopy. For barium/lanthanum/cerium K lines, there is little to choose between the W tube/Gd secondary/Si (Li) detector system and the Am-241/Dy secondary/

intrinisic germanium detector system; they both give good resolution. For osmium/iridi urn/ platinum, both cobalt-57 used as a primary source and cobalt-57 used with uranium as a secondary target give good resolution for the K lines; in the case of the light element

matrix, the Co-57/U secondary combination suffers from a worse peak/background ratio and

from the proximity of the uranium K Compton scattering when compared with Co-57 used as a primary source.

With cobalt-57 as a primary source and an intrinsic germanium detector, lead/platinum

ratios have been measured in the laboratory, through the can, on a test converter after

removal from a car run on leaded fuel. Useful semi -quantitative results for lead are currently being obtained under laboratory conditions using a computer-based multichannel

analyzer program which integrates peak heights and subtracts backgrounds based on an assumed platinum content.

We would like to thank Drs. M. F. Elgart and R. G. Hurley for a number of useful

discussions and Drs. B. E. Artz and J. L. Bomback for commenting on the manuscript.

Reference

[1] Artz, B. E. , X-ray Spectrometry, 6, 165 (1977).

415

National Bureau of Standards Special Publication 604. Proceedings of the Workshop on Energy

Dispersive X-Ray Spectrometry held at NBS, Gaithersburg, Maryland, April 23-25, 1979.

(Issued March 1981)

ENERGY DISPERSIVE DETECTORS - A BIBLIOGRAPHY (1981)

C. E. Fiori and C. R. Swyt

Biomedical Engineering and Instrumentation Branch

Division of Research Services

National Institutes of Health

Bethesda, Maryland 20205

This paper is an expanded and updated bibliography of references relating to energy

dispersive detectors for electron column instruments with more complete cross-referencing

by subject than the previous bibliography by the first author1. The bibliography is not

all-inclusive and is meant only to give the reader a starting point from which to become

familiar with the large body of literature which exists on the topic of energy dispersive

detectors. Papers from the more relevant journals, e.g., X-Ray Spectrometry, Review of

Scientific Instruments, Scanning Electron Microscopy, etc., have been cited most heavily but

a number of historical papers of general interest from many sources are also included.

Both new citations and references from the previous bibliography are listed because of the

nature of the workshop resulting in this volume.

Among the types of detectors covered are Si, Ge, Cd Te, Hg I 2 , windowless Si(Li) and

gas proportional. References treating the physics of Si (Li) detectors and their application to the electron microprobe and scanning and transmission microscopes are extensively

included. Papers treating their applications, such as x-ray fluorescence and proton

induced x-ray fluorescence, are less comprehensively referenced.

Broad areas covered in the cited literature are the physics of energy dispersive

detectors, quantitative applications, characterization and calibration, and applications

in fundamental physical measurement.

REFERENCES

Physics of Detectors, Si: 3, 4, 32, 40, 70, 87, 88, 99, 127, 150, 154, 163, 164, 172, 174,

177, 185, 202, 223, 224, 241, 243, 248, 252, 256, 258, 261, 279, 285, 296, 387, 423, 425,

429, 430, 434, 435, 436, 467, 468, 469, 470.

Physics of Detectors, Ge: 43, 64, 67, 81, 87, 88, 144, 166, 167, 169, 170, 171, 172, 173,

174, 175, 178, 179, 180, 181, 182, 195, 196, 197, 198, 244, 247, 256, 258, 261, 279, 292,

293, 294, 295, 301, 321, 366, 381, 382, 422, 423, 425, 430, 449.

Windowless Si (Li) Detectors: 12, 15, 273, 274, 340, 346, 347, 361.

Cd Te: 76, 123, 124, 201, 261, 373, 426.

Hg I 2 , Hg 02 , Pb I 2 : 75, 77, 137, 201, 254, 255, 299, 311, 373, 374, 388, 426, 427, 448.

