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^fffEAU Of NBS SPECIAL PUBUCATION 460

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

>f Monte Carlo Calculations

ectron Probe Microanalysi.'

anning Electrc Microscop
NATIONAL BUREAU OF STANDARDS

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' Located at Boulder, Colorado 80302.
^ of Stiffl^and Use of Monte Carlo Calculations

in Electron Probe Microanalysis

and Scanning Electron Microscopy

{ ~ ^

Proceedings of a workshop held at the

I National Bureau of Standards

li Gaithersburg, Maryland, October 1-3, 1975

! Edited by

i Kurt F. J. Heinrich

i Dale E. Newbury

Harvey Yakowitz

Sponsored by

Analytical Chemistry Division

Institute for Materials Research

National Bureau of Standards

I

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson. Secretary

Edward 0. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

U, ^.NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

• • •

Issued December 1976

I
Library of Congress Catalog Card Number: 77-608002

National Bureau of Standards Special Publication 460

Nat. Bur. Stand. (U.S.), Spec. Publ. 460, 169 pages (Dec. 1976) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

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

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(Add 25 percent additional for other than U.S. mailing).
Foreword

The Analytical Chemistry Division of the Institute for Materials Research, National Bureau of Standards, seeks to develop new techniques of chemical analysis and improve

existing techniques. Part of the mission of NBS is to disseminate knowledge in the scientific

and technical community. To aid in reaching this objective, the Analytical Chemistry

Division 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 sharply

defined subjects in a wide variety of specialist areas. The objective 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 limits

detailed discussion of special subjects. Past topics of these workshops and the published

proceedings include: Quantitative Electron Probe Microanalysis (NBS Special Publication

298, available from editors). Aerosol Measurements (NBS Special Publication 412), Oil Pollution

Monitoring (NBS Special Publication 409), and Secondary Ion Mass Spectrometry (NBS Special

Publication 427). 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 Division Office, Analytical Chemistry Division, National Bureau of Standards, Washington, DC 20234.

This volume contains the proceedings of a Workshop on the Use of Monte Carlo Calculations

in Electron Probe Microanalysis and Scanning Electron Microscopy. The three-day meeting

involved participants from the United States and Europe. The workshop format consisted of

a keynote talk on each topic followed by extensive discussions. The papers in this volume

are based on the keynote talks augmented with some points raised in the discussion.

The extensive development of electron beam instrumentation for the microscopy and

analysis of samples of diverse scientific interest has resulted in a need for detailed

studies of the origin and characteristics of the signals involved. This publication

provides a detailed view of one approach to the study of such signals, namely the Monte

Carlo technique for electron trajectory calculation. The publication should be of value to the numerous workers who utilize the scanning electron microscope and electron probe

microanalyzer.

John D. Hoffman

Director

Institute for Materials Research

National Bureau of Standards

iii
Preface

The utility of electron probe microanalysis and scanning electron microscopy is demonstrated by the fact that more than 2500 such instruments costing from $25,000 to $200,000 are now in use worldwide, and more instruments are constantly being added.

Moreover, the instruments are at work in fields as diverse as biology, metallurgy, electronics, chemistry, and forensic science. Simultaneously with the development and distribution of the instrumentation, much work has been devoted to the study of electron

interactions with solids with the purpose of characterizing the signals which are utilized

in microscopy and analysis. One approach which has proven especially useful in both

qualitative and quantitative studies of electron interactions is that of Monte Carlo

electron trajectory simulation. Monte Carlo calculations have enabled scientists to visualize the electron interaction volume in solids as to size and shape, to predict the

parameters characterizing the secondary and back-scattered electrons, the x-rays, and the

Auger electrons, and to analyze factors which determine resolution in micrographs.

The purpose of the present workshop was to bring together a number of workers in Monte Carlo trajectory techniques to assess the current state of development and to look

for areas of future development. The list of participants in the meeting does not include

all active workers in the field, but we hope that this document contains a reasonably

complete description of the subject. By examining the areas of application, including

particulate analysis, thin film analysis, and magnetic domain imaging, some idea can be

obtained of the future directions which the Monte Carlo technique may follow.

December 10, 1976 K. F. J. Heinrich

H. Yakowitz

D. E. Newbury

i V
Abstract

This book is the formal report of the Workshop on the Use of Monte Carlo Calculations

in Electron Probe Microanalysis and Scanning Electron Microscopy held at the National Bureau of Standards, October 1-3, 1975. The papers cover a wide range of topics within

the field: the history and development of Monte Carlo methods for use in x-ray microanalysis; the study of the distribution of electron and x-ray signals by Monte Carlo

techniques; the effect of the choice of scattering models on the calculations; techniques

for considering the distribution of energies of the beam electrons propagating in the

specimen; evaluation of ionization cross-section models; and applications of Monte Carlo

techniques to the study of particles, thin films, and magnetic domain images. The contributions include reviews of general interest as well as papers treating specific topics.

The volume should be of wide interest to workers in the fields of scanning electron

microscopy, electron probe microanalysis, electron physics, and other fields involving the

interaction of electrons with solids.

Key Words: Electron probe microanalysis; electron-solid interactions, magnetic domains;

Monte Carlo electron trajectory calculations; particle analysis; scanning electron microscopy; thin film analysis.

V

Contents

Page

Foreword iii

Preface iv

Abstract v

1. The Role of Monte Carlo Calculations in Electron Probe

Microanalysis and Scanning Electron Microscopy - K. F. J. Heinrich 1

2. The History and Development of Monte Carlo Methods for Use

in X-ray Microanalysis - H. E. Bishop 5

3. Studies of the Distribution of Signals in the SEM/EPMA by Monte Carlo Electron Trajectory Cal cul ations--An Outline - D. E. Newbury and H. Yakowitz 15

4. The Effect of Scattering Models on the Results of Monte

Carlo Calculations - L. Reimer and E. R. Krefting 45

5. Characteristics of a Monte Carlo Program for Microanalysis

Study of Energy Loss - J. Henoc and F. Maurice 61

6. Evaluation of Formulas for Inner-Shell Ionization Cross

Sections - C. J. Powell 97

7. NBS Monte Carlo Electron Trajectory Calculation Program - R. L. Myklebust, D. E. Newbury, and H. Yakowitz 105

8. Application of Monte Carlo Simulation to Electron Microprobe

Analysis of Thin Films on Substrates - D. F. Kyser and K. Murata 129

9. Calculation of Type II Magnetic Contrast in the Low-Loss

Image in the Scanning Electron Microscope - 0. C. Wells 139

10. A Study of Type II Magnetic Domain Contrast in the SEM by

Monte Carlo Electron Trajectory Simulation - D. E. Newbury, H. Yakowitz, and R. L. Myklebust 151

vii

National Bureau of Standards Special Publication 450. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

THE ROLE OF MONTE CARLO CALCULATIONS IN ELECTRON PROBE

MICROANALYSIS AND SCANNING ELECTRON MICROSCOPY

Kurt F. J. Heinrich

Analytical Chemistry Division

National Bureau of Standards

Washington, DC 20234

1. Introduction

The emission of characteristic x-rays, on which electron probe microanalysis is based,

depends on the deceleration and scattering of the primary electrons and the x-ray production

as a function of the energy of the primary electron. In the ZAF methods applicable to flat,

electron-opaque targets, Bethe's theories are used to calculate the electron deceleration

and the ionization cross-sections of the target atoms. The effects of scattering on the

generated x-ray intensities are taken into account by an empirical correction factor based

on the measurement of the energy distribution of backscattered electrons, and the absorption

losses of the emerging x-rays are also determined by a semi-empirical approach due to Philibert, or by empirical generalizations of experiments by Green and by Castaing, et al

.

Further corrections must be applied to account for the effects of indirect (fluorescent or photon-photon) x-ray excitation. (For references on various aspects of the ZAF method see reference [1]-.)

Although some aspects of the ZAF procedure are still open to improvement, the achievable

accuracy is of the order of magnitude of the measurement or standardization errors and is considered satisfactory. The application of the ZAF method cannot, however, be extended

with rigor to the analysis of specimens which are not flat, homogeneous and electron-opaque,

because the empirical adjustments in this method cannot be used to treat such geometrical

configurations. Yet, many important analytical situations fall into the categories which

are not considered by the ZAF method. These include the analysis of supported and unsupported thin layers, of inclusions and free particles, and thin sections of biological specimens

(fig. 1).

A related problem arises in analysis by means of x-rays of large wavelengths in which

the observed x-radiation emerges from a shallow region of the excited volume adjacent to the

specimen surface. In this case, most of the x-ray absorption models commonly used in ZAF procedures do not adequately predict the observable x-ray intensities. For the prediction

of the x-ray emission from such targets, we must choose a method which follows more closely

the chain of events within the specimen with particular reference to the spatial distribution of these events. A procedure commonly used to this effect is the Monte Carlo calculation.

The study of spatial distribution of the points of origin of signals is gqually impor- tant to scanning electron microscopy, in which high spatial resolution {^]00 A) is desirable,

and specimen surfaces are not usually flat. With the rapid development and diffusion of

scanning electron microscopy, the contributions of the Monte Carlo method to the knowledge

of the physical foundations of scanning electron microscopy become increasingly significant.

Such contributions are presented in paper by Myklebust, et al . , Wells, and Newbury et al

.

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

1
Figure 1. Some configurations not considered in the usual ZAF procedure. A: inclined

electron beam; B: unsupported foil; C: supported coating; D: small inclusion; E: thin

section of biological material; F: particle mounted on thin foil.

2. The Monte Carlo Model

This evaluation is based on the summation of the events occurring in a large number of

simulated electron trajectories within the specimen. In principle, the interactions between

the electron and the target material should all be treated individually. Where elements of

randomness enter the decisions concerning the occurrence and nature of an event, a random

number is used to choose from among a series of equally possible outcomes. Each trajectory

is terminated when the primary (beam) electron has lost too much energy to produce the

signal of interest, or when the electron leaves the target material (backscattered or trans- mitted electrons). Hence, by changes in the equation representing the specimen surface, the

method can be adapted to diverse geometric configurations.

The observed intensity of any signal--such as x-rays and secondary electrons--depends

on at least four parameters: the distribution in depth of the primary electrons, their

energy distribution, the cross-section for signal generation as a function of electron

energy (see paper 6 for x-ray production), and the attenuation of the signal by the target,

and where applicable, by the surrounding force fields. In some cases, such as secondary

electron and fluorescent (or indirect) x-ray emission, intermediate mechanisms of energy

transmission must also be described. If all mechanisms are known from theory, then the

Monte Carlo procedure can be based entirely on these theoretical premises. If the results

so obtained are not fully satisfactory, empirical adjustments can be made. As will be

shown in the ensuing papers, it is one of the strengths of the Monte Carlo procedure that

such empirical adjustments can lead to simple yet accurate models.

3. Simplifications

It is sometimes assumed in the literature of microprobe analysis that the use of Monte

Carlo techniques circumvents the uncertainties which plague the ZAF method; thus, the results

2
of the Monte Carlo method are often cited as paragons in the evaluation of variants of ZAF. This assumption does not, however, hold. A full event-by-event treatment of the electrontarget interaction would exceed the capacity of most computers and be economically untenable.

Moreover, the interactions of electrons with lattice regions and large molecules are matheI matically intractable. Therefore, all existing Monte Carlo models contain important simplications, and the choice of the simplifications may significantly affect the results (see

paper 5).

For instance, it is common practice to ignore the effects of target structure, and to

^ treat the electron deceleration as a continuous process due to interactions with orbital

' electrons of an amorphous medium consisting of free atoms. When the stopping power equation

^ of Bethe is used for this effect, the uncertainty in the values of the mean excitation

i potentials creates exactly the same problem as ZAF procedure [2]. Further inaccuracies

I which arise from straggling are discussed by Hdnoc and Maurice.

Another problem arises from the choice of scattering parameters. According to Reimer

:i (paper 4), the use of the Rutherford model leads to large errors in scattering angles. Even

. more startling is the observation by Reimer that indirect excitation due to secondary electrons of high energy is a significant source of x-rays (about 30% for AlKal) in aluminum.

I Such a mechanism has not been taken into account in any earlier model for x-ray excitation,

I either of the ZAF or the Monte Carlo type,

i' Further complications may arise in the so-called multiple-scattering models, in which

I the electron trajectory is arbitrarily divided into segments, and the scattering acts within

i each segment are replaced by a single hypothetical scattering event. In some of these

I models, the results depend upon the number of intervals into which the trajectories are divided. The effects of inelastic large-angle scattering are also ignored in most Monte

i) Carlo models. Finally, if an insufficient number of trajectories is calculated, statistical

j errors may arise regardless of the fit of the model.

Only a small fraction of electron trajectories produce characteristic x-ray photons

j (10"2 to 10"^ for Ka lines of elements under typical conditions). For this reason, one must

i' sum fractional probabilities of x-ray production within individual trajectories, to effec1 tively decrease the statistical uncertainties in x-ray production, or use scaling procedures

which, in turn, may produce artifacts (clusters) in the graphic representation of the distribution of sites of x-ray generation.

The effects of simplifications in the model depend to a large degree on the intended

applications. For instance, when the measurement of thin layers or of single large-angle

scattering is of prime interest, the first segments of the trajectory must be treated carefully, and it may be inadequate to assume that the first scattering interaction always

occurs at the mid-depth of the first interval of integrations. On the other hand, such an arbitrary assumption will have little effect on the depth distribution of the short-wavelength x-radiation from electron-opaque targets. It is thus possible, with judicious selection of procedures and fitting adjustments where needed, to obtain models of relative simI plicity which are in satisfactory agreement with a wide variety of experiments (as shown in paper 8)

.

4. Conclusions

We hope that these proceedings will not only illustrate the wide range of objectives

that can be attained by the Monte Carlo calculations, but also the complexity of the problem

and the diversity of possible solutions. An understanding of the options which are available

should help in the selecting the procedure which is most appropriate for a given purpose.

We also trust that we have provided the proper historical perspective and shown some of the

limitations of this versatile technique.

References

[1] Reed, S. J. B. , Electron Microprobe Analysis, Cambridge University Press, London, 1975.

[2] Heinrich, K. F. J. and Yakowitz, H., Mikroehim. Acta, 1970 , 123-134.

3

National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

THE HISTORY AND DEVELOPMENT OF MONTE CARLO METHODS FOR USE IN X-RAY MICROANALYSIS

H. E. Bishop

Materials Development Division

AERE Harwell

Didcot, Oxon, England

Green in 1963 first demonstrated the usefulness of Monte Carlo calculations

to the understanding and development of the theory of x-ray microanalysis. As his calculations was based on experimental scattering data it was of relatively

limited application. By 1965, at the 4th International Conference on X-Ray

Optics and Microanalysis in Paris, papers from Japan and the United Kingdom

demonstrated that a more general approach based on theoretical cross-sections

was possible. Since then many further developments have been reported and a number of Fortran programs have become available.

Monte Carlo calculations may be divided conveniently into two classes; those

aimed at a general understanding of the electron scattering process leading to x-ray production and those intended to solve particular problems. For conventional quantitative microprobe analysis of a flat specimen, uniform over the

analyzed volume, the first class is the most important as it should lead to improvements in the conventional ZAF procedures. In a situation where there are

special boundary conditions, such as thin surface films or particles, the

second class is useful in predicting or interpreting the observed results. In this paper, the various approaches to Monte Carlo calculations are reviewed and

their suitability for different applications is discussed.

Key Words: Electron probe microanalysis; Electron scattering; electron-specimen

interactions; Monte Carlo electron trajectory calculations; scanning electron

microscopy; x-ray microanalysis

1. Introduction

Electron scattering in a solid is a complex multi-dimensional problem that is quite

intractable to analytical solutions. The first attempts to describe the phenomenon were

thus based on empirical relationships and simplified scattering models, leading to a good

qualitative understanding of the scattering process which was sufficiently quantitative for

many purposes. The development of electron probe x-ray microanalysis, however, was found to require a much more exact approach if the technique was to achieve its full quantitative

potential. Fortunately, the demand for more precise theory coincided with the computer

revolution so that direct simulation of electron scattering by Monte Carlo techniques

became feasible, and as increasingly powerful computers have become available, so have more

ambitious Monte Carlo calculations.

The first application of Monte Carlo techniques to the problems of electron probe

microanalysis was by Green [1]-^, who used experimental results from thin films as a basis

for his calculation. His paper showed clearly the potential of the method, but was rather

limited in its scope by the need for detailed experimental data. By 1965 a more general

approach based on theoretical scattering cross-sections was reported at the 4th Interna-

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

5
tional Conference on X-Ray Optics and Microanalysis in Paris by workers from both Japan [2]

and the United Kingdom [3]. This approach, although relatively simple, required what was,

at that time, a large computer. Small computers soon became increasingly common in laboratories and, in 1971, Curgenven and Duncumb [4] described a very much simplified calculation

based on the same general approach that has proved very successful. Since then, the rate of

publication of Monte Carlo calculations has increased, many of these aimed at explaining

contrast effects in the scanning electron microscope rather than at microanalysis problems.

More exact scattering cross-sections have been tried and the effects of replacing the con- tinuous energy loss approximation by individual inelastic events have been explored.

This paper does not set out to be a comprehensive review of Monte Carlo calculations

relating to electron probe microanalysis, but rather to examine the various approaches that

may be taken and to relate them to the problems that the calculations are setting out to solve. In the main, the calculations may be divided into two classes: those aimed at providing a general understanding of the electron scattering processes leading up to x-ray

production; and those intended to solve particular problems. For example, in the conventional microprobe analysis of a flat, uniform specimen, the first class may be used to evaluate and develop the ZAF correction procedures. The second class finds its main application in dealing with situations where there are special boundary conditions such as thin

surface films or particles where generalized empirical relationships, such as are used for

the ZAF correction, cannot easily be applied.

2. General Discussion of Electron Scattering Problem

Before discussing the different Monte Carlo approaches that have been used, it is instructive to discuss the electron scattering problem in general. Electron scattering

occurs through two distinct processes: (1) elastic scattering by the nucleus which is responsible for most of the angular scattering and (2) electron-electron scattering that causes the incident electron to lose energy, but causes relatively little angular deflection.

In a typical case met in x-ray analysis, where one wishes to investigate the characteristic

x-rays produced by an electron beam in a solid, with the beam energy three times the critical ionization potential, each incident electron would suffer several hundred elastic

scattering events and a similar number of inelastic events before its energy dropped below

the ionization potential and ceased to be of interest. Given that the efficiency of characteristic x-ray production per incident electron is usually less than 10"^, a vast computation

would be required if a direct simulation approach were to be used even if all the individual scattering cross-sections were adequately known. The art of setting up a Monte Carlo

calculation is to introduce as many simplifications to the scattering process as possible

without affecting significantly the end result.

The first simplification that is common to all calculations, so far, is to ignore the

structure of the solid. To take full account of electron diffraction effects would be

completely impracticable for the thick specimens of interest in x-ray production, although

some empirical allowance may be possible. The uncertainties introduced by this necessary

approximation will limit the precision that is worth aiming for in any calculation.

Having replaced real matter with a continuum approximation, electron scattering may be

approached as a diffusion problem. That is, the scattering distribution in the target may

be represented as a function, f (_r, v^, E, s) where r_ and y_ are position and direction vec- tors and E and s are electron energy and total distance traveled in the target. No general

solution to this problem has been published but if one introduces the continuous energy loss

approximation based on the Bethe [5] energy loss equation, that is, E is directly related to

s, the scattering problem can be written as a Boltzmann equation [6] which can yield solutions for two particular cases:

(1) thin films in the small angle approximation--the multiple scattering

theories used for electron transmission; and

(2) infinite targets for which Lewis [6] has derived integral equations for the

moments of electron scattering distributions, from which electron energy

loss distributions and the depth dose curve distribution could also be

obtained [7].

6
The significance of these two solutions to Monte Carlo calculations is that the first

allows elastic scattering events to be grouped together giving a fixed number of steps in a condensed path--rather than treating each elastic scattering event individually--with a corresponding saving in computing time. The second gives an accurate solution for the

particular case of an infinite target, against which the Monte Carlo results may be checked

to reveal any systematic errors that may be introduced by the calculation procedure.

The Boltzmann equation may, of course, be solved by purely numerical methods similar to those used for diffusion problems with the added complication of a varying diffusion coefficient. Dashen [8] has applied such an approach but the calculation is very unwieldy and

would appear to be much less efficient than a comparable Monte Carlo calculation. Ogilvie

and Brown [9] have solved a simplified transport equation formulated by Bethe, Rose and

Smith [10] by numerical methods and applied the results to x-ray production. This approach

appears very attractive but essentially neglects large angle single scattering events. It

is likely, therefore, that the method will give reasonable results for electron penetration

(within the limits of the continuous energy loss approximation) but will be less reliable

near the specimen surface and for electron back-scattering where single large angle scat- tering events are very important.

The continuous energy loss approximation is extremely valuable in reducing the com- plexity of Monte Carlo calculations, and for most microanalysis problems it is quite acceptable. Its use produces two main systematic errors:

(1) energy spectra obtained from the calculation are more strongly peaked than

is found experimentally; and

(2) spatial distributions are similarly affected to a lesser extent. The

effect is only really significant in the tail of the distribution and so the continuous loss approximation should not be used if an accurate calculation of x-ray production in the region of boundaries is required. In these cases, it is necessary to take some account of so-called "energy

straggling" effects. On the whole, however, it is usually desireable to retain as far as possible the framework used in the continuous loss calculations rather than using individual scattering cross-sections. Landau

[11] has extended the Bethe approach giving loss distribution rather than mean energy loss. Alternatively, a purely empirical distribution function

may be employed.

3. Monte Carlo Calculation

In this section, the main types of Monte Carlo calculation used in relation to microprobe analysis are discussed. By the time Green started his work, the Monte Carlo technique

was well established and about this time there were a number of publications devoted to the

general problems associated with Monte Carlo calculations [12]. The one of most relevance

was a paper by Berger [13] which reviewed the use of Monte Carlo techniques for high energy

electrons. The development of Monte Carlo calculations for microprobe analysis is considered

below under three headings: experimental scattering; multiple scattering; and finally,

single scattering and other improvements.

3.1 Experimental Scattering Approach

Once the continuous energy loss approximation has been accepted, the main problem in setting up a Monte Carlo calculation is how to represent the angular scattering of the

electron. Green overcame this problem yery simply by taking the experimental data of Thomas

[14] for electron scattering in a 1000 A copper film. With his calculation, Green was able

to get very encouraging agreement with experimental data for electron backscattering and the

depth distribution of characteristic x-ray production. This approach is clearly limited in

its scope because of the need for accurate experimental data, and the main significance of

Green's work v/as to demonstrate the significance of the Monte Carlo technique to the theory

of microprobe analysis. Bishop [15] carried out further work based on Green's approach

again showing good agreement with experiment. In this paper, a comparison was also made

with calculations of the moments of electron scattering distribution from the Monte Carlo

calculation with those calculated from the Lewis equations [5]. Both this calculation and

7
that of Green were made before Cambridge University had acquired a really modern computer.

As a result a great deal of attention had to be given to the efficiency of the computer

program and to presenting the results in as concise a form as possible. In particular, it

would not have been possible to run a full calculation for every different ionization potential, so a set of complete trajectories was produced and stored in the form of a scattering

matrix f(z,s) where z is the depth below the surface and s the actual distance traveled in the target. Electron scattering distributions obey a well-known scaling law [7,16], i.e.,

for a given element the function f(z/r, s/r), where r is the Bethe range, is almost independent of incident beam energy. Owing to this property, the scattering matrix once evaluated

for a particular element and beam specimen geometry contains all the information necessary

to calculate x-ray production for any critical ionization potential over a wide range of

beam energies. An example of such a matrix for copper is shown in Table 1.

3.2 Multiple Scattering Approach

Following Green's pioneering work, the next stage was clearly to replace experimental

scattering data by theoretically based scattering distributions that could then be applied more generally. A suitable multiple scattering formalism had been developed by Goudsmit and

Sanderson [17] and two papers based on this approach were presented by Bishop [3] and by

Shimizu, et al . [2], to the 4th International Conference on X-Ray Optics and Microanalysis

in Paris in 1965. The only substantial difference between these two calculations was that

Bishop [2] retained a fixed step length as this was more convenient to use in conjunction

with the scattering matrix method of recording the results while Shimizu, et al . , used a reducing step length As^, where

AS. = ^ AS^ . 0

This is, in general, a better choice than a fixed step as it allows for increased scattering

as the electron loses energy. As many subsequent calculations are based on the same ap- proach it is worth considering it in rather more detail.

Inelastic scattering is represented by the Bethe equation

dE 2TTe^N Z . 1.165E

where N is Avogadro's number, Z and A are the atomic number and atomic weight of the target,

respectively, and J is the mean ionization potential. Integration of this relationship

gives the Bethe range, r, of the electron and a relationship between s, the distance traveled

in the specimen and residual electron energy, E.

The elastic scattering can be represented by a screened Rutherford cross-section

: (2)

2 4 do Z^e^

16E^(l+2a-cose)^

where a is a screening factor, e the electronic charge and e the scattering angle.

The screening factor a, like the mean ionization potential, may be treated as an adjustable parameter to give a good fit to experimental data. A typical expression is that

used by Bishop

a = 3.4 Z^/^/E (3)

(E is in electron volts).

The angular scattering distribution f(p,e) for each step is determined by the GoudsmitSanderson formula which is expressed as a Legendre series

, T (-p(l-aJ)

f(p,e) = 4^ 2^(2n+l)e " P^(cos e) (4)

n=0

8
-a

<u J3

+-> <ucn

sro

+->

s-(UQ-Q.O

(J

cn

c:

s-<u

+->

••-> lau

1/5

CMo

ID CO LO cn cn CM

CO CM co CO

CM CM CO CO CO CO CO CM O CO CO cn r~

CM CO CO o <n

CM CM CO CO CO CO CO CM

LD CO LD CM CO LO

O o CO

LO LO CM CM CO =3- CM CTl CM CO CO CO CO co CO CM

C\Jm o ^ LD LO CO ^ LO

LO r-^ ^ LO CO CO o 1^

CM CM CO CO CO CO CO CM

<Ti r-. 00 r-- >^ ^ CO

CO LO LD CO "=1- CM CM CO CO co CO CO

CO CO CO ^ *-D CO LD LD LO CO CM

CM CO CO CO CO CO CO CM

OO LD cnO LD

IX) «^ o CM CO ^ CM CM CO CO CO CO CO CM

CM

O CM CM ^ CO CD

CM CO CM CM CO CO CO CO CO CO CM CM LO LO LD

CO CO co CO cn

O CM CM CO CO CO oo CO CO CO

CT< LO CO LO CO cn

00 CTl LO o LO CM

CM CM CO CO CO oo CO CO o r-- LO cn CD

CO LO CM CO o o LD

CM CO CO CO CO CO o LO LO CO LD LO o cn

LO <:;}- 00 CNJ CM o CM CO CO CO

LO <^ CM LO CO LD

CO LD CM

CM CO =3- CO

Ln CO CO o CO I-^ cn cn

CO CM LO

OO CM CO ^ LO LO o O LD cn cn LO CO

CM

O CO LO

CO 1^ LO LO LO LO CM ^ o-i LD CM r-- co J CM CO CO o 00 o CM

CO LO LO LD LO ^

CO CO CM CO r-. CO CM LO

CM CO 00 cn CM LD CO LO LO LD LD LD CM ^ LD LO CO CO cn

CM CM CTi

O ^ 00

CO LO CO CO ^_ CM CM CO LO LO LO o

CM CM

O CM CO

CO LO CO

O 00

CM CTt CO 00 r-- CM 00 CO CM ^ CO ' CO LO 00 ' CO

CO CMl CTi CM 00 CTi O cn

COr

CO

LO

O CM LD CTi CO ' CM

00

O CO CM CT. LO CTi CM

CM CM r~- CM

CM CM O 00

.— CO

LO CM

CO .— CO

Q.

+->

O >— LD

CO LD CM

CM I— I—

"a- 03 CM

CO LO CM

r- CO o ^ cn

CO LD I

CM LO cn

LD ^ cn ^ LD I

CM I— 1 O — CO CM

CM I— I

LD LO cn o r-- CO

CM I— 1

CO CO ' CM cn r— I CM I— .

LD r-- LO

cd- LD

O CM I— 1

LO I I

LO LO cn II— LO

COo o CO

CM CO

LD CO

CM CO LO CM

cn ^ ^ CM

CMl I

CM LO CO CM

CO 00 ^ CO o O LD ^ n- O CMO r— O O CM

CO CO o o o O CNJ O O

I— o O

OO r--

OOO

OO LOCO o o o o o

CO LD cn

LO CO 00

CM .

o CM cn ' CO "^i- LO I CM I— o cn LO I C\J 1— CO

CM 1

O I— LO < o CO ^ 1— o CM

CM CO

CO IDLD

cn

00ocn

CMo

LD

00ocn

00o

00o

CO

00

LD

CO

LO LD r-~ 00 cn

O I— OJ CO LO LD r-~ 00 cn

O 1— C\J CO ^ CM CM CM CM CM 9
where the an are the Legendre coefficients for the single scattering distribution and p is the mean number of elastic scattering events in the step as given by

P=^o^AS (5)

where is the total elastic cross section and p is the density.

The Rutherford cross-section is particularly useful in evaluating eq. (4) because the

Legendre coefficients may be obtained from a simple recurrence relationship, although

precautions must be taken to overcome rounding errors [18]. There is however, no reason why

a more precise cross-section should not be used, although the generation of the Legendre

coefficients will be more complicated.

Equation (4) as it stands can give problems with convergence as it includes the unscattered proportion of the beam. Shinoda, et al . [19], have found that this effect limits the

mininum step that can be used. However, this convergence problem may be simply avoided by

recasting the equation in the form

1 v ^ pa F(p,e) = e"P [6(9) + ^ 2^ (2n+l) (e "-1) P^(cos e)] (4b)

0 where 6(e) is a delta function [18]. Even so, several hundred terms may be necessary for

the light elements where the screening function is small. Equation (4) allows angles of

equal scattering probability to be evaluated for each step in the electron trajectory and

these are subsequently used in the Monte Carlo program.

The multiple scattering approach essentially replaces the effects of a number of single

scattering events that may befall an electron traveling a distance as in the target by a single scattering event occurring at some arbitrary point (selected at random) within the

step. Provided the step length is chosen such that the chances of two large angle scattering

events (say greater than 20°) is small, this approach is a very good approximation. A good

criterion for choosing the step length is to set it equal to the mean free path for single

scattering through an angle greater than say 10 or 15°. It may not be possible to satisfy

this criterion easily for heavy elements or at low voltages, in which case it may be necessary to work out some corrective strategy based on an effective step length.

Calculations based on this approach give very good general agreement with experiment

with the exception that energy spectra are rather too narrow and that the tails of scattering

distributions are much smaller than experimental ones--both predictable consequences of the

continuous energy loss approximation. Agreement is best for the light elements and rather

poorer for the heavy elements. This again is to be expected as the Rutherford cross-section

is a very poor approximation for the heavy elements.

Subsequently, a number of papers based on this general approach^have appeared specifically aimed at microanalysis problems. For instance, Pascal [20], Henoc and Maurice [21]

have used the fixed step length approach, while the paper by Shimizu, et al . [22], using a variable step length, is representative of a number of relevant Japanese papers. Although

this method is successful and relatively efficient, a fairly large computer is required to handle the tables containing angles of equal scattering probability. As small computers

have become more common within the laboratory, there is a need for a much simpler type of

calculation. Such a calculation was described by Curgenven and Duncumb [4] and is probably

the best small computer calculation available. Although often referred to as a single scat- tering calculation, it is more strictly a multiple scattering approach with a fixed step

length. The Bethe equation is again used for energy loss and the Bethe range is divided

into between 25 and 100 steps. The Rutherford cross section in its simplest form may be

written as

cot(e/2) = ^

where b = 1.44 x lO'^ Z/E (E in kV) and a is an impact parameter. In each step, the angular

deflection is determined from a random number n, such that

cot(e/2) = a.

10
The impact parameter a is chosen so the calculated back-scattering coefficient matches the

experimental value.

At first sight, this approach appears over-simplified but, in practice, the approximations are not all poor provided a sufficiently short step length is used. When one bears in mind the approximate nature of the Rutherford cross-section, this greatly simplified approach

may give results very comparable in accuracy to calculations based on the Goudsmit-Sanderson

multiple scattering theory. It would be interesting to see a systematic test of this ap- proach against experiment and other Monte Carlo calculations.

A further development within this general category is the "single scattering" model of

Murata, Matsukawa, and Shimizu [23]. In this model, the step length is chosen equal to the

elastic mean free path and angular deflection selected from the screened Rutherford crosssection. A true single scattering model would have step lengths selected at random. With a fixed step length, a multiple scattering distribution should always be used in principle

I but, in practice, the error involved in using the single scattering cross section is proba-

' bly negligible. Although this approach requires substantially more computing time, the

authors have shown that it is more appropriate for the heavy elements than a full multiple

I scattering approach.

3.3 Single Scattering and Other Improvements

The calculations described in the previous section are subject to two major approximations. First, the continuous energy loss approximation and second, the use of the Rutherford

scattering cross-section. The former is for many applications quite acceptable, but in the

energy range of importance in microprobe analysis the Rutherford cross-section is a very

poor approximation particularly for the heavy elements. Krefting and Reimer [24] have been

able greatly to improve agreement with experiment by using the exact Mott elastic cross- section for scattering greater than 10° using a true single scattering approach for elastic

scattering. Energy loss was also partly treated in a single scattering mode. Low energy

losses were described by a modified Bathe equation, but higher losses (greater than about

100 eV) were treated as single scattering events. Any calculations attempting to achieve

absolute accuracy, particularly for the heavy elements and lower beam energies, will have to incorporate these more precise cross-sections, although the multiple scattering formalism

may still be used if required.

There are a number of ways in which the continuous energy loss approximation may be

replaced. The most obvious is to include all the possible inelastic interactions in a single scattering calculation. This approach however, would lead to a very lengthy calculation and it is doubtful vjhether the cross-sections are well enough known in many cases to justify the expense. A simpler alternative is to take the mean energy loss for each step as predicted by the Bethe formula, but to determine the actual energy loss by sampling from

some distribution about the mean energy loss using either the Landau equation [11] or some

empirical formula. Both these alternatives have been adopted in recent papers, Henoc and

Maurice [25] using the Landau formula, and Shimizu, et al . [26] using an empirical expression.

These two papers do demonstrate clearly the effects of the continuous loss approximation and

where it is acceptable. Shimizu et al . were investigating the energy spectra of electrons

transmitted through thin foils. The spectra given in a calculation based on the continuous

loss approximation are quite unrealistic (although the mean energy loss is correct). By

introducing energy straggling, good agreement was obtained with experiment. On the other

hand, Henoc and Maurice did not find that straggling had a very significant effect on the

overall distribution of characteristic ionization in a solid except in the tail of the

distribution. Thus, for most microanalytical applications, the continuous loss approximation

is satisfactory.

4. Presentation of Results

A Monte Carlo calculation is essentially a statistical experiment. How the results are treated will depend on the aims of the calculations which for microanalysis fall into two categories:

(1) a direct simulation to solve a specific problem; and

(2) calculations aimed at gaining a general understanding of electron scattering

in solids.

n
The first category presents no real problem. The simplest Monte Carlo calculation that

will give an acceptable result is employed to give the desired information directly. In this application, the Curgenven and Duncumb method [4] is the obvious first choice. Monte

Carlo calculations have been incorporated within correction programs to determine atomic

number and absorption corrections, for instance by Shimizu et al . [27]; however, the length

of Monte Carlo calculations makes them unsuitable for general use in such applications.

Improvements to the standard ZAF correction procedures are better approached via the

second oategory using the results of the calculation to test procedures and to suggest the

correct parameters to use, subsequently confirming the absolute value of any parameters by

experiment. Monte Carlo calculations have not had as much impact in this field as they

should in spite of the excellent agreement that has been demonstrated between the calculations

and experiment. This is probably because of the difficulty of presenting the results in such a way that other workers can make use of them without setting up identical calculation.

The author has found the f(z,s) matrix described earlier to be a very convenient method of

saving the results of a Monte Carlo calculation and he has been able in this way to develop

ideas using data from Monte Carlo calculations carried out some eight years previously [28]. A more general use of this means of presenting results would lead to a much more effective

and widespread use of Monte Carlo calculation.

