NBSIR 74-457

Transmission of Electrons Through Foils

Stephen Michael Seltzer

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

April 1974

Prepared for Office of Naval Research

Arlington, Virginia 22217

NBSIR 74-457

TRANSMISSION OF ELECTRONS THROUGH FOILS

Stephen Michael Seltzer

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

April 1974

Prepared for Office of Naval Research

Arlington, Virginia 22217

U. S. DEPARTIVENT OF COMMERCE, Frederick B. Dent. Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

TRANSMISSION OF ELECTRONS THROUGH FOILS*

by

Stephen Michael Seltzer

The transmission of electrons through foils has been studied

by a Monte Carlo method. Cases involving electrons with energies

from 50 keV to 1 MeV normally incident on beryllium, mylar, aluminum, and titanium foils are considered. Good agreement with experimental results has been found for quantities such as the number

transmission, the energy and angular distribution of the emergent

electrons, and the spatial distribution of energy deposited in the

foil.

A comprehensive set of results has been generated for 100, 150,

200, 300 and 400-keV electrons incident on beryllium, mylar, aluminum

and titanium foils that are commonly used as vacuum windows in con- junction with low energy electron accelerators. Quantities given are

the electron number and energy transmission and reflection, the energy

absorbed, and the energy and angular distribution of the transmitted

electrons. It is shown that much of the results can be presented in a scaled form which reduces the explicit dependence on, and facilitates

the interpolation with respect to, the incident energy.

*/ This report is based on a Thesis submitted to the Faculty of the

Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Master of Science 1973.
ACKNOWLEDGMENTS

I wish to thank Dr „ Raymond Hayward, my thesis advisor, for

his many useful comments on this work. I would also like to express

my gratitude to Dr. Martin Berger who provided guidance and suggestions over the course of this work.

Dr. Robert Placious suggested the problem treated in this work,

and conversations with him and Dr. William McLaughlin directed my

attention to the applications of the results presented here.

Finally, I wish to thank my mother, Mrs. Henrietta Seltzer, for

the time and effort she contributed in the preparation of this manuscript.

ii
TABLE OF CONTENTS

CHAPTER PAGE

I. INTRODUCTION 1

II. BACKGROUND 3

A. Energy Loss 3

1. Single Inelastic Scattering 3

2. Stopping Power 4

3. Multiple-Scattering Distribution of Energy Losses.. 7

B. Angular Deflections.......... 10

1. Single Elastic Scattering 10

2. Multiple-Scattering Distribution of Angular

Deflect ions 11

C. Coupling of One-Variable Multiple Scattering Theories.. 13

D. Solutions of the Multi-Variable Transport Equation 15

III. METHOD OF CALCULATION 17

A. Theoretical Distributions 17

B. Transport Model..... 19

1. Definition of Boundary Conditions and Quantities

Calculated 19

2. Division of Electron Tracks into Segments 21

3. Generation of Electron Histories 22

IV. COMPARISON WITH EXPERIMENT 26

V. NEW RESULTS

.

32

VI. COMPARISON WITH SIMPLE THEORY...... 35

111
CHAPTER ^^^^

37

FIGURES o o . o . »

"

54

TABLES , = o o o ........ « • — ' o

APPENDIX. TABLES OF TRANSMITTED ELECTRON CURRENT..., .... 69

REFERENCES. . ..... ........ 128

iv
LIST OF FIGURES

FIGURE PAGE

1. Comparison of experimental and calculated number

transmission coefficients T.^ for aluminum. The

N

foil thickness z is expressed in terms of the

ratio z/Tq, where r^ is the mean range of an electron incident with energy E , The solid curves

are calculated values of T^ and pertain to the

transmitted charge. The dashed curves are calculated such that the emergence of a primary electron

and one or more associated secondary electrons is

treated as a single "transmission event" (corresponding to the experimental conditions of

Seliger) 37

2. Comparison of experimental and calculated number

transmission coefficients T^ for beryllium. The foil

thickness z is expressed in terms of the ratio z/r^,

where r^ is the mean range of an electron incident

with energy E^ 38

3. Comparison of experimental and calculated number

transmission coefficients T„ for 1-MeV electrons

N incident on titanium foils. The foil thickness z is

expressed in terms of the ratio z/r , where r^ is

the mean range of the incident electrons...... 39

4. Comparison of experimental and calculated angular

distributions of electrons transmitted through aluminum

foils of thickness z. The incident energy E^ is 1 l^V;

the angular distributions shown include all electrons

transmitted with energies greater than 100 keV 40

5. Comparison of experimental and calculated energy

spectra of electrons transmitted through aluminum foils

of thickness z. The incident energy E^ is 1 MeV; the

energy spectra shown include all electrons transmitted,

regardless of angle 41

6. Comparison of experimental and calculated energy spectra

of electrons reflected from an aluminum foil. The

Incident energy E^ is 0.5 MeV; the energy spectra shown

include all electrons reflected regardless of angle.,. 42

V
Comparison of experimental and calculated distributions of energy deposition as a function of depth.

The quantity shown is the scaled energy deposition

distribution ^oD(z) , where E is the energy and r ^o

the mean range of the incident electrons. The

results for Be, Al , and Cu for Eq=0.5 MeV are for a broad beam' of electrons incident perpendicularly on

a semi- inf inite medium. The results for Al for Eq=

0.1 MeV are for a plane perpendicular source in an unbounded medium, i.e., with backing material behind

the source plane. The experimental distributions have

been normalized so as to give the same total energy

deposited as the calculations

Universal curves of number transmission and energy

absorption coefficients for titanium foils. The curves are composites drawn through the Monte Carlo

results (shown as points) for the number transmission

coefficient and the energy absorption coefficient

The foil thickness z is expressed as a fraction

of the mean range r^^ of the incident electrons........

Universal curves of the most probable energy and the

full-width at half -maximum of the spectra of electrons

transmitted through titanium foils. The curves are composites drawn through the Monte Carlo results (shown

as points) for the most probable energy Ep and the

full-width at half -maximum W as fractions of the incident energy E^. The foil thickness z is expressed as

a fraction of the mean range r^ of the incident electrons . . . o o . . . . =

Normalized energy spectra of transmitted electrons.

The quantity shown is E^S(E, z )/Tj^, where E^ is the

incident energy, S(E,z) the energy spectrum and

the number transmission coefficient.

a. Dependence on foil thickness; titanium,

Eq=200 keV.

b. Dependence on incident energy; titanium,

z = 1 mi 1 (11.43 mg/cm2).

c. Dependence on foil material; E =200 keV, z=2 mils

vi
FIGURE PAGE

11. Percentile cosines for electrons transmitted through

beryllium, mylar, aluminum, and titanium foils. A

percentage P of the electrons transmitted by a foil

of given thickness emerge with direction cosines

between cosQp and 1.0. Foil thickness z is expressed

as a fraction of the mean range r^ of the incident

electrons. The curves are composites drawn through

the Monte Carlo results shown as points:

<f Eq=100 keV

A 150

200

^7 300 ^ 400 . „ „ . . . 47

12. Equilibrium shapes of the cumulative angular distributions of electrons transmitted through thick foils.

The curves give the fraction of the transmitted electrons that have direction cosines between cosQ and

1.0.. 48

13. Normalized angular distributions of transmitted

electrons. The quantity shown is the angular distribution AC^.z) divided by the number transmission co- efficient Tj,^.

a. Dependence on foil thickness; titanium,

E„=200 keV.

b. Dependence on incident energy; titanium, z=l mil

(11 .43 mg/cm2)

c. Dependence on foil material; E^=200 keV, z=2 mils. 49

14. Current, differential in energy and angle, of electrons

transmitted through a l-mil(11.43 mg/cm^) titanium

foil. The incident energy E^ is 200 keV. The histograms represent calculated values of the transmitted

current, J(E,9,z). The curves give the quantity

S(E,z)A(e ,z)/Tj^, where S(E,z) Is the calculated energy

spectrum (regardless of angle), A(6,z) the calculated

angular distribution (regardless of energy) and T-^ the

number transmission coefficient.

a. Averaged over the interval 0° 9 < 10°.

b. Averaged over the interval 30° ^ 0 < 40°.

c. Averaged over the Interval 60° < 0 < 70°......... 50

vli
Most probable energy of electrons transmitted through

planar silicon detectors. The quantity shown is the

most probable Ep as a fraction of the incident energy

Eq. The detector thickness z is expressed as a fraction of the mean range r^ of the incident electrons. The solid curve is a composite drawn through

the Monte Carlo results (shown as points). The dashed

curves are from Landau's energy loss theory, without

inclusion of multiple scattering detours..............

Comparison of Monte Carlo results and Landau-BlunckLeisegang theory. Given are the most probable energy

Ep and the full-width at half -max imumi W of the spectrum of electrons transmitted through titanium foils

of thickness z. Curves obtained from the Monte Carlo

results are labeled MC; from Landau's theory. L; and

from the theory of Blunck and Leisegang, LBL

Mean deflection cosine as given by the complete Monte

Carlo calculation and as derived from the theory of

Goudsmit and Saunderson. The dashed curves, from the

Goudsmit-Saunderson theory (labeled A), pertain to

the flux of electrons having traveled pathlength z in an unbounded medium. Plotted versus z /r , where r ' o ' o is the mean range of the incident electrons, the

dashed curves represent results for 100 keV ^ E^ ^ 400 keV. The Monte Carlo results pertain to the cur- rent of electrons transmitted by a foil of thickness

z and are represented by the solid curves (labeled MC)

drawn through the points:

tSt Eo=100 keV

A 150

200

V 300

O 400 .... o ... o .

viii
LIST OF TABLES

TABLE PAGE

1. Properties of foils considered in this paper , 54

2. Electron stopping power and range. Powers of ten are shown in parentheses; e.g., 5.21(-3) means 5.21x10"^ 55

3. Dependence of calculated electron transmission on the

step-size parameter n used in the Monte Carlo model. The

average fractional energy loss per step was 0.083/n.