Gas Proportional, Ionization, Scintillation Detectors: 7, 34, 38, 39, 45, 48, 63, 73, 87,

111, 113, 183, 209, 216, 246, 251, 260, 267, 268, 269, 270, 321, 323, 326, 386, 388, 391,

402, 424, 439, 450.

Fano Factor, Ionization Energy: 94, 112, 290, 291, 382, 436, 441, 464.

Calibration and Characterization: 49, 50, 51, 52, 56, 57, 58, 59, 70, 71, 114, 176, 217,

246, 256, 263, 272, 277, 324, 325, 365, 399, 419, 422, 440, 445.

1 Fiori, C. E. , "Energy-Dispersive Detectors: A Bibliography" in Microbeam Analysis-1979,

ed. D. E. Newbury (San Francisco Press, 1979) p. 361.

417
Incomplete Charge Collection, Dead Zone: 4, 11, 79, 103, 116, 120, 127, 141, 165, 191,

201, 261, 262, 272, 301, 431, 436, 455, 463, 467, 468.

Escape Peaks: 64, 116, 120, 277, 313, 387, 410, 436, 455, 457.

Artifacts: 116, 117, 120, 133, 136, 145, 187, 271, 278, 313, 328, 337, 436, 455.

Electronics: 37, 38, 55, 78, 83, 97, 98, 100, 104, 105, 106, 107, 108, 110, 116, 117, 120,

133, 135, 153, 158, 162, 212, 213, 218, 219, 226, 227, 242, 245, 249, 266, 284, 289, 305,

306, 307, 308, 309, 368, 404, 407, 416, 436, 447, 451, 455, 465.

Pulse Pile Up: 68, 116, 120, 135, 281, 319, 359, 412, 436, 451, 455.

Multichannel Analyzers and Accessories: 5, 35, 68, 78, 82, 184, 249, 250, 259, 298, 317.

Automation: 1, 6, 128, 140, 229, 360, 367.

Background Fitting: 9, 16, 42, 40, 69, 84, 116, 119, 120, 228, 253, 277, 280, 312, 320,

334, 348, 349, 364, 372, 383, 396, 397, 408, 409, 417, 433, 443.

Peak Fitting, Determination: 9, 72, 79, 95, 116, 118, 120, 136, 190, 200, 210, 264, 277,

278, 280, 282, 286, 297, 345, 348, 349, 352, 371, 372, 385, 409, 415, 420.

Mathematical and Statistical Procedures: 42, 89, 129, 200, 250, 370, 375, 376.

Trace Analysis: 54, 60, 131, 138, 140, 142, 265, 283, 297, 320.

Microprobe, SEM Application: 1, 12, 15, 20, 22, 27, 65, 69, 89, 90, 102, 109, 116, 117,

120, 128, 132, 186, 222, 230, 231, 232, 236, 244, 265, 268, 271, 274, 281, 287, 300, 303,

304, 314, 315, 316, 330, 331, 332, 335, 336, 339, 340, 343, 350, 354, 355, 356, 357, 358,

369, 378, 392, 393, 394, 395, 396, 399, 405, 406, 411, 413, 428, 444, 455, 456, 459.

TEM Applications: 13, 14, 15, 16, 19, 21, 26, 29, 117, 136, 145, 146, 148, 149, 187, 257,

328, 339, 341, 342, 353, 356, 358, 383, 403, 461, 462.

XRF, PIXRF, Applications. 10, 28, 46, 47, 53, 54, 61, 66, 71, 84, 91, 92, 93, 125, 126,

131, 141, 151, 157, 160, 161, 192, 193, 203, 204, 205, 206, 207, 208, 214, 215, 225, 228,

265, 273, 274, 276, 283, 320, 321, 351, 362, 369, 379, 380, 400, 401, 421, 432, 433, 434,

437, 438, 452, 454, 455, 466.