5. Suggested Aims for Future Work

The basic framework for the ZAF correction for microprobe analysis is well founded and

the inherent approximations involved, for instance using an average value for the stepping

power, are acceptable. However, there are three areas in which Monte Carlo calculations

could make further significant contributions:

(1) effects of the angle of incidence of the electron beam;

(2) choice of the mean ionization potential, J; and

(3) an improved absorption correction.

With the extensive use of energy dispersive x-ray detectors in the scanning electron

microscope, there is a growing requirement for a careful examination of the variation of the

absorption and atomic number correction with the angle of beam incidence. This requires a straightforward application of one of the well -proved calculations.

Of more fundamental importance is the choice of the mean ionization potential, J. There is currently a great deal of uncertainty about what value is best, particularly for

the very light elements and also for the very heavy elements where the incident beam energy

is less than the binding energy of the more tightly bound atomic electrons. A series of Monte Carlo calculations for different elements using the appropriate single inelastic

scattering cross-sections to calculate energy loss as a function of distance traveled

(ignoring angular scattering altogether) could well throw invaluable light on this problem.

The Philibert absorption correction formula owes its success to the fact that f(x) is extraordinarily insensitive to the depth distribution of characteristic ionization for

f(x)>0.8. It may be shown to be quite inadequate, even in its full form, for lower values

of f(\). Monte Carlo calculations have already been used to show the form a more accurate

approach should take [28], but more work is required to establish a procedure for routine

use

.

References

[1] Green, M. , Pvca. Phys. Soa. 82, 204 (1953).

[2] Shimizu, R. , Murata, K. , and Shinoda, G. , X-Ray Optics and Microanalysis, R. Castaing,

P. Descamps, and J. Philibert, eds . , p. 127 (Hermann, Paris, 1966).

[3] Bishop, H. E., see ref. [2], p. 112.

[4] Curgenven, L., and Duncumb, P., Tube Investments Research Report 303 (1971).

12
I [5] Bethe, H. A., Ann. Phys. Leipzig b, 325 (1930).

[6] Lewis, H. W., Phys. Rev. 78, 526 (1950).

[7] Spencer, L. V., National Bureau of Standards Monograph No. 1 (1959).

[8] Dashen, R. F. , Phys. Rev. 134A , 1025 (1964).

' [9] Ogilvie, R. E. and Brown, D. B., see ref. [2], p. 139.

I [10] Bethe, H. A., Rose, M. E., and Smith, L. P., Proo. Am. Phil. Soo. 78, 573 (1938).

j [11] Landau, L., J. Phys. USSR 8, 201 (1944).

j [12] Hammersley , J. M. and Handscomb, D. C, Monte Carlo Methods (Methuen, London, 1964).

i [13] Berger, M. J., Methods in Computational Physics, Vol. 1, B. Alder, S. Fernback, and M. Rotenberg, eds. (Academic Press, New York and London, 1963).

[14] Thomas, R. N., Ph.D. Thesis, University of Cambridge (1961).

[15] Bishop, H. E., Proe. Phys. Soo. 85, 855 (1965).

[16] Gruen, A. E., Z. Naturforsch. 12a, 89 (1957).

[17] Goudsmit, S. and Sanderson, J. L., Phys. Rev. 57, 24 (1940).

[18] Bishop, H. E., Ph. D. Thesis, University of Cambridge (1966); Brit. J. Appl. Phys. 1_8, 703 (1967).

[19] Shinoda, G., Murata, K. , and Shimizu, R., Quantitative Eleotron Probe Microanalysis,

K. F. J. Heinrich, ed., p. 155 (National Bureau of Standards Special Publication 298,

Washington, D.C. 1968).

[20] Pascal, B., X-Ray Optics and Microanalysis, G. Mollenstedt and K. H. Gaukler, eds., p. 137 (Springer, Tubingen, Berlin, 1969).

[21] Henoc, J. and Maurice, F. , Proceedings of the Sixth International Conference on X-Ray

Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinodawa, eds., p. 113 (Univeristy of Tokyo, 1972).

[22] Shimizu, R. , Ikuta, T. , and Murata, K. , J. Appl. Phys. 43, 4233 (1972).

[23] Murata, K. , Matsukawa, T., and Shimizu, R. , J. Appl. Phys. Japan 1_0, 678, (1971 ).

[24] Krefting, E. and Reimer, L., Quantitative Analysis With Electron Mioroprohe and Secondary

Ion Mass Spectroscopy, E. Preuss , ed., p. 114 (Julich: Zentralbibl iothek der KFA,

Julich GmbH, 1973).

[25] Henoc, J. and Maurice, F. , CEA-R-4615, (Commisariat a I'Energie Atomic, France, 1975).

[26] Shimizu, R. , Kataka, Y., Matsukawa, T. , Ikuta, T. , Murata, K. , and Hashinoto, H., j.

Phys. D 8, 820 (1975).

[27] Shimizu, R., Nishigori, N., and Kurata, K. , see ref. [21], p. 95.

[28] Bishop, H. E., J. Phys. V 2009 (1974).

13

National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1975.

STUDIES OF THE DISTRIBUTION OF SIGNALS IN THE SEM/EPMA

BY MONTE CARLO ELECTRON TRAJECTORY CALCULATIONS

AN OUTLINE

Dale E. Newbury and Harvey Yakowitz

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234

Monte Carlo electron trajectory calculations provide a powerful technique for

the study of the characteristics and distributions of the various signals generated

in electron-specimen interactions. This paper is a review of the diverse applications of Monte Carlo techniques to the calculation of signal distributions. The

following topics will be considered: extent of the primary interaction volume;

lateral, depth, angular and energy distributions of backscattered and secondary

electrons; the extent of the x-ray generation volume, and the depth distribution of

characteristic and continuum x-rays. Numerous examples of these calculated distributions are drawn from the literature and compared, when possible, with appropriate

experimental results.

Key Words: Backscattered electrons; electron probe microanalysis; Monte Carlo electron trajectory simulation; scanning electron microscopy; secondary electrons;

x-rays.

1. Introduction

The techniques of Monte Carlo electron trajectory simulation have been applied by many

authors to the problem of determining signal distributions arising from electron-specimen

interactions. In this paper, we shall review examples of calculations of the following

signal distributions: (1) extent of the primary interaction volume; (2) lateral and depth

distributions of backscattered electrons; (3) angular and energy distributions of backscattered

electrons; (4) lateral and depth distributions of secondary electrons; (5) angular and

energy distributions of secondary electrons; (6) extent of x-ray generation volume; and (7)

the depth distribution of x-ray production. Examples of these distributions, as calculated

by Monte Carlo techniques, will be presented, taken from the literature as well as our own

work. This paper is intended as an outline review, highlighting the areas in which Monte

Carlo techniques have been used, rather than a comprehensive review of all work. Therefore,

we will not attempt to list all citations of a particular type of calculation; instead,

representative examples of each calculation will be given.

2. Extent of the Primary Interaction Volume

Direct experimental visualization of the primary electron-target interaction volume is only possible in certain targets such as gases or liquids. Indirect visualization is possible for some solid organic compounds such as polymethylmethacrylate, where chemical

etching following electron beam irradiation produces a pit which reveals the extent of

primary electron interaction. For many targets of interest to the microscopist/analyst,

e.g., solid elemental or compound targets, experimental determination of the interaction

volume can only be carried out indirectly through sequential measurements of a series of

films of different thicknesses so as to determine the fractional electron transmission.

15
The Monte Carlo electron trajectory calculation is directly applicable to the problem

of determining the interaction volume. Since the coordinates (X, Y, Z) of the electron are calculated at each scattering point, a plot of these successive points traces the path of

the electron through the solid. By superimposing many trajectories on the same plot, a picture is built up of the interaction volume. Such a plot must necessarily be two-dimensional; and therefore, the true trajectory path in three dimensions is actually projected

onto the particular section selected.

Curgenven and Duncumb [1]^ presented a series of trajectory plots revealing the extent

of the primary interaction volume (figs. 1-4). The trajectories plotted by the Curgenven

and Duncumb program show sharp angular changes, a result of their use of a multiple scat- tering approximation. Plots of trajectories with single scattering models are much smoother,

a result of the small step length used in these calculations and the small angle of scattering. The interaction volume in copper as a function of increasing beam energy (fig. 1) shows an increase in size. This increase is expected since the elastic scattering cross- section is proportional to l/E^, and the stopping power, which is related to the rate of

energy loss with distance traveled in the specimen, goes as log E/E. Thus, both the elastic

scattering cross-section and the stopping power decrease with increasing energy, leading to

an increase in the size of the interaction volume with increasing energy. Note that the

interaction volume at 20 keV is roughly spherical and more than 1 ym in diameter.

electron x-ray Figure 1. Interaction volume in copper with

20 kV normal beam incidence at various accelerCopper ating voltages (Curgenven and Duncumb [1]).

The effect of the atomic number of the target upon the interaction volume is illustrated in figure 2, where aluminum and gold are compared assuming a beam energy of 20 keV. The

interaction volume has a diameter of approximately 3 ym for aluminum and 0.4 ym for gold.

The reduction in the physical size of the interaction volume with increasing atomic number,

Z, is a result of the dependence of the elastic scattering cross-section upon V- . From figure 2, one can also observe the increased number of trajectories which exit the specimen;

i.e., backscatter, for gold as compared to aluminum. Furthermore, examination of figure 2 shows a basic difference in the shape of the interaction volume in the low Z material (aluminum) as compared to the high Z material (gold). The dense region of interaction volume in aluminum is pear-shaped, with a narrow neck near the surface, whereas the interaction volume

in gold resembles a sphere partially submerged below the specimen surface. The interaction

volume for an element of intermediate atomic number, copper, shows a transition structure

containing the neck characteristic of the pear shape, while showing a more spherical shape

(fig. 3). Note that even in aluminum, the region over which x-rays are produced is spherical, but the density of production varies greatly. In gold, the density is more uniform.

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

16
20 keV

Aluminum

electron x-ray

20 keV

Gold

Figure 2. Comparison of the interaction

volume in aluminum and gold at normal

beam incidence and 20 keV beam energy

(-Curgenven and Duncumb [1]).

The effect of the angle of beam incidence on the interaction volume is shown in figures

3 and 4 for copper bombarded with a beam having energy of 20 keV. As the specimen tilt

increases and the beam approaches grazing incidence, the interaction volume decreases in size and the path density increases near the surface. These effects result from the tendency of electrons to scatter in the forward direction. The angular distribution of elastic

scattering is strongly peaked in the forward direction; at grazing incidence, the forward

scattering keeps the electrons near the surface. The increased tendency for electrons to escape the specimen can also be observed in figures 3 and 4; thus, the backscattering

coefficient increases as a function of tilt.

0.5 ym

20 keV

Copper

x-ray

x-ray

Figure 3. Interaction volume in copper at normal incidence and 20 keV beam energy

(Curgenven and Duncumb [1]).

Figure 4. Interaction volume in copper with

20 keV beam energy at beam incidence angles

of 45° and 10°; compare with figure 3 (Curgenven and Duncumb [1]).

17
The density of plotted trajectories in figures 1-4 makes individual trajectories difficult to discern; the poor resolution results from the superposition of trajectories in two dimensions. By calculating the scattering events within a radial segment, the radial distribution function can be obtained (fig. 5). For quantitative evaluation of the interaction

volume, a plot such as that in figure 5 is more useful than a drawing of superimposed

trajectories

.

1500-

N unit vol

1000-

Copper

Eo = 30 keV

e = 0°

500- Figure 5. Lateral extent of the interaction

volume indicated by the emission of backscattered electrons.

0

0 0.1 0.2 0.3 0.4

r/R^

0.5 0.6 0.7

At normal beam incidence, the sections Y-Z and X-Z (coordinate system: Z perpendicular

to surface, positive going into the specimen; X,Y orthogonal and contained in the surface

plane, X the axis of tilt; Y pointing down a tilted specimen) are equivalent because of the

radial symmetry of scattering. For tilted specimens, the Y-Z and X-Z sections are not equivalent because of forward scattering. This asymmetry is illustrated in figures 6 and 7,

where the Y-Z and X-Z sections for iron tilted 55° (beam incidence 35°) and a beam energy of

30 keV are given. The high path density portion of the interaction volume is more compact

in the X-Z section than in the Y-Z section.

Figure 6. Section of interaction volume perpendicular to the tilt axis for iron at a tilt angle of 55°, and a beam energy of 30 keV (Myklebust, et al . [3]).

18
2

'

' " / . , 30 keV

0.5 ym •„ .' 550

Figure 7. Section of the interaction volume containing the axis of tilt for

iron at a tilt angle of 55° and a beam. energy of 30 keV (Myklebust, et al . [3]).

The analysis of particles is gaining increasing importance. This class of objects

provides one of the most difficult analysis situations, and quantitative analysis is frequently impossible by conventional techniques when the particle size approaches the x-ray

interaction volume. The techniques of Monte Carlo electron trajectory simulation are well

suited to analysis when the geometry of the target becomes important, and calculations for

particles are in progress [4]. The effects of finite particle size relative to the interaction volume are shown for aluminum spheres in figure 8, illustrating the enhanced loss of

electrons from particles.

5 keV 17 keV

'-fete?

Aluminum sphere

30 keV

0.5 ym

Figure 8. Interaction volume in aluminum spheres at different beam energies

(Myklebust, et al. [3]).

When the specimen thickness is below the diameter of the interaction volume, electron

transmission occurs. This transmission causes a change in the lateral dimension because

backscattering is reduced. Curgenven and Duncumb [1] illustrated this effect by plotting

the trajectories of 20 keV electrons passing through 0.25 ym and 1.3 ym thick films of

aluminum (fig. 9). The transmission through the 0.25 ym film is nearly complete and the

interaction volume is small. For the 1.3 ym film, backscattering is significant, and the

interaction volume is quite similar to that observed in the top 1.3 ym slice of the interaction volume of the bulk target. The work of Krefting and Reimer [5] has shown that the

19
0.25 ym

1 ym

Figure 9. Interaction volume as a function

of specimen thickness for aluminum with

normal beam incidence and 20 keV beam

energy (Dunoumb and Curgenven [1]).

Al uminum

20 keV

transmission coefficient, t, can be accurately calculated by Monte Carlo techniques (fig.

10). Shinoda, et al , [6], have shown that the angular distribution of electrons transmitted

through a thin foil calculated by Monte Carlo techniques is in good agreement with the

experiment (fig. 11).

0 200 400 600 800 1000 ug/cm2

Schichtdicke ^

Figure 10. Coefficient of transmission of electrons, t, through thin films,

Monte Carlo and experimental values (Krefting and Reimer [5]).

20
Figure 11. Angular distribution of electrons

transmitted through an aluminum foil of 0.557

ym thickness. Monte Carlo calculations by

Shinoda, et al . [6]; experimental points by

Thomas [7].

0 0.4 0.8 1.2 1.6

e (rad)

Several authors [8,9,10] have carried out experiments involving irradiation of polymethylmethacrylate, followed by etching, to reveal the contours of the energy deposition

(fig. 12). Comparison of Monte Carlo calculations of the energy deposition contours with

the experimental results show good agreement (fig. 13).

Figure 12. SEM images of polymethylmethacrylate following

increasing chemical etching times to reveal deposited

energy contours (Everhart, et al . [10]).

21
Experiment Incident Monte Carlo

electron

Figure 13. Comparison of energy dissipation

profiles in polymethylmethacrylate determined experimentally and by Monte Carlo

calculation (Shimizu, et al . [8]).

3. Properties of Backscattered Electrons

Beam electrons which exit the specimen after scattering through a large angle, or else

after undergoing several small angle collisions which reverse the direction of flight, are collectively referred to as backscattered electrons. These electrons provide an important source of information in scanning electron microscopy. Many authors have used Monte Carlo

techniques to calculate the properties of backscattered electrons. It is well known experimentally that the backscatter coefficient, n , increases with increasing atomic number, Z, of

the target (fig. 14). This dependence can be used as a test to adjust the parameters of the

n

T T- Trill I 1

0.5

0.3

keV

1 •central target

0.1 / 0 massive

1 1 1 1 i _i 1_

target

1 I 1 1

I I 1 1 1 1 1 10 30 50 70 90 Z

Figure 14. Backscatter coefficient, as

a function of atomic number; experimental

data determined by Heinrich [11]. Central

and massive targets refer to the experimental techniques.

Monte Carlo techniques so that realistic values will be calculated for other parameters of

interest. For example, the backscatter coefficient increases as a function of increasing

specimen tilt, and Monte Carlo calculations show good correspondence to experimental mea- surements of this effect (fig. 15).

Of particular interest in determining the depth of sampling of the backscattered electron signal is the escape probability of incident beam electrons which have penetrated to various depths. Calculations of the escape probability for 30 keV electrons incident on

22
0 10 20 30 40 50 60 70 80 90

Tilt e (degrees)

Figure 15. Backscatter coefficient as a function of specimen tilt for

iron with a beam energy of 30 keV; experimental values for iron--3.22%

silicon (Myklebust, et al. [3]).

copper reveal a rapidly decreasing probability of escape with depth (fig. 15). Elaboration

of this calculation gives the fraction of escaped electrons, r\Q, and absorbed electrons, nA^

as a function of depth (fig. 17). Although the absorption of electrons takes place at a greater mean depth than the mean penetration of the backscattered electrons, the absorbed

0 0.1 0.2 0.3 0.4 0.5 0.6

Depth (ym)

Figure 17. Maximum penetration depth of backscattered and absorbed electrons for aluminum

and gold (Murata [2]). dashed line from

Thomas [6].

23
(specimen) current signal does not contain information from deeper within the specimen than

does the backscattered electron signal. Contrast formation in the specimen current signal

depends only on the fact that electrons have excited the specimen and, thus, the specimen current signal has a depth of sampling identical to that of the backscattered electrons. The

depth distribution of backscattered electrons is affected by the proximity of an edge, which

leads to enhanced escape because of the lateral extent of the interaction volume (fig. 18).

Primary Beam

Figure 18. Depth distribution of maximum penetration of backscattered electrons

with a point beam incident near an edge (Shimizu and Murata [12]). The histogram

indicates the number of electrons which have penetrated to a given depth before

backscattering.

Incident

24
In determining the spatial resolution of the backscattered electron signal, the escape

of electrons remote from the beam impact area must be considered as a result of the lateral extent of the interaction volume. Murata [2] has calculated the lateral escape area of

backscattered electrons for different atomic number targets and for conditions of normal and

tilted specimens (fig. 19). For normal incidence, a strong central peak is observed, with

the lateral extent reduced in the case of high atomic number. For a tilted specimen, the

central peak is enhanced and sharper for gold. The area distribution becomes asymmetric,

with the forward lobe enlarged. The extent of the forward lobe is quite similar to the

lateral range observed at normal incidence. All of these observations are in keeping with

the characteristics of the interaction volume noted previously.

The angular distribution of the backscattered electrons can be calculated by recording

the angle of elevation of the electron trajectory above the surface. A histogram of such

exit angles is shown in figure 20. Experimentally, the angular distribution is found to follow a cosine law because of the state of nearly complete diffusion or randomness of the

electron trajectories following many elastic scattering acts. A cosine law distribution is superimposed on the histogram of figure 20, and the correspondence is very good.

Electron Beam

90 60 30 0 a (degree)

Figure 20. Angular distribution of backscattered electrons relative to surface normal and a cosine distribution [12]; experimental results of Kanter [13].

Wells has reported the angular distribution of low energy-loss electrons which scatter

out of the specimen after one or two elastic collisions [14]. The measurements were made

with the specimen highly tilted (i.e., beam incidence nearly grazing) to enhance forward

scattering out of the specimen. Wells' results are shown in figure 21 (barred points) and

are compared with the angular distribution for single-event electrons predicted by a Monte

Carlo technique.

The energy distribution of the backscattered electrons has been calculated by means of Monte Carlo techniques by several authors [3,4,12,15,16,17]; examples of these distributions

are shown in figure 22 along with experimental results. Usually, the correspondence is very

good, although the position of the high energy peak and the shape of the distribution is somewhat dependent on the form chosen for the scattering model.

Wells has demonstrated the great value of forming SEM images with those electrons which

have lost only a small fraction of the incident energy (low-loss electrons) [14]. The image

derived from these electrons, which have traveled a limited distance in the solid, is par- ticularly sensitive to surface structure. The information is localized near the impact

point of the beam, making this signal especially useful for high resolution microscopy.

Murata [19] has carried out Monte Carlo calculations to study characteristics of such backscattered electrons. A comparison is shown of the surface emission function of backscat25
Beam

20°

Wells Low Loss

Simula tion

Eloss <500V

I Weils Experimental —Monte Carlo

Sample : 0.5/i.m Si Og on Si 20 keV

Figure 21. Angular distribution of low loss electrons

with a beam incidence angle of 20°. Experimental

results of Wells [14]; Monte Carlo calculations

by Myklebust, et al. [3].

tered electrons with consideration of (a) all energies (fig. 23), and (b) low energy losses

(fig. 24). Even at normal beam incidence, the low-loss electrons show a stronger and more sharply defined central peak. Murata further calculated the surface emission area of backscattered electrons selected by the angle of elevation along which they exited the specimen

(fig. 25(a)). For a specimen tilt of 45°, the electrons emitted at an elevation angle less

than 30° come from about 25 percent of the area of emission of electrons with elevation

angles in the range 60-90°. Moreover, the mean electron penetration depth was found by the Monte Carlo calculations to be strongly dependent upon the elevation angle of the backscattered electron, especially for tilted specimens (fig. 25(b)). The utility of (a) tilting

the specimen, (b) accepting only a limited angular evelation range, and (c) energy selection

is thus confirmed by the Monte Carlo calculations.

26
Aluminum 30 keV

Monte Carlo

Multiple Scattering Model

Single Scattering Model

Experiment

Kulenkampff and Spyra

dn/dE

1.0 X 10-2

10 20

FeAl3 Alloy

(40.8 - 59.2 wt%)

30 keV

Eg = 30 keV

Monte Carlo

Multiple Scattering Model " Single Scattering Model

10

Monte Carlo

Multiple Scattering Model

Single Scattering Model

Experiment

Bishop (Copper Target)

20

3.0 X 10-2

dn/dE

2.0 X 10-2

1.0 X 10-2

0.0 X 10-2

10 20 30 keV

E

2.0 X 10-2

dn/dE

1.0 X 10-2

0.0 X 10-2

30 keV

E

Figure 22. Energy distributions of backscattered electrons calculated by Monte Carlo techniques with single scattering and multiple scattering

models, Shimizu, et al . [16]; experimental results of Bishop [18].

27
Figure 23. Surface distribution of backscattered electrons of all energies;

copper target, beam energy 20 keV (Murata [19]).

28
Figure 24. Surface distribution of backscattered electrons

which have lost less than 1 keV; copper target, beam energy

20 keV (Murata [19]).

Figure 25. (a) Surface emission area for backscattered electrons which have been emitted

with low (0-30°) or high (30-90°) exit angles relative to the surface, (b) Mean penetration depth as a function of exit angle (Murata [19]).

29
4. Properties of Secondary Electrons

The most common signal used by scanning electron microscopists for obtaining images of

three-dimensional objects has been the secondary electron signal. The low energy of these

electrons (< 50 eV) allows for their efficient collection by means of a suitably biased

grid. Moreover, because of their low energy, they can only escape from near the surface of

the specimen, and thus convey information concerning approximately the top 50 K layer of the

specimen.

The most extensive study to date of secondary electrons by Monte Carlo techniques is that by Koshikawa and Shimizu [20], and most examples given here will be drawn from that

paper. These authors used the energy excitation function of Streitwolf [21] to describe the

energy distribution of the production of secondary electrons. The primary beam was considered

to penetrate--without scattering or significant energy loss--to a depth of 75 A, while

producing secondary electrons along this path. Secondary electrons produced by the exiting

backscattered electrons were ignored. Since secondary electron production by the backscattered electrons is more efficient than production by primary electrons, this simplification

restricts strict interpretation of the results of Koshikawa and Shimizu [20] to the case of

primary electrons incident on thin films where backscattering is negligible.

The energy of the secondary electrons produced along the path of the primary electrons

was then adjusted for energy loss during passage through the solid prior to escape. The

mean free path was deduced from results of Kanter [22] and of Palmberg [23]. Moreover, the

slow electrons undergo energy loss and scattering in collisions with conduction band electrons. The authors considered the loss of energy from the slow electron, its change in direction due to scattering, and the energy and direction of the conduction electron after

the interaction (fig. 26). Finally, they considered the refraction of the slow electrons by

the surface potential barrier.

Surface ////// /u /// \/ /////////////// j / n J J

Ejected electron

Scattered electron

Scattered electron

Figure 26. Schematic of secondary

electron calculation by Monte

Carlo techniques (Koshikawa and

Shimizu [20]).

Taking into account all of these effects, Koshikawa and Shimizu [20] determined the

energy distribution of the emitted secondary electrons from copper. This distribution is compared with the experimentally observed distribution in figure 27. The shapes of the

distributions are similar, with the peak energy slightly higher for the Monte Carlo calculation.

The angular distribution of the emitted secondary electrons calculated by the Monte

Carlo technique is shown in figure 28 with a cosine law distribution superimposed on the

calculated points. Good agreement is found, indicating that the secondary electrons are generated in random directions by the incident beam.

30
I 1 1 1 L_ 0 5 10 15 20

Energy (eV)

Figure 28. Angular distribution of secondary electrons and

a superimposed cosine law (Koshikawa and Shimizu [20]).

Horizontal line is the surface, vertical line is the normal

to the surface. Polar coordinate plot (r,e).

The calculated lateral distribution of the secondary electrons with an incident point

beam is shown in figure 29. The distribution at normal incidence is about 10 K wide, full

|. width, half maximum, indicating the limiting spatial resolution which can be expected. As

' the specimen tilt is increased, the distribution becomes skewed in the forward direction,

and the strong central peak is diminished.

I The distribution in depth of the origin of the emitted secondary electrons is shown in figure 30. The f unction^shows a rapid decrease with only a few percent of the signal origi-

[ nating at a depth of 75 A. Examination of the lateral distribution of the secondary eleci| trons as a function of depth (fig. 31), reveals that the strong central peak observed in

I figure 29 originates with those secondary electrons formed near the surface. This behavior

: occurs because the secondaries produced deeper in the specimen are more likely to scatter

laterally while propagating back toward the surface.

3]
Primary beam

6 = 0°

i.o

FWHM

(b)

Primary beam

9 = 30°

\

-30 -15 0 15 30 Distance x

1.0

Primary beam °^

e = 45°

-30 -15 0 15 30 Distance x (^)

(d)

1.0

Primary beam

: 60°

-1 1 ^ -30 -15 0 15 30 45 60 -30 -15 0 15 30 45 60 Distance x (^) Distance x (^)

Figure 29. Lateral distribution of secondary electrons with a point beam at various incidence angles to the surface (Koshikawa and Shimizu [20]).

Depth Z (A)

Considering the emergence distance from the primary beam impact point and the depth

origin of the secondary electrons, the authors obtained figure 32; a depth distribution

strongly peaked near the surface occurs at the beam impact point. As the distance of emer- gence from the impact point is increased, the depth distribution becomes relatively flat.

Koshikawa and Shimizu also considered the excitation of characteristic Auger electrons,

and a plot of the depth distribution for the copper MMM transition (E = 5§ eV) is shown in figure 33. Virtually all of these electrons originate from less than 20 A in depth since

the energy loss processes rapidly cause deviation from the characteristic energy.

32
sJ3

So

il.O (a)

0.75

0.5

0.25 .J -30 -15 0 15 30

0.5

^^^^^^

10.25

-30 -15 0 15 30

(d)

// \o.25

1 • >H

(e) 0.25

-30 -15 0 15 30 -30 -15 0 15 30 Distance x (^)

(c)

\o.5

\o.25

-30 -15 0 15 30

(f)

-30 -15

0.25

15 30

Figure 31. Lateral distribution of secondary electrons as a function of

excitation depth below the surface, (a) 0-5 A; (b) 5-10 A; (c) 10-20 A;

(d) 20-30 A; (e) 30-40 A; (f) 40-50 A (Koshikawa and Shimizu [20]).

^,^ce ^ 6.0 ^

OA^^^ 9.0 "

Primary beam

Intensity of SE (arbitrary units)

50 100 ISO 200

cX 25'

Figure 32. Depth distribution of secondary electrons as a function of

position of emission relative to beam impact point (Koshikawa and Shimizu [20]),

Shimizu and Murata considered the contribution of secondary electron production by the

exiting backscattered electrons [12]. Their results for the spatial distribution of emitted

secondary electrons are shown in figure 34. The same sharp central peak is again observed,

with the number of secondary electrons falling off rapidly with distance from the beam

impact point. The contribution of backscattered electrons is found to be small, remaining

relatively constant with distance from the beam impact point. Near the beam impact position,

33
the secondaries produced by backscattered appear to be insignificant. But, backscattered

electrons can exit thousands of angstroms from the impact point; and, hence, the secondary

electron intensity integrated over this emission area becomes very large.

Shimizu and Murata [12] calculated the emission of secondary electrons as a function of

specimen tilt; the results yielded the behavior shown in figure 35. The Monte Carlo calculations follow most closely the experimental formula of Bruining [24]. The secant law

usually assumed for tilt behavior underestimates the actual value significantly at tilt of 40°.

Depth Z (K) Distance (K)

Figure 33. Depth distribution of copper MMM

type Auger electrons (E = 58 eV) as calculated by Monte Carlo techniques; and exponential decay law is also shown (dashed

line (Koshikawa and Shimizu [20]).

Figure 34. Laterial distribution of secondary

electrons emitted from copper and aluminum;

contribution of secondaries created by backscattered electrons for copper (hatched area)

(Shimizu and Murata [12]).

2.0

6(a)

6(0)

1 .0

§ Monte Carlo Bruining 's expt 1 formul

a

- - Secant 1 aw

Copper 30 kV

0° 20° 40° 60° Angle of incidence a

Figure 35. Emission of secondary electrons

as a function of specimen tilt. An experimental formula of Bruining [24] and a se- cant dependence are also shown (Shimizu

and Murata [12]).

5. Distribution of X-ray Signals by Monte Carlo Methods

The typical outputs of the Monte Carlo technique as applied to x-rays produced by

electron-solid interactions are: (1) two-dimensional plots of the distribution of ionization in the solid, (2) the computation of the distribution in depth <^{z) of the x-rays generated by the primary electron beam, (3) the computation of the absorption correction

factor f(x) for x-ray microanalysis based on the <i>{z) calculation, and (4) the distribution

in depth of continuum x-radiation of a given energy. The major achievement is the ability

to predict x-ray outputs from geometrical configurations which classical theory is not capable of handling.

34
Plots of the ionization distribution were first presented by Curgenven and Duncumb;

some of their results are reproduced in figures 1-4. The method used was to compute the

probability of an ionization at any location of the electron along its trajectory. If the Monte Carlo model indicated that an ionization should occur at that location, then a dot was

plotted. This procedure was carried out for all electrons at all positions. The resulting

plots give a qualitative idea of the x-ray distribution in the solid as a function of atomic

number, Z, incident beam energy, (keV), and incident beam angle [1,25].

Bolon and Li fshin, and Bolon, et al . , extended the work of Curgenven and Duncumb to include the effects of finite beam size and to analyze thin film specimens with and without

substrates [26,27]. Again, the resulting plots yield a qualitative view of the processes

taking place in the solid as shown in figures 36 and 37.

Figure 36. Monte Carlo calculated electron and x-ray distributions in 500 A gold films on various substrates (Bolon, et al . [27]).

In addition to plotting the results of the ionization distribution, the Monte Carlo

technique can be used to calculate the distribution in depth of the x-rays generated by the

primary electron beam. Several papers have reported the results of such computations [2-

6,8,9,12,15,16,21,28]. Typically, a plot of the parameter (i,{z) against z, the mass thickness, is obtained. The parameter ^(z) represents the number of x-rays generated at depth z normalized by the number of x-rays which would be generated from a "free layer" of the

specimen, i.e., a free standing layer of vanishingly small thickness [29]. A typical plot

of (j)(z) against z for normal electron beam incidence is shown in figure 38. The maximum in this function occurs because the lateral spread of electron paths increases the probability

of ionization per unit of depth, and because some electrons scatter back toward the surface,

producing ionizations along the path. If the incident beam angle is altered, the c[)(z) plot

against z changes; figure 39 shows the results of a Monte Carlo calculation for a copper

target [25]. The peak of the distribution moves to lower z values as the electron backscatter coefficient increases; the peak is primarily dependent on this parameter.

35
20.0 kV , 0.40 ym 1 ^ 1 vacuum w

Ni-10 Cr

(a)

20.0 kV 0.40 ym

TaC Ni-10 Cr

(b)

20.0 kV 0.40 ym

1

1

1?—: 7

1 vacuum "V'?<"= TaC Ni-10 Cr

(d)

Figure 37. Simulated electron and x-ray dis- tributions for: (a,b) focused (0.2 ym)

electron beam and (c,d) defocused (1.0 ym)

electron beam; TaMa x-radiation (Bolon, et al. [27]).

<t>(pz)

0.2

Al umi num

20 keV

• Castai ng-Henoc experiment — Monte Carlo calculation

02 04 Q& OB 1.0 Mass thickness pz (mg/cm^)1.2

Figure 38. Distribution in depth of AlKa

x-rays in Al matrix. Experimental data

[32] are compared with Monte Carlo re- sults calculated using the NBS program

(this volume, p. ). Note that mass thickness is indicated as pz.

36
The ({)(z) function can be used to predict the absorption effect [29] and atomic number

effect [29], thus facilitating quantitative x-ray microanalysis. The generated x-ray intensity is:

00 /0

while the emitted intensity is

CO

Iemit = F(x) = /e-x^*(z)dz (2)

0

where x

= v csc^j;, u being the x-ray mass attenuation coefficient of the specimen for the

monitored x-ray line, and ii the average x-ray emergence angle from the specimen with respect

to the detector. If the specimen has an average atomic number, Z*, while the standard has

atomic number Z, then in the conventional ZAP notation [30]:

C = k kp (3)

where C is the mass fraction, k is IeMIT/^ ' EMIT > > '^^^ measured intensity ratio of sample

to standard, kz is the atomic number correction accounting for differences in electron backscatter and retardation between a sample of atomic number I* and a standard of atomic number

Z, k/\ is the absorption correction factor accounting for differences in absorption of the

x-ray line of interest along its path within the target for the specimen and standard, and

kp accounts for secondary emission (fluorescence) effects. The terms kz and k/\ have tjeen

examined in great detail both experimentally and theoretically by many authors in an effort

to obtain the most accurate quantitative analysis scheme possible as outlined in reference

[30].

The definition of the primary absorption correction factor f(x) (also designated fp) is ^emit/^gen' ^"^^^ (""^

f(x) = F(x)/F(0). (4)

For the specimen, likewise, f*(x) = F*(x)/F*(0) and

* The term kz in eq. (3) is defined as the ratio Ige^)/^GEN' i-e., the ratio of primary

beam x-rays generated in a target of atomic number Z, divided by those generated in target

of atomic number Z*. Hence,

4 = ^

• F(0)/F*(0). (6)

Therefore, the product

k k - ii^Ho) 1- ilM . Fj^ . HO) = lEx) (7)

37
The usefulness of the (|)(z) distribution is now clear; with <t>{z) , one can obtain the number

of photons generated by the primary beam, the terms kj and kA, their product, and the absorption correction factor f(x). Indeed, Castaing pointed this fact out in his thesis [29].

However, relatively few (j)(z) distributions have been determined experimentally; the experiment is tedious, difficult, and experimental errors are potentially very large [31-33].

Therefore, the Monte Carlo method was seen as a means to obtain i,{z) distributions and,

hence, all of the useful by-products for x-ray microanalysis. All of the (t)(z) curves so calculated depend upon the specific Monte Carlo simulation model employed. Thus, while the

(})(z) distributions can be calculated, the result is subject to all of the uncertainties in the model, e.g., scattering law, continuous slowing down approximation, etc. Crucial

questions then, are how sensitive to model changes is the distribution and how well do calculated (t>{z) distributions agree with experimental findings?

Reimer and Krefting have indicated that the calculated (t)(z) distribution is not particularly sensitive to the scattering model selected nor is <t>{z) greatly affected by fairly

large changes in model input parameters [34]. The effect of using one or the other of two different mean ionization potentials to calculate <i>{z) for Au-Ma is shown in figure 40. Henoc and Maurice have indicated that electron straggling effects play only a minor role in altering the calculated <\){z) distribution [35].