Results are for 200-keV electrons incident on a 2-mil

(13.72 mg/cm^) aluminum foil, and are based on a sample

of 20,000 Monte Carlo histories. Quantities given are:

Tj^, number transmission coefficient; Rj^, number reflection

coefficient; T^, energy transmission coefficient;

energy absorption coefficient; Rg, energy reflection

coefficient; E^^/Eq, mean energy of transmitted electrons

as fraction of Incident energy; (cos9)„ , mean deflection

a. V cosine of transmitted electrons... = o....o.. 56

Transmission, reflection and absorption of electron beams

incident on beryllium, mylar, aluminum and titanium foils.

Quantities given are: T-^, number transmission coefficient;

Rjsi, number reflection coefficient; Tg, energy transmission

coefficient; energy absorption coefficient; R^, energy

reflection coefficient. . . <, . . . . <, . . . . . . . . 57

Characteristics of the energy spectra of electrons transmitted through beryllium, mylar, alimiinum and titanium

foils. Quantities given are: E /E^, mean energy as fraction of incident energy, Ep/E^, most probable energy

as fraction of incident energy; W/E^, full-width at halfmaximum as fraction of incident energy; W/Ep , full-width

at half-maximum as fraction of most probable energy 61

Cumulative angular distribution of electrons transmitted

through beryllium, mylar, aluminum and titanium foils.

The quantity given is the fraction of the transmitted

electrons that emerge in directions between 0° and 9 65

ix

I. INTRODUCTION

An electron traveling through a foil undergoes a succession of

inelastic and elastic collisions, causing changes in its energy and

direction. Thus, a beam of electrons, originally monoenergetic and

monodirect lonal , can have broad energy and angular distributions after

passing through a foil; if the foil is of sufficient thickness, a

significant fraction of the incident electrons will be completely

absorbed. The purpose of this work is to report on Monte Carlo calculations of these effects for electrons incident with energies from

1 MeV down to about 100 keV. Our goal is two-fold: first, to verify

the Monte Carlo model through comparisons with appropriate experiments,

and second, to generate a comprehensive set of transmission data for

beryllium, mylar, aluminum, and titanium foils for incident electron

energies from 100 to 400 keV.

Such foils are frequently used as windows for bringing out of the

vacuum region a beam of electrons generated by an accelerator. These

windows may be present for a variety of reasons: a) They are used in

many experimental configurations in order to avoid the problems

associated with oil contamination of a target in the vacuum system or

with the changing of targets, b) In order to obtain broad beam irradiation geometries, a foil may be purposely introduced to "spread"

the beam by broadening its angular distribution. c) A gaseous target

must be enclosed, with a window present to allow entrance to the electrons. An interesting example of this is in the recently developed

1 2 electron-beam-controlled gas-discharge laser. ' d) Electron ac3

celerators are finding growing use in industrial radiation processing.
In these situations, in which large samples are moved through an

electron beam, practicality dictates the use of a window,

A foil a few mils* thick is nearly transparent to high energy

electron beams. In such cases, the effects of scattering are small

and can be calculated using available analytical theories. However,

for electron energies below about 0„5 MeV, the angular deflection and

energy loss begin to be appreciable. For this reason, incident

energies between 100 keV and 400 keV were emphasized in the present

calculations. So that the results would be more directly applicable

to situations of interest, the calculations were made for foils of

readily available thickness: 2, 3 and 5-mil beryllium; 1/4, 1/2 and

1-mll mylar; and 1/2, 1 and 2 mils for both aluminum and titanium. The

quantities obtained include the fraction of the incident current and

energy that is transmitted, the amount of energy absorbed by the foil,

and the energy and angular distributions of the transmitted electrons.

These results will aid in the selection of an optimum window by a

balance of mechanical and thermal requirements, on the one hand, and

desired transmitted current and spectral and angular properties of the

transmitted beam, on the other. In addition, for a given window,

knowledge of the characteristics of the transmitted beam is important

in the design and analysis of experiments employing these beams.

*The mil is commonly used as a unit of thickness for commercially

available foils in the United States. One mil = 1/lOOOth of an Inch = 25.4 pm.
II. BACKGROUND

The basic interactions that govern the passage of electrons

through matter have been reviewed by several authors including

4 5 6 7 8 9

Evans, Rossi, Be the and Askin, Fano, Birkhoff, Berger , and

Zerby and Keller.''"^ In these articles, various theories of single and

multiple Coulomb scattering are summarized and compared with the results of numerous measurements. Here we will consider the pertinent

single-scattering cross sections and discuss multiple scattering

theories with an eye toward developing an accurate description of the

electron penetration process.

A. Energy Loss

1, Single Inelastic Scattering

Simple classical considerations lead to a cross section for the

scattering of electrons by free electrons

4^:= -4 • (1)

where € is the fractional energy transfer in units of the incident

kinetic energy E,

Miller,''""'' using a relativlstic theory which Includes spin and

exchange effects, derived the cross section

= 2TTeir I ^ _L_ +f__I_f„ 2rii ] , (2)

2 where T = E/mc . With the energy transfer ^ is associated the angular

deflection (JU^ (In the laboratory system) of the primary electron

C05^<*)p = 0-^)CT-f2) (3)
The angular deflection uu^ of the secondary electron is obtained from

cos'^oJs = ^(r+z) . (4)

The probability per unit pathlength of an inelastic scattering

with fractional energy transfer greater than a given cut-off value,

^ > ^ , can be obtained from the Miller formula:

c

A J

4g: dl€ = HaZ 2i^e^ U - -L_ + /-x_) a , ^ \ (5)

where N is Avagadro's number, Z the atomic number of the medium, and

A the atomic weight. The integral in Eq.(5) extends only to ^ = 1/2

because the outgoing electron of higher energy is, by definition, the

primary electron.

2. Stopping Power

The mean collision energy loss per unit pathlength - the collision

loss stopping power - is defined as

'A

L,(E) ^ iLz r ea^-dc . (6)

Eq.(6) is evaluated by separating the integral into the two intervals

0 ^- ^ € ' and s € < 1/2, where E is a small energy but one that

is large compared to that of the atomic electrons. In the first

interval, the binding of the atomic electrons is taken into account

12 through Bethe stopping power theory:

A J

4
where I is the mean excitation energy for the tnedium.

In the second interval, 6 ^ €

'

, the binding of the atomic electrons is assumed negligible, and the Miller cross section is applicable. When the integral over the Miller cross section is combined

with Eq.(7), the parameter drops out, and the final result is

The quantity 6 in Eq.(8) is the density effect correction factor

which gives the reduction in the collision loss due to the dielectric

13 polarization of the mediimi. Sternhelmer and Peierls give a general

expression for 6, based on a fit to calculated values:

(8)

o

(9)

where

(10)

and

(11)

In Eq.(ll), hv is the plasma energy for the medium,

«/2 '/2 (12)

5
3

where p is the density of the medium (in g/cm for the second equality in

Eq,(12)). The parameter a is obtained from the expression

a= -Q--f.CoipX. , (13)

and X and X, for solids and liquids* are determined as follows:

For I 100 eV, d = 3, X^ = 2 and

0.2 ,-C<3.6?J

(14a)

For I - 100 eV, d = 3, ^1 ^ ^

0 (14b)

.The mean energy loss by bremsstrahlung was evaluated using a

package of bremsstrahlung production cross sections discussed in

14 detail by Berger and Seltzer. This package consists of Bornapproximation formulas combined with screening. Coulomb, and highfrequency correction factors. Although it is included, the bremsstrahlung energy loss is of minor importance in the present calculations; in the cases considered here, less than about 27o of the

initial kinetic energy of the electron is converted into bremsstrahlung

photons , :

*The parameters for gases and liquid hydrogen are given in

Reference 13.

6
The total stopping power L(E), the sum of the collision and

bremsstrahlung mean energy losses per unit pathlength, is used to

define the mean range r :

(15)

Defined as above, the mean range is the total pathlength* travelled

by an electron in the continuous-slowing-down approximation. The

collision energy loss formula ceases to be valid at energies comparable

to the binding energies of Che atomic electrons; so the zero lower

limit of the integral in Eq.(l5) was replaced by a cut-off energy of

about 1 keV, and a small estimated residual range was added to the

result. For initial energies E greater than about 10 keV, this

residual range is negligible. The calculations of stopping power and

range are similar to those in previous tabulations .

' Parameters

entering into the energy loss formula are given in Table 1, and

stopping powers and ranges are given in Table 2 for beryllium, mylar,

aluminum, and titanium.

3. Multiple-Scattering Distribution of Energy Losses

17 18 Williams and Landau developed multiple scattering theories

which give the distribution of the energy losses A resulting from

successive Inelastic collisions along a given pathlength s - the

so-called energy-loss straggling distribution. In both theories.

*Note that the mean range refers to pathlength and not to depth

of penetration, which is largely determined by multiple elastic

scattering detours.