Quantitative Applications: 6, 17, 18, 19, 20, 21, 22, 24, 25, 27, 65, 74, 85, 86, 89, 93,

95, 96, 115, 123, 128, 129, 138, 139, 140, 142, 143, 148, 149, 199, 220, 222, 229, 231,

233, 234, 235, 253, 268, 275, 277, 278, 288, 300, 303, 304, 314, 315, 316, 317, 329, 331,

332, 333, 338, 340, 345, 346, 347, 356, 357, 365, 367, 378, 384, 390, 392, 393, 394, 395,

396, 404, 405, 406, 411, 414, 417, 418, 436, 444, 455, 459.

Physical Applications: 60, 101, 189, 190, 221, 237, 238, 239, 397, 398.

Light Element Analysis: 12, 96, 111.

Health, Biological Applications: 13, 26, 46, 47, 53, 60, 141, 143, 160, 194, 203, 205,

208, 215, 283, 303, 304, 322, 341, 357, 390, 418, 454.

Geological: 6, 89, 228, 314, 390, 393, 395, 400, 418.

Metallurgical: 10, 69, 86, 90, 143, 149, 228, 253, 288, 383, 390, 418, 438, 453, 454, 462.

Particle: 14, 19, 21, 115, 205, 207, 390, 418, 428.

Additional Topics: 1, 62, 70, 123, 194, 195, 310, 460.

Si, Ge Review Papers, Books: 2, 8, 23, 30, 31, 33, 36, 41, 44, 80, 116, 120, 121, 122,

127, 130, 134, 147, 152, 155, 156, 159, 163, 168, 172, 188, 211, 230, 236, 240, 302, 318,

327, 344, 350, 355, 363, 377, 436, 442, 446, 453, 454, 455, 458, 471.

418
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439

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NBS-114A irev. 2-8C) U S DEPT OF" C O MM* 1 PlIRI ICATIONI OR Pprfnrminp C\tqt\t\ Rpnnrf Kin 3. Publication Date

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REPORT NO. SHEET (See instructions] MBS SP fi04 JunP lQfil

4. TITLE AND SUBTITLE

Energy Dispersive X-Ray Spectrometry

Proceedings of a Workshop on Energy Dispersive X-Ray Spectrometry held at the

National Bureau of Standards. Gaithersbura. MP. April ?3-?5. 1979

5. AUTHOR(S)

K.F.J. Heinrich, D.E. Newbury, R.L. Myklebust, and C.E. Fiori

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered WASHINGTON, D.C 20234 Final

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Same as above.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 81-600047

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or /ess factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) This book is the formal report of the Workshop on Energy Dispersive X-Ray

Spectrometry held at the National Bureau of Standards, April 23-25, 1979. The

papers included in these proceedings cover a wide range of topics in the field:

(1) Physics of detectors -- Properties of silicon (lithium), germanium (lithium),

intrinsic germanium, and mercuric iodide detectors are described and compared.

(2) Windowless x-ray detectors -- The application of windowless energy dispersive

x-ray detectors to the measurement of soft x-rays below 1 keV is described,

including special electronic techniques for processing pulses produced by low

energy x-rays. (3) Electronics for energy dispersive x-ray spectrometry ~ The

signal processing chain is discussed with special emphasis on state-of-the-art

multichannel analyzers. (4) Spectral processing and statistics -- Fundamental

statistical considerations in spectral processing are described. A variety of curve techniques is discussed, including sequential simplex and least squares fitting with

a digital filter. (5) Artifacts in energy dispersive x-ray spectrometry in electron

beam instruments -- Artifacts arising from the basic detection and amplification

processes and instrument-related artifacts observed in electron beam instruments

operating in the range 10-1000 keV are described. (6) Applications -- Applications

of energy dispersive x-ray spectrometry are given in quantitative x-ray microanalysis and x-ray fluorescence analysis.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Electron probe microanalysis; energy dispersive x-ray spectrometry; germanium

(lithium) detector; scanning electron microscopy; silicon (lithium) detector;

Sppct.rfll riPPOnvnlutinn; v-ray analyst;; y-ray fj nnrg^gnre

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