Agreement of calculated (}){z) distributions with experimental results is fairly good,

especially at low z values [3,5,6,15,28]. Hence, the Monte Carlo results for (()(z), f(x)>

k/\, and are probably no worse than those derived by other methods. Indeed, Shimizu, et

al . [16], used the Monte Carlo method for correcting raw data for quantitative x-ray microanalysis. The results were comparable to ZAF results for a variety of materials and experimental conditions.

So far as f(x) vs x curves derived from Monte Carlo methods are concerned, Duncumb

compared Monte Carlo results for Cu-Ka with those calculated by the Phi 1 i bert-DuncumbHeinrich relation [25]; results are shown in figure 41. This figure indicates the effect of

tilt on the result as well thus illustrating the potential value of the Monte Carlo method

in dealing with a wide variety of experimental arrangements. Figure 42 shows l/f(x) plotted

against x for Ta-La as calculated by Monte Carlo procedures, but using two sources for mean ionization potential [36,37] for comparison with Green's [38] experimental results. It is difficult to believe that, since the Monte Carlo (})(z) distribution is not greatly sensitive

to model, but is affected by input parameters, significant improvement in conventional x-ray

microanalysis will result from Monte Carlo calculations. However, intractable experimental

arrangements can be attacked by means of the Monte Carlo method.

38
1 .0

0.8

f(x)

0.6

0.4

0.0

Monte Carlo

Z-Mr a L i7U

2000 4000

X

6000 8000

Figure 41. Variation Monte Carlo calculated absorption correction factor

f(x) vs. X 9S a function of tilt

(Duncumb [25]). The dashed lines indicate the effect of varying the mean ionization potential.

3.5

3.25

3.0

2.75

1 2.5

2.25

2.0

1.75

1 .5

1 .25

1.0

n 1 1 r Ta-La

19.75 keV

© Green, experimental

Duncumb

J value.

Berger-Sel tzer

J value

J I I I I I L 2000 4000

X (cmVg)

6000 8000

Figure 42. Monte Carlo calculated value of l/f(x) vs. x '''or TaLa using two values of mean ionization potential. Experimental results of Green [38] are shown as wel 1

.

Among these difficult specimen geometries is the case of a thin film on a substrate.

Kyser and Murata have used Monte Carlo calculations to derive intensity against thickness

curves for Si, Cu and Au on AI2O3 substrates for a wide variety of incident beam energies

[9]. Agreement with experimental results was very good (fig. 43). Furthermore, these

authors were able to quantitatively analyze binary thin films of Co-Pt and Mn-Bi on Si02

substrates; neither composition nor film thickness was known. The Monte Carlo method was

used to establish calibration curves for several thicknesses as shown in figure 44; the

final results were graphically interpolated from these curves and the measured intensities.

Agreement with nuclear backscatter measurements was good [9].

Bolon and co-workers were able to measure Ta content in square TaC rods a few micrometers across in a di rectional ly solidified TaC-NiCr eutectic alloy. Monte Carlo calculations

agreed with experimental findings within a few percent. These authors also successfully

investigated thin film specimens with the aid of Monte Carlo computations [26,27].

39
100

Au Ma

ij; = 52.5

y/p =1124 cm2/g

Monte Carlo

• Experiment

Figure 43. Monte Carlo intensity ratios as a function of thickness and incident beam energy

for Au films on AI2O3 as compared with experimental results (Kyser and Murata [9]).

1000

Thickness {A)

10,000

0.3

0.2

0.1

1 1

CoPt on SiOj

1

20 kV CoKa

(// = 52.5° PtMa

.^_^^^pt= 100 (/jg/cm^)

pt=100 ^

—-____50 ^^^Xr'^ 50^ ——

=--c£^]^~=^C\^ 25

~r 1 i

0.3

0.2

0.1

Figure 44. Monte Carlo calculated calibration curves for Co and Pt x-ray

emission from thin films on Si02 substrate (Kyser and Murata [9]).

X-ray computations based on the Monte Carlo method have been made for spherical and

cylindrical particles [3,4]. Results are in good agreement with experimental findings (fig.

45), thus indicating that the Monte Carlo method can be used to handle cert/Jin of the problems

involved with particulate analysis. . ,

Apparently, only one attempt to derive the distribution of continuum x-rays as a function of z by means of Monte Carlo methods has been carried out. Reed obtained this distri40
1 I

1 \ \ 1 f

Sphere diameter (ym)

Figure 45. Comparison of Monte Carlo calculated intensity ratios from Al spheres with experimental findings of Bayard [43]. Ikyij^ the intensity

from pure, bulk aluminum. I^j-jg^ is the intensity from a sphere of a particular diameter.

bution for 8.04 keV continuum x-rays in a copper target assuming Eq = 29 keV [40]. The

cross-section for continuum x-ray production was obtained from results reported by Kirkpatrick

and Weidmann [39]. The Monte Carlo results were obtained from the work of Bishop [44]. The

combination resulted in the distribution reproduced as figure 46. Apparently, the distribution of characteristic and continuum x-rays of like energy is similar— not an altogether

surprising result.

41
The study of the continuum distribution would be useful in studying background effects

in energy dispersive x-ray detectors [41]. Monte Carlo calculations of this distribution

might also be useful for analytical situations in which a small quantity of heavy material

(high atomic number) resides in a light (low atomic number) matrix. In such cases, most of

the observed x-ray intensity from the heavy component will be produced by continuum x-rays

[42].

6. Summary

Monte Carlo electron trajectory calculations can be used to obtain diverse information

about the distribution of signals in the SEM/EPMA. In the examples reviewed in this paper,

the characteristics of the interaction volume, backscattered electrons, of secondary electrons, and of x-rays have been calculated. Where experimental measurements are available,

the agreement between the Monte Carlo calculations and experimental values is generally very

good. The Monte Carlo calculations can be carried out in many cases where experiments are difficult or intractable. Monte Carlo calculations provide a powerful tool for both the

microscopist and the analyst.

References

[1] Curgenven, L. and Duncumb, P., Simulation of Electron Trajeotovies in a Solid Target,

by a Simple Monte Carlo Technique, Report 303 (Tube Investments Research Laboratories,

Saffron Walden, Essex, England, July 1971).

[2] Murata, K. , J.A.V. 45, 4110 (1974).

[3] Myklebust, R. L., Yakowitz, H. , Newbury, D. E. , and Heinrich, K. F. J., this volume,

P-

[4] Yakowitz, H. , Newbury, D. E. , and Myklebust, R. L. , in SEM/1975, Proceedings of the 8th

Annual SEM Symposium, 0. Johari , ed. , p. 93 (IITRI, Chicago, 1975).

[5] Krefting, E. R. and Reimer, L. , in Quantitative Analysis with Electron Mioroprobes and

Secondary Ion Mass Spectrometry, E. Preuss, ed. , p. 114 (Kernforschungsanl age , Julich,

1973) .

[6] Shinoda, G. , Murata, K., and Shimizu, R. , in Quantitative Electron Frobe Microanalysis,

K. F. J. Heinrich, ed. , p. 155 (National Bureau of Standards Special Publication 298,

Washington, D.C. , 1968)

[7] Thomas, R. N., Ph.D. Thesis (Cambridge University, 1961).

[8] Shimizu, R. , Ikuta, T. , Everhart, T. E., and Devore, W. J., J. A. P. 46 (1975).

[9] Kyser, D. F. and Murata, K. , Monte Carlo Simulation of Electron Beam Scattering and

Energy Loss in Thin Films on Thick Substrates, IBM Research Report, RJ 1404 (June

1974) .

[10] Everhart, T. E., Herzog, R. F. , Chung, M. S., and DeVore, W. J., in Proceedings Sixth

International Conference on X-ray Optics and Microanalysis, G. Shinoda, K. Kohra, and

T. Ichinokawa, eds., p. 81 (University of Tokyo Press, 1972).

[11] Heinrich, K. F. J., in X-ray Optics and Microanalysis, R. Castaing, P. Deschamps, and

J. Philibert, eds., p. 159 (Hermann, Paris, 1966).

[12] Shimizu, R. and Murata, K. , J. A. P. 42, 387 (1971).

[13] Kanter, W.,Ann. Physik 20, 144 (1957). •

[14] Wells, 0. C, in Sem/2975, Proceedings Eighth Annual SEM Symposium, 0. Johari , ed.,

(IITRI, Chicago, 1975).

42
Murata, K. , Matsukawa, T. , and Shimizu, R., Jap. J. A. P. 10, 678 (1971 ).

Shimizu, R. , Nishigori, N., and Murata, K. , in Proaeedings Sixth International

Conference on X-vay Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa,

eds., p. 95 (University of Tokyo Press, Tokyo 1972).

Shimizu, R. , Ikuta, T., and Murata, K. , J. A. P. 43, 4233 (1972).

Bishop, H., Brit. J. A. P. series 2, 673 (1968).

Murata, K. , in SEM/1973, Proceedi-ngs Sixth. Annual SEM Symposium, 0. Johari , ed., p. 268 (IITRI, Chicago, 1973).

Koshikawa, T. and Shimizu, R. , J. Phys. D. : Appl. Phys. 7, 1303 (1974).

Streitwolf, H., Ann. Phys. Lpz. 3, 189 (1959).

Kanter, H., Phys. Rev. Bl, 522 (1970).

Palmberg, P., A-nal. Chem. 45, 549A (1973).

Bruining, H. , Physiaa 1046 (1936). -

Duncumb, P., in Electron Microscopy and Analysis,. W. C. Nixon, ed., Conf. Series 10,

p. 132 (Inst. Phys., London, 197l).

Bolon, R. B. and Li f shin, E., in SEM/73, Proceedings Sixth Annual SEM Symposium, 0. Johari, ed., p. 285 (IITRI, Chicago, 1973).

Bolon, R. B., Li f shin, E., and Ciccarelli, M. F., in Practical Scanning Electron Microscopy, J. I. Goldstein and H. Yakowitz, eds., p. 299 (Plenum Press, New York, 1975).

H^noc, J. and Maurice, F., in Proceedi-ngs Sixth International Conference of X-ray

Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds., p. 113

(University of Tokyo Press, Tokyo 1972).

Castaing, R., Ph.D. Thesis, University of Paris (1951).

Yakowitz, H., in Practical Scanning Electron Microscopy, J. I. Goldstein, and H. Yakowitz, eds., p. 327 (Plenum Press, New York, 1975).

Castaing, R. and Deschamps, J., J. Phys. Radium, 1_6, 304 (1955).

Castaing, R. and Henoc, J., in Proceedings Fourth International Congress on X-ray

Optics and Microanalysis, R. Castaing, P. Deschamps, and J. Philibert, eds., p. 120

(Hermann, Paris, 1966).

Brown, J. D. and Parobek, L. , in Proceed.ings Sixth International Conference on X-ray

Optics and Microanalysis, G. Shinoda, K. Kohra, and T. Ichinokawa, eds. p. 163

(University of Tokyo Press, Tokyo 1972).

Reimer, L. and Krefting, R. , this volume, p.

Henoc, J. and Maurice, F., this volume, p.

Duncumb, P. and Reed, S. J. B., in Quantitative Electron Prohe Microanalysis, K. J. Heinrich, ed. , p. 133 (National Bureau of Standards Special Publication 298,

Washington, D.C. , 1968),

[37] Berger, M. J. and Seltzer, S. M., /VatZ. Acad. Sci. , Natl. Res. Council Publ. 2133,

205 (Washington, D.C, 1964).

43
[38] Green, M. , Ph.D. Thesis, University of Cambridge (1962).

[39] Kirkpatrick, P. and Wiedmann, L., Phys. Rev. 67, 321 (1945).

[40] Reed, S. J. B., X-ray Spectrometry 4, 14 (1975).

[41] Fiori, C. E. , Myklebust, R. L. , Heinrich, K. F. J., and Yakowitz, W.,Anal. Chem.

48, 172 (1976).

[42] Myklebust, R. L., Yakowitz, H., and Heinrich, K. F. J., Proaeedings Fifth National

Conferenoe MAS, paper No. 11 (New York, 1970).

[43] Bayard, M. , in Mioroprobe Analysis, C. A, Andersen, ed., p. 323 (Wiley, New York,

1973).

[44] Bishop, H. E., Proa. Phys. Soc. 85, 855 (1965).

44
National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

THE EFFECT OF SCATTERING MODELS ON THE RESULTS OF MONTE CARLO CALCULATIONS

L. Reimer and E. R. Krefting

Physikalisches Institut

Universitat Munster, Germany

A special Monte Carlo model is presented considered Mott scattering for

scattering angles c>10°, a mean angular deviation caused by multiple scattering

for c<10°, individual inelastic scattering events with energy loss aE>200 eV and

a continuous energy loss for aE<200 eV. Results of the Monte Carlo calculations

are compared with experimental results of the backscattering coefficient, the

angular and energy distribution of backscattered electrons and the transmission

of thin films. Some calculations of the depth distribution of x-ray emission

are reported. Parts of the program are changed to see which parts of the model

influence the results.

Key Words: Monte Carlo electron trajectory simulation; scanning electron microscopy; electron probe microanalysis; electron-specimen interactions; electron

microscopy; x-ray generation.

1. Introduction

The results of Monte Carlo calculations (MC) depend on the model used. It is not possible to consider all collisions individually, especially the inelastic ones, because

this increases the computation time. One has to use approximations, for example, a Bethe

law, for the energy loss along the electron trajectories.

Our complete model consists of the following parts:

(1) use of single elastic collisions with the exact Mott scattering cross-sections

for large angle scattering > 10°) instead of the Rutherford cross-sections;

(2) treatment of small angle scattering < 10°) by a mean angular deviation of the

electron trajectory with a formula of Lewis for multiple scattering;

(3) taking into account single inelastic electron-electron collisions and the generated fast secondary electrons (FSE) with energies larger than 200 eV, using a formula of Gryzinski; and

(4) use of continuous energy loss along the electron trajectory by the Bethe

formula, subtracting the energy losses caused by the single inelastic collisions

considered in (3).

If one wants to calculate the backscattering coefficient, n, of bulk material at normal

incidence, the result is rather insensitive to the model. One obtains, for example, reasonable values if only Rutherford scattering at angles ? > 10° and a continuous energy loss by

the Bethe formula are used [1]^. But if one tries to get agreement between MC and experiments for the increase of n with increasing film thickness or the energy and angular distribution of backscattered electrons, one obtains systematic deviations. In Section 6, we report calculations omitting or changing some of the above-mentioned parts of the program and

how such omissions influence some results.

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

45
2. Single and Multiple Scattering Data

2.1 Single cross-sections for scattering angles c > 10°

Figure 1 demonstrates the relative differences if Mott cross-sections (da/dfj)^ calculated by Buhring [2], Badde, et al . , [3], and Reimer, et al . [4], are used instead of

Rutherford cross-sections (da/dfi)^. These cross-sections were verified in experiments on Hg

lOkeV

^0= 20keV

Figure 1. The ratio of Mott to Rutherford cross- section as a function of the scattering angle ^ and electron Eq for Al , Ge, and Au. The re- sults are obtained using Byatt potentials. The

influence of another approximation for a screened

Coulomb potential (Tietz) is demonstrated for Ge

at 10 keV and 100 keV.

120^ 750^ IW

46
vapor (40-200 keV) by Kessler and Weichert [5]. Because of the rapid decrease, proportional

to l/sin'* c/2, of the Rutherford cross-section, the results are represented as a ratio of

(da/dfi)n/|/(da/dfi)fj. The relativistic Rutherford cross-section is

\ do.

E„ + m^c^

0 0 + 2ni^c' 0 0 sin^ 5/2

(1)

Values of da/dn for scattering angles c < 10° are strongly influenced by the screening of

the Coulomb potential of the nucleus; these have been determined by WKB calculations [6].

In addition to this influence of screening at low scattering angles, deviations occur at large scattering angles increasing with increasing atomic number Z. The screening is expressed by exponential terms in the Coulomb potential [7]. Figure 1 for Ge shows no large

deviations (dotted lines) if the screening parameters proposed by Tietz [8] are used.

One needs the total cross-section

^el 10J {%) 2. sin ,d,

no ^Uii/|^ (2)

for the calculation of the free path, s, between collisions. These collisions with scat- tering angles 5 > 10° were calculated individually. They represent a fraction of about 10 percent of all elastic collisions.

2.2 Small angle elastic scattering

We used a mean scattering angle for collisions with scattering angles <; < 10°. The mean value (cos ?) which depends upon the free electron path, s, was calculated by means of

a formula of Lewis [9]

<cosc> = exp

s / 10° da

dn ( 1-cos c)sin cdc 1 ds

0 \

'0

with N = number of atoms per mass thickness (s in g«cm"2),

(3)

2.3 Single inelastic scattering

Electron-electron collisions cause scattering as well as energy losses, W. We considered all inelastic collisions as single events resulting in an energy loss W larger than a limit Wmin

W > W^. = E„T + 200 eV

mi n nl (4)

(Ep-] = ionization energy of an atomic shell with quantum numbers n,l). This means that the

generated fast secondary electrons (FSE) reach at least a kinetic energy E|^-jp > 200 eV. We

used the semi-empirical formula of Gryzinski [10]

da

dW

e^E

nl

3 W^E

E+E

nl

3/2 , ,x^nl/^l^W

1-^

-nl

1/2

(5)

47
for the differential cross-section as a good approximation for nonrel ativistic energies.

One obtains for the classical collision between two free nonrelativistic electrons

, 4 da = !ie_ - • (5)

and the incident electron is then scattered through an angle

. 2 W (7)

sin ^ = I

The classical formula (6) for large W is a good approximation to formula (5) of Gryzinski

for large W. We therefore used eq. (7) for calculating the scattering angle.

The FSE with Ei^-jp, > 200 eV can contribute to electron backscattering , the transmission

of electrons through thin films and K-shell ionization. We treated the diffusion of such

FSE in the solid by means of a simplified model: the FSE are assumed to be emitted isotropically and to penetrate in a straight line up to a c range r from their origin (point P) of

generation. If a sphere of radius r around P is drawn, all those electrons lying on the area F of the sphere outside the specimen will leave the specimen and the probability of

escape is

^ ( (r-z)/2r for z < r w = F/4^r^ = . (8)

( 0 for z > r

The range r is calculated by the empirical formula [11]:

r = a E^ivg-cm'^) with a = 10, b = 1.43 (E in keV). (9)

2.4 Continuous energy losses

Bethe [12] obtained the formula

f) = ^

Z f„, in i^)' m

^VBethe '^^ n,l \ nl /

for the energy loss per unit path length. We used this formula with the approximation fpi = Zpi for the oscillator strength f^^] . A portion of the inelastic collisions was however

considered in 2.3:

N, ^ E. ^ (11)

in

^ ^nl /

W dW

^single n,l

min

Therefore, only

dE /dE\ /dE

" ^^^4ethe V^^single

(12)

48
is used as a continuous decrease of energy proportional to the mean free path, s, between

the two single collisions considered (elastic or inelastic). The energy loss by single

inelastic collisions is of the order of 50% of the whole energy loss. We included no further angular deviation by this kind of inelastic scattering because the inelastical ly scat- tered electrons having small energy losses are concentrated at very small scattering angles.

3. The Organization of the Program

In the MC, one obtains a free path s, scattering angle azimuth angle \p, or energy

loss W by means of a random number 0 < R < 1. If n(x) is the probability distribution

(x = s, ii, or W) the following relationship exists:

R = 7 n(x)dx /n(x)dx. (13)

0 0

The free path, s, between two collisions, for example, is obtained from the total

cross-section

a

t

= ^el °iner (^4)

and exp[-N0t s] is the probability that the electron experiences no collisions along the

path s. The probability that s has values between s and s + ds is therefore

n(s)ds = - Na^ exp[-Na^ s]ds (15)

and (13) becomes

R = 1 - expf-No. s]. (16)

I

1-

Since R' = 1 - R is also a random number, solving for s gives

- In R' No,

(17)

In the case of the azimuth angle ^, one gets the simple relation

^ = 27rR. (18)

Analogous equations are obtained for c and W, respectively, by using the expressions (do/dfi)^ or (5) for n((;) and n(W). Solving for c or W can only be done by numerical methods.

Figure 2 shows the flow chart of a MC program for the calculation of the backscatteri ng coefficient n and the transmission t in terms of film thickness z and angle of incidence CqThe calculation starts at the target surface with the initial conditions x,y,z = 0,

E = Eg and = 0. The electron has the position r^ = (xn,yn,zn) at the n^h collision and

the direction (cp^. i>n) after this collision, and a free path s is calculated by eq. (17).

The new position rp+i = (Xfi+l '^n+l '^n+1 ^ ^'^ electron following collision is

= + s sin cos

y,+l = Vn + ^ *n Vl = ^n " ^ ^n- ^^^^

49
start

New particle with

? = (x,y,z) = 0

C

= 0

E = Eo

Determination of free path s New coordinates fn+i = rp + s Mean energy loss (dE/ds)'S

< cos c ) on the path s

too

Q-O

O

O

I

Loop over all angles of incidence eg Ask for eq. (20) and storage of data

for reflection and transmission

Yes Nuclear collision ? No

Angle deviation by

nuclear collision

with c > 10°

Inelastic electron-electron

collision. Energy loss and

angle deviation. Contribution

of FSE to back-scattering and

transmission

J

Yes

N = N + 1 No

E > E

7 mi n

•

No

N > No ?

Yes Output

End

Figure 2. Monte Carlo Program.

Then < cos c) on the free path s caused by collisions with <: < 10 is calculated by eq.

(3) and (5,1];) are changed for this small angle contribution. The mean energy loss (dE/ds)-s

on the free path s (eq. (12)) is subtracted from E. Then interrogation for penetration

through different thicknesses z* or for backscattering follows:

'n+1 S - 'n+1 ^0

> z* : transmission

(20)

< 0 : backscattering

(Co = angle of incidence; ?o ^ 0- "ormal incidence). If this is the case, the desired

parameters such as energy, exit position or angle, are stored.

A further random number, R, selects either a nuclear collision (? > 10°) or an inelastic electron-electron collision:

50
< R: nuclear collision

°^el '^inel > R: electron-electron collision.

(21)

In the case of a nuclear collision, a scattering angle c is determined by a random

number R. ^-values are stored for 1000 R-values in 10 keV intervals of energy values E. This high number--1000--is the minimum required to obtain enough scattering angles in the

interval ? > 90°. For R-values near 1, there must be a more dense distribution of c(R).

An azimuthal angle ii is obtained from eq. (18) and the sine and cosine of the new directions

(Sn+-|. ^j^+i)

a>^e calculated by

(22) cos

^n+1

= cos cos z - sin c cos sin

^n sin

^n+1

= n - cos2 ,^^^)l/2

cos Vl = [cos ii^ (sin c cos ii cos

^n ' cos c

- sin ii^ sin ? sin 4;]/sin ^n+1 sin ^+1 = [sin ijj^ (sin c cos ij; cos

^n ' cos ?

+ cos sin c sin ij;]/sin ^n+1

•

In the case of an electron-electron collision the first random number determines the

ionized shell; the second one, the energy loss W and the scattering angle c. by eqs. (5) and

(7); with an azimuth angle ip, one obtains a new direction (cp+i, ij^n+i ) analogously to (22).

The kinetic energy Ej^^.^ of the generated FSE is

^kin = W-^r (23)

The range r follows from eq. (9), and using the probability w of eq. (8), a contribution to backscattering or transmission is obtained.

If the remaining energy of the electron is large enough (E > Efp-jf^), the calculation

will be continued with the next collision; otherwise, the contribution of this primary

electron to backscattering, transmission and ionization of an atomic shell is treated like a FSE. Usually, the energy E . is 0.25 E . mi n 0

4. Results of the Monte Carlo Calculations and Comparison with

Experiments on Backscattering and Transmission

A few examples will be shown to demonstrate the agreement between experiments and MC calculations for Al and Au as typical representatives for materials with low and high atomic

numbers.

4.1 Backscattering of films at normal incidence

If we assume single scattering only, the backscattering coefficient of very thin films

due to single collisions with scattering angles c > tt/2 is:

N, z

//2

1^2. sine d,. (24)

51
Values calculated with eq. (24) and experimental values of the initial slope dn/dz are listed in table 1, which shows that the experimental values are much larger. Therefore, one must consider multiple scattering even in very thin films.

Table 1. Values of dn/dz for very thin films of Au (£^=40 keV)

- 1 2 Single scattering calculated with (24): dn/dz (g -cm ) Rutherford, non-relati vistic 195

Rutherford, relativistic 210

Mott cross-sections 380

Experiments

Niedrig and Sieber [13]

Seidel [14]

Drescher [15]

465

560

520

Figures 3a and b show MC calculations and experimental results for relatively thin

films of Al and Au at Eq = 40 kev and normal incidence (cos = The use of Mott scat- tering cross-sections alone cannot explain the increase of n with increasing film thickness

z. One gets much better agreement if the generated fast secondary electrons (FSE) are also

considered. There remains a constant difference between MC and experimental values, which

is nearly independent of film thickness. Therefore, we assume that this difference is caused by FSE with energies between 50 and 200 eV, taecause only secondary electrons with

E < 50 eV are suppressed in the retarding field experiment.

Al UOk^

cos d: =1jO

.<>'Mott*SE

Motf^

^ o

Experiment: —«

Monte -Carlo: —o--

100 200 fjg/cm^ 300

Figure 3. Monte Carlo and experimental

values of the backscattering coefficient

n as a function of film thickness z.

Monte-Carlo:

Rutherford *konst. Buhring*konst

Buhring*konst.*Sek.El.

20 AO 60 80 /jgton^

Figures 4a and b show the calculated values and experimental data obtained for greater

thicknesses. The indicated values in percent are added to the MC values to give a best fit

to the experimental data. The need for this adjustment is attributed to the FSE with

E = 50-200 eV. Figure 4 shows three curves for comparison for Au at 40 keV: (a) a straight

line obtained with eq. (24) using relativistic Rutherford cross-sections; (b) MC calculations

with Rutherford cross-sections for c > 10°; and (c) calculations using Mott cross-sections.

52
OOO xoo-xo0,5 Vo ^

Al

Experiment: ^

Monte-Carlo: o

(a)

° /

7.

40^

n

200 400

Schichtdicke

600 600 1000pg/cm^

50\

E.: r

I lOkeV

o20

'3Vo

M.-C.:+l5Vo

200 400 600

specimen thickness pt

Experiment

o Monte -Carlo -— Rutherford (MC) — —single scattermg

600 lOOOpg/cm^

:b)

Figure 4. Comparison of MC and experimental values of the backscattering

coefficient n as a function of film thickness for thick specimens.

53
4.2 Backscattering of bulk material for oblique incidence

Figures 5a and b show the backscattering coefficient n of bulk material for different

angles of incidence Cg versus cos Co (cos co 1 » normal incidence). There is quite good

agreement between MC and experiment for the increase of n with increasing cq- The curves

for Al show no large differences for different electron energies. MC calculations for both

Al and Au show somewhat larger values at 100 keV.

. -, , .

1 4Uf ' 1 ' • 02 Oji 06 Od 10 0 Q2 0,4 0,6 0,6 1p

cos(yj ^' co5(yj

^

Figure 5. Dependence of backscattering coefficient n of bulk material on the angle

of incidence l, for different electron energies.

4.3. Angular distribution of backscattered electrons

The angular distribution dn(;;)/dfi of backscattered electrons for bulk material shows

roughly a cos^ dependence. This is a circle in a polar diagram or a straight line if

plotted against cos c(cos c = 1: backscattering into ? = tt). There should be larger deviations for thin films at low cose near c = 90°, caused by pronounced single elastic scat- tering into these directions. Figure 6 shows experimental results and the MC calculations.

Figure 6. Angular distribution dn/dfi as a function of cos c for thin films of:

(a) Al , and (b) Au.

54
4.4. Energy distribution of backscattered electrons

The energy distribution of backscattered electrons shows for Al a broad maximum with a most probable energy Ep/Eg = 0.7 (see experiments of Kulenkampff and Spyra [16] in fig. 7).

The MC calculation in figure 7 gives results for the contribution of different depths to this energy distribution. For bulk material, our MC values show a shift of Ep/Eo to some- what smaller values. This will be further discussed in Section 6.2.

Figure 7. Energy distribution of

backscattered electrons for Al

MC calculations about the contribution of different depths. Full curve: experiments of Kulenkampff

and Spyra [16].

4.5. Transmission of films

A good agreement is also obtained (fig. 8) between experiments and MC values of the

transmission x through thin films measured over the range 0 < c < tt/Z.

Figure 8. Transmission x as a function of film thickness.

55
5. The Calculation of X-Ray Emission

Characteristic x-ray emission can be calculated by two methods. In the first, the

numbers of electron-electron collisions with E^] are summed. To obtain absolute values, one must also consider that there are several possibilities for filling the nl-shell; for example, emission of Ka- and Kg-lines. FSE also can contribute to ionization if their energy

E > E -j . There is a probability

for ionization. The contribution of FSE to the ionization of the K-shell of Al is of the

order of 30°^. To a first approximation this does not, however, cause a change in the depth

distribution and can be neglected for relative calculations. It is a disadvantage of this

method that the probability of an ionization, especially of the K-shell, is too low. The

tallying of K-shell ionizations in the form of integer numbers requires very large numbers

of incident electrons in the MC calculation. Therefore, it is better to sum up the probabilities of K-shel 1-ionization along the path s between two collisions as real numbers

smaller than 1. In this case, the single electron-electron collisions with electrons of the

nl-shell must be taken into consideration only for their contribution to the electron diffusion and not for their contribution to x-ray emission. In both cases, the absorption can be taken into account by multiplying with an exponential term exp[-xz] (x = y/sin g,

6 = take-off angle). One obtains also the f(x) correction curves used in microanalysis.

Figures 9-11 show calculated results for Al and Au assuming Eq = 20 and 40 keV and

different values of the angle of incidence Co- The results for Al in figure 9 are calculated for the K-shell ionization while those of Au in figure 11 are for different ionization

energies Ep] of spurious elements in an Au matrix. A more extensive program for comparison

of experimental data of x-ray emission and MC calculations is being undertaken in our labora-

(25)

tory.

2 cmmg Al

cmmg

0 250

Schichtdicke 500 750 ijg/cm^ 1000

Figure 9. Calculated depth distributions of k-shell ionization for Al at Eq = 20 and 40 keV and different angles of incidence Cq.

56
Figure 10. Calculated f(x) correction factors for Al Ka at Eq = 20 and 40 keV

and different angles of incidence CqFigure 11. Calculated depth distributions for x-ray emission of trace elements

with an ionization energy En] in an Au matrix for: (a) normal, and (b) 53°-incidence.

6. Influence of Changing Different Parameters of the Monte Carlo Model

We changed some parts of our normal program (see Section 1) to see which parts of the

model influence the results of the calculation.

6.1. Backscattering of Au for 40 keV incident energy (fig. 12)

The full curve represents the results of the normal MC program including the FSE. The

experimental values are somewhat higher because in this diagram no additional percentages

were added to account for the FSE between 50 and 200 eV as in figure 4. In the curve with

(dE/ds)x 1.5, the continuous Bethe energy losses are increased by a factor 1.5. Because we considered about 50% of the total mean energy loss as single electron-electron collisions,

the mean energy loss is increased by 25%. This change has no influence for thin films, but

saturation is reached at smaller film thicknesses because the total path length is shorter

than before. Increasing the energy loss also causes more frequent elastic collisions because

da/dQ increases with increasing energy. If the cross-section for elastic scattering with

5 > 10° is increased by a factor 1.5, the backscattering coefficient of thin films increases

by the same factor because this coefficient is built up by single and multiple elastic

scattering. The curve, indicated by a(Ruth), was calculated with unscreened Rutherford

57
cross-section (c > 10° only). For very thin films, the backscattering is smaller because

(do/dfi)R<(da/dfi)fvi for c > 90°. There is a steeper increase with increasing film thickness

caused by multiple scattering because (da/dn) r>( da/dfi)[v| far medium scattering angles.

Compared to the results of Rutherford scattering in figure 4b, the FSE are included in this

calculation, so that n for bulk material is larger than the experimental value.

Corresponding changes occur in the transmission curve (fig. 13).

6.2. Energy distribution of backscattered electrons (fig. 14)

There is a good agreement between our normal model (RE in fig. 14) and the experiments

of Kanter [17]. One gets the same "most probably energy" E^/Eq = 0.97. Deviations occur at small E/Eq, since we only considered the FSE in the scattering model but not in the calculation of the energy spectrum. The latter requires further assumptions about the energy

distribution of the FSE. If this contribution to ionization by single electron-electron

collisions is omitted (o(ioni) = 0 in fig. 14), a shift to Ep/Eg = 0.93 occurs. Because

omitting single electron-electron collisions increases the continuous Bethe losses, (see eq. (12)), a factor of 1.5 for the Bethe losses results from the same tendency, whereas Ep/Eg

increases to 0.96 if the single elastic scattering is increased by a factor 1.5.

Figure 12-14. Influence of changing different parts of the MC program on the re- sults of the backscattering coefficient n (fig. 12), the transmission t (fig. 12),

the transmission t (fig. 13), and the energy distribution of backscattered electrons for Au and Eq = 40 deV (fig. 14).

58
6.3 X-ray emission (fig. 15)

Figure 11 shows the depth distribution of shells with different ionization energies E^-] in a gold matrix assuming normal electron beam incidence. Figure 15 demonstrates for

Ep-] = 1 and 30 keV, the influence of the previously discussed alterations of the model. The

corresponding f{x)-curves are very insensitive to these relatively large changes of the

depth distribution and show differences of only 2-5 percent. This means that one should not calculate depth distributions from measured values of f(x).

Au 1*0 keV

cos ^o=W

30keV

normal

(dE/ds)^1.5

O(»>10°)'1.5 a (Ruth)

Figure 15. Influence of changing different

parts of the MC program on the results of the

depth distribution for x-ray emission with an ionization energy of: (a) 1, and (b) 30 keV

in a gold matrix.

7. Monte Carlo Calculations of Low Angle Scattering Problems

There are many problems in conventional transmission (CTEM) and scanning transmission

electron microscopy (STEM) in which one needs all the elastic and inelastic scattering

collisions. If one only looks at the angular distribution of transmitted electrons through

thick films, it is possible to solve the problem by means of multiple scattering integral

[6,18]. Also, if the spatial distribution normal to the electron probe after passing a thick film is required, a corresponding multiple scattering integral can be calculated

[19,20]. But if one wants information about the spatial broadening of the electron beam for

a limited detector aperture in STEM, one must know the electron distribution as a function

of angle as well as the spatial distribution. This is not possible by a multiple scattering

formula. Therefore, MC calculations are useful for this problem and also for the calculation of multiple energy losses (chromatic error in CTEM).

Because only low scattering angles are of interest, the MC calculations can be simplified. With the calculated scattering angle ? and the azimuthal angle ij), two angular coor- dinates Cx = ccosij; and c = ?sin4< are obtained and eqs. (19) and (22) become, respectively.

59
^x,n+l " ^x,n ^x' S,n+1 " S,n ^^^^

The elastic and inelastic cross sections are described by the Lenz formula [18], but with

empirical parameters Xg and [6], which are also in good agreement with WKB calculations.

The energy losses can be calculated by a formula fitting experimental distributions of

plasma energy losses. Results of this method have been reported for the spatial broadening

of the electron probe [20,21] and the chromatic error [22].

References

I] Reimer, L., Optik, 27_, 86 (1968).

2] Biihring, W. , Z. Physik, 212_, 61 (1968).

3] Badde, H. G., Drescher, H., Krefting, E. R., Reimer, L., Seidel , H., and Biihring, W. , Proceedings 25th Annual Meeting of EMAG, p. 74 (Institute of Physics, London, Bristol,

1971).

4] Reimer, L., Badde, H. G. , Seidel, H., and Biihring, W. , Z. Angew. Physik, 3]_, 145 (1971 ).

5] Kessler, J. and Weichert, N. , Z. Physik, 212_, 48 (1968).

6] Reimer, L. and Sommer, K. H., Z. Naturforschung, 23a , 1569 (1968).

7] Byatt, W. J., Phys. Rev. 104, 1298 (1956).

8] Tietz, T., Nuovo Cimento, 36, 1365 (1965).

9] Lewis, H. W., Phys. Rev. 78, 526 (1951).

10] Grysinski, M. , Phys. Rev. 238, A336 (1965).

II] Drescher, H. , Reimer, L., and Seidel, H., Z. Angew. Physik, 29_, 331 (1970).

12] Bethe, H., Ann. Physik, 'o_, 325 (1930).

13] Niedrig, H. and Sieber, P., Z. Angew. Physik, 31, 27 (1971).

14] Seidel, H., Ph.D. thesis (Miinster, 1969).