7
the single scattering cross section of Eq.(l) was used,

where k is the energy transfer (not fractional energy transfer). The

cross section is assumed independent of A along s (p^ = p^) implying

the restriction A <^ E. The use of this cross section does not take

into account the binding of the atomic electrons, implying the further

restriction A » €'e. Both authors arrive at similar distributions,

but Landau's results - presented in a more convenient form - have

wider use . »

Landau allowed possible energy transfers in a single collision

to extend to Infinity. His result is a distribution that can be

expressed in terms of a universal function of a single scaled energy

variable,

(16)

where

and

The universal function of cr(X) is given by

(17)

(18)

(19)

19

and has been accurately evaluated and tabulated by Borsch-Supan.

According to his tabulation, cp(X) has a maximum at \ = -0.225 and a
full-width at ha If -maximum of 4.02 § . The same values of the most

probable \ and width of the distribution have been obtained by

20 21 Maccabee and Papworth. Rohrlich and Carlson, using the M(<ller

cross section, Eq.(2), for the energy transfer in single electronelectron collisions, found that the corrections for relativistic , spin

22

and exchange effects are practically negligible. Vavilov used a

finite maximum energy transfer in obtaining the energy-loss distribution of heavy charged particles. His result was a distribution in

terms of two variables. However, when applied to electrons traveling

short pathlengths, the Vavilov distribution reduces* to the simpler

Landau distribution.

23

Blunck and Leisegang, by including a higher order term in an

expansion used in solving Landau's transformed equation, derived a

method to take into account more details of atomic excitation. Their

result, in the form of a correction to the Landau distribution, is

given by

f (A.s) j

^U'.s) ex^[{Az4f] dA' - (20)

24

According to Blunck and Westphal, the variance of the Gaussian in

Eq. (20) is given by

= /£>il Z eV , (21)

where S is the mean collision energy loss (in eV) for pathlength

(A = Lc(E)s).

*Differences in the large energy-loss, essentially single-scattering, tall can be minimized by truncating the Landau distribution at

the appropriate maximum energy loss. The remaining differences have

little practical affect, since these large energy losses are very rare,

9
B. Angular Deflections

1. Single Elastic Scattering

25

The Rutherford cross section for the scattering of a nonrelativistic electron by a bare nucleus of charge Ze is

;

-- p'v^'-le)' • (22)

The screening of the nuclear charge by the orbital electrons can be

accounted for by replacing (l-cos9) with (1-cosQ + 2^) . From the

work of Moliere

7=;^(^'[ll3-3-7^(2^'j . (23)

where a is the Thomas -Fermi radius

2 1/3 (L- O.^SS JL^ 2. . (24)

The angular deflections due to inelastic scattering by the orbital

2 electrons can be approximately accounted for by replacing Z with

Z(Z + 1) in Eq.(22)

27

Mott derived the cross section for scattering by an unscreened

nuclear charge which displays explicitly the modifications to the nonrelativlstic Rutherford cross section by relativistlc effects and the

intrinsic spin of the electron^ The Mott cross section <^j^(^) is given

as a slowly converging Legendre series in 6. For small angles,

(e) ^ I I + E

^^^^ cos y ( I - cos

9)'^"*

] , (25)

10
where

Combining the Mott cross section with the screening correction

and the inelastic scattering correction leads to the final form for

the single scattering cross section for angular deflections

- '1-COS

p'lr-'d-cosO^aT^^^'l ^ *c (27)

where h(6) represents the corrections to the small angle approximation

to give <j(^) for arbitrary G and must be evaluated numerically* from

2. Mult iple -Scatter ing Distribution of Angular Deflections

28

A solution to this problem was given by Williams based on a

fitting together of a Gaussian describing the central small-angle

portion of the multiple scattering angular distribution and a single29 scattering tail. Goudsmit and Saunderson developed a theory which

is exact for any angle and which can be evaluated with any single

scattering law.

The Goudsmit-Saunderson distribution can be written

A^,(e,i)^ :E 2jp e"^ V/ccs©) , (28)

*In our calculations, values of h(&) were obtained using a FORTRAN computer code written by Dr. J. Coyne of the National Bureau

of Standards.

11
where

.+ ) = 2tt (ne)[\-?^(Cvze)^(iCcose) . (29)

The variation of (t(6) due to energy loss along the pathlength s

can be taken into account by replacing the exponent sG^ in Eq.(28) with

Gii(s')ds', Changing the variable from s' to the fractional

residual mean range

t = -S^ , (30)

the Integral becomes

: I

Qg(s')ds' = ^ J ^fCt)di . (31)

Spencer"^*^ has demonstrated that

QAt) ^ JLLL_ ^/i) > (32)

* tct+b) ^

so that

S

The constant b can be easily determined from Eq.(32) with knowledge

of the G^ ' s for the end-points of the pathlength s.

From the integral properties of Legendre polynomials it can

easily be seen that the mean multiple-scattering angular deflection

for pathlength s is given by

<cos e>^^=J 0OS6 A^s^O,s)dCcosQ) = exp|^-J ^i^sO^l^' j . (34)

12
<. 3 1 32

The work of Moliere and Snyder and Scott was done in the small angle

approximation; their work and modifications to it has been reviewed by

33

Scott. The relation between the small-angle, Moliere theory and that

34 3 5 of Goudsmit and Saunderson have been discussed by Lewis and Bethe,

who show that at small scattering angles the theory of Goudsmit and

Saunderson reduces to that of Moliere. In some applications the use

of Moliere 's angular distribution may be preferred because it is more

easily numerically evaluated than the distribution of Goudsmit and

Saunderson.

C. Coupling of One- Variable Multiple Scattering Theories

The multiple scattering theories mentioned above are solutions

to one-variable problems. The difference between pathlength and depth

of penetration is disregarded; energy loss is either neglected or

treated in the continuous-slowing-down approximation*; and the theories

are, strictly speaking, valid only for an unbounded medium^

The solutions of the one-variable problems can be patched together or overlaid to account for both energy loss straggling and the

effects of angular deflections. In order to extend Landau's theory

to describe the shape of the spectrum of electrons transmitted through

thicker foils, several author s"^^ have combined the Landau energyloss distribution with a simple foil- thickness correction derived from

41

the work of Yang, Yang obtained the distribution of actual pathlengths in a foil of a given thickness in two cases: for all particles

^In the continuous-slowing-down approximation, the energy loss per

unit pathlength at every point along the electron track is assumed to

be given by stopping power theory. Thus, the statistical fluctuation - straggling - in the energy loss is ignored.

13
regardless of angle; and only for those particles emerging at 0.°. His

solution is based on the use of the small angle Gaussian approximation

to multiple scattering, but can be improved by substituting in it a

mean value derived from a more exact theory, A more complete approach

42 43

has been used ' based on the folding of the Landau and the Yang

distributions. However, this method ignores the correlation between

pathlength and energy-loss straggling; it cannot account for the

attenuation of the incident beam due to complete absorption; and it

can only be applied for the two cases treated by Yang,

These objections can be surmounted by successive convolutions of

multiple scattering energy-loss and angular distributions over a

series of short path segments for which the one-variable theories are

valid„ Performing the convolution by random sampling is the basis of

Monte Carlo electron transport calculations. This approach has been

9 rather thoroughly outlined by Berger, and is distinguished by its

ability to account for many types of effects (e.g. the emission and

transport of secondary radiation) and to treat complex boundary con44 45-47 ditlons. The Monte Carlo calculations of Sidei et al, Berger,

48 49 50

Mar, Bishop and Berger et al, were done in the continuousslowing-down approximation, convoluting only successive multiple

42 scattering angular distributions; the work of Hebbard and Wilson,

51 52 53 54

Leiss et al, Schneider and Cormack, Perkins, l^issner, and

Berger and colleagues'*"^' include also the effects of energy-loss

straggling. A non-stochastic, numerical integration technique - the

63

phase-space time evolution method - was used by Cordaro and Zucker

to perform the convolutions. Their calculation was done in the

continuous-s lowing-down approximation, but in principle it can be

14
extended to Include energy-loss straggling,

D. Solutions of the Multi-Variable Transport Equation

Solutions of the multi-variable transport equation have been

obtained in some cases. The moments method developed by Spencer"^^

64

has been used by him to calculate the depth distribution of energy

deposition for a wide range of electron energies and absorber materials.

6 5

The coverage has been widened by the calculations of Adawi and

Kessaris^^ to include 10 to 20-MeV electrons in water. Kessaris^^ has

extended the moments method to obtain the spectra of electrons as a

function of depth inside the medium. However, the moments method

applies only to unbounded media, and calculations so far have employed

the cont inuous-slowing-down approximation.

6 8 Using a discreet ordinates method, Bartine et al have integrated

the transport equation with foil boundary conditions, but with only

moderate success. For low initial electron energies - below, say, 50 kev

it is feasible to accumulate the affects of successive individual inter69 actions using a Monte Carlo method. This has been done by Maehlum,

Stadsnes and Maehlum, Berger and Seltzer,^''" Wedde.'^^ McDonald et al,^"^

74 and Mclntyre, who treat single elastic collisions but assume energy

loss to be continuous. Individual inelastic collisions have been

included also in the work of Berger.

One should also mention the application of age diffusion theory

to electron transport problems. This approach has been taken in the

work of Be the et al,^^ Weymouth, Roesch,^^ Meister.'^^ Archard,

81 82 83 Tomlin and Kanaya and Okayama ; Cosslett and Thomas discuss its

application to problems involving electrons with energies below

15
30 keV. As pointed out by Spencer, age diffusion theory can offer

only a very approximate description of electron penetration.

In summary, for electron penetration problems in bounded media

for v^7hich both energy loss straggling and multiple scattering angular

deflections are important, the methods which offer the most accurate

solutions are the Monte Carlo, the discreet ordinates and the phasespace time evolution. Of these, the last two are more potential in

nature, work having just been started using them.

Ifi
III. METHOD OF CALCULATION

The path of an electron is represented by a series of short,

straight line path segments, and a Monte Carlo technique is used to

perform the segment-by-segment convolution of applicable multiple

scattering distributions. The procedures are incorporated in the

84

computer code ETRAN, the latest version of which has been used in

the present calculations. The efficacy of this method depends on two

considerations: the accuracy of the underlying theoretical distributions used, and the precision of the transport model.

A. Theoretical Distributions

The limits of validity for the distributions governing energy

loss in a path segment are determined primarily by the importance of

the effects of the binding of the atomic electrons. The stopping power

formula, Eq.(8), is not correct for electron energies not considerably

greater than atomic binding energies. The use of the Miller cross

section, Eq.(2), for governing the production of knock-on electrons

ignores binding completely. The Landau energy-loss distribution,

Eq.(19). loses its validity as the electron energy approaches atomic

binding energies. Including the Blunck-Leisegang resonance broadening

correction, Eq.(20), improves the Landau distribution at these lower

energies; however, the width of the Gaussian correction is only approximately known, so that at very low energies or for extremely short

pathlengths - for which the correction is large - the resultant distribution becomes unreliable.

A conservative estimate of the lower limit for the validity of

the energy-loss formulas is the K-shell electron binding energy for

17
the tnedium. It is true that for the K-shell electrons the binding

effects will be very important, but the K-shell electrons usually

represent two out of many, the remainder of which have much smaller

binding energies.

Since the Goudsmit-Saunderson distribution, Eq.(28). is exact, its

validity depends only on that of the single-scattering cross section,

26

Eq.(27). Moliere stated that his screening calculation is valid for

electron kinetic energies E ^ lOOz'^^"^ eV. Zeitler and Olsen^^ found*

that the error in the Moliere approximation is of the order of

4 / o 2 2 / Y = 2Z ^Ce"/hc) /(pp); thus an error of 57o or less implies a low

energy limit of E ^ 600 Z '3 eV, This restriction can be relaxed in

33

light of the fact, as shown by Scott, that the multiple scattering

distribution is rather insensitive to the exact value of the screening

angle. As regards the separate treatment of the screening and the

relativlstic and spin effects used in constructing the single-scatter85 ing cross section, Zeitler and Olsen pointed out that the error

Incurred in this factorization is also of the order of y. The cor2 rection for inelastic deflections, replacing Z by Z(Z +1), is somewhat approximate. It is correct for large angles, but fails within

the screening region - for angles near zero - where it underestimates

the effect. This is relatively unimportant, however, since these very

small angular deflections will have little effect on the multiple

scattering distribution. An additional factor for correcting for

*Zeitler and Olsen used a different potential than that (based

on the Fermi-Thomas model) used by Moliere, but they employed the same mathematical approximations.

18
inelastic deflections has been proposed by Fano . This factor

usually has little effect and has not been used in the present calculation.

The multiple' scattering angular distribution describes only the

statistical effects of many scatterings by individual atoms. No account

is taken of quantum mechanical interference (diffraction) effects.

Possible evidence of diffraction effects for electrons with energies

87 88 from 60 to 100 keV has been reported ' for thin metallic foils.

Based on the points brought out in the preceding paragraphs, and

various comparisons with experiments, the over-all low energy limit

of the calculation is estimated to be of the order of Z keV. In all

of the cases considered, here , the incident energy is larger than this

limit. In some cases the electron histories are continued to energies

below this limit, but by then the electrons are quite diffuse and

little error is expected in the final results.

B. Transport Model

1. Definition of Boundary Conditions and Quantities Calculated

Electrons are incident along the z-axis, normal to the parallelplane target (6^ = 0°), with kinetic energy E^. The target is assimied

to have infinite extent in the x and y- directions, and only the spatial

variable z is considered. The computer code is arranged so that

several targets of different thickness are treated simultaneously.

The basic quantity calculated is the current of electrons transmitted (or reflected) by a target, as a function of emergent energy

and angle. This quantity will be denoted as J(E,9,z) and has the

dimensions of energy ^ sr ^, normalized to one incident electron.

Other quantities, such as the distribution of energy deposition D(z)

19
(^n units of MeV/(g/cm^). noTMiized to one incident electron), are

also calculated but do not constitute the chief results of this report.

Various integrals over J(E,9,z) are useful quantities and are

defined as follows:

(35)

is the angular distribution of emergent electrons (in units of sr );

i2TT

( tUCcosQ) T(E,@,-r) (transmission)

o ^^^^

211 j atose) (reflection)

-I

is the energy spectrum of emergent electrons (in dimensions of energy )

_6o 1

is the number transmission coefficient;

% 0

R^(z>^f as [ decode) J(t,0.^) (38)

is the number reflection coefficient;

Tg(^) = - [ f

dCCose)T(E,a,z) (39)

is the energy transmission coefficient;

t o

o r* R^(z) = J. j

EflE

J

dcc^sd) J(E,0,2^ (40)

is the energy reflection coefficient;

20
$^(x> * 1 -T^tz) -Rg£T) (41)

is the energy absorption coefficient;

Eavf*) = ^o"^€^^V^H^^^ (42)

is the mean energy of the transmitted electrons; and

<cos©>^^ J— dccose) cose Ace,z) (43)

is the average deflection cosine of the transmitted electrons,

2. Division of Electron Tracks into Segments

The length of the straight-line segments, or steps, are chosen

so that the electron kinetic energy decreases by the factor 2 TM per

step, on the average. That is, the size of the k*"^ step is

where

Eve = 2 "^t^.^ . (45)

The logarithmic energy grid was chosen for two reasons. The

Landau distribution was derived under the assumption that the single

collision cross section does not change over the pathlength considered,

The energy dependence of the cross section Eq.(l') is in the multipli2 cative factor l/o2. With the choice of m = 8 in Eq.(45), p changes

P by less than 9% over any step*.

The logarithmic energy grid has the further advantage that the

mean multiple-scattering angular deflection per step changes little

*The energy dependent parameters defining the distribution are evaluated at (E^ ^ + ^k^

'^^ s^^P- so that the error in

p2 is, on the average, less than 5%.

21
froTTi step to step. In order to reduce the mean angular deflection per

step, the major steps defined by Eq.(44) are divided into _n equal

sub-steps. The effect of sub-step size is shown in Table 3 for a

200-keV electron beam incident on an aluminum foil. A decrease in

step size beyond n = 4 does not significantly change the results if

the statistical uncertainties of the entries in Table 3 are taken into

account. Because the computer time needed for the Monte Carlo calculation is roughly proportional to n, the smallest value for which the

final results have converged is used. For aluminum, using n = 4

results in a mean angular deflection per sub-step of about 20°, The

same, or smaller, mean deflection per sub-step leads to the choice of

n = 2 for beryllium and mylar and n = 6 for titanium.

3„ Generation of Electron Histories

The multiple scattering collision energy loss A for a major step*

is sampled from the Landau-Blunck-Le isegang distribution, Eq.(20).

The rate of energy loss is assumed to be A/s for purposes of estimating

the electron energy along the step.

The multiple scattering angular deflection for the sub-step is

sampled from the Goudsmit-Saunder son distribution, Eq.(28). The distribution is assumed to be the same for all sub-steps within a given step.

The sampled polar deflection, uu, is in a coordinate system whose polar

axis is along the direction of motion at the beginning of the sub-step.

The change in direction from 9 to 9

'

, measured with respect to the

*The Landau distribution is evaluated for the major step, rather

than the sub-step, so that A is not too small. As A approaches the

energies of the atomic electrons, the Landau distribution becomes

increasingly invalid, and the energy loss depends strongly on the

rather uncertain Blunck-Le isegang correction factor.

22
fixed z-axis, is obtained from the transformation

Cos COS d COS 6l) -V Sin 9 S.n 6J COS A(f > (^6)

where A''o is the azimuthal deflection in the 'j>system. Electron spinpolarization is not considered, so is assumed to be random^

The production of knock-on electrons, whose energies are above a

specified electron cut-off energy, is sampled according to the Miller

formulas. Eqs.(5) and (2), The production point is chosen randomly

along the sub-step; the direction of the primary electron is taken to

be f' or depending on whether the production point is in the first

or second half of the sub-step, respectively; and the direction of the

knock-on electron is determined by the conservation of energy and

momentum. Eq.(4)^ Because the energy loss and the angular deflection

have been accounted for, on the average, in the Landau and the

Goudsmit-Saunderson distributions, no change in the energy and direction of the primary is made as a result of a sampled knock-on production event. The energy, spatial, and directional coordinates of the

knock-on electron are stored and used to initialize a later electron

history.

The production of bremsstrahlung photons is sampled according to

14

the cross section package described elsewhere. The sampled photon

energy is subtracted from the energy of the electron, thus contributing

to energy-loss straggling. If the energy of the bremsstrahlung photon

is greater than a chosen photon cut-off energy, the electron history

is interrupted while the photon history is traced using conventional

89 sampling techniques. The photon production point and the direction

of the primary electron are chosen as in the case of knock-on electrons

23
The Jntrinslc polar angle of emission and a random relative azimuthal

angle are sampled; these are then combined with the direction of the

electron. The cross sections for photon interactions were taken from

90

the tabulation of Hubbell„ The coordinates of photo -, Compton and

pair electrons produced in the course of the photon history are saved

for later electron histories, if the energy of these is greater than

the electron cut-off energy. No angular deflection is applied to the

primary electron in a sampled bremss'irahlung photon production event;

that deflection is assumed to be included in the nuclear scattering

cross section, Eq.(27). used in the Goudsmit-Saunderson distribution.