15] Drescher, H., Ph.D. thesis (Miinster, 1970).

16] Kulenkampff, H. and Spyra, W. , Z. Physik, 137, 416 (1954).

17] Kanter, H., Ann. Physik, 20_, 144 (1957).

18] Lenz, F. , Z. Naturforschung, 9a_, 185 (1954).

19] Jost, K. and Kessler, J., Z. Physik, 176, 126 (1968).

20] Reimer, L. , Gilde, H., and Sommer, K. H., Optik, 30, 590 (1970).

21] Gentsch, P., Gilde, H., and Reimer, L., J. Microscopy, 100, 81 (1974).

22] Reimer, L. and Gentsch, P., Ultramicrosoopy, ]_, 1 (1975).

23] Krefting, E. R. and Reimer, L., Quantitative Analysis with Electron Microprobes and

Secondary Ion Mass Spectroscopy, Report Jul-Conf-8, p. 114 (E. Preuss Kernforschungsanlage, Jiilich, Germany, 1974).

60
National Bureau of Standards Special Publication 450. Proceedings of a Workshop on the Use

of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

CHARACTERISTICS OF A MONTE CARLO PROGRAM FOR MICROANALYSIS

STUDY OF ENERGY LOSS

J. Henoc

C.N.E.T.

Bagneux, France

and

F. Maurice

C.E.N. Saclay, France

The x-ray depth distribution curves obtained by the Monte Carlo method

are in disagreement with the experiment. For this reason. Bishop [2]^ proposed

in his thesis the use of the theory of Landau, which takes into account the

statistical nature of the energy loss of the electrons. This procedure gives

a more realistic description of the events. We have taken into account, in this way, the effects of electron straggling upon the function (f>(pZ) and the

energy loss distribution of electrons passing through thin films. The general

solution of the problem of electron trajectory simulation is straightforward,

but it requires lengthy and expensive calculations. The procedure can be

reduced in length by means of two simplifications, one of which was proposed

by Bishop. 2 Different treatments affect in different ways the statistical

distributions of the variables which are being studied. In addition, the

importance of the parameters and physical models which govern the diffusion of

electrons (step length, diffusion model, screening parameter, etc.) will be

discussed.

Key Words: Electron energy loss; Landau electron deceleration theory; microanalysis; Monte Carlo electron trajectory calculations; scanning electron

microscopy; x-ray microanalysis.

1. Introduction

It is well known that, while traveling through a target, electrons lose their energy

by creating ionizations which are the origin of radiative transitions; x-ray photons

resulting from these transitions provide the signal used in x-ray microanalysis. It is also known that primary electrons are progressively scattered from their initial direction;

consequently, x-ray emission is a three-dimensional phenomenon. Because of the small

penetration of the electrons in matter, x-ray microanalysis is considered to be a surface

analysis technique; thus, it is easy to understand why, initially, the theory has dealt

only with the calculation of the total number of ionizations produced by electrons as their energy decreases to the minimum excitation energy. The spatial distribution of the

phenomenon is only implicit in the absorption correction [4], but becomes explicit in the

fluorescence correction. The simplest model appears quickly to become inadequate when the

field of application of the method widens, because each theoretical treatment of x-ray

emission implies the knowledge of the spatial distribution.

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

^H. E. Bishop, personal communication.

61
Experiments made by Castaing and Descamps [5] on the depth distribution of characteristic x-ray production were the first to clearly show the limits of the original method,

while they also suggest directions for further improvement.

Some semi-empirical attempts have been made at analytical treatments, such as that of

Philibert [16], later improved by Heinrich [10]. However, the Monte Carlo method introduced

by Green [9], and developed independently by Bishop [2] and Shimizu [18], appears to be the

only way to take into account the different variables in competition in the x-ray production

of targets of any shape.

This method consists of a sampling of electron trajectories determined by means of a certain number of random variables; the statistical distribution of these variables being

known and usually deduced from the theoretical cross-section. The expression "Monte Carlo

calculations" covers a great number of calculation techniques with the same common basis.

In the present work, we have chosen to deal with the statistical energy distribution of

electrons transmitted through a thin metal foil. This example allows the comparison between

the Monte Carlo statistics and a simple probability calculation.

From this basic concept, we describe the simulation method of electron trajectories

in any target, in order to derive the matrix that gives the energetic and spatial distribution of the electrons. Such a matrix supplies the necessary information which is needed in microanalysis.

To show the flexibility of the model, after using the classical continuous Bethe

deceleration law [1], we will turn to the Landau statistical energy loss theory [14].

2. Basis of the Monte Carlo Technique

2.1 Background (principle)

The Monte Carlo technique consists in sampling a random variable x by selecting at random from a set {x.} of values of equal probability of this variable. The values of

equal probability must be previously determined as follows:

(i) let f(x)dx be the probability density function, i.e., the probability for x to be included between x and x+dx

(ii) let F(x) be the distribution function that represents the probability that x, whose extreme values are x-j and X2, is lower than a given value x

F(x) = jf(x)dx . ' (1)

^1

This function takes its values on the [0-1] interval, which is divided into N equal

intervals (fig. 1). One element x^. of the set of equal probability values is defined by:

F(x.)=^.2^ (2)

where i is an integer and 1 < i < N.

2.2 Example of application

In this paragraph, we put the emphasis on the statistical behavior of the energy of

electrons transmitted through a thin metal foil. The physical principles of the Landau

theory will be discussed in Appendix 1. The probability density function, f(A), for the

electrons of initial energy E, to suffer an energy loss included between A and A+dA,

62
F(x)-

Figure 1. Determination of the set of

values of equal probability.

while crossing a foil of thickness psCg/cm^), may be expressed by means of an integral in terms of a universal parameter A:

f(x) = ^

f

_ exp {u[ln(u) + A]}du

•'-i oo+a (3)

i provided that f(A)dA = f(x;

dx

dA

dA.

A is linked to the initial energy E of the electron and to the characteristics of the

ij specimen by:

A = A - g[l - C + In (4Ec / J )] (4)

with

N e PS ^

in which Z is the atomic number of the target element, A is the atomic mass, and J is the mean ionization potential. C, e, & N are, respectively, the Euler constant, the electron

charge, and the Avogadro number.

The associated distribution function is then defined by:

F(A) = /f(A)dA (5)

A set {A.} of N values of equal probability has been derived by integrating (5), then

Interpolating by using (2). Afterwards we have made a sampling of x using a pseudo-random

number generator of uniform distribution,

corresponds to an energy loss A.. According to (4), each of these X. values

In figure 2 are shown two histograms of the energy losses suffered by primary electrons

of 1000 keV passing through a 4.5 ym thick aluminum film. Sevely, et al . [15], used the same physical conditions in their experiment, also quoted in the same figure. The sampling

63
n(A)

(a.u.)

A(l<eV)

Figure 2. Energy loss distribution of 1000 keV electrons transmitted through a 4.5 ym

thick aluminum foil. experiment; Monte Carlo cal cul ation--5000 trajectories

128 A values of equal probability; Monte Carlo cal cul ation--5000 trajectories

512 A values of equal probability; discrete probability distribution— 50 eV class

width; continuous probability distribution.

is made with 5000 trajectories and the number of values of equal probability is either 128 or 512; the width of a class is always 50 eV.

In this particular case, which is a single probability problem, we can also obtain

the histogram of energy losses by calculating:

/•^+1

•^x. f(x)dx

n(A.) = (6)

1^+1 - ^l

n(A.) represents the relative weight of every class and |A^-^]-Aj| has the same value as in the^Monte Carlo calculation. The distribution n(A) is also plotted in figure 2. The

continuous curve represents n(A) when the width of the class vanishes. If the difference

between two adjacent values of x of equal probability is spread over several energy classes,

as it is the case for strong energy losses, the histogram produces accumulations and

holes. The histogram shown in figure 3 is obtained by increasing the width of the class

by a factor of 3; the above-mentioned shortcoming disappears, but the accuracy for weak

losses is poor. The width of the class must be adjusted for the number of values of equal

probability in order to get a good sampling of the variable. If this condition is fulfilled,

the direct calculation and the Monte Carlo approach will fit, provided that the number of

trials is sufficiently high. In our case, 512 values of x of equal probability and a class width of 50 eV are suitable enough to reach the goal. With regard to the number of

64
n(A)

(u.a.)

0 0.5 1 15 2 2.5 A(keV)

Figure 3. Energy loss distribution of 1000 keV electrons transmitted through a 4.5 ym

thick aluminum foil. Monte Carlo distribution 5000 trajectories--l 28 A values of equal

probabil i ty--150 eV class width.

trials, we must keep in mind that it should be optimized in order to achieve the required

accuracy at the lowest cost.

In practice, when many random variables compete, so that only the Monte Carlo technique

can be used, it is obvious that the choice of the working conditions is quite arbitrary.

The justification can only be made "a posteriori. "

3, Description of a Versatile Monte Carlo Calculation for Microanalysis

3. 1 Main Outl ines

We know that along its trajectory in the target, an electron is slowed down and

scattered from its original direction. The trajectory is then simulated as a succession

of broken lines such as AA'A" (fig. 4) generally called steps; ds-j=ds^ 1+ds-j 2 denotes

the path length of AA'A". The energy decrease along the elementary patf^, depends only

upon the path length dsj , assuming the Bethe continuous energy loss law holds; furthermore,

a one-to-one relationship may be established between the path length

s = ds

' j=l ^

65
Figure 4. Elementary path.

and the mean electron energy E^. for a given step:

S^. =S(E.)

In particular, the Bethe range R which is taken as unity is defined by R = S(0).

To consider the statistical part of the treatment we will start with Shimizu's point

of view [19], which uses a single scattering cross-section. This means that the elementary

path must have such a length ds. that the electron undergoes, on the average, only one scattering event while traveling through. In other words, ds-j must be equal to the mean

free path A-j of the electron for the actual energy E^-. The point A' where this event effectively takes place is selected at random (1st random variable). The new direction is self-determined by two angles selected at random (fig. 5): a has an uniform distribution

in (0,27r) (2nd random variable) and e is taken out of a set {Q^.} of angles of equal

probability built from the single scattering cross-section. Tpie scattering cross-section

is indeed the probability density function for an electron to be scattered in the e direction in solid angle dfi.

Bishop's method [2] is a simplification of this elementary model because it uses the

convolution of Go'jdsmit and Saunderson [8], which allows the calculation of the angular

deviation resulting from multiple interactions. The elementary path ds. may thus be

increased and the cost is lowered at the same time. The previous schem^ is retained for

the calculation even if the justification is less obvious than in the single scattering

approach.

66
Physical constants of the component(s)

Number of steps in the trajectory

Number of trajectories

Energy of the primary electrons

Excitation potential of analyzed element(s)

Calculation of the Bethe range

Calculation of energies of each step

Mean energy at the middle

of each step

/

Mean energy at the end of

each step

= E (S.)

Set of scattering angles

of equal probability for

each step

Calculation of the value

of ionization cross- section for each end of

step (jj (Si)

N = 1

Sampling of the trajectories

n(Si, Z, ) : number of electrons which

11 energy Et^ = E (Si) in the stab Z,^

<

N = N+1

> NMAX^

F

Calculation of the X

4)

-Ray depth distribution

Figure 6. Schematic flow chart of a conventional treatment for the

electron trajectory simulation with continuous energy losses.

67
3.2 Diagram of the Monte Carlo calculation

The main lines of Bishop's Monte Carlo calculation which we have used (1972) are shown in figure 6. The process contains three phases. Preliminary operations are per- formed in the first of these: calculation of the Bethe range and subdivision of this

range into a certain number of steps, computation of energies for every half-step, and end

of step. Some mean values may also be computed, for instance, the mean value of the

ionization cross-section. This phase is ended when the set of scattering angles of equal

probability has been determined. The second phase is the actual Monte Carlo calculation

with the setting of the matrix n(S-j,Z|^) which gives the number of electrons with a known

energy (subscript i) belonging to a given slab of the target (subscript k). In the last

phase the computation of any particular quantity that is significant in microanalysis is made.

3.2.1 Preliminary operations

3.2.1.1 Range calculation

The Bethe range is defined by Bethe's deceleration law [1], which links the mean energy

loss dE to the path length d(ps) with familiar symbols:

dE

2^ e^ N Z

E I

In (l.66 ^) d(ps).

Inversion of this formula gives the range by integrating between E^ and E. = 1.03 J; the

upper limit is chosen so that the integral is convergent: ^

E

,P

, 2 ^e^ N Z . dE / —J ^ 1 n (l.66 f)]

For practical purpose, we use:

.2 R(g/cm'

2, 7.85-104 Z EI 1^2 In (l .166 - EI |^2 In

^1 166

where EI is the exponential integral function and energies are given in keV. R, which is used as the unit path length, is divided into I intervals of constant length A(ps) = R/I.

After the^i^'i step-jthe electron has traveled the distance Sji = "ij iA(ps) we, also define

S|2 = (i-^)^ = (i--^) A(ps) as the distance traveled up to the midale of the i step.

3.2.1.2 Half-step and end-of-step energies

Assuming continuous energy loss, the Bethe equation holds and we can find for the mean energy at the middle and at the end of the i^'' step:

/-(i-])A(ps) jA(ps)

Ei2 = E +

^' dE/d(ps) Ell = E + / dE/d(ps).

i 0 i 0

These values are found by successive approximations, applying the RUNGE-KUTTA method; they

are stored for further use.

68
3.2.1.3 Mean value of the ionization cross-section

We wish to calculate the mean number of ionizations when the electron has traveled

through (i-2)R/I < ps < (i+2-)R/I with 1 < i < I. Let i|j (E) be the ionization cross- section; thus:

= / T '^(E) dE/d(ps) d(ps) .

For the ionization cross-section, the expression:

U

is commonly used, U being the overvoltage.

3.2.1.4 Setting of scattering angles of equal probability

The multiple scattering is treated by the Goudsmit and Saunderson theory as the

convolution of the angular distribution given by the single scattering cross-section.

This theory is summarized in Appendix 2. Let us simply say that the probability F(e,p)

for an electron to be scattered within angle e after any number m of scattering events is F(9 ,p) , given by:

00 F(e,p) = E fm(e)^(m)

m=0

where f„(e) is the probability for an electron to be scattered with an angle e after m collisions and TT(m) is the probability for the electron to undergo m collisions. Tr(m) obeys a Poisson distribution law:

•T(m) = ^ exp (-p)

with p denoting the mean number of collisions suffered by the electron along the elementary

path; p depends on the inverse of the square of the energy. For every energy El^ at the

half-step, we get a distribution function:

G(e,Ei2) = /V{e)dfi . (7)

The equal probability angles e. are calculated from (7) according to eq. (2); sin e:| are

! stored in the memory or in a s?)ecial file. The subscript i refers to the energy.

3.2.2 Monte Carlo calculation

When a new electron impinges onto the target, its energy is Eg and its coordinates

are Xq = Yq = Zq = 9o=0. But, as far as the depth distribution is concerned, the Z coordinate is sufficient to locate the electron in the forward direction. In addition, we must make sure that the electron number is not greater than N, the maximum number of

trajectories to be described; otherwise the control is transmitted to the next phase. At

69
the end of the (i-l)th step, (fig. 5) the electron behavior is characterized by the dis- tance travelled S^-_| = (i-l)R/I, the energy Eli-j, the cartesian coordinate Z^-_i and the

angle e^-_i between the trajectory and the forward direction. At the beginning of the i^"

step, a randomly selected fraction r of the elementary path is added to the electron path

length without any change in direction; consequently, the new coordinate is:

Zl = Z._^ + r cos 0^._., R/I

The remaining part (1-r) of the elementary path then takes place after the electron. has

been scattered in a direction given by selecting at random. two angles ej and a.,

is taken from the set of angles of equal probability {6'^} determinedMn 3.2^1.4 Vor E E|2. The new electron coordinates are calculated using

cos G.j = cos

0^._i cos 6'^ + sin 0.j_.| sin e'j cos

Z. = z; + (1-r) cos e. R/I = Z._.| + r cos e._i + (1-r) cos e.] R/I.

If we can find a value of k, with 1< k < I such that:

(k - I) R/I < Z. < (k + 1) R/I,

the matrix element n (S.j, Z|^) is increased by 1.

If Z^ < 0, the electron is backscattered and the n5(S.) component of the ns vector i incremented by 1

If 0 < Z^ < R/I, the matrix element n(S-i , Zq) is incremented by 1,

The vector n^ (S-j) gives the energy distribution of the backscattered electrons and

the n(S.j, Z^) row is used to determine the scattering properties of the surface layer.

3.2.3 Characteristic x-ray depth distribution

As a result of the previous paragraph the number of ionizations produced in the k^*^ slab is proportional to i n(S., Z,),.

_ i = l

Under the same physical conditions, the same number, N, of electrons striking a selfsupported film of thickness R/I, made of the same material would produce a number of

ionizations proportional to N ij; (Eq). Therefore the relative intensity of the characteristic radiation (p^ emitted by the embedded slab may be expressed by:

I n(S,, Z^)»,

^

For the emission at the surface layer, the enhancement produced by both the progressive

diffusion and the backscattering is given by:

I n(S. Z )

70
Figures 7, 8, and 9 show the depth distribution of Ka radiation emitted by a copper target,

the acceleration voltage being 29 kV, and by an aluminium target, the acceleration voltage

being 15 and 29 kV.

Figure 7. Characteristic x-ray depth distribution for a copper target at 29 kV. experiment (Castaing and Descamps,

(1955); conventional calculation,

15000 trajectories, 45 steps; conventional calculation with the mean Landau

energy substituted for the Bethe energy,

5000 trajectories, 25 steps; full

treatment, 2000 trajectories, 45 steps;

alternative treatment, 15000 trajectories, 45 steps.

Figure 8. Characteristic x-ray depth distribution for an aluminum target at 15 kV. experiment (Castaing and Henoc, 1966);

conventional calculation, 15000 tra- jectories, 20 steps; conventional calculation with the mean Landau energy substituted for the Bethe energy, 5000 trajectories, 25 steps; full treatment,

2500 trajectories, 20 steps; alternative treatment, 15000 trajectories, 20

steps.

logq(pZ)

15 PZ(mg/cm^J

W \

Figure 9. Characteristic x-ray depth distribution for aluminum target at 29 kV. experiment (Castaing and Henoc, 1966);

conventional calculation, 15000 tra- jectories, 25 steps; full treatment,

2000 trajectories, 45 steps; alternative treatment, 15000 trajectories, 45 steps.

71
4. Introduction of Statistical Energy Losses

into the Monte Carlo Calculation

It has been shown both by experiment and theory that the energy losses are of statistical nature. Before describing a general approach of electron penetration with regard

to the statistical energy losses, we will try to keep the scheme given above.

4.1 Mean energy losses deduced from the Landau distribution

In an exploratory Monte Carlo calculation. Bishop [2] had already determined the

electron mean energy loss using the Landau theory [14]. The elementary path length being

R/I, the residual energy E] of the electron after traveling R/I, is computed as in paragraph 2.2. This has been done by selecting at random from the set of x of equal probability

which allows a E] value to be found when Eq is known. The process is continued to find

E2 from E] for the second step and so on until Ej vanishes to the minimum excitation

potential. We must note at this point that the maximum value of j may be greater than I. If Ej ^ denotes the energy of electron number n at step number i, the mean energy Eip for tfie step may be defined by:

N^ ^i n .m n=l

i N

The mean energy defined above is then substituted for the Bethe energy without changing

anything else in the schematic diagram of figure 6. The x-ray depth distributions obtained are also plotted on figures 7, 8, and 9; they fit closely with those obtained with

the Bethe law. This is not surprising because it seems that the mean value of the Landau

distribution must agree with the Bethe formula. This method relates the continuous deceleration law to the mean of the statistical distribution; however it lacks homogeneity.

Moreover, the notion of mean energy is not significant in the neighbourhood of the excitation potential

4.2 General treatment of the simulation of electron trajectories

We now consider the diffuse penetration of electrons into matter in a more realistic

fashion by introducing a new random variable, i.e., energy. Figure 10 shows a diagram of

the employed method. The first block represents the input of, data: physical constants of

the target component(s) , number of trajectories, energy of primary electrons, and minimum

excitation voltage of the calculated element(s). The calculation of the Bethe range

(taken as unity) and the length of each step is performed in the second block. The values

of A of equal probability, which have been already calculated and saved cn a file, are input into the computer memory. Represented next in the diagram is the trajectory description. When the electron impinges onto the target, the coordinates are initialized to zero as usual and the number of trajectories already made, contained in a storage register,

is tested. If this number is greater than the input maximum value, the trajectory description is ended. Otherwise, one more trajectory is started. Let us now suppose that the

electron coordinate is Z. -, at the end of the (i-U^h step, the trajectory making a Q-j_]

angle with the forward direction; the electron energy El^ at the half i^" step is com- puted by selecting at random a A value and using the relationship which links x, El^i

and El2. jhe set of scattering angles {e^} of equal probability is calculated for

E = eI^; one of those elements 9'] is selicted at random and so is a. Furthermore another

selection at random of a new X value gives the end-of-step energy EP. If EP is lower

than the minimum excitation potential, the trajectory is ended and ^the control is trans- mitted to the beginning of one more trajectory. Otherwise, the continuation of the i^"

step proceeds to get I-\ , Qj , as previously mentioned in paragraph 2.2.2. The electron is backscattered if Z-j < 0 or Z-j is located in the appropriate slab of subscript k in which

the target has been divided, the contribution of this electron to the (}>, (pZ) depth distribution is calculated for the actual value i)j(Ei^) of the ionization cross-section. The

results of this general procedure are also plotted in figures 7, 8 and 9.

72
Physical constants of the component (s)

Number of trajectories

Energy of primary electrons

Excitation potential of analyzed element (s)

Calculation of the Bethe range and path length

of a step

Input of \ of equal probability

I

N = 1

STOP

Sampling of half step energy E.

Calculation of the distribution {Q-j.}

selected at random

i

The end step energy eJ^ selected at random

E < E^ > N = N + 1

New coordinates Z^. , 9^ and i'^ (pz)

1=1 + 1 Z]

Figure 10. Schematic flow chart of the full treatment for the

electron trajectory simulation with statistical energy losses.

73
4,3 Some alternatives

The general method we have just described is lengthy and computer-time-consuming

because the angular scattering distribution depends upon energy which is a random variable. A simpler alternative consists in defining a discrete set {£^1 of energy values for

which the angular distribution is computed once and for all and saved in a file. Let us suppose the electron has the actual energy El^; the scattering angle is thus selected at random in the distribution corresponding to the energy value provided that |£ -E^^| is minimum for E=e„. As a matter of fact, this induces a slight modification of statistical

distribution. Another way to use the set {e^} and the related set {eJ} of equal probability scattering angles is to determine the elementary path length A(ps) so that the

energy decreases from e-j.] to e-j during the i^'^ step. Consequently, the step lengths are different and the selection at random of a small x leads to a rather large A(ps) and,

consequently, to an overestimate of the penetration depth when the multiple scattering

theory is used. Finally in a extreme simplification, Shimizu [19] uses an exponential

distribution for the statistical energy losses.

On figures 7, 8, and 9 are plotted the depth distributions corresponding to the model

first described.

Programs of the different methods used in the present paper are listed and described

by Henoc and Maurice [12].

5. Conclusion

We have reviewed some possibilities of simulating the penetration of electrons in matter according to the Monte Carlo technique. The validity of this method may be tested

by comparing theoretical values and experimental curves (Castaing and Descamps, [5];

Castaing and Hence, [6]) of the characteristic x-ray depth distribution. The general

agreement which is observed allows the use of this technique with some confidence in spite

of the differences existing for great depths. Moreover, these differences are somewhat

reduced by introducing the statistical energy losses.

The accuracy requirements in routine microanalysis do not justify the expense of the more detailed treatment, and the ZAP method is quite convenient. Nevertheless, the Monte

Carlo method is an essential tool for thin specimen analysis. In this field, the computer

cost will always be lower than the amount of labor required to get experimental calibration

curves.

References

[1] Bethe, H. A., Handb. Phys. 24, 273, (Springer Verlag, Berlin, 1933).

[2] Bishop, H., Thesis, Cambridge University (1966).

[3] Borsch-Supan, W. , NBS J. of Research, 6^n°4, (Oct. -Dec.) 245 (1961 ).

[4] Castaing, R. , Thesis Paris, Publication ONERA N°55 (1951 ).

[5] Castaing, R. and Descamps, J., J. Phys. Radium 18, 304, (1955).

[6] Castaing, R. and Henoc, J., 4° Congres International sur I'Optique des Rayons

Microanalyse, Orsay, 120 (Hermann Paris 1966).

[7] Fano, U., Phys. Rev. 33, 117 (1954).

[8] Goudsmit, S. and Saunderson J. L. , Phys. Rev. 57, 24 (1940).

[9] Green, M. , Proc. Phys. Soc. 82, 204, (1963).

[10] Heinrich, K. F. J., The Electron Microprobe, 296 (John Wiley, New York, 1966).

74
[n] Henoc, J. and Maurice, F. , Proceedings of the 6th International Conference on X-Ray

Optics and Microanalysis, Osaka, 113 (University of Tokyo Press, Tokyo 1972).

[12] Henoc, J. and Maurice, F. , Rapport CEA N°4615 (1975).

[13] Humbert, P.^and Colombo, S., Introduction Mathematique a

1 'etude des theories

el ectromagnetiques , ]_, 81 (Gauthiers-Vil lars , 1949).

[14] Landau, L. , J. Phys. USSR 8, 201, (1944).

[15] Sevely, J., Perez, J. and Jouffrey, B. , High Voltage Electron Microscopy Proceedings

of the 3rd International Conference, Oxford, 32, Academic Press, London 1974).

[16] Philibert, J., 3rd International Symposium on X-Ray Optics and Microanalysis, Stanford,

379 (Academic Press, New York, 1963).

[17] Schiff, L.I., Quantum Mechanics, 162 (McGraw-Hill, New York, 1955).

[18] Shimizu, R, et al . , 4^ Congres International sur I'Optique des Rayons X et la Microanalyse, Orsay, 127 (Hermann Paris, 1966).

[19] Shimizu, R. , et al . , J. of Phys. D (Appl . Phys.) 8, 1 (1975).

[20] Spencer, L. V., Phys. Rev. 98, 1597, (1955).

75
Appendix I

Electron Deceleration - Landau Theory (1944)

Let us consider an electron of initial energy E traversing the path length s. Let f(s,A]

be the probability that this electron suffers an energy loss between A and A + dA along

this path length s.

Let to(e) be the probability of an energy loss e per unit path length.

The relative number of electrons having the energy E-A after traveling through the

distance ds is the difference between the number of electrons of energy E-A+e which lose

the energy e along the path length ds and those whose energy was originally E-A, but have

similarly lost energy e in traversing the same path. The resulting integral equation is the fol lowing:

^%^= [f(s,A-E) - f(s,A)] de. (1)

3S .0

The Laplace transform of the function f(s,A) is written:

4>(s,p) = / e-P^ f(s,A) dA.

•'o

The inverse transform gives:

f(s,A) =

/+i<x>+a

.PA _1_ / e^^" 4>(s,p) dp

2vi •'-i«'+a

a - General solution for (|)(s,p)

The derivative of the expression (j)(s,p) with respect to s gives:

94>(s.p) _ / 8f(s,A) -pA ^. (2)

9S •'o dS

Upon multiplying the integral equation (1) by e'^^ and then integrating with respect to A, we obtain:

r g-pA afii^ ^ r J r e-p^[f(s,A-£)-f(s,A)] dAl de.

•'o 3^ -'O J

Comparing with (2), 3(j)(s ,p) appears in the left-hand side of the above equation.

9S

The innermost integral of the right-hand side is calculated by using the characteristics

of the Laplace transform for a linear change of the variable.

Let us calculate f e"P^ [f(A-£) - f(A)] dA

0

(In the calculation of this integral the parameter s is omitted in order to simplify the

notation.

76
Upon a change of variable A = t - e , dA = dt, we have:

f

e-P^ f(A) dA = f

e-P(^-^) f(t-e) dt

/

e-P^ f(A) dA = e^P^ /

e-P^ f(t-e) dt

since f(A) = 0 for A < 0, it follows that f(t-e) = 0 for t < e.

Thus, we can replace the lower limit by 0.

By multiplying both sides of the equation by e'P^, we obtain:

e-P^ r g-pA ^ r^^-pt

•'o •'o

thus: e"P^ [f(s,A-e) - f(s,A)] dA = (e'P'^-l) f e"P^ f(s,A) dA = (e"P'-l) <l.(s,p)

•'o -^0

It remains to perform the last integration:

r U3{£) (e-P^ -1) *(s,p) de = $(s,p) f

a)(E) (e-P^ -1) de

•'o •'o

Let us come back to the initial equation:

- *(s,p) f

.(e) (1 - e-P^) de

ds / <.(£) (1 - e"^") de

(f>(s,p

We obtain by integrating with respect to s

In (f>(s,p) - In K = - s / a)(e) (1 - e'P^) de

The boundary conditions give:

f(0,A) = 6(a)

i since all the electrons have the initial energy Eq at the surface of the thin foil, and

{I can deduce:

^(o,p) = f f(o,A)e"P^ dA = f e"P^ 6(a) dA = 1 'o •'o

77
By comparing (4) and (5), we find K = 1 and finally:

s a^ie) (1 - e"P^) dc

ct>(s,p) = e

0 (6)

b - Approximation for a particular solution of (ti(s,p)

We intend to evaluate oj(e) (1 - e"P') de. (7)

0

The solution of the problem requires the knowledge of ^(e). ^(e) has been determined only

for losses which are large with respect to the binding energy of atomic electrons. With

the trick used by Landau, a complete knowledge of ui{e) is not necessary. Let us set:

e - the characteristic energy of the atom, of the same order of magnitude as the

° binding energy of the atomic electrons.

^m,.. - the maximum energy which can be transferred in an ionization.

In the integral (7) under consideration, we suppose that only the values of p for which

pe^ << 1 and p^_^^ >> 1 are important.

Then, we can define such that:

£1 >> e 0

P£l « 1

The integral is written:

Resolution of the first integral

1 - pe since pei << 1

CO (e) e de expresses the mean energy loss along the path length ds. Then, we can apply

the Bethe formula; consequently, we have:

78
Resolution of the second integral

For e <<£<<£ , we can use for uj(e) the familiar expression:

0 max

, X _ ^ N e^ Z 1 a.(e) = —^

p ^- —

}\{e) (1-eP^) ds = ^4^p I f

1 - e -pe

£1

n e"* H 1 I n

A 7

TT e A L £1

SI Ci £2 J

£ 1 £ J

Pei

1 - p£i because p£i << 1

w e^ N Z „ n , f" e"P^ ,^ n

^4^p l-p [1 4- Ei (P£i)]

The expansion of Ei (x) gives:

10 / ^^ X Ei (x) = - C - In X + E (-1)^""'^ —

n=l

X* X.

where C is the Euler constant.

The expansion can be limited to the two first terms where x << 1

Ti e'* N Z

E " A

p [1 - C - In (p£i) ]

Finally: J

^{z) (1 - e"P^)de

^

^'^ p I

p [in jl + In El + 1 - C - In El - In pj

Upon returning to the expression (6) of (})(s,p), we obtain:

_ N e^ Z , „ / L E A

P i

1 - C - 1n (^))]

(()(s,p) = e

(8)

79
c -Probability density

Replacing (t)(s,p) by (8) in the expression of the inverse Laplace transform, we obtain the probability density f(s,A).

J- i OO+0

setting u = with 5 = — ^ ps

f(s,A]

1 1 f + loo+O gU(ln u+x)^^ = i P (a)

where:

C . ,n(i|0)

tion:

By introducing the Bethe mean energy loss written according to the Landau nota7= 2 in (1.166 i)

we obtain; A - A + In (|) - 1.502

The constant 1.502 is different from that of Spencer as reported by Bishop. The

origin of this difference is due to the fact that the deceleration function used by

Spencer is applied to energies of a different order of magnitude.

Borsh-Supan [3] has given another form of P(x) which is more convenient for computation. Detailed calculations are given in Appendix III.

P(A)

2 i TT exp (u(ln u + x)) du

ioo+Q

is plotted in the figure.

80
I

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -10.00 -6.00 -2.00 2.00 6.00 10.00 14.00 18.00

LflnBDfl

d - Distribution function for the energy loss

The distribution function F(x) is obtained by integrating the probability density

P(x)

F(x) = f P{x) dx

forX <x< + ^

0 where x is the value of x corresponding to A = 0.

This integral is calculated numerically according to Gauss method. The figure below

is the plot of F(A).

81
82
Appendix II

Electron Scattering

As it has been pointed out above, the elementary path length we have used for practical

purposes is roughly an order of magnitude larger than the mean free path. Consequently a multiple scattering theory must be employed. The multiple scattering is derived from the

single scattering by a convolution process; the aim of this appendix is to give an outline

of this theory.

1 - Elastic single scattering cross-section

In quantum mechanics, the incident particle is represented by a plane wave \l) (r)

this plane wave is scattered by the atoms of the medium into a wave which behaves°as a spherical wave at a great distance r from the scattering center. The total electron wave

function is

^ir) - ^^(7) . f(e,,) ex^iJlir)

where f(e,d) is the scattering factor. The density of scattered particles at M(r,e,(j)) is then

4 |f(e,*)|^

r

The differential scattering cross-section a is the probability for an electron to be scattered into the direction {Q,<t>) in the solid angle do. thus:

a(e,c^) = |f(e,*) 1^ / da = \f{e,<^)\^

r

Scattering obviously has a cylindrical symmetry, so that we can write

o(e) = |f(e)|2

For elastic scattering^ the conservation law implies that the moduli of the inci^dent

and scattered wave vectors k and !<' are equal. The scattering wave vector q = k' - k is related to the scattering angle by

where x is the wavelength associated with the particle.

83
Under typical experimental conditions of microanalysis, the perturbing potential

introduced by the scattering atom is small with respect to the energy of the incident

electron. The diffusion amplitude is calculated according to the Born approximation and

it can be shown that f(e) is the Fourier transform v(q) of the interaction potential v(r)

between the incident electron and the atom (Schiff, [17]):

f(e) = -iL v(q)

where m is the electron mass and f) = t^-, with h the Planck constant

v(q) = iv(r) exp (i q-r) d^r

To take into account the screening of the nuclear charge Z by atomic electrons, a screened

coulomb potential is chosen for v(f)

v(r) = ^ exp (- ^)

The screening parameter R is essentially the atomic radius

R = k a,-^/3

= first Bohr radius of the hydrogen atom

k = 1

The Fourier transform calculations of the v(r) potential can be found in Appendix IV. The final result is

v(q)

Then:

f(e)

and

a(e)

4 TT Z e'

m 4iT Z e

R^

^ 2 ^2 4

4 m Z e

f,4 , 1 ^ 2,2

^ + q )

84
By substituting 2 k sin for q and factoring 4 k in the denominator, the electron

kinetic energy E can be introduced as

ts2 ,2

^

^

n k_

2 m

I) = 4 .2 z2 Z2 e^

16,V f^+sin^l)' 16e2 (sin2|.-|^''

4R"k' 4R"k'

setting = j-^ or = |^ 2;fR = kR

a(e ) = e^

16 e2 (sin2

2 4 /

a((j)) is the differential single scattering cross-section. The total cross-section o^- is obtained by integrating a(e) over the whole solid angle

o(e) dfi = / a(e) 277 sine de Jo

^ 16 E'

^2 4

Z e sin

.2 \ 2 .26^ "o sin 2+ ^

de

setting a

= Then:

sine de ^

/"

Jo (sinH+ a )2

' io

TT 2 sin 2" cos 2"

(sin^ 1+ « )2

setting u = 2"-

The integral becomes:

I

^/2

4 sin u cos u (sin^

du U+a

setting v = sin u V 4 V dv

3 3

•'0 (v +a)

2 a(l+a)

85
the ratio gives the scattering form factor

According to Spencer [20] and Fano [7], the total cross-section is multiplied by

the factor Z + 1 in order to take into account the contribution of ineTastic scattering.

Z

2 - Distribution function for multiple scattering

The calculation is performed according to the Goudsmit and Saunderson method [8] using

the convolution of the single scattering cross-section expanded in terms of Legendre

polynomials.

2.1 - Expansion of the single scattering cross-section

The f(e) expansion as a Legendre series can be written:

f(e) = TT- T, (2n + 1) a P (cose]

4 IT

^ n n

^ n=o

with: a^ = [ 2.