Electrons usually cross a target boundary in the middle of a

sub-step. To determine the energy at crossing, the collision energy

loss is re-evaluated by sampling from the Landau-Blunck-Le isegang

distribution for the portion of the major step taken to the boundary.

The direction at crossing is determined by sampling a deflection from

an exponential approximation to the Goudsmit-Saunderson distribution

for the fraction of the sub-step to the boundary. Upon crossing, a

score is added to the appropriate energy and angular histogram bin for

the array J(E,9,z).

This process is repeated, step by step, until the desired number

of primary histories and resultant secondary histories have been

traced, A history is terminated if the electron (or photon) leaves

the target or if its energy falls below the cut-off value. Also, if

the residual mean range of an electron is smaller than the distance

to any boundary of interest, the history is no longer followed.

The approximations inherent in the transport model mainly involve

the neglect of certain correlations. The correlation of large

24
deflections with the production of either energetic bremss trahlung

photons or knock-on electrons is ignored. Similarly, the correlation

between large energy losses by the primary electron and the appearance

of energetic knock-on electrons is neglected. An additional approximation is involved in the production of a bremsstrahlung photon or

knock-on electron due to the lack of detailed knowledge of the energy

and direction of the primary electron within the step. The approximations could be eliminated in a single scattering Monte Carlo model,

but this would result in a large increase in the already considerable

amount of computer time necessary for the calculations. For example,

an ETRAN calculation of the histories initiated by 10,000 1-MeV electrons incident on a thick titanium target requires about 15 minutes on

an IBM 360/91.

As can be seen, the overall accuracy of the calculation depends

on the interplay of a number of complicated factors. The validity of

the calculat ional procedure is perhaps best judged through comparisons

with experiments.

25
IV. COMPARISON WITH EXPERIMENT

Numerovis comparisons of the results of ETRAN calculations with

those of experiments have appeared in the literature during the past

91 few years. Rester and Rainwater. for an incident energy E^ = 1 MeV,

and Berger and Seltzer, for energies of 1.0. 2.66 and 8.2 MeV,

compared measured and calculated distributions in energy and angle of

electrons transmitted through aluminum foils. Comparisons of energy

92 and angular distributions were also given by Lonergan et al for

beryllium, aluminum, and gold foils for incident energies of 4 and 8 MeV,

93

and by Rester and Derrickson for 1.0 and 2.5 MeV electrons incident

on aluminum, tin, and gold targets. ETRAN results were compared to

measured energy spectra of reflected electrons by Rester and Der94 rlckson for 1-MeV electrons incident on aluminum. Iron, tin and gold

targets. The agreement found in the above mentioned references was

generally good; in a few cases large statistical fluctuations in the

calculated results, due to a small sampling base, prevented conclusive

f Indlngs . -

.

Good agreement was found between measured and calculated thick

target bremsstrahlung spectra by Berger and Seltzer and Rester

95

et al. These two references provide comparisons for beryllium,

aluminum. Iron, tin, and gold for electrons incident with energies in

the range from about 0.1 to 2.8 MeV. At higher energies, 10 to 40 MeV,

14 Berger and Seltzer found good agreement between calculation and

measurement for tungsten targets. Although bremsstrahlung spectra are

not of importance in the present work, these spectra are sensitive to

the distribution In energy and angle of the primary electrons in the

target, and thus offer indirect evidence as to the accuracy of the

26
cal c u] a(. ion.

Distributions in depth of energy deposition have been presented

in Rerger et al''"'' for 5 to 5^ keV electrons incident on an air target;

96

in McLaughlin and Hussman for 0,1 to 3.0 MeV electrons incident on

97 polystyrene and aluminum targets; in Rosenstein et al for 2 MeV

98 electrons incident on a polystyrene target, and in Eisen et al for

?. MeV electron incident on aluminum, copper, and tin targets. Generally

good agreement between calculation and experiment was found in all

of these cases.

Another class of problems provides evidence of Che accuracy of

I hr calculations. The response of NaT detectors to high energy gamma

rays depends on the escape of. and radiation by, electrons produced in

the detector. Good agreement between calculated and experimental Nal

6 0

response functions was found in Berger and Seltzer for incident gamma

rays with energies up to 20 MeV. A more direct test of the electron

5 8

transport calculation is found in Berger et al, in which calculated

and experimental response functions for silicon detectors have been

compared for Incident electron energies of 0.25, 0.50. 0.75 and 1.00 MeV.

The agreement was found to be generally good. Certain discrepancies

occurring for energies of 0.25 and 0.50 MeV are possible evidence of

channelling effects, presumably caused by diffraction*.

There have been few measurements of electron transmission in

cases directly pertinent to the immediate problem: that of electrons

with energies of a few hundred kilovolts, and foils with atomic number

*The silicon detectors used in the experiment referred to were highly ordered crystals. Diffraction effects, if they indeed exist in

the cases mentioned, would probably be smaller in a less ordered

mater ial .

27
Z 30. The most comprehensive coverage has been for aluminum. In

Figure 1, the calculated number transmission coefficient T^(z) , is

compared to measured values for electrons incident on aluminum targets

with energies from 0.05 to 1.0 MeV. In order to greatly reduce the

dependence of the results on the incident energy E^, the number transmission is plotted not against the target thickness z, but rather the

ratio 7./r , where r is the mean range of the incident electrons (see

o o Table 2), This "scaling", in addition to being intuitively appealing,

99 1 00

has a strong theoretical basis ' and has often been used in the

presentation of electron penetration results. In this way. results for

electrons of slightly different energies can be compared in a meaningful way

The experimental values shown in Figure 1 were obtained by Miller

J „ J . I 101 102 , ,103 ,104 . and Hendricks, Miller, Agu et al , and Horikiri et al m

measurements using a Faraday cup that subtended 85% or more of 2Tr.

Failure to include the very large angle transmission events would cause

the reported T^^-values to be at most 1 or 2% lower than obtained in a

full 2Tr-measurement . The Faraday cup used by Dupouy et al ''"'^^ ' '^'^^

appears to subtend 80% of 2r( (see Figure 1 of Reference 105). Their

,-1.0 measured values were corrected bv multiplying by j A(t^ , z ) d( cosS ) U.O ^ 'V

I j A(6 , z ) d(cos9 ) , where A(ti,z) was obtained from the Monte Carlo

0.2

calculations. This correction amounted to no more than 37o in any case.

Note that Tj^(z) attains values larger than unity due to the transmission of knock-on electron set in motion by the primary electron.

Se 1 iger ''^'^^ used a 2Tr Geiger counter that recorded transmission "events"

in which the emergence of a primary electron and one or more associated

secondary electrons resulted in a single count. Calculated results

28
pertaining to such a situation are also given in Figure 1. Overall,

the agreement between the calculated and experimental data is good;

the differences are no larger than the differences among the various

sets of experimental points themselves.

The number transmission coefficient for beryllium foils is given

108

in Figure 2. The data of Agu et al fall well below those of

Dupouy et al'''^^ and the calculated values. The rather sketchy data of

Dupouy and colleagues manifest the same dependence on as that of the

calculated curves, and on an absolute basis the agreement is judged to

be reasonably good.

The only experimental data available for titanium foils are those

102

of Miller. For E =1.0 MeV. the number transmission coefficient,

o as a function of titanium foil thickness, is given in Figure 3. The

agreement between calculation and experiment is good. The calculated

curves in Figures 1, 2 and 3 were based on the analysis of 10,000 Monte

Carlo histories.

In order to test the validity of the calculated energy and angular

distributions of transmitted electrons, a case considered by Rester

93 and Derrickson has been re-examined*. Calculated and experimental

angular distributions of transmitted electrons are given in Figure 4,

energy spectra in Figure 5, for 1 MeV electrons incident on aluminum

foils. Rester and Derrickson estimate their experimental error to be

about 10 per cent; the statistical errors in the calculated results are

2 about 5 per cent. The discrepancy in the energy spectra for 0.10 g/cm ,

r *A comparison between experimental results and those obtained with

ETRAN was given by Rester and Derrickson. However, their calculations

were based on 15,000 to 20,000 Monte Carlo histories, whereas 50,000

are used here

29
according to a private communication from Dr. Rester, may be due to uncertainty in the numerical integration over angle of the experimental data.

On the basis of these considerations, the agreement between calculation

and experiment is satisfactory.

Spectra of electrons reflected from a thick aluminum foil are

given in Figure 6, for = 0,5 MeV. Calculated results agree well

109 with the measurements of Jakschik and Jungst.

Calculated and experimental distributions of energy deposition

as a function of depth, D(z) , are given in Figure 7 for electrons

incident on thick targets of beryllium, aluminum and copper. In the

experiments, the electrons passed first through a vacuum window before

striking the target. Huffman et al"'""'"'^ used a 0.994 mg/cm^ aluminized

111 2 mylar window; Aiginger and Gonauser, 3.8 mg/cm aluminum; Trump

112 2 113 2

et al, 17.2 mg/cm aluminum; Frantz , 15.5 mg/cm aluminum; and

108 2

Agu et al , 15 mg/cm beryllium. A procedure suggested by Spencer

was used to account for this initial penetration: If the window was of

the same material as the target, its thickness was simply added to the

depth in the target at which the measurement was taken. In the cases

in which the window and target materials differed, an equivalent thickness of target material, one which produces the same mean deflection

cosine as calculated from the Goudsmit-Saunderson distribution, was

added.