Jo

f(e) (cose) sine de

in which:

f(( a(1+a) 1 (l+2a-cose)'

with: a = 3.4'10'-^ Z-^^/E

f 2.^

Jo

(l+g) 1 (l+2a-cose)

2

' n

P (cose) sine de

a„ = 2a(l+c

1

TT (l+2a-cose)

P (cose) d(cose) = 2a(l+a)

2 n

+1

1

-1 (l+2a-U ,2 n

P (u)du

setting x = 1 + 2a

J^/2a (1+a) = j

+ 1

1 (x-u

.2 n PJu) du

86
then:

a^/2a (l+a)

^1 p,(u;

dx /

1 x-u

du = -2 d Pn(^:

dx / 1 2(x-ul

du

so that a /2a(l+a) =-23— Qp(x) where Q^(x) is the second kind Legendre function. The

Q (x) functions are linkea^by the two relations (Humbert and Colombo, [13]):

n Q = X Q' - Q' ^n ^n-1

(n+1) Q = Q' T - X Q'

^ ' n ^n+1 ^n

by multiplying the first of these by (n+1), the second one by -n, and by then adding

n Q^^^ = (2n + 1) X Q;, - (n+1)

Q;^_^

Then sucessive are deduced by recurrence from the first two;

from Qq(x) = 1 L (^) we get Q^(x) -

- Ix+l)(x-l) " 4a(l+a)

from Q^(x) = f

L (^) - 1 we get Q'(x) = 1 In (^) - ^^^^^

1 , ^x+1

Qi(^) =4^) 0+2a+2a(l+a) In (^))

a = - 4a (l+a) Q' = 1 0 '

' ^0

a^ = - 4a (l+a) = 1 + 2a + 2a (l+a) In (^)

The different a^ can be defined from the same recurrent law:

n a^^^ = (2n+l) (l+2a) a^ - (n+1) a^_^

the initial value of n being 1

87
2.2 - Convolution of two functions expanded in terms of Legendre series - Generalizations

Let us consider an electron beam with the given angular distribution

fl(ei) = 47 E (2n+l) a^ (cos e^) n=o

After one collision (see figure), the angular distribution is

F(e2) dfi2 = /^i^S]) ds^i f2(w) dfi^

where:

^zi^-) = 47 E (2n+l) (coso)) n=o

denotes the single scattering cross-section after one collision; by substituting this

expression for f^^'^'i 9^^-

F(e2) = Jf^(e^) dQ^ ^ E (2n+l) b P (coso)) n=o

from the addition theorem of Legendre polynomials:

P^(cosa)) = P^(cos6^) P^(cose2) ^ 2 (n+k)! Pn^^ose-,) p[j(cose2) cos k{^^ -

<p^)

the last term vanishes when the summation is performed over all the possible values of

(<l>2 - 't']

thus:

but

""(e?) = 47 E (2n+l) b P (cose2) /^fi(ei) P^lcose^) 2^ sine n=o

de P f-j(6l) P^(cose-j) 2tt sine de = a^; then

•'0

F (62) =

4^^ E (2n+l) b^ (cose2)

n = o

Generalizing, we can say that, when the primary beam distribution is:

f (e) = ^ E (2n+l) a P (cose)

0 ' 4tt ^ ^ n n

^ n=o

88
J. I- and when the single scattering cross-section after the i collision is:

= 47 E (2n+l) i^ P, (cose)

n = o

the angular distribution after m collisions will be;

1 ye) E (2n+l) i, P, (cose)

n = o

In the following particular case, when the primary beam is a 6(e) Dirac distribution:

= |6(l-u) P^(u) du = 1

and when after any number of collisions the single scattering cross-section is the screened

Rutherford expression: b^ = c^ =....= i^^ , we obtain the simpler form:

2.3 - Scattering distribution function

The mean^,number p of collisions suffered by an electron which has traveled through

A(ps) is p = ^

A(ps). If we assume that the distribution n(m) of the collision

number m obeys the Poisson law, we can write;

n(m) = ^ exp (-p)

The scattering probability density function is then:

F(p,e) = E f (e) n(m) m=o

-1 oo r" °° m n

= 47 E ' (2n+l) E p"^ exp (-p) ^^m 1

p n=o Lm=o m! nj n ,cose,

1 4^ E (2n+l) exp (p b^) exp (-p) P^ (cose)

^ n=o

i: (2n+l) exp(-p(l-b^)) P^ (cose) n=o

exp (-p) fjl E (2n+l) exp (p bj P^ (cose)"]

L n=o J

89
According to Bishop, the unscattered fraction of the electron beam must be taken

separately in order to get a faster convergence:

F(p,e) = exp (-p) [5(6) + 4^ L (2n+l) {exp(p b )-l) P (cose)l

L n=o J

1 °° with 6(e) = J2 (2n+l) (cose) n=o

The distribution function G(e) is obtained by integrating the above scattering probability

density F(p,e)

2.3.1 - Distribution function related to a Legendre series probability density function

It will now be shown that in this case the distribution function can be

simply written in terms of the Legendre coefficients.

Setting: F(6) = t- (2N+1) b P (cose) n=o

as the Legendre series, the distribution function G'(e) becomes:

1 re -

'^'(s) " 4^ / ^ (2n+l) b P (cose) 2tt sine de

^0 n=0 ' r> ^

I n n

CO f G'(e) = J E / (2n+l) P^ (cose) sine de

n = o

•'0

fb /" P (cose) sine de + f

b (2n+l) P (cos9) sine del

L ° ^ ° n=l ^ " " J

tb^ / sine de + b^ f

(2n+l) P (cose) sine de~| ° Jo n=l ^ Jo " J

setting u = cose

1 r rcose -,

Z K (^-Pi(^)) - E j (2n+l) P^(u) duj

90
The recurrence formulae give:

by adding:

by integrating (2n+l) j

P^(u

(2n+l) P^(u) = P;^^(u) - P;^_^(u]

by substituting in the above value of G'(e), we get

G'(9) = J

by adding term to term

G'(e) = i

1 KO-Pl(^))- E (Vt(u) - P^.^(u))-

L n = l .

i ^b^(l-P^(u))+b^P^(u) .b2P^(u) . E (b,,2A) Vl(")]

= 1 po^t^l Po(^) ^(V^)Pl(^) - £ (W2)f'n.l(^)]

2.3.2 - Numerical computation

By replacing the b coefficients by their expressions while taking into account the

unscattered part of the beam, we get:

G{e) = exp(-p)jl + y^expipa^)-}) + ^(exp(pa^ )-l) - L ^(exp(pa^_^) - exp(pa^^^

)) P^ (cosa)

This form is particularly well adapted to computation.

91
Appendix III

Calulation of: p(a;

1 /°+ia>+a

> y -! 4.

^u(ln u + a) , e ^ ' du

a may have any value provided that the singularities of the integrand

are located on the left of the vertical axis going through a. According to Borsh-Supan [3], we may take for a the real value of the

variable for which the integrand has an extremum. The derivative of

the function equals zero for ^ = g-C^+A)-

Then we set: o = e

The following change of variable is made:

is u=a+iy or u=pe

The integral is separated into two parts

/

L

/

9-

\

+ i CD

-ioo

P(X) = ° ^(a+iy)(lnpe^^+x)

, j'- ,(a+iy)(lnpe^^+x)dyj

In the first integral, transforming y into - y, p into p and e into - e, we obtain:

iy)(lnpe'^®+x)^ (a+iy)(lnpe'

p{x;

•'o _

since a = 6"^^"^^^ Ina = - (1+x) and X = -1 - Ina

p(^) ^ _! ^^"j^g(o-iy)(lnp-ie)-(l+lna)(a-iy)+ ^(a+iy) ( Inp+i e)-( 1+lna) (o+iy)j

We separate the real and imaginary exponents of e

p(x: 2^ /o

;alnp-ye-a-alna) -i(ylnp+ea-y-ylno)^ {alnp-ye-o-alno)g+i (ylnp+ea-y-ylno

dy

We set: Ina = J

Ina^, 9 = arctg > ^"p ^ 7 Y^l

P(A) = ^

Inp-^i^)- y arctg ^

iy(i ln(^-)-l)+ oarctg J

-i(^ ln(^)-l)+ aarctg J

+ e dy

92
The first form for P(x) is:

cos

j y^^ ln(l+ '^)-"']+ oarctg J j dy

This form is particularly well-suited for large negative values of x. On the contrary,

for large positive values, the vertical axis through the point o approaches the origin; in this case it is better to follow the equivalent path of integration shown on the scheme.

In fact, the integral on the external arc of the circle equals zero.

On the lower part of the cut, we may write u = | u

|

e' iTT

On the upper part u = |u|e It:

and on the circle C u = e e

Then, the integral is decomposed into three parts.

P(^) = 2l7[/° e^ln|u|-i.)t-xu gU( In | u |+iTr)+xu ^ J

^u(ln | u |+ie)+xu

^^J

/gU(lnu+x)^^ = 0: since the quantity under the integral sign is bounded

P(,) = [/_° e"^"l"l^'^ du 4- y^'" e^""^ guln|ul+xu

J

Upon substituting v for - u

P(A) = 2^ [ /:

-irrV Vln V -XV , , , e '

' dv +

+00 -iiTV -vln V - XV , e e ' ' dv

]

P(x) 2U I— •'0

,g+inv _ g-invN g-vClnv+x,^^ -v(lnv+x) . , e ^ 'sin ttv dv .

A second form of P(x) is

e-^^^"^^') sin .V dv .

In practice, in the calculation of P(x), we shall take the first form for x < - 1 and the

second one for A > - 1

93
Appendix IV

2 Fourier Transform of the v (r) Potential: v(r) =

The transform v(q) of v(f) is defined by:

2 4

v(q) = / ^ e'^" d3 r

with ( q. r = q. r cose

2 dj r = r sin e de do dr

v(^) = r Z|!,2 ^" R dr e^"^"^ "^^^^ sine de

•'0 •'0

1 - Integration with respect to $

r v(^) = 2. Ze^ Tr e" ^ dr e^'^" sine de

•'o •'o

2 - Integration with respect to

by writing cose = u.

e'"^' sine de e^ du = qr

= i_ (p-'"'^'' p"^^'^'^) = 2 sin qr ^ c ;

the integral becomes:

,2 r o-r/R

= 47TZe / e sin qr dr

3 - Integration with respect to r

We integrate twice by parts:

94
First integration

-r/R . e sin qr dr - sin qr

,-r/R

17^ + qR / e"'^^'^ cos qr dr

qR / e^'^'^'^cos qr dr

Second integration

I

-r/R

e cos qr dr

r/R

1/R cos qr qR I e"'"'''^ sin qr dr

0 0

r/R = R - qR

I e''"" sin qr dr

0

Coming back to the initial integral

j

e~^^^ sin qr dr = qR - qR j e"'"^'^ sin qr dr)

e"''^'^ sin qr dr = qR" - q''R 2 2o2 -r/R . , e sin qr dr

-r/R . , e sin qr dr

1 . q2R2

•

Finally, the Fourier transform of the potential is:

v(q)

? ?

^ 4tt Ze^ qR

? 2

1 + q'^R^

v(q)

4Tr Ze

95

National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

EVALUATION OF FORMULAS FOR INNER-SHELL IONIZATION CROSS SECTIONS^

C. J. Powell

National Bureau of Standards

Washington, DC 20234

An evaluation is presented of various formulas that can be used to describe cross sections for inner-shell ionization by electron impact in the

electron-probe microanalyzer. Owing to lack of experimental data, most of the

discussion pertains to the case of K-shell ionization. For incident energies

greater than four times threshold (Uj<^ > 4), the Bethe equation can describe

adequately the observed cross-sections and recommended values are given for

the two Bethe parameters. For (Uk < 4), the classical result of Gryzinski or the expressions of Lotz or Fabre should be useful.

Keywords: Cross sections; electron-probe microanalysis; electrons; innershell; ionization.

1. Introduction

The models and procedures for converting x-ray intensities observed with the electronprobe microanalyzer (EPMA) into elemental concentrations have been described in a number

of recent reviews [1-10]^ and in other papers included in these proceedings. Although, in

practice, x-ray intensities from a sample are often compared with those from a standard

with known composition (ideally approximately that of the sample), the accuracy of the

analysis will depend in detail on the chemical similarity of sample and standard, the

degree of validity of the correction procedures for the particular analytical situation,

and on the accuracy of the parameters used in the correction procedure [4].

The purpose of this paper is to evaluate formulas for one parameter used in the so- called atomic-number correction, the cross section Qp] for ionization of electrons in the

nl shell by electron impact. This parameter is often thought to be not important in the

simpler analytic approaches [1] because the cross-section term "cancels out" when intensities

from sample and standard are compared. The inaccuracy introduced by this apparent cancellation depends upon the implicit assumption that the rate of ionization can be related to the average electron energy in the sample and standard through the electron stopping

power; in practice, the rate of ionization will depend on the distribution of electron

energies in sample and standard. The variation of Qnl with electron energy should be

known, however, for Monte Carlo calculations of x-ray generation as a function of depth

[11]. Absolute values of Qnl are required when fluorescent emission is significant [4] and

when the sample has an unusual topography (e.g., when particulates are being analyzed) [1 2]

A detailed intercomparison of calculated, semi -empirical and experimental inner-shell

ionization cross-section data has been presented elsewhere [13]. We will review in

Section 2 some of the cross-section formulas that have been used previously in EPMA pro- cedures and others that have been proposed recently. Measurements of Qk will be discussed

briefly in Section 2 and a comparison will be made in Section 4 between this data and the

formulas presented in Section 2.

^Work carried out at the National Bureau of Standards under the partial sponsorship of the

U.S. Energy Research and Development Administration, Division of Biomedical and Environmental Research.

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

97
2. Review of Formulas

The most widely used formula for Qp] is that due to Bethe [14]:

In this equation, E is the electron energy, Epi the binding energy of electrons in the nl shell, and Zpl the number of electrons in the nl shell. The parameters h^i and c^] are often assumed to be constants for a particular subshell but they could be a function

of Z. Finally, eq. (1) is expected to be valid only when E >> Ep] , but a satisfactory

lower limit for E has not been defined previously.

It is convenient to rewrite eq. (1) as a function of Up] = E/Ep-] so that cross- section data for different elements can be readily compared:

Qnl^nl' = ' ''''' ^nl^ll"(^nlUnl)/Unl ^"^'^^^ . (2)

The problem now is to determine optimum values of the parameters b^-j and Cp] . Mott and

Massey [15] analyzed some calculations of Qp] by Burhop [16] and concluded that b|^ w 0.35,

b| ~ 0.25, and C|^ = C[_ « 2.42. These values of bp] and c^l have been widely used in EPMA, but as we will point out later, further analysis of Burhop's results leads to ap- preciably different values for these two parameters.

Equations (1) and (2) are not only invalid near threshold (Up] =1), but also do not give a zero cross-section in this limit (unless Cnl =1). To circumvent both difficulties,

Worthington and Tomlin [17] modified the Bethe equation in an arbitrary, but plausible

way, as follows:

6.51x10

-14

^nl nl ^nl^l

U

nl '"[it

4U

nl 65+2.35exp(l-U

nl

2 2 cm eV . (3)

For large Up], eq. (3) becomes identical to eq. (2) (with Cp] = 2.42). The Worthington

and Tomlin equation with the Mott and Massey values of bp] has also been used extensively

in EPMA.

Green and Cosslett [18] assumed for simplicity that ck = 1 and modified the value of

bK to agree with experimental cross-section data for Ni and Ag in the vicinity of = 3. Their result.

= 7.92x10"^^ ln(U^)/U^ cmW. (4)

corresponds to = 0.61. Equation (4) has been used by Brown [19] as the basis of a more general formula:

Q^^eJ - 7.92x10-14 d^]ln(U^l)/Up] cmW . (5)

The parameter dp] was determined from absolute measurements of x-ray intensities for six

elements and with electron transport in the sample being described by a Boltzmann equation. Brown's values of dp] were

0.85 + 0.0047 Z

2.19 + 0.0098 Z

and

-23

98
which correspond to

and

= 0.52 + 0.0029 Z

b, = 0.44 + 0.0020 Z ^23

in eq. (2).

Hutchins [5] has used a variation of eq. (4):

Qnl^nl'^^^(U^l)/Unr ' (6)

where the exponent m was stated to be between 0.7 and 1.

The formulas listed above are those which have been used in EPMA correction procedures,

Although various simplifications and modifications have been made to the original Bethe

formula (eqs. (1) and (2)) and have been found useful, it is often difficult to judge the

validity of particular expressions for particular shells and ranges of Uni and to select

the most suitable parameters.

Other calculations of Qnl have been made, and other formulas for Q^^] have been

proposed [13]. We quote here the result of a calculation by Rudge and Schwartz [20] for

a fictitious hydrogenic ion with Z = 128:

2 1.626x10-^^ Z^^S^^ln(U^^ ) ^ ,2 . ^nl^l

=

' {7a;

where

S,(UJ = 2.799 -V^.O^ (7b)

and

c ^^^ \ - o ICO 4. 1 .147 0.212 i-.^ S. (U. ) = 2.168 +

-fi p- . (7c)

h ^1 ^L, U, ^

' '-1

These results for Qk and

Q^i have been obtained from calculations for Up] between 1.25

and 4. Kolbenstvedt [21] has derived a simple approximate formula for K-shell ionization

applicable to relativistic electrons and heavy atoms. His result can be simplified for

nonrelativistic energies to become:

2 3.590 X 10-^^ ln(2.38 U,) 1.^93 x 10-^^ T

^^In

cm^eV^ . (8)

McGuire [22] has calculated K-shell and L-shell ionization cross-sections for low-Z atoms

and his numerical results can be fitted to the Bethe formulas, as described in Section 4.

Drawin [23] has proposed the semiempirical formula

Q^^E^^^ = 4.32 X 10"''^ Z^^ f^(U^^ - l)ln(1.25 f2 U^^)/U^/ cm^eV^, (9)

99
where and f2 are parameters estimated to have values in the ranges 0.7 to 1.3 and 0.8

to 3.0, respectively, but which are often assumed to be unity. For Un] >> 1, eq. (8)

reduces to eq. (2) with bn] = 0.66 and Cpi = 1.25. Another semiempirical formula has

been proposed by Lotz [24]:

^^nl^n/ = ^nl ^nl ^^1 {

^ " ^1 ^^Pt-^nl^^nl " ^^^}/\^ (10)

The parameters an], bp] and Cpi (which should not be confused with the Bethe parameters

in eq. (2)) were selected to fit cross-section data for the removal of outer-shell electrons

of atoms to produce ions and may not be appropriate for inner-shell ionization.

Finally, Gryzinski [25] has derived a formula for the inner-shell ionization cross- section based on a classical argument. His result is:

Q .E .2 = 6.51 X 10"^^ Z , g(U ,) cm^eV^ , (Ha)

where

9(Uni) = I {Uf^ -y 1" ^2.7 + (U-D^/^^J . (lib)

3. Experimental Data

Various measurements of inner-shell ionization cross-sections have been reviewed

elsewhere [13]. Quite a few measurements have been made of cross-sections for K-shell

ionization, but relatively few measurements exist for L-shell ionization.

Figure 1(a) shows some recent measurements of K-shell ionization cross-sections for

C, N, 0, and Ne by Mehlhorn and his students [26,27]. These cross-sections are plotted

in the form suggested by eq. (2) and it can be seen that there is good internal consistency,

in shape and absolute magnitude, in the data for these four elements. These cross- sections, which are based on absolute measurements of Auger-electron yields from gas- phase targets under electron bombardment, agree well with most of the previous x-ray

yield measurements for Z 29. (For medium- and high-Z atoms, relativistic corrections

to the formulas for Qk become necessary.) The Auger-yield measurements cover a range of

Uk from near theshold to 25, and it is convenient to use this data for comparison with

the formulas presented in the previous section.

4. Comparison of Formulas with Experiment

Figure 1 shows a graphical comparison of the various formulas presented in Section 2 with experiment. A smooth curve has been drawn through the experimental data shown in figure 1(a) and this curve has been replotted in the other panels as a solid line. For

> 4, this curve also represents eq. (2) with bx = 0.9 and = 0.65.

Figure 1(b) shows a comparison of the experimental curve with plots of the WorthingtonTomlin relation (eq. (3)), using the Mott and Massey value of b^ = 0.35, and the Green

and Cosslett result (eq. (4)). The experimental curve peaks at a larger value of Uk

than for these two formulas; also, the experimental cross-section is appreciably greater

than those predicted by the formulas for Uk ^6.

Plots of the Drawin (eq. (9)), Lotz (eq. (10)), and Gryzinski (eq. (11)) expressions

are shown in figure 1(c). The Gryzinski expression gives a good fit to the experimental

curve for Uk < 6. All three expressions give similar cross-section values for Uk > 10. The Lotz expression would agree well with experiment if the parameter aK was increased by

about 25%.

100
0 10 20 0 10 20 30

Figure 1. Plot of Qk^k^ against Uk = E/E|<. (a) Measurements of Glupe and Mehlhorn and Bekk

[26] for C (*), Ne (x), N (), and 0 (•) . The solid line is a smooth curve through the

experimental points and has been redrawn in the other three panels. The dashed curve is the Bethe equation (eq. (2)) with bK = 0.9 and Ck = 0.65. (b) The short-dashed curve denoted WT is the Worthington-Tomlin equation (eq. (3)), and the long-dashed line denoted GC

is the Green and Cosslett equation (eq. (4)). (c) The short-dashed line denoted L is the

result of Lotz (eq. (10)) with aK = 4 x 10"^'+ cm^eV^, bK = 0.75, and C|/ = 0.5), the longdashed curve denoted G is the result of Gryzinski (eq. (11)), and the aot-dashed curve denoted D is the result of Drawin (eq. (9)). (d) The short-dashed curve denoted M is the

result of calculations by McGuire [13,22] for Be, C, and 0, the long-dashed line denoted

RS is the result of Rudge and Schwartz (eq. (7)), and the dot-dashed curve denoted K is the result of Kolbenstvedt (eq. (8)).

101
Figure 1(d) shows a smooth curve based on McGuire's [13,22] calculations for Be, C,

and 0, the Rudge and Schwartz result (eq. (7)), and the Kolbenstvedt relation (eq. (8)).

McGuire's result agrees moderately well with experiment, particularly near threshold.

There is reasonable agreement of the Rudge and Schwartz equation with experiment for

5. 5, but closer to threshold (the specific region for which the calculations were

made), the agreement is less satisfactory. Kol benstvedt ' s result does not agree as well

with experiment as the other two calculations.

Similar comparisons can be made with Los-shell cross-section data [13]. The data

base is here rather limited, but it also appears that the Worthington-Toml in relation

(eq. (3) with bL,3 = 0.25) again underestimates the ionization cross section (by at least

50%). The Drawih (eq. (9)) and Gryzinski (eq. (11)) results agree well with each other

and with Christofzi k' s [27] cross-section measurements for Ar. Lotz's result (eq. (10)),

with aL^^ = 2.6 x 10"^^ cm^eV^, bL23 ^ 0.92, and ci ,3 = 0.19, however, differs substantially in shape and magnitude when compared with the other data.

Tests have been made to determine whether the inner-shell ionization cross-section

data can be fitted to a linearized form of the Bethe equation (eq. (2)) in order to find

the region of Unl over which the equation is valid and to derive values for the Bethe

parameters bnl and Cp] [13]. Almost all of the cross-section data could be satisfactorily

fitted over the range 4 < Un] ^. 30. (It is interesting to note here that analysis of

Burhop's [16] calculated cross-sections yields results [13] (e.g., bx = 0.50 and ck = 1.41 for Ni , and bLj = 0.32 and C|^3 = 1.85 for Hg) appreciably differ from those quoted

by Mott and Massey [15]). These fits indicated that the parameters b^ w 0.9 and

c^ ~ 0.60-0.75 could describe Q|< adequately for light atoms in the range 4 < Uk 25; the dashed curve in figure 1(a) shows eq. (2) plotted with bK = 0.9 and ck = 0.65. The

situation for L23-shen ionization is less certain, but the values bL2 3 ~ 0.5-0.6 and

CL23 ~ ^ appear to be reasonable for UL23 ~ while the values bL23 ~ 0.6-0.9 and

C|_23 - 0.6 seem appropriate for 4 < UL23 ~ ^'^ attempt has been made here to fit the cross section data to analytic formulas near threshold in the manner described by Green

and Stolarski [28].

Values of the parameter b^^] have also been derived from photoabsorption data and were found to be consistent with the ionization data if account was taken of the expected

distribution of differential oscillator strength with respect to excitation energy.

Specifically, b^] should, in principle, be regarded as a function of Uf^] for Up] < 20. The values of b^] and Cnl derived from the fits to the cross-section data should therefore

be considered as empirical parameters and should not be used outside the range of each

fit.

5. Summary '

The foregoing comparison of experimental data with the predictions of various cross- section formulas together with the previous analysis [13] of inner-shell ionization

cross-section data can be used to draw the following conclusions.

(1) Useful empirical fits of cross-section data can be made with the Bethe equation

(eqs. (1) and (2)) over a wide range of Unl (typically from 4 to 30).

(2) The present data base does not indicate a significant variation of the Bethe

parameters with Z. This conclusion, which differs from that of Brown [19], has

also been independently reached by Quarles [29] for K-shell ionization.

(3) A better choice of the Bethe parameters can now be made than those recommended

many years ago by Mott and Massey [15]. Recent calculations and measurements

show that the inner-shell ionization cross-sections are appreciably (at least

40 to 50%) higher than those calculated with the Mott and Massey parameters.

Specifically, it is suggested that the values bK ~ 0.9 and ck - 0.65 are appropriate for 4 < Uk < 25.

(4) The results of Lotz (eq. (10)) when increased by 25% (that is, aK = 5 x lO-i"* cm^eV^, bK = 0.75, and ck = 0.5 and of Gryzinski (eq. (11)) for Qk agree well

with experiment near threshold (1 < Uk < 4) . These expressions could therefore

be useful for improved atomic-number correction procedures in EPMA; see also. Note Added in Proof.

102
(5) More experimental measurements are required to establish cross-section formulas

(particularly near threshold) for shells other than the K shell.

Note Added in Proof :

Fabre [30] has derived a modification to the Bethe formula [eq. (1)] that could be

useful near the threshold for ionization. His formula is:

6.51 , ^o-''i„,Muj , , . ;

where kp] and xnl ^'^^ parameters; kpi is equivalent to l/bpi in eqs. (1) and (2), and xnl is expected to be greater than 1. Fabre has fitted eq. (12) to the K-shell ionization

cross-section data [26] shown in figure 1 and finds that kK = 1.18 and XK ^ 1-32.

Equation (12) fits these measurements of Qx within 6%, with these values of k|^ and X|^, in the range 1.5 < U|<; < 25.

References

[I] Quantitative Electron Probe Microanalysis^ K. F. J. Heinrich, ed. (National Bureau of

Standards Special Publication 298, Washington, DC, 1968).

[2] Birks, L. S., Eleatron Probe Microanalysis (Wi 1 ey-Interscience , New York, second

edition, 1971).

[3] Reuter, W., Surface Sci. 25, 80 (1971).

[4] Heinrich, K. F. J., Anal. Chem. 44, 350 (1972).

[5] Hutchins, G. A., in Characterization of Solid Surfaces, P. F. Kane and G. B. Larrabee,

eds., p. 441 (Plenum Press, New York, 1974).

[6] Lifshin, E., in Scanning Electron Microscopy, by 0. C. Wells, A. Boyde, E. Lifshin,

and A. Rezanowich, p. 243 (McGraw-Hill Book Co., New York, 1974).

[7] Yakowitz, H., in Practical Scanning Electron Microscopy, Electron and Ion Mioroprobe

Analysis, J. I. Goldstein and H. Yakowitz, eds., p. 373 (Academic Press, New York,

1 974) .

[8] Ogren, J. R., in Systematic Materials Analysis, J. H. Richardson and R. V. Peterson,

eds.. Vol. 1, p. 143 (Academic Press, New York, 1974).

[9] Philibert, J. and Tixier, R., in Physical Aspects of Electron Microscopy and Miorobeam

Analysis, B. M. Siegel and D. R. Beaman, eds., p. 333 (Wiley, New York, 1975).

[10] Reed, S. J. B., Electron Microprobe Analysis (Cambridge University Press, London,

1975) .

[II] Bishop, H. E., private communication.

[12] Heinrich, K. F. K. , private communication.

[13] Powell, C. J., Rev. Mod. Phys. 48, 33 (1976).

[14] Bethe, \\.,Ann. Physik 5, 325 (1930).

[15] Mott, N. F. and Massey, H. S. W. , The Theory of Atomic Collisions, 2nd edition, p. 243 (Oxford University Press, London, 1949).

[16] Burhop, E. H. S., Proc. Comb. Phil. Soc. 36, 43 (1940).

[17] Worthington, C. R. and Tomlin, S. G., Proc. Phys. Soc. (London), A69, 401 (1956).

103
[18] Green, M. and Cosslett, V. E., Proa. Phys. Soa. (London), 78, 1206 (1961).

[19] Brown. D. B., in Handbook of Spectroscopy, Vol. 1, J. W. Robinson, ed., p. 248 (CRC

Press, Cleveland, 1974),

[20] Rudge, M. R. H. and Schwartz, S. B., Proc. Phys. Soc. (London), 88, 563 (1966).

[21] Kolbenstvedt, H., J. Appl. Phys. 38, 4785 (1967).

[22] McGuire, E. J., J. de Phys. C4, 37 (1971); Phys. Rev. A 3, 267 (1971); private

communication.

[23] Drawin, H.-W., Z. Physik 'm, 513 (1961); ibid., \n, 429 (1963).

[24] Lotz. W., Z. Physik 232, 101 (1970).

[25] Gryzinski, M. , Phys. Rev. IJS, A336 (1965).

[26] Glupe, G. and Mehlhorn, W. , Phys. Lett. 25A, 274 (1967); Glupe, G. and Mehlhorn,

W., J. de Phys. C4, 40 (1971); Bekk, K.-J., Diplom-Thesis , University of Freiburg

(1974).

[27] Christofzik, H.-J., Diplom-Thesis, University of MLinster (1970).

[28] Green, A. E. S. and Stolarski, R. S., J. Atmos. Terr. Phys. 34, 1703 (1972).

[29] Quarles, C. A., in Electronic and Atomic Collisions, Proceedings of the Ninth

International Conference on the Physics of Electronic and Atomic Collisions, Seattle

1975, J. S. Risley and R. Geballe, eds., p. 898 (University of Washington Press,

Seattle, 1975); Phys. Rev. A U, 1278 (1976).

[30] M. Fabre de la Ripelle, J. Phys. (Paris), 10, 319 (1949); private communication; and

to be published.

104
Il National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use

|! of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

ij held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

NBS MONTE CARLO ELECTRON TRAJECTORY CALCULATION PROGRAM

Robert L. Myklebust, Dale E. Newbury and Harvey Yakowitz

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

A number of Monte Carlo programs for obtaining information of interest to x-ray microanalysts and scanning electron microscopists have been developed; the

outputs from many of these programs are discussed elsewhere in this volume [1]^.

The purpose of this paper is to present the Monte Carlo program for microanalysis

currently in use at the National Bureau of Standards. This program is a considerably extended treatment of the Curgenven-Duncumb formulation [2]. A large

number of electron trajectories can be simulated rapidly--about 1200 per minute

of central processing unit time--at a reasonable cost. The electron scattering

calculation is an approximation to multiple elastic electron scattering in an amorphous solid. Details of other assumptions made in constructing the program,

as well as a number of comparisons of calculated and experimental data, are given in the text.

Key Words: Electron trajectories; magnetic effects; Monte Carlo calculations;

multiple scattering x-ray generation; particle analysis; quantitative x-ray

microanalysis; thin films; x-ray emission.

1. Introduction

In 1971, our group acquired a copy of the compact Monte Carlo electron trajectory

program devised by Curgenven and Duncumb [2]. In adapting this program for use with our computer, v/e introduced a number of changes necessary to facilitate the execution of the program. Moreover, in subsequent studies, it became necessary to make further extensive

changes to the program. The resulting program, hereafter referred to as the NBS Monte Carlo

Program, has been adapted to the study of a proposed mechanism of Type II magnetic contrast

in the scanning electron microscope, to the study of x-ray emissions from particles, and to problems of quantitative x-ray microanalysis from tilted specimens.

While the NBS Monte Carlo program uses the basic structure devised by Curgenven and

Duncumb [2], the extensive changes made in the original program necessitate a detailed

description, which is the subject of this paper. Modification of the step length between

elastic collisions and the elastic scattering angle distribution will be discussed. The

techniques of adapting the program to the particular applications will be described in this

paper and in a second paper in these proceedings. The results of the testing of the program

against experimental determinations of electron-solid interaction parameters will be given.

2. NBS Monte Carlo Program

We will describe a typical trajectory calculation to illustrate the computational

techniques in the NBS Monte Carlo program. Prior to impact on the specimen, the electron

beam parameters are its direction cosines, beam diameter, energy, and the intensity distribution within the beam. The right-handed, orthogonal, coordinate system used in the program

is shown in figure 1. The Z-axis is defined to be normal to the surface of the specimen and

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

105
Figure 1. Coordinate system used in Monte Carlo

electron trajectory calculations.

the positive sense is into the specimen. The X-axis is further defined to be the axis of

specimen tilt. All tilt angles are specified about the X-axis, and 0° tilt corresponds to normal beam indcidence. The Y-axis lies in the plane of the surface of a flat specimen; the

positive sense is downward from the beam impact point for a tilted specimen.

The beam diameter is a variable which is important primarily for the characterization

of small particles. The intensity distribution within the beam depends on the beam focus.

An ideal finely focused beam has a Gaussian intensity distribution along its diameter [3]. A defocused beam may have a nearly constant intensity distribution along its diameter. Both cases are considered in the NBS program through the selection of either a Gaussian or linear

random number generator for intensity distributions along the beam diameter.

For the case of a Gaussian intensity distribution, the beam diameter is arbitrarily

defined, in accordance with the convention used in electron microscopy, as the diameter of

the circle containing 80 percent of the current [4]. To scale the Gaussian distribution,

the chosen beam diameter is divided by 2.56 to give the radius corresponding to one standard

deviation, a, of the distribution. This radius then serves to scale the electron impact

points along a given diameter vector. All of the electrons are used in the calculations

except that electron impacts at a distance from the center greater than 4o are disregarded.

A linear random number generator is used to rotate the diameter vector through 2 tt radians.

For normal beam incidence on a flat specimen, the coordinates of the beam impact points are given by:

X = cosIR^tt) Rg a • (1)

Y = sin(RL^) Rg a (2)

Z = 0

- (3)

where Rl is a random number from a linear distribution (-1 £ R|_ £ + 1) and Rg is a random

number from a Gaussian distribution (-4 £ Rg ;! + 4).

A plot of a typical Gaussian beam cross-section produced by this procedure is shown in figure 2. A uniform beam density is obtained by substituting a linear random distribution

(Rl) for Rq in eqs. (1) and (2) and replacing the scaling factor, a, by the specified beam

radius. For a non-planar surface, the Z-coordinate of impact is calculated according to the

form of the surface of the specimen. For spheres, the equation is:

Z

= , - Vr2-x2-y2 (4)

and for cylinders oriented with the cylinder axis along the X-axis:

Z = r - Vr^-y^ (5)

where r is the radius of the sphere or cylinder.

The electron trajectory within the specimen is defined as follows. The total length of

electron travel within the specimen is assumed to be the Bethe range, Rg, computed by integrating the Bethe energy loss law [5,6]

106
Figure 2. Distribution of electron impact

points generated by Gaussian random num- ber scheme. The beam diameter contains

80 percent of the impact points. For calculations, all impacts with radii up Beam Diameter to 4a are used,

/;•

03J dE/ds

dE (6)

dE -78,500 Z 1 [ 1-166 E d?= —EA ^" \—J

keV cm

(7)

where Eq is the initial beam energy (keV), E is the instantaneous energy (keV), s is the mass length (g/cm^), Z is the atomic number, A is the atomic weight (g/mol), and J is the mean ionization energy (keV). J may be obtained from either the Duncumb equation [7]:

J = Z|14(l-e-0-^M.

(Z/7.5)' 100+Z ]

(8)

or the Berger-Seltzer equation [8]:

J = 9.76Z + 58. 5Z -0.19

(9)

The lower limit of integration of eq. (6) was arbitrarily selected as 1 .03J following Curgenven and Duncumb [2], since the integration of eq. (6) cannot be evaluated when E £ J.