Given in Figure 7 is the dimensionless quantity r^ D(z)/E^, as

a function of z/r^. Plotted this way. the dependence of the distribution on E is minimized, and the area under the calculated histoo grams is Because the measurements were relative, the experimental

data have been normalized so as to give the area obtained in the

30
calculation. The agreement between calculation and experiment is

fairly good. In particular, the agreement displayed for aluminum

for incident energies of 100 and 400 keV indicates that diffraction

effects are not important in these cases, at least in the determination of the energy deposition distribution.

31
V, NEW RESULTS

Calculations were made for 100, 150, 200, 300 and 400 keV electron beams incident on beryllium, mylar, aluminum, and titanium foils.

The results are based on a sample of N = 20,000 Monte Carlo histories,

except in the cases of E^ = 200 keV for which N = 50,000. Table 4 gives

the number and energy coefficients T^, R^^, T^, and R^. As defined

in Chapter III, T^^ and R^ are the fraction of the Incident electrons

which are transmitted and reflected by the foil, respectively; T $

E A

and R^, when multiplied by E^, give the energy transmitted, absorbed

and reflected by the foil, respectively. The standard deviation of

the and R^ results is equal to [f(l-f)/N] upon substitution of

T., or R -values for f

N N

Table 5 gives the spectral characteristics of the transmitted

electrons. In addition to the mean energy of the transmitted electrons

E^^, the most probable energy E and the full-width at half maximum W

of the spectrimi* of transmitted electrons S(E,2) are also given. The

spectral width is presented in two ways: the width referred to the

incident beam energy, W/E^; and as a characteristic of the beam now

available for subsequent experimental use, W/E^.

The quantities of Tables 4 and 5 can be interpolated to other

energies and foil thicknesses by plotting them v£. z/r^. As an example,

this is done in Figures 8 and 9 for titanium, where it can be seen that

the residual dependence in Incident energy is difficult to separate

*In order to obtain smooth distributions from the histogram

representations, a least-squares f it of a cubic spline was made to the

histogram using a method devised by Powel 1 32
from statistical fluctuations. This apparent universality is partly

illusory, however, because some real residual dependence on incident

energy should remain, as can be seen, for example, in Figure 2.

Rather, the composite curves of Figures 8 and 9 represent the higher

energy results at small values of ^/r^ and lower energy results at

larger values of z/r^^ The curves are useful, however, because interpolation to values of E and z within the ranges listed in Tables 4 and

o

5 places one in the correct portion of the curve. Energy spectra of

transmitted electrons S(E,Z) are given in Figure 10, where the dependencies on foil thickness, incident energy and foil material are shown.

Cumulative angular distributions of transmitted electrons,

1 cosQ ). are given in Table 6. The fraction of incident

••' cos"

electrons transmitted in directions within a specified cone can be

obtained by multiplying the appropriate value from Table 6 by that for

T^, The angular distributions may also be scaled to facilitate interpolation with respect to E and z. In Figure 11, percentile cosines of

the cumulative angular distribution are plotted —vs. z /r for beryllium

o

'

mylar, aluminum, and titanium foils. As can be seen, beyond a certain

value of z /r the shape of the angular distributions no longer changes.

These equilibrium shapes are given in Figure 12 where they are compared

to that due to differential angular distributions proportional to cosQ

2 and to cos 9. It appears that the equilibrium angular distributions

can be closely represented by cos9 (a + b cos6) in agreement with the

76 theoretical results of Bethe et al. The approach to equilibrium, as

well as the dependence on foil material, for the differential angular

distribution A(6,z) can be seen in Figure 13.

33
To a large extent the energy and angular dependence of the transmitted current can be separated, and J(E,6,z) can be approximated by

the quantity S (E , z ) A(0 , z ) /T^^ (z) . This is shown in Figure 14 for

200 keV electrons incident on a 1-mil titanium foil^ The histograms

are J(E,^'.z) averaged over the angular intervals 0° ^ 6 ^ 10°.

30° ^ Q 40° and 60° < 9 <: 70°; the curves are the corresponding

approximate quantities obtained using the results found in Table 4 and

Figures 10 and 13. For many applications the approximation may be

sufficient. It is constructed so as to be correct on the average, but

It cannot account for all details. As can be seen in Figure 14, the

distribution J(E,9,z) peaks at lower energies, and is wider, for greater

angles of emergence. This is due to the fact that the electrons emerging at larger angles travel, on the average, a longer pathlength. For

those instances for which more detailed information is desired, tables

of J(E,^\z) in histogram form are given in the Thesis on file at the

University of Maryland.

34
VI. COMPARISON WITH SIMPLE THEORY

The Monte Carlo calculation incorporates the multiple scattering

distributions of Landau (for energy loss) and of Goudsmit and

Saunderson (for angular deflections) . It is of interest to determine

the conditions under which the use of these two theories by themselves,

without recourse to a Monte Carlo calculation, may suffice to describe

electron transmission through foils.

According to Landau's theory, the most probable energy of transmitted electrons can be written

]

• (47)

Eq.(47) is obtained from Eq<,(l7) using for the value of the most

probable >, >^ = -0.225. The pathlength s in the definition of ?

(see Eq,(]8)) is interpreted to be the foil thickness.

The predictions of Eq.(47) are compared to Monte Carlo results in

Figure 15. The Monte Carlo data are from the calculations used to

generate the silicon response functions which were verified experi58 mentally in a previously mentioned comparison. The Landau curves

have slopes that are proportional to terms which are essentially the

collision loss stopping power; plotted vs_. z/r^, the Landau values of

E /E are not independent of E . p o o The full-width at half-maximum of the Landau distribution is

W - 4.02 ^. The FWHM of the Blunck-Le isegang Gaussian correction is,

1/2

Wj^j =(8 ln2;a, where rr is given by Eq.(21), Assuming the peak region

of the Landau distribution is sufficiently Gaus sian- 1 ike , the width of

35
the convolution of the Landau and Blunck- Leisegang distributions is

2 2 1 /2

i W + W 1 . The most probable energy and the width of the energy

spectrum of transmitted electrons, obtained from these theories are

compared with the Monte Carlo results for titanium foils in Figure 16.

As in the case of silicon in Figure 15, the agreement is best for foil

thickness that are small fractions of the mean electron range - for

which the effects of multiple scattering angular deflections and energy

loss are unimportant.

The mean deflection cosine obtained from the Goudsmit- Saunderson

distribution is given by Eq.(34). The Goudsmit- Saunderson results

pertain to the flux of electrons in an infinite medium crossing a

small spherical probe in any direction, whereas the Monte Carlo results

are for the current of electrons crossing the plane exit face of the

foil in the forward direction only. The flux and current are related

by a factor of cos*^ , but for distributions which peak strongly at small

angles the differences between the two will be small. The GoudsmltSaunderson and Monte Carlo mean deflection cosines are given in

Figure 17. The agreement is good for small foil thicknesses for which

small angular deflections predominate.

36
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42
Table 1. Properties of foils considered in this paper.

Material Z/A I (eV)

P

3

(g/cm )

Mass Thickness

of 1-mil foil

(mg/cm )

Be 0 .4438 60 1.85 4.70

Mylar. (C^^HgO,)^ 0 . 5204 73 1.38 3.51

Al 0 .4818 163 2.70 6.86

Ti 0 .4593 247 4.50 11.43

Z/A : ratio of atomic number to atomic weight.

I : mean excitation energy, from Ref . 15.

p : density.

43
Table 2. Electron stopping power and range. Powers of ten are shown In

parentheses; e.g., 5.21(-3) means 5.21x10"-^.

a Stopping Power, LCE) Mean Range i^^(E)

E (MeV CTn2/g) (g/ctn^)

(MeV) Be Mylar Al Ti Be Mylar Al Ti

0.05 5.46 6.23 5.07 4.30 5.21(-3) 4.59(-3) 5.71(-3) 6.52(-3)

0.06 4.79 5.47 4.47 3.97 7.l7(-3) 6.3l(-3) 7.82(-3) 8.90(-3)

0.08 3.93 4.49 3.68 3.29 1.18(-2) 1.04(-2) 1.28(-2) 1.45(-2)

0,10 3.40 3.88 3.20 2.86 1.73(-2) 1.52(-2) 1 .86(-2) 2.10(-2)

0.15 2.67 3.05 2.54 2.28 3.42(-2) 2.99(-2) 3.64(-2) 4.09(-2)

0.20 2.30 2.64 2.20 1.98 5.45(-2) 4.77(-2) 5.77(-2) 6.46(-2)

0.30 1.94 2.22 1.86 1.69 1.02(-1) 8.94(-2) 1 .08(-l) 1.20(-1)

0.40 1.76 2.03 1.71 1.55 1.57(-1) 1.37(-1) 1.64(-1) 1.82(-1)

0.50 1 .66 1.92 1 .62 1.48 2.15(-1) 1.88(-1) 2,24(-l) 2.48(-l)

0.60 1.60 1.85 1.57 1.43 2.77(.l) 2.41(-1) 2.87(-l) 3a7(-l)

0.80 1.53 1.78 1.51 1.39 4.05(-l) 3.51(-1) 4.17(-1) 4.59(-l)

1.00 1 .50 1.75 1.49 1.37 5.37(.l) 4.65("1) 5.51(-1) 6.04(-l)

a/ c.s.d.a. range computed in the continuous-slowi ng-down approximation.

44
Table 3. Dependence of calculated electron transmission on the

step-size parameter n used in the Monte Carlo model.

The average fractional energy loss per step was 0»083/n.