At present, we are using a multiple scattering model in which the electron step length

varies directly as the electron energy. This is a modification of the first Monte Carlo

program used at NBS based on the Curgenven-Duncumb program [2], which used equal steplengths regardless of the electron energy. These step- lengths v/ere computed by arbitrarily

dividing the calculated Bethe range of the electron into 60 equal steps. Each step contains

a number of scattering acts depending on the electron energy. The energy of an electron

anywhere within a given step is assume^ to be equal to that at the mid-point of the step. A table of these energies is calculated from Bethe's continuous energy-loss relation by a fourth order Runge-Kutta solution [9] of the differential eq. (7).

The Curgenven-Duncumb program used an empirical Rutherford-type expression to describe

the electron scattering within the solid. This expression is based on the path of a charged

particle passing near to an atomic nucleus and assumes that the energy of the particle is not appreciably altered as it passes through the atom. The particle is subjected to a repulsive force which varies inversely as the square of the distance between the nucleus and

the particle. The path of the particle is then a hyperbola having the nucleus as its external focus [10]. Under these conditions, Rutherford showed that

107
cot(e/2) = ^ (10)

where 6 is the scattering angle,

p is the impact parameter (i.e., the distance of closest approach of the

particle to the nucleus), and

b is equal to 2ZeQ/m^ , I being the atomic number, e, m, and v being the

electron charge, mass and velocity, respectively, and Q the charge on the nucleus.

Curgenven and Duncumb [2] modified the expression given by Rutherford, and introduce the

concept of a statistical variation in 6 as follows:

where E is the instantaneous energy and F-j = 0.0072 Z-j/EgP-j. F-j embodies the parameters p and b in eq. (10), and Rl is a random number such that (0 < R|_ £ 1). is a function of Z (Pi = 0.263Z?-^) where the coefficient and exponent are adjusted such that the program will

compute the correct backscatter coefficient for a flat specimen.

The energy distribution of the backscattered. electrons which results from this model is shown in figure 3, curve A. Its shape is in reasonable agreement with experimental determinations of the energy distribution [11], curve B, except for the electrons of highest energy.

Because the initial scattering point in this model was assumed to be at the surface, and

because the multiple scattering distribution produces too many scattering angles greater

than 90°, the model overestimates the number of backscattered electrons of high energy in the first step.

Equation (11) truncates the angular distribution of 6 to a minimum value of about 10° to 15°, depending on the parameter P], since as Rl approaches unity, tan(i3/2) approaches

F-j/Ei rather than zero. While the results obtained with this program generally agree with

the available experimental data, the energy distribution deviates considerably from that

observed experimentally. To overcome this deficiency, an energy-dependent variable steplength was introduced. With the table of energies obtained for the case of equal step

lengths, the Bethe range was calculated for each of these energies. The variable step

length was then defined as 1/31 of the Bethe range appropriate to each energy. This fraction was selected because with it, the sum of sixty steps would approximately equal the

Bethe range for the incident beam energy. The energy for each step was again computed to be

equal to the energy at the mid-point of the step, and P-j was also adjusted to give the

correct backscatter coefficients. The energy distribution of the backscattered electrons in this case is, however, quite different from the experimental values (see fig. 3, curve C).

The statistical variation in the angle 6 used by Curgenven and Duncumb (see eq. (11)) was then modified to allow small-angle scattering as follows:

When the random number (Rl) approaches one, the angle 6 approaches zero. This scattering

model produces an energy distribution of backscattered electrons with a shape similar to the

experimental results, but still too many electrons backscatter in the first step (see fig.

3, curve D). In addition, we can now reproduce the angular distribution of low-loss backscattered electrons as measured by Wells [12] (see fig. 4). Only electrons backscattered

during the first multiple scattering step are counted as low-loss electrons.

The initial scattering point has been moved from the surface to a variable depth within

the solid. This depth is an exponential function of the step-length which allows only 0.1 percent of the incident electrons to travel a full step length before scattering.

(11)

(12)

108
w

Figure 4. Angular distribution of low

Figure 3. Backscattered electron energy distri- energy-loss electrons as measured by

butions calculated with various step lengths Wells [12]; Monte Carlo calculations

and scattering parameters; experimental data for electrons emitted after the first

of Bishop [11]. multiple scattering interaction.

In reality, an absorbed electron will undergo from several hundred to several thousand

elastic scattering events. Since these events are all combined in our multiple scattering

model into 60 steps, the modification of the single-scattering model must be made so that

all scattering acts within each step will be replaced by an equivalent scatter through an angle larger than that corresponding to a single scattering event. A frequency histogram of

scattering angles generated by our modified scattering function is shown in figure 5 for

aluminum with E = 20 keV, curve A. In the same figure is the distribution of angles of

electrons transmitted through a thin foil calculated by the program, curve B, and compared

with the data of Thomas [13], curve C. The calculated distribution is similar to the ex- perimental distribution, i.e., Gaussian, but is shifted to higher angles (see curve D). The

109
f {B)

01 0.2 03 0.4 05 06 0.7 08 0.9 9 (Radians)

13 14 1.5

Figure 5. Angular distribution of electrons transmitted through an aluminum film

(0.56 ym thick); experimental data of Thomas [13]. The angular distribution

generated by the scattering law used in the Monte Carlo calculation is shown,

along with the angular distribution calculated for the film.

0 6

0 5

c

2 0 4 o

o

LL.

om

0 2

1 1 1 1 1 1 1 1

,,0^^^"^ Backscatter Fraction, 77 -^y^ vs

/X

Atomic Number, Z

Heinrich (Composite Target) / /

// //

— Heinrich (Massive Target) -— Monte Carlo

//

//

//

X 1 1 1 1 1 1 1 1 10 20 30 40 50 60

Atomic Number , Z

70 80 90

Figure 6. Backscattering coefficient versus atomic number for a beam energy of

30 keV; data of Heinrich [14].

110
magnitude of this shift is related to the number of single scattering events combined in a multiple scattering step and controlled by the impact parameters, eq. (10). Selection of

the impact parameters is made so that the relationship of the backscatter coefficient vs. atomic number agrees with measured values. Such a fit was made for the experimental values

at aluminum and gold, and very good agreement was obtained at intermediate atomic numbers

(see figure 6). Tests of other electron properties with the selected impact parameters are described in a later section.

The Bethe range, rate of energy loss, step-lengths, and the scattering angles determine

the behavior of the electrons within the specimen. The exact calculation of the spatial

coordinates of the electron within the specimen is shown in Appendix A.

Since the exact position of the electron (X,Y,Z) is known after each scattering act,

backscattering can be determined by comparing the electron coordinates with the equation of

the locus of the free surface of the specimen. The current program can accommodate flat,

semi-infinite specimens at any tilt, and simple shapes such as spheres and cylinders. Other

shapes could be handled through the appropriate surface locus equations. Films on substrates

can also be studied, with the electron behavior controlled by the material in which the

electron is instantaneously located.

The effect of a magnetic field within the specimen can be simulated through the use of

the Lorentz force equation. A detailed discussion of magnetic studies is presented elsewhere in this volume.

3. X-Rays

In addition to the various electron signals produced by the Monte Carlo program,

information concerning x-ray signals also is computed. The generation of x-rays by electrons

is a relatively inefficient process yielding only one characteristic photon for every 1,000

to 10,000 primary electrons. For Monte Carlo computations, however, this inefficiency may

be avoided by considering the probabilities of x-ray excitation as appropriate fractions of

a photon to be excited for each electron scattering event. For K- or L-shell ionizations,

this probability may be written, according to Bethe as:

n - In U. T ^K,L = \,L ^j— (13)

where Qk,L is the ionization cross-section for K- or L-shell electrons, Ek,l is the minimum

ionization potential, qK,L is a constant, and Ui is the instantaneous overvoltage (Ei/EK,L)-

The x-ray generation in our program is normalized by dividing the probability at any step by

the probability at Eq:

9 In

0 0 which results in:

\ ^i 1" ^0

0

(15)

(Note that this equation is strictly applicable only at large U. In the present calculations, we need to calculate the ionization cross-section down to the excitation energy, and

in the absence of a definitive cross-section at U 1 , we employed eq. (15).)

The cross-sections are first summed at the depths in the specimen where they are generated. This summation produces a distribution in depth of x-ray generation, i.e., a plot of

(})(Z) versus Z (where Z is depth). The number of photons generated per electron is then

computed by numerical integration of the distribution (})(Z) and division of the result by

Ek. If the specimen contains several elements, the cross-section for each element is multiplied by its mass fraction before summing so that x-rays are computed for all elements at each electron scattering. When the specimen is a film on a substrate of a different material, the generated x-rays for either the film or the substrate are computed depending on

111
whether the electron is in the film or the substrate. Otherwise, the x-ray generation is computed in the same way for all specimens irrespective of their shape as long as the electron is within the specimen. The second summation of the same cross-sections is performed

versus the path length for an x-ray leaving the specimen in the direction of the detector.

The probability of generation of x-rays in each increment of the path length is multiplied

by the factor which indicates the loss through absorption along the path:

I' = I e-^1^ (16)

where I' is the emitted x-ray intensity, I is the generated x-ray intensity at each increment of the path length, y-j is the linear absorption coefficient for the x-rays, and s is the path length out of the specimen in the direction of the detector. This procedure pro- duces a distribution in depth of emitted x-rays which is then numerically integrated to obtain the number of photons emitted per electron. The shape of the specimen and the x-ray

emergence angle determine the intensity of x-ray emission to a large extent since the path

lengths vary as a function of these parameters. The effect of emergence angle on a sphere

is shown in figure 7. It is easily seen that for low emergence angles, the path becomes

very long for spheres of large diameter. If the specimen is a film on a substrate, both the

x-ray emission from the film and from the substrate are computed.

Beam

SEM

Figure 7. Emergence effects on the x-ray path length

for a particle of intermediate size (1-20 ym diameter); typical x-ray detector positions for the

scanning electron microscope and the electron probe

microanalyzer are shown.

The absorption in the specimen, fp, is then computed by dividing the number of emitted

x-ray photons by the number of those which were generated. In the case of flat specimens,

it is also possible to compute the absorption function, fp vs. x> where x

= (y/p)csc 'V , (y/p)

being the mass absorption coefficient and the x-ray emergence angle, by calculating the x- ray emission at various values of "i* in the expression:

r = I e'^^dz. (17)

The ratio of the the x-rays emitted from a specimen and from a pure element standard

can also be computed by the Monte Carlo technique. Relative emitted x-ray intensities (k)

obtained in this way are only corrected for absorption. No corrections for secondary

fluorescence have, as yet, been programmed into our Monte Carlo program.

4. Comparison of Results of the NBS Monte Carlo Procedure and Experiment

A number of tests of the NBS Monte Carlo program have been carried out. Both the

properties of electron interaction and x-ray generation have been examined. The variation

of the coefficient of backscattering, n, as a function of atomic number, specimen tilt_,

specimen shape and internal magnetic field in the specimen has been studied. In addition,

the transmission of electrons through thin films and the energy distribution of backscattered

electrons have been computed.

112
The x-ray computation tests include the calculation of photons generated per incident

electron, the distribution in depth of electron-excited x-rays in the specimen, the absorption of x-rays, the construction of analytical calibration curves for both normal and

oblique electron beam incidence, and of x-ray intensities from thin films and particles.

Considering the potential for errors in the model itself, as well as in input parameters

(e.g., mean ionization potentials required for the model), the agreement with experimental

findings was good in each of these tests.

4.1 Electron Interactions with the Target

The program is constrained by providing a fit to the experimental backscatter coefficients at normal electron beam incidence. This condition was set for aluminum and gold,

with reference to Heinrich's measured n values [14]. Agreement over the entire range of

atomic numbers is good, as shown in figure 6.

A further series of tests of the applicability of the Monte Carlo procedure to electron

backscattering was carried out. Heinrich showed that the backscatter fraction, n, varied

discontinuously with atomic number, Z, in the range of 22 <_ Z <_ 29 [15]. The effect was especially pronounced for an incident electron beam energy of 40 keV. A comparison of the

measured and calculated results is shown in figure 8. The Monte Carlo results reproduce the

observed trend very well, and the magnitude of the backscatter fraction is within a few

percent of the measured data.

0.31

0.30

029

1 1 1 1 1 1 1

/y

y /

Eo = 40 keV

Heinrich Expt'l — Monte Carlo

022 1 1 i 1 1 1 1 21 22 23 24 25 26 27

Atomic Number, Z

28 29 30

Figure 8. Backscatter coefficient versus atomic number for the elements at the

end of the first transition series; experimental data of Heinrich [14].

The backscatter fraction increases as the specimen tilt angle increases from zero,

i.e., normal beam incidence. At 90° specimen tilt, the value of n should be unity since the

electrons skim the surface without penetrating. The value of n as a fraction of specimen

tilt, e, was measured for an iron-3.22 weight percent silicon alloy. The measurements were

made with the specimen current signal. The incident current and absorbed current were measured using a high-stability current amplifier. The specimen was biased to +30 V in

order to suppress secondary electron escape. The region of the scanning electron microscope

specimen chamber above the specimen was covered with a carbon-coated plastic shield so as to minimize re-backscattering onto the sample. Measurements were carried out at 10° intervals

in the range 0° to 70° of specimen tilt. The corresponding curve was calculated by the Monte Carlo method. The comparison of experimental and calculated values is shown in figure

9; the curves are in reasonable agreement.

Electron backscattering from tapered needles of pure nickel and from a homogeneous

alloy of 40 weight percent gold-copper (SRM-482) [10] was measured as a function of needle

diameter. Figure 10 shows the comparison of the calculated and measured results for the

gold-copper alloy.

113
0.8

0.6

0.4

0.2

-r

^ Measured for Fe-3.22 Si O O Calculated for Iron by Monte Carlo

o- :

10 20 30 40 50 60 70 80 100

Tilt, B (Degrees)

Figure 9. Backscatter coefficient versus specimen tilt (beam energy 30 keV); ex- perimental data for Fe-3.22% Si; Monte Carlo calculations for iron.

1.0

1 1 1 1 1

40Au-60Cu (SRM-482)

0 9

® \

30 keV # Monte Carlo Calculations 08 ® Measurements

0,7

®\

0 6

0 5 04 ® IS bulk

0.3

0.2

0,10 1

1 1 1 1 0 12 3 4 5

Diameter (/^.m)

Figure 10. Backscatter coefficient (all emitted beam electrons) versus diameter

for a copper-gold alloy needle.

The relationship for the electron backscattering fraction as a function of specimen

composition has been shown experimentally to be [11]

m n-EC,n,

114
where C^. is the weight fraction of element i,

T]. is the backscatter fraction of element i, and

n* is the backscatter fraction of the specimen which consists of m components.

The backscatter coefficients for gold-copper and gold-silver alloys and the pure

elements were calculated. Experimentally, the backscatter coefficient follows a simple rule

of mixtures, so that the Monte Carlo method should yield a straight line through each com- position point connecting the pure end members. As figure 11 shows, the computational

scheme correctly predicts the electron backscattering fraction as a function of composition.

0.52

o 0 28y^

CD

0.24

0.20

0.16 J L

Monte Carlo

Backscatter, rj vs. wt. % Au

25 keV Cu- Au Alloys

10 keV Ag- Au Alloys

J L

Au- Weight Percent

10 20 30 40 50 60 70 80 90 100

Figure 11. Calculated backscatter coefficients versus composition for copper-gold

and silver-gold alloys.

The effect of a magnetic field within the specimen on the incoming electron beam has

been investigated exhaustively with this Monte Carlo procedure. The details can be found

elsewhere in this volume [17]. We comment here that agreement of predicted and observed

results is very good.

Cosslett reported a series of measurements of electron transmittance and backscatter

fractions as a function of thickness for copper films bombarded with 10 keV electrons [18].

The agreement between these fractions and those predicted by Monte Carlo calculations is shown in figure 12.

Wells' technique of energy filtering the backscattered electron signal so as to accept

only those electrons which have suffered a low energy loss provides better effective reso- lution in scanning electron microscopy [20]. In order to obtain these low-loss electrons,

the specimen is tilted 50° to 70° with respect to the electron beam. Then, many electrons

strike the surface and re-emerge having lost only a small fraction of their initial energy.

Wells has measured the angular distribution of the low-loss electrons [3].. The Monte Carlo

computed distribution agrees well with the experimental results as is shown in figure 4.

115
Film Thickness (A) 1000 2000 3000 4000 5000

1.0

II II II II III

Copper

Eo = IOkV

08

®

\ Transmission tj^

/Experimental Data\

\from Cosslett

/

0.6 \ fit)

0.4 \® Backscatter -n /

3 ca (2

0.2

^jyr"— \

^ ^

®\.

0

/

(2 II II 1 _T'~^^— 1 ®—^—1—1—1—

0 100 200 300 400 2 500 600

Film Mass Thickness (/xg/cm^)

Figure 12. Transmission and backscattering coefficients for copper films;

experimental data by Cosslett [18].

For all of the many cases subjected to testing, the agreement between predicted and

measured electron properties is satisfactory. Such agreement indicates that the electronsolid interaction is adequately modeled by the described form of the Monte Carlo procedure.

4.2 X-Ray Production

The first test of the x-ray portion of the program involves calculation of the number

of photons generated per electron. If the program fails to correctly predict the x-ray

generation, then all x-ray computations become suspect. As a consequence of the good fit of

the backscattering parameters, there is very good agreement between the calculated and

measured values for K x-ray lines as a function of atomic number and beam energy (figures 13 and 14). For L x-ray lines, agreement is not as good. We believe that this may be due to the use of an incorrect ionization cross-section for the L series. Powell has pointed out that these cross-sections seem to be in substantial error [21]. We intend recalculating our

L series data with Powell's suggested cross-sections.

Castaing and Henoc have measured the x-ray generation distribution as a function of

depth, (f)(z), in aluminum for several incident beam energies [22]. Figure 15 shows the

comparison of these data to those calculated by the Monte Carlo method. The Monte Carlo

model appears to predict the distribution <p{z) quite well. However, the x-ray calculation

is sensitive to input parameters such as mean ionization potential [23] and x-ray mass attenuation coefficients [24]. The effect of varying the mean ionization potential J on a predicted (^{z) curve for Au-Ma (beam energy of 10 keV) is shown in figure 16.

We have compared Monte Carlo calculations of l/f(x) vs. x for a wide variety of elements

and incident beam energies with empirical results [16]. A comparison is shown in figure 17; however, the agreement is not always this good. The effect of the uncertainty of J on one

set of results is shown in figure 18, showing one such difficulty. Effects of the model and

details of input parameters cannot be clearly separated, and until the input parameters are better known, this situation will be difficult to resolve. Moreover, the uncertainty in the

experimental data is not known.

116
T" T" 7

- Ka Photons Generated Per Electron, Mg vs. Atomic Number

Assumes U=Eo/E^^ of 2.0 Monte Carlo (Duncumb J Values) Green (Experimental)

Figure 13. Photons generated per electron

as a function of atomic number; experimental data of Green [19].

15 20 25

Atomic Number, Z

30 35

2x10 "I—

n I r

Photons Generated Per Electron, f Gen. vs. U-l (U=Eo/Eq)

M.Green Expt'l NBS Monte Carlo

Figure 14. Photons generated per electron

as a function of beam over-voltage;

experimental data of Green [19].

10 (U-l)

20 50

I I I 100

117
T 1 1 1 r
f (x)

6.0

5.0

4.0

3.0

2.0

Absorption Correction Factor l/f(x)

vs. Parameter x For Al, 20 keV Incident Electrons # Castaing-Henoc Experimental Results — Monte Carlo Calculation

_L _L _L 1000 2000 3000 4000

X ( cm^ /g )

5000 6000 7000

Figure 17. Absorption correction factor l/f(x) versus x (absorption coefficient

multiplied by the cosecant of the take-off angle (v) for aluminum, beam energy

20 keV; experimental data of Castaing and H^noc [22].

The effect of uncertainties in the mean ionization potential on the analysis of gold in gold-silver alloys (SRM 481) was tested. The Monte Carlo program was given certified alloy

compositions (homogeneity is also certified) [16], and the expected intensity ratio

I binary/Jpure was calculated (k/\). The calculated value was then compared with the measured

value obtained from the Au-Ma line for analysis with v = 52.5° and an incident beam energy

of 10 keV. Two sources for mean ionization potentials were utilized. Results are shown in table 1. The urgent need for accurate input parameters is clearly pointed out by these and

similar results testing effects of different x-ray mass attenuation coefficients [25].

We carried out a set of measurements on copper-gold alloys (SRM 482) with tilted specimens. The ratio of the intensities of CuKa and Aula in alloys of 20, 40, 60 and 80 weight

percent copper in gold to the intensity of CuKa and Aula from pure copper and gold was measured. The experiment was performed in a commercial scanning electron microscope equipped

with an energy-dispersive x-ray spectrometer (EDS). Specimen tilts of 30°, 45°, and 60°

with respect to the normal position of the electron beam were used. Integrated peak intensities were taken; the region of interest was the full peak. At least 5x10** counts were

recorded for peaks of interest. Background substraction was carried out by the method

described by Fiori, et al . [25]. The electron beam energy was 20 keV. The specimen-detector

configuration was carefully arranged so that the x-ray emergence angle was equal to the tilt

angle; specimen height was not changed between sample and standard. The beam current was checked between each analysis with the aid of a Faraday cup. Stability within two percent

was maintained throughout the course of the experiment. The total count rate into the EDS system was maintained well within the linear input-output region. Standards were measured

before and after analysis of the alloys; two regions on each sample were analyzed. The

intensity ratios for copper and gold, called k^u and k^^ , respectively, were plotted against

the ratio k/C. The results for the 45° tilt are shown in figure 19. If the plot of k/C

against k is scattered, then the experimental results are internally inconsistent [16]. The

119
f(x)

0 1000 2000 3000 4000 5000 6000 7000 8000

X (cm2/g)

Figure 18. Effect of varying the value of the mean ionization potential, J, on l/f(x) versus xTable 1. Comparison of intensity ratios (K) calculated by Monte Carlo with measured values.

Au-Ma in Au-Ag at 10 kV Cu-La in Cu -Au at 25 kV

c Km K(D-S) K(B-S) C Km K(KFJH) (KAMEN)

0.2243 0.201 0.221 0.200 0.1983 0.097 0.083 0.073

.4003 .362 .349 .363 .3964 .219 .188 .178

.6005 .599 .549 .563 .5992 .387 .345 .337

.8005 .771 .759 .771 .7985 .632 .561 .584

KAMEN: M(Cu,CuLa) = 1907; M(Au,CuLa) =

KFJH: M(Cu,CuLa) = 2079; M(Au,CuLa) =

C is concentration, is measured intensity ratio, K(K-S) is calculated with the DuncumbShields mean ionization potential (J), K(B-S) is calculated with the Berger-Seltzer J.

120
1.38

1 r 1 1 \—I r

k /c vs k

0.66 I I

I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

k

Figure 19. Intensity ratio, k, divided by certified concentration, c, plotted

against k for copper-gold alloys.

lack of such scattering implies that these results are internally consistent. The corresponding Monte Carlo results are shown in figure 19 as well. The largest relative difference

between the measured and Monte Carlo results is about five percent. Considering the input

parameter uncertainties, this agreement is satisfactory. Similar results were obtained at

30° and 60° tilt.

Quantitative analysis of particulates is not feasible with conventional electron probe

microanalysis data reduction schemes. However, the Monte Carlo procedure is capable of

dealing with the geometric configurations prevalent in particulate analysis. We have tested

the NBS Monte Carlo program by calculation for both spherical and cylindrical shapes.

Bayard measured the ratio of AlKa x-ray intensity from spheres ranging in diameter from

1 to 10 ym to that of pure bulk aluminum. The nominal x-ray emergence angle was 52.5° with

respect to the substrate; the incident beam energy was 17 keV [27]. Calculations of the

intensity ratio were made with the Monte Carlo program and the ratio of emitted intensities

was computed. The results are shown in figure 20; agreement of Monte Carlo calculations

with Bayard's experimental results is very good.

121
Sphere diameter ((jm)

A needle consisting of a 40 weight percent gold-copper alloy was fashioned from SRM

482. The intensity ratios for CuKa and Aula were measured with pure copper and gold as standards. The experiment was performed with a scanning electron microscope in which the

standards were tilted with respect to normal electron beam incidence by 30°. The x-ray

emergence angle was 30°. Incident beam energy was 30 keV. Apparent cylinder diameters were determined assuming that the magnification indicator on the scanning electron microscope was correct. The comparison of experimental and Monte Carlo results are shown in figure 21. The agreement for the gold is excellent, but the copper results show a systematic deviation.

The analysis of a thin film on a bulk substrate is important in many instances. Several techniques have been devised to carry out such analyses [28,29]. The Monte Carlo method

is well suited to deal with such cases [30]. For our test, the data of Bishop and Poole for

a copper film on a nicke] substrate [31] were selected. The nominal copper film thickness

was 0.035 mg/cm2 or 390 A. The ratios of CuKa and NiKa line intensities from this specimen

to pure bulk CuKa and NiKa intensities were measured. The incident electron beam energy was 10.75 keV; the x-ray emergence angle was 75°. For copper, the experimental result was 0.476

and the Monte Carlo result was 0.445. For nickel, the experimental result was 0.570 and the

Monte Carlo result was 0.560. We conclude from the foregoing series of tests that the Monte Carlo program described

here reproduces a wide variety of experimental findings with good agreement usually within

ten percent relative to experimental values. Therefore, this program can be used to predict,

to a good approximation, similar data under a wide variety of conditions. Since the cost of

the Monte Carlo procedure is relatively high, the ultimate use of the Monte Carlo method

will probably be to determine if empirical tests can be employed in explaining observed

phenomena. In other words, the Monte Carlo results will be used as a referee test for

empirical formulae. When the results of both agree, then the empirical formula may be

considered as being reliable. Monte Carlo methods have already been used for such purposes,

e.g., the testing of an analytical expression for <t>{z) and an analytical method for thin

film analysis [32]. The Monte Carlo procedure can also be used to decide on experimental

conditions. An example is shown in figure 22, which shows the predicted SiKa intensity

ratio for a cylinder measured under the two conditions illustrated. Clearly, the geometrical

effect of a 0° emergence angle produces undesirable results due to high absorption; hence,

for particle analysis, one requires a high effective emergence angle, v. Other possible

experimental configurations may be first tested by the Monte Carlo procedure.

122
T I I I

I I I I I I r

Apparent Cylinder Dianneter {/im)

Figure 21. Intensity ratio Idiam/^bul k versus diameter for a copper-gold alloy

needle.

Detector Bearn

Cylinder Diameter (/inn)

Figure 22. Effect of take-off angle on the intensity ratio

Icliam'''^bulk silicon K- radiation from an iron-3.22% Si alloy needle.

123
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Electron Frobe Microanalysis and Scanning Electron Microscopy , this publication (p. 105).

[2] Curgenven, L. and Duncumb, P., Simulation of Electron Trajectories in a Solid Target by

a Simple Monte Carlo Technique, Tube Investments Research Laboratories, Report No. 303

(Hinxton Hall, Saffron Walden, Essex, England, 1971).

[3] Wells, 0. C. Scanning Electron Microscopy, p. 71 (McGraw Hill, Nev/ York, 1974).

[4] Broers, A. N., in Microprobe Analysis, C. A. Anderson, ed., p. 88 (Wiley, New York,

1973).

[5] Bethe, H., Annalen der Phys. LPZ. S, 325 (1930).

[6] Bethe, H., Rose, M. , and Smith, L. , Proc. Am. Phil. Soc. 78, 573 (1938).

[7] Duncumb, P., Shields-Mason, P. K. , and daCasa, C, Proc. Fifth international Congress on X-ray Optics and Microanalysis, p. 146 (Spri nger-Verl ag , Berlin, 1969).

[8] Berger, M. J. and Seltzer, S. M. , Natl. Acad. Sci . , Natl. Res. Council Publ . 1133, p. 205 (Washington, D. C. , 1964).

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[II] Bishop, H. E., in X-ray Optics and Microanalysis, IV International Congress on X-ray

Optics and Microanalysis, Orsay, 1965, R. Castaing, P. Deschamps, and J. Philibert,

eds., p. 153 (Hermann, Paris, 1966).

[12] Wells, 0. C. in SEM/1975 , Proc. of the 8th Annual SEM Symp., 0. Johari , ed., p. 43

(IITRI, Chicago, 1975).

[13] Thomas, R. N., Ph.D. thesis, Cambridge University (1961).

[14] Heinrich, K. F. J. in X-ray Optics and Microanalysis , IV International Congress on X- ray Optics and Microanalysis, Orsay, 1965, R. Castaing, P. Deschamps, and J. Philibert,

eds., p. 159 (Hermann, Paris, 1966).

[15] Heinrich, K. F. J., in Quantitative Electron Probe Microanalysis, K. F. J. Heinrich, ed. Natl. Bur. Stds. Spec. Publ. 298, p. 5 (NBS, Washington, D. C, 1968).

[16] Heinrich, K. F. J., Myklebust, R. L., Rasberry, S. D., and Michaelis, R. E., Natl. Bur. Stds. Spec. Publ. 260-28 (NBS, Washington, D. C. , 1971).

[17] Newbury, D. E. , Yakowitz, H. , and Myklebust, R. L., A Study of Type II Magnetic Domain

Contrast in the SEM by Monte Carlo Electron Trajectory Simulation, this publication

(p. 151).

[18] Cosslett, V. E., in X-ray Optics and Microanalysis , IV International Congress on X-ray

Optics and Microanalysis, Orsay, 1965, R. Castaing, P. Deschamps, and J. Philibert,

eds., p. 85 (Hermann, Paris, 1966).

[19] Green, M., Ph.D. thesis. University of Cambridge (1962).

[20] Wells, 0. C, in SEM/1974 , Proc. of the 7th Annual SEM Symposium, 0. Johari, ed. , p. 1 (IITRI, Chicago, 1974).

[21] Powell, C. J., Evaluation of Formulas for Inner-Shell Ionization Cross-Sections, this

publication (p. 97).

124
[22] Castaing, R. and Henoc, J., in X-ray Optics and Miaroanalysis , IV International Congress

on X-ray Optics and Microanalysis, Orsay, 1965, R. Castaing, P. Deschamps, and J. Philibert, eds., p. 120 (Hermann, Paris, 1966).

[23] Heinrich, K. F. J and Yakowitz, H., Mikroahim. Acta 123 (1970).

[24] Yakowitz, H. and Heinrich, K. F. J., Mikrochim. Acta 182 (1968).

[25] Yakowitz, H., in Practical Scanning Electron Microscopy, J. I. Goldstein, and H. Yakowitz,

eds., p. 327 (Plenum, New York, 1975).

[26] Fiori, C. E. , Myklebust, R. L. , Heinrich, K. F. J., and Yakowitz, H., Anal. Chem. 48,

172 (1976).

[27] Bayard, M. , in Microprobe Analysis, C. A. Anderson, ed., p. 323 (Wiley, New York,

1973).

[28] Colby, J. W., in Advances in X-ray Analysis, Vol. II, p. 287 (Plenum, New York, 1968).

[29] Oda, Y. and Nakajima, K. , J. Jap. Inst. Met. 37, 673 (1973).

[30] Kyser, D. F. and Murata, K. , IBM J. Res. and Dev. 18^, 352 (1974).

[31] Bishop, H. E. and Poole, D. M. , J. Phys. D: Appl. Phys. 6_, 1142 (1973).

[32] Yakowitz, H. and Newbury, D. E. , in SEM/1976 , Proc. of the 9th Annual SEM Symposium, 0. Johari, ed. p. 151 (IITRI, Chicago, 1976).

125
Appendix A

The specimen surface is contained in the X-Y plane, with the positive direction of the

Y-axis pointing down a tilted specimen and away from the beam impact point.

The exact technique of calculating the point coordinates is as follows (figure 23):

• P(L-11

V L

Figure 23. Schematic of Monte Carlo calculation.

P(L*1)

With the electron at a known point P(L), having come from a known point P(L-l), calculate P(L+1).

The electron has traveled from P(L-l) to P(L) along a line (L) whose direction cosines

(a,b,c) can be calculated from the coordinates (X,Y,Z) (L-1) and (X,Y,Z) (L). If the electron were not scattered at point P(L), it would continue along line (L). The act of scattering

the electron through an angle 6, calculated from the screened Rutherford model, sends the

electron to some point P(L+1) on the perimeter of the base of a cone whose altitude is the

extension of line LI, (fig. 23).

To calculate P(L+1), the point (Xo,Yo,Zq) at which the extended line LI intersects the

plane containing the base of this cone is first determined: the path on the same surface

that the electron travels from P(L) to P(L+1) has a length equal to the step-length for that

scattering act, D(L), which is known. The altitude of the cone, H, is given by:

H = D(L) cosB. (^-"I)

Similarly, the radius of the base of the cone is:

R = D(L) sine. ^^-2)

The coordinates of the center of the cone base (Xq'Yq,Z^) are given by:

Xq = X(L) + (H-a) (A-3)

Yq = Y(L) + (H-b) (A-4)

Zq = Z(L) + (H-c). (A-5)

We next establish a new set of coordinate axes, X',Y',Z' (fig. 24) with X'Y' plane

containing the base of the cone, and the Z' axis coincident with extended L'.

The direction cosines of the primed axes in terms of the original axes must be determined. For the Z' axis, the direction cosines are the same as line LI:

a^i = a, b^, = b, c^i = c. (A-6)

126
Trace of Y= Y

Figure 24, Schematic of Monte Carlo calculation

showing choice of coordinate axes.

P(L*1)

j The X' axis is arbitrarily chosen to be the line of intersection of the plane of the

' base of the cone with the plane Y = Y^. From the knowledge of a point in the plane of the

[ cone base, (Xo,Yo'^o)' direction cosines of the normal to the plane, i.e., those of

i line LI, the equation of the cone base is:

aX + bY + cZ - (aX^+bYp+cZ^) = 0.

The line of intersection with the plane Y = Y^ is:

aX + cZ - (aX^+cZ^) = 0.

(A-7)

(A-8)

Thus, the line has the form:

aX.

Z = ^ X + K ; K

0 + Z, C 0

and the direction cosines of the X' axis with respect to the original axes are:

a^, = cos (arctan (-a/c))

b^, = cos 90° = 0

'X' cos (arctan (-c/a)).

(A-9)

(A-10)

(A-11)

(A-12)

Knowing the direction cosines of X' and Z', we can obtain the direction cosines of Y' by the "direction number device." Since the original X,Y,Z axes were chosen to form a righthanded coordinate_system, we maintain parity, making X',Y',Z' right-handed by forming the

cross product (Z'XX' ):

b^.-c^,

^Z' *^X'

a^.-b^,

(A-13)

(A-14)

(A-15)

In the coordinate system X,Y,Z, the point P(L+1) on the base perimeter of the cone is chosen randomly. The azimuthal angle is calculated with a random number RN such that

0.0 £ RN £ 1.0. The coordinates of the point P(L+1) are

127
X'(L+1) = R-cos (RN-2^) (A-16)

Y'(L+1) - R-sin (RN-2^) (A-17)

Z'(L+1) =0. (A-18)

The coordinates X ' ,Y ' ,Z ' (L+1 ) can be transformed back to the space X,Y,Z to give the

desired point P(L+1) since we have the direction cosines of the X',Y',Z' axes:

X = a^, • X'(L4l) + ay, • Y'(L+1) + Xq (A-19)

Y = b^, • X'(L+1) + by, • Y'(L+1) + Yq (A-20)

Z = c^, » X'(L+1) + Cy, . Y'(L+1) + Zq. (A-21)

128
National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

APPLICATION OF MONTE CARLO SIMULATION TO ELECTRON MICROPROBE

ANALYSIS OF THIN FILMS ON SUBSTRATES

David F. Kyser

IBM Research Laboratory

San Jose, California 95193

and

Kenji Murata

University of Osaka Prefecture

Osaka, Japan

A Monte Carlo simulation procedure has been developed for high-energy (1-50

keV) electron beam scattering, energy loss, and x-ray production in targets consisting of thin alloy films on thick substrates. The method utilizes experimental x- ray intensities referenced to thick standards, and avoids the problems associated

with preparation and characterization of thin film standards. The chemical analysis C-j (weight percent) and mass thickness pt (g/cm^) of the alloy film can be

deduced separately; specific examples are given of alloy films MnxBii_x and

COxPti_x on Si02 substrates. The chemical analysis obtained with the Monte Carlo

procedure is correlated with that obtained from nuclear backscattering energy

analysis.

Monte Carlo-generated calibration curves of total electron backscatter yield

n are presented for the case of Al films on Au substrates and Au films on Al substrates. These curves show a smooth transition of n versus pt between the asymptotic limits of n(Al) and ri(Au). The shape of the curve depends on the electron

beam voltage Eq (keV). Further experimental work is necessary to confirm the

accuracy of these calculations for n, and then the method may be useful for a nondestructive, localized thickness measurement of thin films on substrates.

Key Words: Electron probe microanalysis; Monte Carlo electron trajectory

calculations; nuclear backscattering analysis; scanning electron microscopy;

thin films; x-ray analysis.

1. Introduction

Quantitative electron microprobe analysis of thin alloy films on thick substrates is an important and challenging problem for which there are two procedures available. If the film

is sufficiently thick to stop the electron beam before it penetrates into the substrate, then

the conventional "ZAF" model [1]^ can be utilized to transform the measured x-ray intensities

into chemical weight fraction. This approach often requires low overvoltages ratios, Uq = Eo/Ec> where Eq is the critical excitation energy or extremely low beam energy, Eq, coupled

with soft x-ray measurements. These operating conditions can lead to serious quantitative

errors due to surface effects (roughness, corrosion) or due to limitations in correction

procedures (e.g., absorption, atomic number).

Another approach is to maintain high beam energy and high overvoltage Uq coupled with

hard x-ray measurements and thick standards. However, corrections must still be made for:

(a) atomic number effects in the film sample and thick standards, (b) loss of x-ray inten-

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

129
sity from the film sample due to electron penetration into the substrate, and (c) enhanced

x-ray intensity from the film due to electrons backscattered from the substrate. There may

also be some absorption and fluorescence corrections. In traditional style, these corrections for thin film targets could be called the "ZAP" model (Z = atomic number, A = absorption, P = penetration). Due to the complex target configuration and variety of experimental

parameters, it is very difficult to write an analytic function and separate solution for

these effects without numerous assumptions. We have instead, attacked this problem with a Monte Carlo simulation of electron scattering, energy loss, and subsequent ionization distribution with depth into the film. We calculate an intensity ratio, k, for each characteristic x-ray line from the film, referenced to a thick standard target. This ratio k is con- ceptually the same as that utilized for electron microprobe analysis of thick targets. No film standards are required . The method assumes that there are no elements common to the

film and substrate, and that the substrate composition is specified. Specific examples of

MnxBii_x and Co^Pt^.x alloy films on Si02 substrates will be described.

In the second part of this paper, we present some calculations of electron backscatter

yield from thin films on thick substrates. The backscatter yield is calculated with the same basic Monte Carlo simulation theory as that used for the x-ray calculations. This type of

calculation would be useful for nondestructive and highly localized measurement of thin film

thickness. The method can be utilized even when there are elements common to the film and

substrate, but requires the film and substrate composition to be specified. The usefulness

of this technique lies in the unique dependence of backscatter yield on atomic number of the

target, and the thickness sensitivity depends upon the difference in atomic number between

the film and substrate. Specific calculations are made for thin films of Al or Au on sub- strates of Au or Al , respectively.

2. Quantitative Microprobe Analysis of Thin Alloy Films on Substrates

In this paper, we restrict ourselves to analysis of thin alloy films on substrates, and

exclude freestanding foils. Foils are also an important application which requires quantitation but that was not the primary emphasis in this work. The special case of foil targets

can easily be treated by elimination of the substrate within the present model.

Quantitative analysis of alloy films on substrates with the electron microprobe has

received some attention already [2-11]. These references can be separated into two broad

classes, namely those dealing with analytic function techniques [2,4,6,8,10], and those

dealing with Monte Carlo simulation techniques [3,5,7,9,11]. The technique of Monte Carlo

simulation is ideally suited for application to complex target configurations such as a thin

film on a thick substrate. The boundary conditions of the particular target can be easily

accommodated within a single model for electron scattering and energy loss, in contrast to the analytic techniques which require additional assumptions and approximations to treat the

interface effects. The following treatment describes our success with Monte Carlo electron

simulation and quantitative analysis of thin alloy films on thick substrates.

2.1 Monte Carlo Simulation

The details of the Monte Carlo simulation for electron scattering, atomic ionization,

and x-ray production have been described previously [9]. A single-scattering model is used

which has been described by Murata et al . [12,13], in applications to thick targets. The

single-scattering model utilizes the Rutherford equation to describe elastic angular scat- tering of the primary electrons by the (screened) atomic nuclei in the target. The Bethe

equation is used to describe continuous energy loss between the elastic scattering events. The energy-dependent mean free path of the electron (for elastic scattering) is the step

length between angular scattering points. When the electron path being simulated crosses

the prescribed boundary between film and substrate, the ionization rate for elements within

the film is set equal to zero unless the electron is subsequently backscattered into the

film. Scattering and energy loss, characteristic to the substrate, continues as the primary

electrons decelerate in the substrate.

Because the angle of scattering is chosen by a computer-generated random number and

analytic probability distribution with angle, no two simulated electrons will have identical

three-dimensional trajectories in the target. To simulate a real electron beam, a large

number of electrons paths (typically lO^-lO^*) are obtained in order to permit a meaningful

statistical analysis. The target is divided into differential volumes or layers, and a

130
histogram of energy deposition and x-ray production is then produced by the Monte Carlo

technique. This is possible because the computer program maintains the energy and spatial

coordinates of each simulated electron. The computer program also generates the energy

distribution, angular distribution, and total yield of backscattered electrons. The program

also generates the absorption correction f(x) where x

= (u/p)csc ii; and f(x) is the Laplace

transform of the ionization depth distribution ?(pz). This same procedure is utilized for a specified film thickness ot(yg/cm2) and specified film-substrate composition, as well as for

each thick elemental standard target required. The final result is a set of calibration

curves (k value) for each x-ray line, with pt as a parameter. The calibration curves (drawn

through a series of calculated points) apply to only one set of parameters such as beam

energy Eq, x-ray take-off angle iIj, and substrate. An example of these theoretical calibration curves is shown in figures 1 and 2 for MnBi and CoPt films on Si02 substrates.

0.3

0.2

as 1

bo

0.1

0

0 25 50 75 100%

C^^ 100% 75 50 25 0

Figure 1. Theoretical calibration curves for MnKa and BiMa x-ray emission

from thin films on Si02.

131
2.2 Experimental Measurements

Two sets of alloy films were obtained for test cases of binary analysis. Alloy films of

MnxBii_x and Co^Pti_x were fabricated on SiOj substrates. All x-ray measurements were made

at Eq = 20 keV in an ARL electron probe microanalyzer with \p = 52.5° and normal beam incidence. The k-values were formed after correction of the observed intensities from sample

and standard for spectral background and detector dead-time. These particular alloy systems were chosen because of insignificant secondary fluorescence effects and the current technical interest ig their magnetic properties. The thickness range of these films was approximately 200-600A. The experimental k-values are given in tables 1 and 2.

Table 1. Experimental k values for MnBi films on SiO^ at 20 kV,

Sample k(MnKa) k(BiMa)

X115-1 0.0 0.0766

X115-3 0.0226 0.0515

X115-5 0.0293 0.0220

X115-7 0.0329 0.0067

X115-9 0.0610 0.0

Table 2. Experimental k values for CoPt films on SiO^ at 20 keV.

Sample k(CoKa) k(PtMa)

10- 2 0.0177 0.0581

11- 2 0.0305 0.114

12- 2 0.0148 0.0675

13- 2 0.0067 0.0465

2.3 Analysis of Binary Films with Theoretical Calibration Curves

The analysis procedure requires that we iterate the experimental data of tables 1 and 2 within the theoretical calibration curves of figures 1 and 2, respectively, for each film

sample in order to converge to a unique solution for composition C-j and total mass thickness

pt. There can be only one unique solution for each film, and a graphical procedure for

convergence is described in reference 9. The results of this graphical procedure are shown

in tables 3 and 4, along with some results of chemical analysis by a nuclear backscattering

energy analysis method [14]. The chemical analyses agree within 5% relative, but the total mass thickness differs in a systematic fashion, i.e., the nuclear backscattering analysis

results for pt are consistently low compared to the Monte Carlo results. The source of this

difference is not known at the present, and further work is required to establish the ac- curacy limits for both of these analysis methods.

2.4 Discussion

The Monte Carlo simulation can also be applied to ternary alloy films and even higher

order systems. However, then a graphical procedure for iteration and convergence is no longer practical, and a computer method must be used after the theoretical calibration

curves have been generated. One suggestion is to curve-fit the binary film calibration

curves with a-coeff icient expressions such as those utilized for thick targets, and then

combine the binary a-coeff icients for application to high-order systems such as ternary

alloys [15,16]. After the theoretical calibration curves have been generated for a particular alloy system and substrate, they can be re-used to decrease the computer cost per

analysis. The initial cost of a calibration curve depends on many factors such as statistical precision desired (i.e., number of trajectories simulated), concentration and thickness

132
Table 3. Monte Carlo and nuclear backscattering analysis of MnBi films.

OuMip 1 c

1 lUil Lc

rirl p

Cdrl 0

2 Hyg/cm ;

Nuclear Backscattering

^Mn^°^) Pt(ug/cm2)

A i 1 J~ I nu n, u n 0

X115-3 25.5 53 26.6 48

X115-5 51.0 38 53.5 34

X115-7 80.0 29 80.1 26

X115-9 100.0 42 100.0 40

Table 4. Monte Carlo and nuclear backscattering analysis of CoPt films

Monte Carlo Nuclear Backscattering

^Co^"'^) p t(yg/cm ) Cc,(%) p pt(yg/cm )

10-2 18.5 53 19.4 49

n-2 16.3 89

12t2 14.0 56

13-2 10.1 40 9.6 34

resolution desired, beam energy Eq, and cutoff energy in the trajectory termination. For a typical point on the calibration curves of figures 1 or 2, 1200 electron trajectories were

simulated on our IBM S/360-model 195 computer. In figure 1, five concentrations C^. and four

thicknesses pt were chosen, in addition to the two thick standards. The total CPU time was under 30 minutes. Improvements in the computer code have now reduced this by a factor of 2.

3. Monte Carlo Calculation of Electron Backscatter Yield

The Monte Carlo computer program used for quantitative chemical analysis with x-ray

production can also be used to generate theoretical calibration curves of electron backscatter yield. The electron backscatter yield n from thick targets, has a unique relationship

with atomic number Z of the target [17-19]. At small values of Z, n = Z/100 and is not very

sensitive to incident beam energy Eq. At large Z, n approaches an asymptotic limit for

normal (90°) beam incidence.

If the target is composed of a thin film with average atomic number Zi and mass thickness pjti on top of a thick substrate with average atomic number I2, then the effective

backscatter yield n will depend on: (a) the relative value of p^t^ and the total electron

range Re in the complex target, and (b) the relative value of Zj, Z2. The dependence of n on Eq is contained in Rg, and this dependence can be strong if the difference IZ1-Z2I

is large. If pitj >> Rg, then n will be representative of Z^, and if pitj << Rg, n will be

representative of I2. At intermediate values of Pit^, n will also be intermediate and vary

smoothly with pit^ between the limits n = ri(Zi) and n

= n(Z2). It is just this sensitivity

of n to piti and Eq which can be used for a local, nondestructive measurement of Pit^, if a continuous calibration curve can be obtained theoretically without the need for experimental

calibration standards.

The special case of a freestanding foil can easily be simulated by elimination of the

substrate backscattering or assigning a very low value to Z2. However, the primary interest

133
was in films on substrates, and hence, we will not discuss the Monte Carlo simulation of

results from foils.

The subject of electron backscatter yield from thin films on thick substrates has

already been of some interest [20-24], primarily from an experimental viewpoint. References

[21] and [23] propose a simple analytic expression for n based on semi-empirical reasoning.

Reference [25] deals with freestanding foils without a substrate and shows that good agreement was obtained between experimental values of n and theoretical values calculated with a similar Monte Carlo model. As discussed in previous sections of this paper, Monte Carlo

simulation is a very powerful tool for application to discontinuous targets such as thin

films on thick substrates, with a minimum of assumptions. The following describes some calculations of n for Al films on Au and Au films on Al

3.1 Theoretical Calibration Curves with Monte Carlo Simulation

The Monte Carlo computer program was used to calculate n for thin films of Al on Au substrates and Au films on Al substrates with normal (90°) beam incidence. The results of

these calculations are shown in figures 3 and 4. As described previously, the Monte Carlo

program calculates one value of n for a particular target configuration (piti,Zi,Z2) and

beam energy (Eq). The continuous curves shown in figures 3 and 4 have been drawn through a series of calculated points. For each point, a total of 10'+ incident electrons were simulated,

and hence the statistical precision of the calculation is given by the variance = ri(l-ri)/N,

where N = number of electrons simulated [26]. Bulk values of mass density pCg/cm^) for Al and Au were used to convert mass thickness pt to thickness t.

There are several things to notice in these theoretical calibration curves. For the case of Al films on Au substrates shown in figure 3, the asymptotic limits of n at large pt are 0.161 (10 kV) and 0.154 (30 kV). This magnitude and weak energy-dependence of n = n(Al)

is in good agreement with that measured by Bishop [17], Heinrich [18], and Wittry [19]

independently on thick targets of Al . As pt(Al) becomes vanishinqly small, the calculated

asymptotic limit of n

= n(Au) becomes 0.520 (10 kV) and 0.482 (30 kV). The average value of

n(Au) is in good agreement with that measured on thick Au [17-19], but the weak energy

dependence of ri(Au) is not. At large Z, experimental results on thick targets show that n increases slowly with increasing E , and at small Z it decreases slowly with increasing E . At intermediate values of Z, n is experimentally found to be independent of Eg. The wrong

0.7

0.6

o 0.5 >

0.4

o

00 0.3 co^ 0.2

LU

0.1

"'

' I 1 1 1 I 1 1 1

aO-^ Films on Au^^Substrate

1

^^^~~^^~-~~-.,^0kV

^\20kV

Eq = lOkV

1 1 1 1 1 1 1 1

1 6 8 10 12 14 Film Thickness (lO^A)

16 18 20

Figure 3. Monte Carlo calculations of electron backscatter yield

from Al films on Au substrate.

134
0.7

0.6 D

1 0.5 >

H 0.4

om 0.3 c

oi_ 0.2

UJ

0.1 -

1 1 1 1 1 1 1 1

Au''' Films on Al '-^Substrate

EQ-10kV

' - ^

30kV

-

1

1 1 1 I I

1 6 8 10 12 14

Film Thickness (10^ A)

16 18 20

Figure 4. Monte Carlo calculations of electron backscatter yield

from Au films on Al substrate.

energy-dependence predicted for ri(Au) is probably due to the failure of the Born approximation in the Bethe energy-loss equation of the Monte Carlo model, with decreasing Eq and high

Z targets such as Au.

For the case of Au films on Al substrates shown in figure 4, the dependence of n (calculated) versus t is linear with t for t £ 250A (10 kV), t <_ 1 ,000A (20 kV), and t <_ 2,000A

(30 kV). For the case of Al films on Au, there is not a large range of linear dependence.

However the calibration curves, if they are accurate, can be used in the nonlinear range

also. Such a technique depends upon a large value of IZ1-Z2I for maximum sensitivity, but

can accommodate elements which are common to both the film and substrate in compound targets.

The latter case cannot be treated as easily with x-ray measurements described earlier in this paper.

3.2 Experimental Results and Comparison with Theory

Holliday and Sternglass [20] have measured n for Al and Au films on Au and Al substrates.

Their experiment was performed with the use of bias grids placed between the target and the

collecting electrodes for backscattered electrons. Holliday and Sternglass (H-S) made no correction for secondary electron emission or backscattering from the grids, and hence,

their results are in error. Bishop [17] has discussed the experimental errors introduced to the measurement of n when grids are used, and he proposed the use of a double target without

grids for which the experimental errors due to secondary electrons and electrons scattered

from sample chamber walls can be eliminated. For the experimental configuration which H-S used, the experimental errors should result in a low value for n, and that is observed in the comparison between the present Monte Carlo calculations and their experimental values.

The asymptotic limits of the H-S data for thick films show n(Al) = 0.13 and n(Au) = 0.45.

These values are also too low compared to those measured by Bishop [17], Heinrich [18], and

Wittry [19], in which the secondary electrons and rescattered electrons were taken into

account. Hence, the discrepancy between the Monte Carlo calculations and the experimental

data of H-S can be explained qualitatively. The asymptotic limits of the Monte Carlo calculations (i.e., thick films) are approximately correct, with ri(Al) being more exact than

Ti(Au) for the reasons already given.

135
3.3 Discussion

There is a need for more measurements of n obtained in a systematic fashion and for a variety of films and substrates in order to compare with, and fully confirm or modify, the

existing Monte Carlo model. The angular and energy distribution of backscattered (or trans- mitted) electrons would allow us to make another valuable comparison. The general accuracy

of the Monte Carlo model will probably be indicated better through direct electron distribution measurements rather than x-ray emission measurements or other atomic relaxation processes

This is due to the additional modeling and input parameters required for the calculation of

atomic relaxation processes (e.g., ionization cross-sections, fluorescent yield, absorption,

etc.) versus calculation of the primary electron scattering and energy loss alone.

The experimental results of reference [21] are not complete enough for comparison with

the Monte Carlo theory and, in addition, they differ significantly from those of H-S for the same target configuration. The experimental results of references [22] and [24] were obtained

at very low values of Eq (^ 2 keV). The present Monte Carlo model cannot accurately simulate

target behavior at electron energies in that range and, hence, that information is not

useful to us now. In the future, we plan to extend the simulation model to lower energies,

perhaps by switching (at some particular value of electron energy along the trajectory) from

the Bethe equation for energy loss to an equation such as that proposed by Rao-Sahib and

Wittry [27].

If n is quantitatively measured in an electron probe microanalyzer (EPMA) or scanning

electron microscope (S£M), great care must be taken to eliminate the effects of secondary

electrons and rescattered electrons. We have attempted to utilize the technique of Heinrich

[18], which employs a measurement of collected target current and incident beam current for

calculating n- However, in our EPMA [28], the target current is found to be a complex

function of target bias, target elevation (i.e., distance below final magnetic lens), and

also final lens current. These observations strongly suggest that secondary electrons from

the target and chamber walls, as well as rescattered electrons backscattered originally from

the target, are producing extraneous collected target currents. We are continuing to search

for an accurate method of measuring n in our EPMA.

4. Conclusions

Monte Carlo simulation of electron beam scattering, energy loss, and x-ray production

has been developed and experimentally verified for targets consisting of thin alloy films

on thick substrates. Theoretical calibration curves of intensity ratio k-\ versus weight

fraction Ci were generated for alloy films MnxBii_x and Co^Ptj-x on Si02 substrates, with

total mass thickness pt as a parameter. Experimental data from alloy films were graphically

iterated; the results converged within the calibration curves to provide a unique analysis

of both Ci and pt. These elemental analyses agreed well with analyses obtained from nuclear

backscattering energy analysis. The Monte Carlo model provides a general procedure for

quantitative and nondestructive elemental analysis of thin films on thick substrates, while

retaining the high lateral spatial resolution provided by the electron microprobe.

The Monte Carlo simulation model is set up to utilize measured x-ray intensities from

the film that are referenced to thick standard targets and, hence, does not require thin

standard films. The use of thick standards greatly simplifies the experimental procedure,

and also eliminates a potential source of error due to thickness uncertainties if film

standards are utilized [29].

If the average atomic number of the film and substrate are sufficiently different, then

a measure of total electron backscatter yield n can also provide information about total mass thickness pt of the film. The Monte Carlo model was used to generate calibration

curves of n versus pt for Au films on Al substrates and Al films on Au substrates as an example. The asymptotic limits of these curves are approximately correct, with n(Al) in better agreement with experiment than ri(Au). Further work is necessary to fully confirm

the Monte Carlo prediction of n at intermediate thickness values. This work will require

careful elimination of experimental uncertainties due to secondary and re-scattered electrons

136
We thank IBM, Japan, Ltd., and the IBM World Trade Corporation for financial support of one of the authors (KM) as a WTC Post-Doctoral Fellow during 1973. J. Baglin (IBM Watson

Research Center) performed the nuclear backscattering energy analysis, and K. Lee and T. Jacobs (IBM San Jose Research Laboratory) provided the MnBi and CoPt films, respectively.

Stimulating and helpful discussions about parts of this work were held with K. Heinrich, W. Reuter, and E. Sternglass.

References

[I] Beaman, D. and Isasi , J., ASTM Special Technical Publication No. 506 (1972).

[2] Colby, J. W., Advances in X-ray Analysis, Vol. 11, p. 287 (Plenum Press, New York, 1968).

[3] Henoc, J., J. de Microscopic 1_5, 289 (1972).

[4] Warner, R. and Coleman, J., Micron 4, 61 (1973). Also see Micron 6_, 79 (1975).

[5] Bishop, H. and Poole, D., J. Phys. D6, 1142 (1973).

[6] Oda, Y. and Nakajima, K. , J. Japan Inst. -Metals 37, 673 (1973).

[7] Bolon, R. and Lifshin, E., Proceedings of Sixth Annual SEM Symposium, p. 285 (IITRI,

Chicago, 1973).

[8] Stine, P., and Hruska, S. and Liedl , G., Advances in X-Ray Analysis, Vol. 17, p. 487

(Plenum Press, New York, 1974).

[9] Kyser, D. and Murata, K. , IBM J. Res. Develop. 1_8, 352 (1974).

[10] Eshel , A., Micron 5, 41 (1974).

[II] Yakowitz, H. and Newbury, D'. , Proceedings of Ninth annual SEM Symposium, p. 151 (IITRI,

Chicago, 1976).

[12] Murata, K. , Matsukawa, T., and Shimizu, R. , Japan J. Appl. Phys. }0_, 678 (1971 ).

[13] Murata, K. , Matsukawa, T. and Shimizu, R. , Proceedings of Sixth International Conferences

on X-Ray Optics and Microanalysis, p. 105 (Univ. of Tokyo Press, Tokyo (1972).

[14] Nicolet, M., Mayer, J., and Mitchell, I., Science 177, 844 (1972).

[15] Ziebold, T. and Ogilvie, R. , Anal. Chem. 36, 322 (1964).

[16] Laguitton, D. , Rousseau, R. , and Claisse, F., Anal. Chem. 47, 2178 (1975).

[17] Bishop, H. , Proceedings of Fourth International Conference on X-Ra Optics and Microanalysis, p. 153 (Hermann, Paris, 1966).

[18] Heinrich, K. , ibid., p. 159.

[19] Wittry, D. , ibid., p. 168.

[20] Holliday, J. and Sternglass, E., J. Appl. Phys. 30, 1428 (1959).

[21] Nakhodkin, N., Ostroukhov, A., and Romanovskii, V., Sov. Phys. -Solid State 4^, 1112

(1962).

[22] Thomas, S. and Pattinson, E., J. Phys. D3, 349 (1970).

[23] Reuter, W. , Proceedings of Sixth International Conference on X-Ray Optics and Microanalysis, p. 121 (Univ. of Tokyo Press, 1972).

137
[24] Goto, K. , Ishikawa, K., Koshikawa, T., and Shimizu, R., Surface Soienoe 47, 477 (1975).

[25] MacDonald, I., Lamki , A., and Delaney, C, J. Phys. D4, 1210 (1971).

[26] Bishop, H., Brit. J. Appl. Phys. 18, 703 (1967).

[27] Rao-Sahib, T. and Wittry, D., J. Appl. Phys. 45, 5060 (1974).

[28] Applied Research Laboratories model EMX-SM, equipped with magnetic beam deflection and

a "pancake" style magnetic objective lens.

[29] Philibert, J. and Tixier, R. , Physical Aspects of Electron Microscopy and Microbeam

Analysis, Ch. 21 (Wiley, New York, 1975).

138
National Bureau of Standards Special Publication 450. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

CALCULATION OF TYPE II MAGNETIC CONTRAST IN THE LOW-LOSS IMAGE

IN THE SCANNING ELECTRON MICROSCOPE

Oliver C. Wells

IBM Thomas J. Watson Research Center

P. 0. Box 218

Yorktown Heights, New York 10598

The backscattered electrons which are often most useful in the scanning

electron microscope (SEM) are those which have lost the least energy. (We have

called these the "low-loss" electrons.) The question is whether they can be

analyzed by a simple electron scattering model, or whether the Monte Carlo

method must be used. Previously, it has been shown that a simple model based

jointly on the Bethe energy loss law and a single Rutherford wide-angle scat- tering event can account for both the angular distribution and the absolute

magnitude of the low-loss electron emission from amorphous Si02 with oblique

incidence with an accuracy of a few percent provided that the energy loss is less than about 1%. This simple [iiodel has been extended to allow for curvature

of tne electron trajectories in the specimen, so as to calculate the contribution

to Type II magnetic contrast in a suitable specimen caused by those backscattered

electrons which have been scattered by a single wide-angle scattering event. Some preliminary theoretical results are given.

Key words: Contrast calculations; energy filtering; low-loss electron image;

magnetic contrast; Monte Carlo electron trajectory calculations; scanning electron microscopy.

1. Introduction

Type II magnetic contrast arises in the backscattered electron image from a solid

specimen because the incident electrons are deflected inside the specimen by the internal

magnetic field of the material [1-12]^. The magnitude of Type II magnetic contrast calculated has been explained by two different theoretical models. First, both the

magnitude of the contrast and the way in which it depends on the angle of incidence

have been successfully analyzed by the Monte Carlo method [3,9]. Also, Fathers, et al. [2], have analyzed the problem by means of a simplified electron scattering model.

In this paper, the electron scattering model approach is followed, with particular

emphasis on calculating the angular distribution of the backscattered electrons, and on predicting the dependence of Type II magnetic contrast on the angle of incidence. The

model used here is an extension of one which had previously been successful in explaining

the scattering of low-loss electrons from an amorphous Si02 target [13-17]. This is referred to below as "The Simple Model." This model has now been extended to calculate

the contribution to Type II magnetic contrast which arises from backscattered electrons

which have been scattered by only a single wide-angle event (in addition to small-angle

inelastic events, of course). The extent to which there will also be a contribution

from electrons which have suffered more than one wide-angle event is not covered by this

theory. (This is a point which may need to be investigated by the Monte Carlo method.)

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

139
INCIDENT

Figure 1. Penetration model for the escape

of low-loss electrons [15].

The three results which must be explained by this model were reported by Newbury,

Yakowitz, and Myklebust [6], and are the following:

(1) type II magnetic contrast calculated by Monte Carlo technique to be in iron is 0.18% at 30 kV when the glancing angle of incidence Gi is 30° (see fig. 1 for the

definition of Qi) (compared to an experimental value of 0.3%);

(2) the contrast varies as E

' (calculation and experiment) (the "three-halves power

law"); and

(3) the contrast is zero for = 0° and 90°, with a broad maximum between = 20° and 60° (calculation and experiment).

2. The Simple Model

This is the model which was originally developed to explain the scattering of low-loss

electrons from a solid, amorphous target [13-17]. It is based on the idea that the distance

which an electron can travel in the target before losing Joo much energy to be accepted by

the energy filter is limited, in a typical case, to 1000 A or less. Under these conditions,

it can be shown that the deflection of typically 30° or more which it must acquire to escape

from the target within such a short total penetration path is most likely to occur as a result of a single wide-angle event rather than as the sum of two or more deflections through

a small angle. (The question of how long the penetration path can be before multiple wideangle scattering will become significant is discussed in Appendix 2.) In the simple model,

therefore, it is assumed that the electron penetrates in a straight line into the specimen

while losing energy by small-angle inelastic events at an average rate given by the Bethe's

energy loss law and with a single wide-angle Rutherford's event to bring the electron out of

the target before it has lost too much energy to be accepted by the energy filter. The

properties of the simple model will be briefly described before discussing how it must be

modified to allow for the internal magnetic field of the Type ll sample.

In the simple model, the incident electrons are assumed to penetrate straight into the

specimen along the line AB in figure 1. The energy loss is assumed to be constant and given

140
by the Bethe's energy loss law. At a distance Sj, along AB, there is a probability P((|))clsdfi, as given by the Rutherford's scattering law, that the electron will be scattered in the

incremental distance ds into an incremental solid angle dfi in the direction BC. The distance

AB + BC must be smaller than some value Lmax or the electron will have lost too much energy

and will be rejected by the energy filter. This gives rise to the following formula for the

low-loss conversion efficiency [13]:

where P((j))dsid(^ is the scattering probability defined above, and where F(Eioss) ^^"^^ ^^"^ a perfect energy filter, unity or zero depending on whether AB + BC is less than or greater

than Lmax- The numerical evaluation of this integral for a practical energy filter is

discussed in Appendix 1. A comparison between the theoretical and the measured low-loss

angular distribution curve is shown in figure 2. The bars show the measurements for an energy

loss of less than 1%, while the full line was calculated numerically from eq. (1). (The

measured values were reduced by 6% to improve the fit with the calculated curve.) Certainly,

for this low-loss region and for this particular specimen, the agreement between the theory

and the measurements is very encouraging. Some further calculated results are shown in figure 3. The agreement between the simple model and the measurements is not so good if the

energy loss is increased. The effect of increasing the energy loss on the observed angular

distribution is shown in figure 4. The experimental curve becomes wider and the peak lifts

away from the surface as the acceptable energy loss is increased. (According to the simple

model, it would be expected that the angles in figure 1 should not change as the energy loss

is increased, so that the geometric form should scale and these curves should all have the same shape.) The energy distribution curve for the electrons which leave the specimen in three different directions is shown in figure 5. (According to the simple model these curves should be flat.) In both of these sets of measurements, there is a departure from

(1)

(± 10" ABOUT L)

Eo = 20 kv LOSS < 100 V

TARGET =Si02

^C0LL = '°'^20o

Figure 2. Comparison between theory (full line) and experiment (bars)

for low-loss emission from amorphous Si02. The experimental values were all reduced by 6% to obtain the best fit with the theoretical curve [16].

141
low -loss

0 10 20 30 40 50 60

GLANCING ANGLE OF INCIDENCE 9

Figure 3. Theoretical low-loss conversion efficiency for flat amorphous

targets calculated for the takeoff angle shown and n,. = 0.5; E^, = 20 kV; ^filter = Eresn = 200 V [13].

Figure 4. Experimental angular distribution

curves from flat amorphous Si02 target as ^loss ^'^ increased: Eq = 20 kV [17].

the predicted values as the acceptable energy loss rises above about 1%, and it is presumed

that this is caused by additional wide-angle scattering events. At present, it is not clear

whether the simple model can be modified to allow for these extra events, or whether the

Monte Carlo approach must be used. This model can, however, be modified in a simple way to allow for curvature of the electron trajectories in the specimen, and it was therefore

decided to calculate the contribution to Type II magnetic contrast which can be expected

from electrons which have suffered only a single wide-angle event.

142
Filter grid potential (kV)

Figure 5. (a) Directions in which the energy distributions were measured,

(b) Observed energy distributions from Si02 with Eq = 20 kV [17].

3. Electron Scattering Model with Curvature of the Electron Trajectories

in the Specimen

The modified model is shown in figure 6. The electron trajectories in the specimen

have a radius of curvature which can be negative (fig. 6a), infinite (fig. 6b), or positive

(fig. 6c). In a practical magnetic material, the radius of curvature is quite large; for

example, with 21,000 Gauss in iron and E^ = 30 kV, then R = 277 m.

R=oo

(a)

Figure 6. Extension to the electron scattering model to allow for

curvature of the electron trajectories in the specimen.

For a given Si and ASi, the contribution to the low-loss conversion efficiency, dn,

which arises within a specified range of exit angles, 62, is given by

dn = ds^d,^ f

PM |i- F(E^^33) de^ (2)

143
where (i<t>^ is a specified small acceptance angle at right angles to the paper plane, and

where the term 8(i)/902 allows for the fact that the elemental collector solid angle do. as measured outside the specimen is no longer the same as the solid angle into which the

scattering event occurs.

The numerical evaluation of eq. (2) is discussed in Appendix 1. Some calculated results

are shown in figures 7 and 8, where the three curves shown for each value of incident angle

in figure 7 correspond to values of L^iax/f^ O'^ -0.1, zero, and + 0.1. Unexpectedly perhaps,

the shape of the angular distribution curve is not greatly affected even by such a severe curvature, and the major change is in the absolute magnitude rather than the shape of these

curves.

Figure 7. Calculated angular distributions of low-loss electrons as modulated by

type-2 magnetic contrast for = 20°, 30°, 50° and 70°. For each value of incident

angle the three curves corresponds to Imax^^ "0-^ (inner curve), 0.0 (middle curve)

and +0.1 (outer curve). The areas inside the four middle curves are shown here as being equal; in practice they would vary with the incident angle as shown in figure 3.

Figure 8. Showing the modulation to be expected

from type-2 magnetic contrast as a function of

the glancing angle of incidence Oj and the

curvature of the electron trajectory in the

specimen.

144
The calculated modulation of the low-loss conversion efficiency by the curvature

parameter Lmax/R is shown as a function of the glancing angle of incidence Qi in figure 8. The vertical axis shows the ratio of the areas inside the calculated angular distribution

curves for the value of L^gx/'^ shown on the horizontal axis to the value when R = oo. These curves were plotted by running the program for Lmax/R = ± 0. 1

, ± 0. 08, ± 0.06, ± 0.04,

± 0.03, ± 0.02, ± 0.01, and zero. For L^ax/R less than about 0.04, these curves can be

approximated by the equation

.^"^^R - T

, ^ "-max ^ ^

- 1 + k . (3)

where the numerical values for k are shown by one of the curves in figure 9. Thus, ac- cording to this model, if L^ax ""s fixed, then Type II magnetic contrast will progressively

01

Figure 9. Showing the dependence of k, L^ax

and the product k-L^ax the glancing angle

of incidence Qi (see text).

10 30 50 70 90

Glancing angle of incidence Qi

increase as the angle of incidence becomes more nearly glancing. One thing must be noticed

in figure 9. These results were obtained with a collector angle of 70° relative to the

specimen surface, and it is the asymmetry of the collector which accounts for the modulation

of the collected signal when = 90°. With a symmetrical collector solid angle, k will be zero at Gi = 90°.

4. Discussion of Results

In the low-loss situation, i^ax ^'^ determined by the maximum energy loss as given by

the potential on the filter grid of the energy filter. For low-loss topographic studies, an energy loss of 1% is appropriate, which leads to Lp^gx = 590 A for iron bombarded ^y 30 keV

electrons. (This number was calculated using the slowing-down value of 0.056 ev/A for iron

at 30 kV as given by Berger and Seltzer [18].) The radius of curvature R at 30 kV in iron

is 277 ym. The Type II magnetic contrast as calculated from eq.(3) for a 1% energy loss and

01 = 30° is therefore 0.106%. In other words, it should only be necessary to increase the

energy loss of the collected electrons to 1.7% for the modulation depth as given by eq. (3) to equal the value of 0.18%^as specified by Newbury, Yakowitz and Mykelbust [6]. (This

corresponds to Lmax = 1000 A at 30 kV.)

Of course, it must be noted that the contrast given by eq. (3) will be reduced by the

presence of an unmodulated background of slower backscattered electrons. The present model

gives data that is relevant either to the filtered case or, alternatively, the model should

predict the conversion efficiency between the incident beam current and the maximum-tominimum current swing in the collected signal.

'

To analyze the non-filtered case, it is necessary to define an "equivalent" value for

Lmax in the case when all of the backscattered current is collected. For example, in the case of electrons scattered from Si02 with Qi = 30°, the energy distribution curve of electrons

145
in the forward peak shows that these electrons have lost, at most, only 2 or 3 keV energy

with Eq = 20 keV, as is shown in figure 5. A natural filtering action is therefore taking

place for the electrons in the forward lobe.

As a hypothesis, we can define the equivalent value of Li^^^ being the path. di stance within which a wide-angle Rutherford scattering event through an angle greater than Gi is equal to the probability of two consecutive scattering events, each through an angle

greater than half of that amount. It is shown in Appendix 2 that this gives rise to

Lmax

= 1-07 X 10-^2 ^(0^)e2 (4)

where

2

cot^(e/4)

(Lmax is in cm if Eq is in eV). This, therefore, gives a value for L^^^ which will depend

on both the incident energy and angle of incidence. Physically, it corresponds to the case when the majority of the escaping backscattered electrons have suffered only a single wideangle scattering event.

The values of \-max calculated from eqs. (4) and (5) for iron with a beam at 30 keV are plotted as a function of Qi in figure 9. The general tendency is for lmax to ^ecome larger

as the beam comes closer to normal incidence. When Qi^= 30°, then lmax ^ ^ calculated

from eqs. (4) and (5). This is smaller than the 1000 A which was calculated above as being

needed to provide the observed contrast, so presumably a higher degree of multiple scattering

can be tolerated in comparison with the present hypothesis.