Results are for 200-keV electrons incident on a 2-mil

(13.72 mg/cm^) aluminum foil, and are based on a sample

of 20,000 Monte Carlo histories. Quantities given are:

Tj^, number transmission coefficient; R^, number reflection

coefficient; Tg, energy transmission coefficient;

energy absorption coefficient; Rg, energy reflection

coefficient; E^^/E^, mean energy of transmitted electrons

as fraction of incident energy; (cos6) . mean deflection

cosine of transmitted electrons.

n T

N

E /E av o <cosO)

av

1 0.911 0.061 0.726 0.240 0.034 0.797 0,695

2 0.872 0.083 0.682 0.270 0.048 0.783 0.713

4 0.845 0.100 0.660 0.282 0.058 0.781 0.726

6 0,841 0.107 0.652 0.286 0.062 0.775 0.731

8 0.841 0.104 0.650 0.288 0.062 0.772 0.733

Uncertainty^ ±0.002 +0.002 ±0.004 ±0 . 004 ±0.002 ±0.005 ±0,004

a/ Estimated standard deviation; magnitude approximately the same for all

values of n.

45
Table 4. Transmission, reflection and absorption of electron beams

Incident on beryllivmi, mylar, aluminum and titanium foils.

Quantities given are: T^^, number transmission coefficient;

Rjj, number reflection coefficient; T^, energy transmission

coefficient; energy absorption coefficient; Rg, energy

reflection coefficient.

Beryl

1

lum

Foil thickness Incident Energy, keV

2

(mils) (mg/cm ) 100 1 50 200 300 400

T

N

2 9.40 0.730 0.980 0.992 1.005 1.005

a ; 14.10 : 0.205 (J .yuz n Q 7 Q u . y /o 1.003 1.006

.5 23.50 0.000 0.471 0.881 0.996 1.005

: 9.40 0.018 0.013 0.006 0.003 0.002

3 14.10 ; 0.018 (J .UL 1 n m o U . UiZ 0.005 0.003

5' 23.50 0.018 0.017 0.018 0.009 0„005

.2. . 9.40 : 0.407 0.782 0.876 0.938 0.959

3 14.10 0.068 0 . 606 0 . 795 0,902 0.936

5 23.50 0.000 0.197 0.573 0.822 0.889

2 9.40 0.584 0.211 0.120 0.061 0.040

3 . 14.10

' 0.923 0.386 0.199 0.095 0.063

5 23.50 0,991 0.795 0.419

\

0.174 0.109

2 9.40 0.009 0.007 0.004 0.001 0.001

3 14.10 0.009 0.008 0.006 0.003 0.001

5 23 . 50 0.009 0.008 0.008 0.004 0.002

46
Table 4. Continued.

Mylar

Foil thickness Incident Energy, keV

(mils)

2 (mg/cm ) 100 150 200 300 400

T

N

1/4 0,88 1.002 1.003 1.003 1.002 1.002

1/2 1.76 0,998 1 .003 1.004 1.004 1.004

1 3.51 0.970 1.000

i

1.002 1.004 1.005

1/4 0„88 0.001 0.000 0.000 0.000 0.000

1/2 1.76 0.005 0.001

f 0.001 0.001 0.001

1 3.51 0.023 0.006 0.003 0.001 0.001

1/4 0.88 0.965 0.982 0.988- 0.994 0.996

1/2 1.76 0.920 0.964 0.977 0.988 0.992

1 3.51 0.805 0.919 0.950 0.975 0.983

1/4 0.88 0.035 0.018 0.012 0.006 0.004

1/2 1.76 0.076 0.036 0.023 0.012 0,008

1 3.51 0.182 0.078 0.048 0.025 0.017

1/4 0.88 0.000 0.000 0.000 0,000 0,000

1/2 1.76 0.004 0.000 0.000 0.000 0,000

1 3,51 0.013 0.003 0.002 0.000 0.000

47
Aluminum

Foil thickness Incident Energy, keV

(mils) ^' m2: / cm ) 100 150 200 300 400

T

N

1/2 3.43 0.910 0.987 0,992 1.002 1.003

1i D . oD 0.602 W . 7 1 O U . 70 o 0.997 1.004

0.046 nU . Amou i 0.979 0.996

1/2 3.43 0.076 0.018 0.008 0.005 0.002

U , OD 0.124 u . u / u u . UJ i 0.010 0.005

0.125 n 1 1 R n 1 on 0.030 0.014

1/2 3.43 0.7 66 0,918 0.955 0.978 0.987

D . OO 0.393 n 7a A U . otsU 0.949 0.969

1 J . 72 0.017 0.384 0,660

$

'A

0.873 0.926

1 /2 3 .43 0.184 0.06 8 0, 040 0.020 0.013

i A Q A 0.536 n 1 QQ 0.044 0.029

' .; i J . / <i 0.911 U . J jU 0.106 0.065

1/2 3.43 0.050 0.014 0.005 0.002 0.000

1 6.86 0.045 0.022 n nn7 \j « \J\J Cj

2 13.72 0.072 0.066 0.058 0.021 0.009

48
Table 4. Continued.

Titanium

Foil thickness Incident Energy, K.C V

^TTl lis) v."ig/CTn ; 100 i jU 300 400

T

N

1/2 5.72 0.599 0.887 0.949 0,995 1.000

1 11,43 0.105 0.597 0.808 0,957 0,985

2 22.86 0.000 0,095 0.430 0.796 0,918

1/2 5.72 0.216 0.112 0,052 0.013 0.007

1 11.43 0.224 0.213

f 0.161 0,053 0.024

2 22.86 0.224 0.220 0.219 0.166 0,091

1/2 5.72 0.430 0,768 0,881 0.957 0.974

1 11.43 0.050 0.419 0,662 0.873 0.931

2 22.86 0,000 0,043 0,265

A

0.632 0,799

1/2 5.72 0.431 0.152 0,079 0.034 0,022

1 11.43 0.808 0.448 0.231 0.088 0,051

2 22.86 0.858 0.821 0.601 0.261 0,137

1/2 5.72 0.139 0.080 0,040 0.009 0.004

1 11.43 0.142 0.133 0,107 0.039 0.018

2 22.86 0.142 0.136 0.134 0.107 0.064

49
Table 5. Characteristics of the energy spectra of electrons transmitted

through beryllium, mylar, aluminum and titanium foils.

Quantities given are: E^^/E^, mean energy as fraction of

incident energy, E /E^, most probable energy as fraction of

incident energy; W^Eq, full-width at half -maximum as fraction

of Incident energy; W/E^, full-width at ha If -maximum as fraction of most probable energy.

' " ^

• Beryllium

Foil thickness

2

(mils) (nig/cm ) 100

Incident Energy,

150 200

keV

300 400

o- 0.557 0 7QQ

E /E

av o n R83 0.933 0.954

3' 14 10 0.332 n 672 0 81 3 0.900 0.930

5 23 . 50 0.418 0 . 650 0.825 0.885

0.642 n R4i

E /E

P o

V/ » 7 W O 0.952 0.969

0.362 n 74S 0.925 0.950

-5

. 23.50 0.479 0.713 0.865 0.911

9.40 0.186 0.078

W/E

o 0.040 0.021 0.014

14.10 0.284 0.123 0.073 0.033 0.020

- 23.50 0.259 0.147 0.062 0.036

2 _ , 9.40 0.290 0.093

W/E

P

0.045 0.022 0.014

3 * - - 14.10 0.783 0.166 0.086 0.036 0.021

5; . 23.50 0.541 0.206 0.071 0.040

50
Table 5. Continued.

Mylar

Foil thickness Incident Energy, keV

(mil s)

2 (mg/cm ) 100 150 200 300 400

E /E

av o

11/, U . oo 0.963 0.990 0.994

1/2 1 . 76 0.961 0.97-3 0 9 84 n 988

1 3.51 0.919 0.948

VSL // VEl P 0

0 971 n 979

1/4 0.88 0.976 0.987 0.992 0.996 0.997

1/2 1 .76 n Q45 0.974 0.983 0 9Q1 n 994

1 3.51 n R75 0.944 0.965

T T / U W/ E

o

V-* . -7 O J- n 9R7

1/4 0.88 0 OPT 0.012 0.007 0.005 0.003

1/2 1.76 0.019 0.013 nu . noR uu o

1 3.51 0.032 0.025

W/E

P

1/4 0.88 0.022 0.012 0.008 0.005 0.003

1/2 1.76 0,038 0.019 0.014 0.008 0.005

1 3.51 0.090 0.033 0.026 0.013 0,008

51
Table 5. Continued.

Aluminum

Foil thickness Incident Energy, keV

(mil s

(mg/cm ; 100 1 jO 200 300 400

E /E av o

1/2 3.43 0.842 0.930 0.963 0.976 0.984

1 6.86 0.653 0.835 0.909 0.952 0.965

•2 13.72 0.115 0.638 0.781

E /E

p o

0.892 0.929

1/2 3.43 0.901 0.954 0.971 0.985 0.990

'1

' 6.86 0.754 0.894 0.936 0.967 0.978

13.72 0.392 0.742 0.849

W/E

o

0.925 0.953

1/2 3.43 0.091 0.038 0.026 0.015 0.010

6.86 0.185 0.076 0.048 0.025 0.016

13.72 0.326 0.197 0.102

W/E

P

0.046 0.026

1/2 3.43 0.101 0.040 0.027 0.015 0.010

1 6.86 0.245 0.085 0.051 0.025 0.016

..2., 13.72 0.832 0.266 0.120 0.049 0.028

52
Table 5. Continued.

Ti tanium

Foil thickness Incident Energy, keV

(mils) (mg/cm ) 100 150 200 300 400

E /E

av o 1/2 5.72 0,718 0.866 0.928 0.961 0.974

1 11 .43 0.476 0.702 0. 819 0.912 0.945

2 22.86 0.458 0.617 0.794 0.870

1/2 5.72 0.821 0.920

E /E

P o 0.953 0.976 0.984

1 11.43 0.538 0.810 0.890 0.947 0.965

2 22.86 0.504 0.724

W/E

o

0.873 0.920

1/2 5.72 0.183 0.074 0.044 0.023 0.018

1 11 .43 0.370 0. 174 0.099 0.045 0.031

2 22.86 0.359 0.237 0.102 0.060

1/2 5.72 0.223 0.080

W/E

P

0.046 0.023 0.018

1 11.43 0.688 0.214 0.111 0.047 0.032

2 22.86 0.713 0.327 0.116 0.065

53
Table 6„ Cumulative angular distribution of electrons transmitted

through beryllium, mylar, aluminum and titanium foils. The

quantity given is the fraction of the transmitted electrons

that emerge In directions between 0° and Q.