The 3/2 power law follows immediately from eq. (4). Thus, in the Monte Carlo results

obtained by Newbury, Yakowitz, and Myklebust [6], it was found that Type II magnetic contrast

varied as Eg^'^. From eq. (3), we would expect it to vary as L^a^/R while Lmax will vary as

Eq^ from eq. (4) and R will vary as Eq^"^. The ratio Imax/^ will therefore vary as Eq^*^.

The dependence of Type II magnetic contrast on Gj is illustrated in figure 9. According to eq. (3), the contrast is proportional to k-L^^ax- The values for k as measured from figure

8 decrease from k = 15.0 at = 10° to k = 1.1 at 0i = 90°. The variations of Lmax withGi

are given by eq. (5), and these are also plotted in figure 9. Here, Lmax increases from 45 a at 01 = 10° to 2,840 A at 0i = 80°. The product kLmax increases from a low value at Gj = 10° to a flat maximum for Gi > 50°. As noted above, if the collector solid angle was to be

symmetrical about the incident beam, then this would make k = 0 at 0i = 90°, and the curve

of kLmax would then be zero for 0i = 0 and 0^ = 90°, in agreement with the published curve

of Newbury, Yakowitz, and Myklebust [6].

5. Conclusions

This model has correctly predicted both the approximate magnitude of the contrast

and the way in which it depends on both the incident electron energy and angle of incidence.

This model cannot, however, be fully evaluated until some further numerical data can be obtained using an energy filter.

I would like to thank D. J. Fathers of the Metallurgy Department at Oxford University

for discussions concerning his model, and to acknowledge that this paper was put into final

form following helpful discussions with the other participants at the Monte Carlo Workshop.

References

[1] Philibert, J. and Tixier, R., Effects of crystal contrast in scanning electron microscopy, (in French), Micron, 1_, 174 (1969).

[2] Fathers, D. J., Jakubovics, J. P., and Joy, D. C, Magnetic domain contrast from cubic

materials in the scanning electron microscope, Phil. Mag. 2J_, 765 (1973).

146
[3] Fathers, D. J., Jakubovics, J. P., Joy, D. C, Newbury, D. E., and Yakowitz, H., A new method of observing magnetic domains by scanning electron microscopy, I. Theory of the

image contrast, Fhys. Stat. Sol (a), 20, 535 (1973). Also, same authors and main

title, ...II Experimental confirmation of the theory of image contrast, Phys. Stat. Sol

(a), 22, 609 (1974).

[4] Fathers, D. J., Joy, D. C, and Jakubovics, J. P., Magnetic contrast in the SEM,

IITP.I/SEM/1973^ 259 and 242 (1973).

[5] Newbury, D. E. and Yakowitz, H., Magnetic domain studies in iron - 3.25 weight percent

silicon transformer sheet using the scanning electron microscope, in Proa. 19th Conf. on Magnetism and Magnetic Materials, (AIP, Boston, Nov. 1973).

[6] Newbury, D. E. , Yakowitz, H., and Myklebust, R. L., Monte Carlo calculations of magnetic

contrast from cubic materials in the scanning electron microscope, Appt. Phys. Lett.,

23, 488 (1973).

[7] Newbury, D. E., Yakowitz, H., and Yew, N. C, Observation of magnetic domains in nickel

using the scanning electron microscope, Appl. Phys. Lett., 2A_, 98 (1974).

[8] Shimizu, R., Ikuta, T. , Yamamoto, T. , Kinoshita, M., and Murayama, T., Direct observation

of the magnetic domain structure in Mn-Zn ferrite using the 200 kV high voltage scanning

electron microscope, Phys. Stat. Sol. Ca) , 26, K87 - K89 and 2 plates (1974).

[9] Ikuta, T. and Shimizu, R., Magnetic domain contrast from ferromagnetic materials in the

scanning electron microscope, Phys. Stat. Sol. (a), 23, 605 (1974).

[10] Ikuta, T. , Dynamic and high resolution observations of the magnetic domain, PhD Diss.,

Osaka Univ. (1974).

[11] Yamamoto, T. , Nishizawa, H. , and Tsumo, K. , High voltage scanning electron microscopy

for observing magnetic domains, J. Phys. D: Appl. Phys., 8^, L113-L114 and 1 plate

(1975).

[12] Yamamoto, T. , Tsuno, K. , and Nishizawa, H., Utilizability of high voltage SEM for

observing magnetic domains, JEOL News, 13e , No. 1, Z(1975).

[13] Wells, 0. C. , Boyde, A., Li f shin, E., and Rezanowich, A., Scanning Electron Microscopy

(McGraw Hill, New York, 1974). Since this book was written, some additional data on the low-loss image has been published as follows: first, in ref. [16] below, some additional micrographs have shown that the topographic information for the smallest

surface features is contained in the low-loss electrons that leave the specimen with 62

< 20°. Second, in the proceedings entitled. Ninth Annual Conference, Microbeam Analysis

Society, July, 1974, Ottawa, Canada, pp. 2A-2E, it is shown that a low-loss image can

be obtained with a loss energy of only 25 eV, which corresponds to an escape depth of

10 A (This final result was obtained by C. G. Bremer.)

[14] Wells, 0. C, Low-loss image for surface SEM, Appl. Phys. Lett., 19, 232 (1971 ).

[15] Wells, 0. C, Explanation of the low-loss image in the SEM in terms of electron scattering

theory, llTHl/SEM/1972, 169 (1972). (This material is more extensively analyzed in

reference [13] above.)

[16] Wells, 0. C. and Bremer, C. G., Low-loss image from a solid specimen, Eighth Internat.

Cong. El. Mic, Canberra, Aus. Acad. Sci., ]_, 216 (1974).

[17] Wells, 0. C, Measurements of low-loss electron emission from amorphous targets,

IITRl/SEM/1975, 43 and 132 (1975).

[18] Berger, M. J. and Seltzer, S. M., Tables of energy losses and ranges of electrons and

positrons, NASA Report number SP-3012 (Washington, D. C, 1964.)

147
Appendix 1

Numerical Evaluation of Integrals

The procedures used to evaluate the integrals given in eqs. (1) and (2) were the following.

In each case, the expression for P{i>) was

1.29 X

10' 15 nZ

2 (A-1)

3 where n = number of atoms/cm ,

= 0.852 X 10^^ for iron

Z = atomic number, "

= 26 for iron, and

= incident electron energy, eV.

P((t>) is expressed above in sterad" - cm' . The expression for FCE-, ) was the same in each

case:

'

F(E

loss

= 0 for E, > E^.,^ loss filter

^filter "loss^ °

"resn

for E, < E^.,, < E loss f 1 1 ter — resn (A-2)

where E loss energy lost by the electron in the specimen, eV;

E^^.1|.g^ = potential of energy filter grid relative to the SEM

cathode, expressed in equivalent eV;

'resn

= resolution of the energy filter, expressed in equivalent

eV; and

and n^ are constants of the energy filter (typically ~ "o

~

The equations used for the simple model were

COSct) ' = COS<i> - C0S(t)^ (A-3)

where <|)' is the actual deflection angle which was submitted into eq. (A-1) and <i>t is the angle of the exit ray in figure 1 above or below BC at right angles to the paper plane.

sin e-j L = S-, + $2 = + s^ sin (4,-0^) cos<t,^ (A-4)

where Gi, Si, and i, are defined in figure 1 and L is the total path length of the electron

in the specimen.

148
For the simple model, this leads to

^low-loss - ^ ^° ^

\

E^^^^ / I I(s') ds' (A-5)

5.2 X 10 L_ nk

" F. 0 I

where s' = s^/L^^^,

/- /• / sin 0, \"o coS(t>^. d(|) d(f), I(s') H l-s'-s- ^ 5 2- ^^'^^

-^.^^t \

sin((f. - e^) cos 4.^ / (1 - cos coS(j)^)

(L is in cm; E is in eV; see eq. (3.29) in ref [13]). m3 X 0

In the model with curvature, the integral on was not carried out (eq. (2)). The

values of ^ and S2 were given by

4) = O-j + 02 if R

=

s, + S2

=0-, + 02 - R if R -

sin 0-, So = S. TT^TTT if R = -

2 1 sin ©2

= R02 - R arc cos r if R

^1

(A-7)

(A-8)

where r = cos - cos 0-| + cos

(©i ^ ),

sin 0

30 1 . (A-9)

2 >/7T7

In evaluating this integral, an addition boundary condition based on the need to establish

the existence of a solution for S2 is |r| £ 1. The numerical values shown in figures 7 and

8 were obtained by substituting the above expressions into eq. (2).

149
Appendix 2

Derivation of Equations (4) and (5)

In the simple model, it is assumed that the total penetration path AB + BC in figure 1 is sufficiently small so that the probability of a second wide-angle event also occurring

can be ignored. The value Lp^g^ which rejection occurs is determined by the energy filter.

If there is no energy filter present, then the forward lobe of the backscattered electron

distribution will still consist of electrons which have lost only 2 keV or 3 keV of energy

when the take-off angle lies close to the specimen surface (fig. 5). This presumably occurs because the slower backscattered electrons have been scattered by additional wide-angle

events. In the present calculation, it is desired to find an "equivalent" value of L^g^

such that the simple model can be applied even though there is no energy filter present.

Accordingly, as a hypothesis, it is required to find the path length L^ax fo*" which the

probability of single wide-angle Rutherford deflection through an angle greater than ej is equal to two consecutive deflections through angles which are greater than half of this amount. (It is presumably at about this point that multiple wide-angle scattering will

become the dominant process.) The probability of a Rutherford deflection through an angle

>0 in a path length L (where L is small) is given by

Prob (>e) =4cot2| (^-10)

^0

where k = 1.62 x

10"'^'* nZ^ = 9.33 x

10''^^ for iron, if E is in eV and L is in cm (see also

equation (3.6a) in reference [13]). The relationship

Prob (>0) = [Prob(>G/2)]^ ' (A-11)

then leads to

'

L .MlZxJoll . ^2 . cpt!(0Z2)

_ (^_^2)

nZ^ ° cot^(e/4)

For iron, n = 0.852 x 10^^ atoms/cm^ and Z = 26, which gives rise to the numerical coefficient

given in eq. (5). -

150
National Bureau of Standards Special Publication 460. Proceedings of a Workshop on the Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy

held at NBS, Gaithersburg, Maryland, October 1-3, 1975. Issued December 1976.

A STUDY OF TYPE II MAGNETIC DOMAIN CONTRAST IN THE SEM

BY MONTE CARLO ELECTRON TRAJECTORY SIMULATION

D. E. Newbury, H. Yakowitz, and R. L. Myklebust

Institute for Materials Research

National Bureau of Standards

Washington, DC 20234

Magnetic contrast in the scanning electron microscope occurs by two distinct

mechanisms: Type I magnetic contrast arises from the interaction of secondary

electrons which have exited the specimen with leakage magnetic fields above the

surface and Type II magnetic contrast arises from the interaction of beam electrons

with the internal magnetic field. The observed properties of Types I and II magnetic contrast are reviewed. The proposed mechanism of Type II magnetic contrast,

that of internal beam-magnetic field interactions, is modeled with the Monte Carlo

electron trajectory technique. Details of the modifications to the standard Monte

Carlo technique to account for magnetic deflection are given. Monte Carlo results are given for the following cases: contrast as a function of specimen magnetization, tilt, rotation, beam energy and energy fraction. The spatial resolution of

domain edges is also considered. Calculated results are compared with experimental

values, and the correspondence is excellent.

Key Words: Iron-silicon transformer steel; magnetic contrast; magnetic domain

images; magnetic domains; Monte Carlo electron trajectory calculations; scanning

electron microscope.

1. Introduction

Magnetic domains were first observed in the scanning electron microscope (SEM) by

Dorsey [1]^, who studied magnetic signatures written on ordinary iron oxide recording tape.

Such a material has a magnetic structure similar to that illustrated in figure 1(a). The

chief characteristic of these materials is the existence of a magnetic field outside the

specimen--the leakage field. The contrast discerned by Dorsey, which is now referred to as Type I magnetic contrast by reason of chronology, arises because of the interaction of the

secondary electrons which have exited the specimen with the exterior leakage magnetic field

(fig. 2(a)). Deflection of secondary electrons either toward the Everhart-Thornley detector

of the SEM or away, depending on the sense of the magnetic field, produces a black-white

contrast. Extension of this technique to crystalline materials of uniaxial magnetic character, such as cobalt and yttrium orthoferrite, was accomplished by Joy and Jakubovics [2] and

by Banbury and Nixon [3]. Such materials have a structure similar to figure 1(a). An example of a Type I image of a magnetic signature on recording tape is shown in figure 2(b).

The characteristics of Type I magnetic contrast may be summarized as follows [4]:

1. Type I magnetic contrast is obtained with secondary electrons only. No contrast

is obtained with the high-energy primary (backscattered) electrons or specimen

current signals

2. The contrast is a form of pure trajectory contrast, arising from the deflection

of emitted secondary electrons by leakage magnetic fields external to the specimen.

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

151
1 V t i V i

(a)

(b)

Figure 1. Schematic diagram of: (a) uniaxial

magnetic domain structure, and (b) cubic

magnetic domain structure.

(a)

0.5 mm

(b)

Figure 2. (a) Mechanism of Type I magnetic contrast, and (b) SEM Type I magnetic

contrast image of magnetic signatures in recording tape.

3. The magnetization vector must be out of the plane of the specimen.

4. The magnitude of the contrast can be as high as aS/S = 20X, where S is the

signal at the specimen, and is dependent on the geometry of secondary electron collection as well as the magnetization strength.

5. The contrast is dependent on specimen (magnetization direction) rotation

relative to the Everhart-Thornley detector.

6. The contrast is independent of accelerating voltage, although an improved

signal-to-noise (S/N) ratio may be obtained at low accelerating voltage due

to enhanced secondary electron production.

7. The resolution limit is determined primarily by characteristics of the con- trast mechanism rather than probe size; the limit is typically several micrometers due to the diffuseness of the leakage fields above domain walls.

152
8. The contrast across the domain is a function of the magnetic field integral

and is non-uniform with respect to position.

Attempts to obtain Type I magnetic contrast from anisotropic magnetic materials of

cubic symmetry, such as iron and nickel, were unsuccessful. Due to the multiplicity of

possible magnetization directions in such cubic materials, the material can establish a lowenergy configuration by forming surface closure domains in which the magnetic field lies

parallel to the surface (figure 1(b)). Such a state is characterized by a negligible leakage field, thus virtually eliminating the possibility of Type I magnetic contrast.

Figure 3. SEM Type II magnetic contrast

image of Fe-3 1/4% Si; beam conditions:

30 keV, 2 X lO-'^ A, specimen tilt: 55°,

Nevertheless, characteristic fir-tree magnetic domains were observed in iron by Philibert

and Tixier by means of a scanning electron beam instrument [5]. The characteristics of this new form of magnetic contrast, shown in figure 3 and now referred to as Type II magnetic

contrast, were elucidated by Fathers, et al . [5,7], and can be summarized as follows [4]:

1. Type II magnetic contrast is obtained with the primary (backscattered)

electron and specimen current signals. No appreciable contrast is obtained

with secondary electrons.

2. The contrast is pure number contrast, i.e., different numbers of electrons

are backscattered from domains of opposite magnetization. Since the con- trast can also be obtained with the specimen current signal, the origin of

the contrast is clearly associated with events inside the specimen.

3. The magnetization vector must lie in the plane of the specimen or have a significant component resolved in that plane.

4. The contrast is strongly dependent on the specimen tilt, with zero contrast

at a tilt of 0° (i.e., specimen normal to the beam) and a maximum at about

55° tilt.

5. The contrast is dependent on the specimen magnetization direction relative

to the beam; the maximum contrast occurs when the magnetization vector is parallel to the tilt axis.

6. The contrast increases with increasing accelerating voltage.

153
7. The contrast is very weak; e.g., for an optimum situation in a conventional

SEM (iron, beam energy 30 keV, 55° tilt, magnetization parallel to the tilt

axis), the measured contrast is aS/S = 0.3%.

Fathers, et al . , proposed a possible mechanism for the contrast [6]; the contrast

formation occurs because of the deflection of the primary beam electrons by the internal

magnetic field during scattering within the specimen (fig. 4). Thus, the deflection by the

magnetic field results in the electrons remaining closer to the surface in a domain with one magnetization sense, and further from the surface in the domain of opposite magnetization.

A difference in the backscattering coefficient would therefore be observed between the two domains. An analytic model was developed in an attempt to explain characteristics of the

contrast and, simultaneously, a Monte Carlo electron trajectory simulation study including

magnetic field effects was carried out [7,8]. We shall be concerned here with the Monte

Carlo calculations only.

Case I Case II

Figure 4. Schematic diagram of proposed

mechanism of Type II magnetic contrast.

2. Monte Carlo Trajectory Simulation in the Presence of a Magnetic Field

The specific details of the general Monte Carlo calculation program used in the present

study are given elsewhere in this volume [9]. Briefly, the general model makes use of an empirical Rutherford model of elastic scattering and the Bethe continuous energy loss approximation. The energy is fully depreciated in 60 scattering acts. The step length, or distance

between elastic scattering events, is variable and related to the instantaneous electron

energy in such a way that the total distance traveled in 60 steps is equal to the Bethe

range for the incident beam energy. The program has been extensively tested with satisfactory results with regard to experimentally observed electron properties; e.g., backscattering

coefficient as a function of atomic number, specimen tilt, and beam energy; transmission,

backscattering, and absorption in thin foils and particles; and angular distributions of

scattered electrons.

The coordinates (X,Y,Z) after each scattering act are calculated relative to a righthanded set of coordinate axes centered on the beam impact point. The coordinate system,

figure 5, consists of the positive-going Z axis into the specimen and normal to the surface,

the X-axis parallel to the tilt-axis, the specimen surface contained in the X-Y plane, and

the positive-going Y-axis pointing down a tilted specimen and away from the beam impact point.

The exact technique of calculating the point coordinates is as follows (fig. 6). With

the electron at a known point P(L), having come from a known point P(L-l), we must calculate

154
V

Figure 6. Illustration of coordinate values

used in calculation of point P(L+1) from

point P{L)

P(L+1). The electron traveled from P(L-l) to P(L) along a line (L) whose direction cosines

(a,b,c) can be simply calculated from the coordinates (X,Y,Z)|__-| and (X,Y,Z)l. If the

electron were not scattered at point P(L), it would continue along line (L). The act of

scattering the electron through an angle 6, calculated from the Rutherford relation, sends

the electron to some point P(L+1) on the perimeter of a cone whose altitude LI is the

extension of line L, figure 6. To calculate P(L+1), the point (Xq,Yq,Zq) where the extended

line LI intersects the plane containing the base of this cone is first determined. The path

on this cone surface that the electron travels from P(L) to P(L+1) has a length equal to the

step length for that scattering act, D(L), which is known. The altitude of the cone, H, is given by:

H = D(L) C0S6. (1)

Similarly, the radius of the base of the cone is:

R = D(L) sine. (2)

The coordinates of the center of the cone base (Xq, Yq, Zq) are given by:

^0 = X(L) + (H-a) (3)

^0 = Y(L) + (H-b) (4)

^0 = Z(L) + (H-c). (5)

We next establish a new set of coordinate axes, X', Y', Z' (fig. 7) with the X'Y' plane

containing the base of the cone, and the V axis parallel to extended LI. The direction

cosines of the primed axes in terms of the original axes must be determined. For the V

axis, the direction cosines are the same as line LI:

a^i = a, b^i = b, c^i = c. (6)

155
Figure 7. Primed coordinate system used in the point calculation scheme.

The X' axis is arbitrarily chosen to be parallel to the line of intersection of the plane of

the base of the cone with the plane Y = Yq. From the knowledge of a point in the plane of

the cone base, (Xq, Yq, Zq) and the direction cosines of the normal to the plane, i.e.,

those of line LI, the equation of the cone base is:

aX + bY + cZ - (aX + bY + cZ ) = 0. (7) 0 0 0

The line of intersection with the plane Y = Y^ is:

aX + cZ - (aX + cZ ) = 0. (8)

0 0

Thus, the line has the form

Z = (-a/c)X + K; K = (a/c)XQ + Z^ (9)

and the direction cosines of the X' axis with respect to the original axes are:

a^^i = cos(arctan(-a/c) ) (10)

b^, = cos 90° = 0 (11)

c^i = cos (arctan(-c/a) ) . (12)

Knowing the direction cosines of X' and Z', we can obtain the direction cosines of Y' by the

"direction number device" [10]. Since the original X, Y, Z, axes were chosen to form a right-handed coordinat^e system, we maintain parity, making X', Y', V right-handed by forming

the cross product (Z'k x X'i):

156
ay. = b^.-c^, (13)

Cyi = -axi-b^,. (15)

In the X', Y', V coordinate system, a point P'(L+1) (identical with P(L+1)) on the cone base perimeter is chosen randomly. The azimuthal angle is calculated with a uniformly

distributed random number RN chosen such that 0.0 £ RN £ 1.0. The coordinates of the point

P' (L+1 ) are

X'(L + 1) = R * cos(RN * 2^) (16)

Y'(L + 1) = R * sin(RN * It.) (17)

Z'(L + 1) = 0. (18)

The coordinates (X', Y', Z')|_+i can be transformed back to X, Y, Z space to give the desired

point P(L+1) since we have the direction cosine's of the X', Y', V axes:

(19)

(20)

(21)

a^, • X (L + 1) + ay, • Y (L + 1) + X 0

b^. • X (L + 1) + by, • Y (L + 1)

c^, • X (L + 1) + Cy, . Y (L + 1)

Effect of a Magnetic Field:

The effect of an internal magnetic field on the electron trajectory is simulated as follows. The trajectory calculation is first performed as described above to obtain the

scattering step from P(L) to P(L+1) without the influence of the magnetic field. The effect

of a magnetic field will result in a deflection of the electron trajectory so that a curved

path of flight is followed, bringing the electron to a point Pm(L+1) instead of P(L+1)

(fig. 8). The magnetically-deflected trajectory is determined through the action of the

Figure 8. Effect of a magnetic field on the

electron trajectory during a single scattering

step

P(L+1) p'

F = -e(v X B

157
Lorentz force, F, on the electron:

F = -eV X

e (22)

where V is the electron velocity, B is the magnetic field vector, a is the acceleration, e

is the electron charge, and mp is the mass of the electron. The direction cosines of the

line of flight from P(L) to P(L+1 ) in the nonmagnetic case are used to give ti^e initial

velocity components for the magnetic calculation. Thus, the velocity vector V is:

V = aVi + bVj + cVk

where V is the magnitude of the electron velocity which is determined from the electron

energy for the particular scattering step. The Lorentz acceleration vector a becomes:

:eV\

(bB -B c)i + (cB -B a)j + (aB -B b)k

y

(23)

(24)

The magnetic deflection of an electron under the conditions used for the calculation is small; hence the time-of-f 1 ight will be virtually the same as in the nonmagnetic trajectory.

This time-of-f 1 ight is

D(L)/V (25)

where D(L) is the step length for the particular step being calculated. This time-of-f 1 ight

is then used to calculate the displacement components, aS^ , under magnetic deflection:

,2 AS.

1 V .t + (l/2)a .t^ 01 ^

' ' 01 (26)

where i represents the component along X, Y, or Z. Thus, in the absence of a magnetic

field, the acceleration term is zero and the displacement components calculated by eq. (26)

would place the electron at point P(L+1). When a magnetic field is present, the acceleration

term acts to deflect the electron to a new point Pm(L+1). For the next step, the direction

cosines are calculated from P(L) and Pf^(L+l). For the field strengths used (up to 50 times

the saturation value of iron, as described below), the adjustment due to magnetic deflection

on a given step is small.

3. Characteristics of Type II Contrast Predicted by the Monte Carlo Model

To apply the Monte Carlo program described above to study Type II magnetic contrast,

the backscattering coefficient was calculated for selected conditions of beam energy,

specimen tilt, and magnetization direction. However, since the experimentally observed

magnetic contrast has a maximum of only about 0.3%, definite statistical limitations are imposed on the Monte Carlo calculations. To obtain adequate statistical confidence, at least 10^ electron trajectories would have to be calculated. However, the backscattering

coefficient must be calculated for each of two senses of magnetization in order to calculate

the contrast, C, observed between domains of opposite magnetization. Thus, between two domains, C = (m - n2)/nave where n is the backscattering coefficient. Such Monte Carlo

calculations proved to be prohibitively expensive. Therefore, to reduce the number of

necessary trajectories, use was made of the fact that the complementary analytic model [6,7]

predicted a linear contrast dependence on magnetization strength. This suggested that a valid calculation could be performed using a higher value of magnetization than actually

found in iron so that the "contrast" was of the order of 5%. The contrast value appropriate

to iron (or another material) could then be obtained by appropriate scaling.

To check the validity of this procedure, a series of Mont^e Carlo trajectory calculations

was carried out with the induction, B, as high as 100 Wb/m^ (B at saturation for iron is 2.16 Wb/m^). A linear relation between contrast and magnetization was indeed observed

(fig. 9). Calculations for other conditions of beam energy and specimen tilt were performed

with B = 60 Wb/m2 giving adequate statistical confidence with 20,000 to 75,000 trajectory

calculations for the conditions of interest. In the figures that follow, the calculated con- trast has been reduced to that appropriate for iron.

158
The behavior of the contrast as a function of tilt was calculated for an accelerating

voltage of 50kV (fig. 10). As the tilt increases from 0° (normal beam incidence) the con- trast increases, reaching a maximum between 50°-60°, and then decreases as the tilt is further increased toward grazing incidence. This behavior can be understood according to

0.6

0.5

^ 0.4-

e 0.3

0.2

0.1

ok

Iron:

" ^-^^ ^^^^ gauss) Tilt axis parallel to B^^^ 5000O Voltage = 50 kV ^" . 70000

30000

\

\

30000

'

30000 30000 /

/

\

X 20000

\

I I L

0 10 20 30 40 50 60 70 80 90 Tilt (degrees)

Figure 10. Calculated and observed magnetic contrast versus tilt angle,

159
the model (fig. 11). At 0° tilt, the effect of the magentic field of opposite sense is simply to change the direction of the curving path of the electrons, i.e., either clockwise

or counterclockwise (fig. 11 (a)). There is no effective difference in depth of the electron distribution between the two domains, and therefore no contrast. As the tilt increases,

the clockwise or counterclockwise curvature does result in a difference in the depth of the

electrons (fig. 11(b)). At high tilts, the electrons tend to scatter out of the specimen

in a few scattering acts, thus minimizing the length of path within the specimen over which

the internal magnetic field can act upon the electron. Experimental measurements of C show

a similar behavior as a function of tilt [11].

Doma i n wal 1 Tilt axiso

(a) (b)

Figure 11. Electron interactions in a specimen

with an internal magnetic field under special

conditions: (a) 60° tilt and B parallel to tilt axis; (b) 0° tilt; (c) tilt and B perpen- dicular to tilt axis.

The absolute value of the calculated contrast falls below that measured experimentally

by approximately a factor of two. Thus, for iron at 30 keV and a tilt of 60°, the calculated contrast is 0.15% and the measured contrast is 0.35%. This discrepancy may occur because of errors in the simplified model of electron interaction used in the Monte Carlo

calculation, as well as experimental errors in the measurement arising due to the considerable difficulties in measuring such low contrast values.

The behavior of the contrast as a function of the accelerating voltage is shown in figure 12. The contrast increases as the 1.44 power of the accelerating voltage. Experimental measurements performed at voltages up to 30 keV supported this trend with an observed

voltage exponent of approximately 1.4 [10]. Recent meas-urements performed on a high voltage

SEM gave measured contrast values of 1.3% (100 keV), 2.8% (150 keV), and 5.2% (200'keV)

[12]. These values correspond reasonably well to 1.1% (100 keV), 2.1% (150 keV) and 3.1%

(200 keV) predicted by the Monte Carlo calculations. In addition, the recent measurements

were made with a detector for emitted electrons, which is both somewhat directional and

energy-selective, and therefore might produce higher measured values. The increase in contrast with increased accelerating voltage is understandable since the Lorentz force is given by eV x B. As the electron energy, and hence, velocity increases, the Lorentz force

increases.

One of the most interesting results obtained with the Monte Carlo calculation concerned

the distribution of contrast as a function of the energy of the backscattered electrons. A histogram of backscattered electron energy for the domains of opposite magnetization is shown in figure 13. The contrast carried by each energy fraction is:

160
Energy AS/S Total

45.4 - 50 keV 0.126

40.1-45.4 0.130

34.1-40.1 0.129

27.2 - 34.1 0.121

18.4 - 27.2 0.116

2.7 - 18.4 0.110

N

Figure 13. Histogram of the number of

backscattered electrons versus energy

from domains of opposite magnetization,

(Black region gives difference between

opposite domains.)

2000-

1000-

0 10 20 30 40

E, keV

161
Thus, accepting electrons which have lost more than about 20% of the initial beam energy

actually results in a decrease in contrast. This result implies that suitable energy filtering of the backscattered electron fraction should result in increased contrast. Experimental verification of this prediction has been obtained with a special forward scattering

experiment [11]. The energy dependence can be explained by the following argument: When the

magnetization vector is parallel to the tilt axis, the Lorentz force is greatest on those

electrons traveling in the original beam direction, since the angle between the velocity and

magnetization vectors is 90°. Electrons which have lost energy due to inelastic scattering

are also likely to have been elastically scattered out of the incident direction. For_ these

scattered electrons, the Lorentz force_is decreased due to the reduced angle between V and

B as well as the reduced magnitude of V which results from energy loss. However, the electron

must spend some time under the influence of the internal magnetic field; and hence, those

electrons which scatter out after only one or two elastic scattering steps are probably not carrying much contrast information. This requirement for some length of path within the

specimen is apparent from the contrast behavior at high tilt. The decreasing contrast at tilts above 65° corresponds to decreasing path length within the specimen and higher back- scatter.

Monte Carlo calculations of the 0° tilt case of figure 11(a) verify that no contrast

occurs. Moreover, the situation illustrated in figure 11(c), where the magnetization vector

is perpendicular to the tilt axis, i.e., B lies along the Y-axis, also is found to produce

no contrast despite the optimum tilt.

Monte Carlo calculations also provided information on the edge resolution to be expected with Type II magnetic contrast. Figure 14 shows an image in which domain walls run parallel and perpendicular to the tilt axis. It can be observed that the domain walls which

are perpendicular to the tilt axis appear more sharply defined than the walls parallel to the tilt axis (e.g., circled area in fig. 14). Plots of electron trajectories obtained

with the Monte Carlo technique for a tilted iron specimen at 30 keV beam energy can be used

Figure 14. Image of magnetic domains in Fe-3 1/4% Si; specimen tilt 55° (axis A);

beam 30 keV; note domain wall resolution in circled area.

to understand this observation of differential wall resolution. The dense trajectory

region of a plot showing a Y-Z section (fig. 15(a)) is larger and more elongated than the

dense trajectory region of the X-Z section (fig. 15(b)). Thus, the 'difference in edge

resolution is a result of the different dimensions of the interaction volume which sweeps

across the domain walls during the scanning action. The image of a domain wall running

perpendicular to the tilt axis is affected by the smaller dimensions of the interaction

volume in the X-Z plane; and thus, the wall position is more sharply defined than the case of a wall parallel to the tilt axis which is affected by the dimensions of the Y-Z section.

162
Figure 15. Monte Carlo trajectory plot for iron

at 55° tilt, 30 keV beam energy: (a) Y-Z section,

(b) X-Z section.

4. Conclusions

Monte Carlo simulations of electron trajectories under the influence of an internal

magnetic field verify all aspects of the proposed mechanism for Type II magnetic contrast.

The contrast arises from the action of the Lorentz force of the internal magnetic field on the scattering electrons. Contrast is maximized by three conditions: (1) a specimen tilt

of 50° to 60°, (2) high accelerating voltage, and (3) magnetization direction lying in the

plane of the specimen and parallel to the specimen tilt axis.

References

[1] Dorsey, J. R. , Woo. 1st National Conference on Electron Probe Microanalysis (College

Park, Md., 1966); Adv. Electronics and. Electron Phys. Suppl. 6, "Electron Probe Microanalysis," A. J. Tousimis, and L. Marton, eds. p. 291 (Academic Press, 1969)

[2] Joy, D. C. and Jakubovics, J. P., Phil Mag., 17_, 61 (1968).

163
[3] Banbury, J. R. and Nixon, W. L., J. Sat. Instrum. 44, 889 (1967).

[4] Goldstein, J. I., Yakowitz, H., Newbury, D. E., Lifshin, E., Colby, J. W., and Coleman,

J. R. , Practical Scanning Electron Microscopy, Chapter 5 (Plenum Press, New York,

1975).

[5] Philibert, J. and Tixier, R. , Micron, 1, 174 (1969).

[6] Fathers, D. J., Jakubovics, J. P., and Joy, D. C, Phil. Mag. 27_, 765 (1973).

[7] Fathers, D. J., Jakubovics, J. P., Joy, D. C, Newbury, E. D., and Yakowitz, H., Phys.

Stat. Sol. (A) 20, 535 (1973).

[8] Newbury, D. E., Yakowitz, H., and Myklebust, R. L., Appl. Phys. Lett. 23, 488 (1973).

[9] Myklebust, R. L., Yakowitz, H., Newbury, D. E., and Heinrich, K. F. J., this publication (p. 105).

[10] C. H. Lehmann, Analytic Geometry p. 289 (Wiley, New York, 1942).

[11] Fathers, D. J., Jakubovics, J. P., Joy, D. C, Newbury, D. E., and Yakowitz, H., Phys.

Stat. Sol. (A), 22, 609 (1974).

[12] Yamamoto, T. , Nishizawa, H., and Tsuno, K. , J. Phys. D. Appl. Phys. 8, LI 13 (1975).

164
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS SP-4602. Gov't Accession

No.

3, Recipient's Accession No.

A T IT \ IT A\'r\ CIICT'TT'I U ^. 1 I 1 Lt. n.i\U D 1 1 I Lt. Use of Monte Carlo Calculations in Electron Probe

Microanalysis and Scanning Electron MicroscopyProceedings of a workshop held at the National Bureau of

Standards, Gaithersburs, Maryland, October 1-3, 1975

5. Publication Date December 1976

6. Performing Org^n ization Code

7. AUTHORlS)

K. F. J. Heinrich, D. E. Newbury, H. Yakowitz (eds.)

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WAinlNOIUN, D.L. /UzJ4

10. Proiect/Task/Work Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as No. 9.

13, Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 77-608002

16. .ABSTRACT (A 200-word or less factual summary ol most sigxiificant information. If document includes a significant

bibliography or literature survey, mention it here.) This book is the formal report of the Workshop on the

Use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy held at the National

Bureau of Standards, October 1-3, 1975. The papers cover a wide range of topics within the field: the history and

development of Monte Carlo m.ethods for use in x-ray microanalysis; the study of the distribution of electron and x-ray

signals by Monte Carlo techniques; the effect of the choice

of scattering models on the calculations; techniques for

considering the distribution of energies of the beam electrons

propagating in the specimen; evaluation of ionization crosssection models; and applications of Monte Carlo techniques

to the study of particles, thin films, and magnetic domain

images. The contributions include reviews of general

interest as well as papers treating specific topics. The

volume should be of wide interest to workers in the fields

of scanning electron microscopy, electron probe microanalysis,

electron physics, and other fields involving the interaction

of electrons with solids.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) ElectTon probe microanalysis; electron-solid interactions; magnetic domains; Monte Carlo electron trajectory calculations;

particle analysis; scanning electron microscopy; thin film analysis.

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of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent

of Documents, Government Printing Office, Washington,

D.C. 20402. Order following NBS publications—NBSIR's and FIPS

from the National Technical Information Services,

Springfield, Va. 22161. Federal Information Processing Standards Publications

(FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. Register serves as the official source of information in the Federal Government regarding stand- ards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended.

Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

"he following current-awareness and literature-survey

iibliographies are issued periodically by the Bureau:

'ryogenic Data Center Current Awareness Service. A

literature survey issued biweekly. Annual subscription: Domestic, $20.00; Foreign, $25.00.

^quified Natural Gas. A literature survey issued quar3rly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the pre- ceding bibliographic services to National Bureau of Standards, Cryogenic Data Center (275.02) Boulder,

Colorado 80302.
U.S. DEPARTMENT OF COMMERCE

National Buraau of Standards

Washington. D.C. 20S34

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