Beryl 1 ium

Foil

(mil

thickness

2

s) (mg/cm ) 10°

Cone Angle, 6 15° 20° 30° 45° 60°

E =100 keV

0

2 9.40 0.012 0.050 0.111 0.189 0.378 0,659 0.863

3 14.10 0.013 0.051 0.111 0.182

E =150 keV

0

0.368 0.650 0.858

\

' 2- 9.40 0.024 0.086 0.185 0.301 0.539 0.802 0.934

„ 3 14.10 0.015 0.061 0.129 0.218 0.421 0.702 0.888

23.50 0.013 0.050 0.107 0.176

E =200 keV

0.360 0.645 0.853

^ '2- 9.40 0.034 0.130 0.269 0.420 0.693 0.912 0.977

3 14.10 0.025 0.093 0.196 0.317 0.558 0.820 0.942

23.50 •• 0.014 0.056 0.122 0.208

E =300 keV

0.404 0.691 0.880

2 9 . 40 0.074 0. 261 0 486 0 678 0. 896 0.978 0.993

3 14.10 0.050 0.185 0.365 0.546 0.806 0.949 0.986

23.50 0.027 0.104 0.218 0.354

E =400 keV

0

0.617 0.860 0.957

9 AO 0.135 0.411 0.663 0.826 0.947 0.985 0.994

14,10 0.071 0.255 0.481 0.675 0.897 0.979 0.994

5 23.50 0.040 0.152 0.317 0.486 0.756 0.937 0.985

54
Table 6. Continued.

Mylar

Foil thickness Cone Angle ,e (mils)

2 (mg/cm ) o

5 o 10 15° 20

- o 30o 45° o 60

E =100 keV

o 1/4 0.88 0.056 0.208 0.412 0.613 0.881 0.990 0.999

1/2 1.76 0.035 0.134 0.273 0.436 0.707 0.919 0.979

1 3.51 0.021 0.077 0.160 0.262 0.484 0.760 0.915

E =150 keV

o 1/4 0.88 0.113 0.395 0.678 0.866 0.986 0.997 0.999

1/2 1.76 0.064 0.222 0.430 0.630 0.893 0.991 0.998

1 3.51 0.040 0.143 0.290 0.447 0.717 0.920 0.980

E =200 keV

o 1/4 0.88 0.175 0.545 0.827 0.953 0.996 0.997 0.999

1/2 1.76 0.092 0.320 0.581 0.786 0.966 0.996 0.998

1 3.51 0.065 0.240 0.446 0.632 0.857 0.962 0.988

E =300 keV

o 1/4 0.88 0.327 0.793 0.970 0.995 0.997 0.998 0.999

1/2 1.76 0.179 0.542 0.828 0.955 0.995 0.996 0.998

1 3.51 0.096 0.325 0.586 0.789 0.968 0.996 0.998

E =400 keV

o 1/4 0.88 0.449 0.904 0.992 0.996 0.996 0.997 0.999

1/2 1 .76 0.257 0.692 0.931 0.987 0.996 0.997 0.998

1 3.51 0.137 0.447 0.733 0.902 0.991 0.996 0.998

55
Table 6. Continued.

Aluminum

Foil thickness

(mils) (mg/cm^) 5° 10°

Cone Angle

15°

e

' 20° 30° 45° 60°

E =100 keV

0

1/2 3.43 0.013 0.052 0.112 0. 191 0.381 0.654 0.855

1 6 .86 0.010 0.038 0.087 0. 152 0.317 0. 591 0. 816

2 13.72 0.011 0.041 0.084 0. 150 0.314 0.605 0.815

E =150 keV

o

: 1/2 3.43 0.022 0.088 0.186 0.305 0.560 0.826 0.943

6.86 0.013 0.052 0,112 0.190 0.374 0.645 0.849

13.72 0.011 0.039 0.090 0.158 0.320 0.593 0.821

E =200 keV

o 1/2 3.43 0.034 0.128 0.262 0.415 0.694 0.915 0.977

1 6 .86 0.018 0.071 0.152 0.255 0.480 0.757 0,909

2

• 13.72 0.011 0.044 0.097 0.165 0.339 0.615 0.834

E =300 keV

o 1/2 3.43 0.091 0.305 0.528 0.707 0.894 0.970 0,989

1 6.86 0.041 0.150 0.302 0.462 0.719 0.910 0.973

2 13.72 0.020 0.077 0. 164 0.267 0.487 0.749 0.907

E =400 keV

o 1/2 . 3.43 0.080 0.296 0.548 0.756 0.954 0.997 0.999

1 6.86 0.050 0.197 0.381 0.569 0.826 0.958 0.988

2 13.72 0.025 0.097 0.209 0.343 0.611 0.859 0.953

56
Table 6„ Continued.

Titanium

Foil thickness

2 (mils) (mg/cm ) 5° 10°

Cone Angle,

y

15° 20° 30° 45° 60°

E =100 keV

o 1/2 5.72 0.009 0.035 0.078 0.141 0.301 0.573 0.803

1 11.43 0.011 0.039 0.087

E =150 keV

o

0.145 0.314 0.583 0.821

1/2 5.72 0.012 0.047 0.103 0.172 0,346 0,624 0,831

1 11 .43 0.011 0.040 0.084 0.149 0.304 0.572 0.808

2 22.86 0.009 0.036 0.077

E =200 keV

0

0.143 0.309 0.572 0.802

1/2 5„72 0.016 0.063 0.136 0.226 0,427 0,699 0.880

1 11.43 0,010 0.041 0.088 0.155 0.319 0.588 0.814

2 22.86 0.010 0.038 0.084

E =300 keV

o

0.147 0.307 0,577 0.807

1/2 5.72 0.031 0.116 0.243 0.385 0.637 0,863 0.956

1 11.43 0.014 0.057 0,125 0.211 0.414 0,698 0,881

2 22.86 0.011 0,040 0,090

E =400 keV

o

0.155 0.322 0,588 0,810

1/2 5.72 0.042 0.155 0.311 0.481 0.751 0.936 0.983

1 11.43 0.022 0,089 0,188 0.302 0.539 0.797 0,929

2 22.86 0.012 0.048 0.102 0.180 0.364 0.634 0,841

57

0 0.2 0,4 0.6 0.8 1,0 0 02 0.4 0.6 0.8 1.0

Figure 1
O CD tl) ^ CJ o

c> 6 d

d

Figure 2
Figure 3
Figure 4
A9W '(z*3)S

Figure 5
Figure 6
2<

>

»

"1 l£> «1 d o m d

(TUJ 03 _ . i2o K (r o <o H li. <

Figure 7
0 0.2 0.4 0.6

z/r.

Figure 8

Figure 10
Figure 11
1.0 0.8 0.6 0.4 0.2 0

COS 6

Figure 12
Figure 13
E. keV '

I Figure 14 •

j
O CO iO St — o o o

0.1 o

UJ |LiJ

Figure 15
Figure 16
'Xb soo>

Figure 17

N _ ; I ! 4 A II. t v , •- 7 .)

U S. CI . OF COMM. BIBLIOGRAPHIC DATA

SHFET

1. IH IM.U A 1 l(i\l (IK Kl' PORT

NBSIR 74-457

2. ( -DV 't I C' S-'- loll

.\',>.

3. l-'.r. ,;.K!,l'-. A,-< , . ,u :, .V,..

5. T';|l>h. .,l!.Mi D.ilc

6 . Pi i f 1)1 1 n ( .)i ; . 1 n 1 7

.

h ! '' n ( n .1

1

4. Ii!l.l{ AM) SI' HI 11 LI ^ . . J- -^T ^ 'I'l. t- -c • 1 Transmission of Electrons ihrough Foils

7. AU i llOR(S) Stephen M. Seltzer

8. Pi rliTiiiiiii; OnMii. Krpoir N.-, NBSIR 74-^57

9. Pi;iu-oKML\(; oiu;am/,a 1 ION .\a.ml; anii ai^ori ss NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. i'roi' CI /T.i-sk/Work Uini 2400439

1 1 . Coir.i .u t Grant N'<i.

12. .Spiinsorinj; Or,i<a;i iz;n i cm N.imc .iml C^ompli^tc AJdri s,s (Street, City, Stutc, ZIP)

Office of Naval Research

Arlington, Virginia 22217

13. 1 ypc- oi KLi^ort .\: Lrriod

>. -ovc rra

14. .^'poii •- or ing Aj,',c-ncy f ooi

15. .SbTPI.l- .MF.NTARY NOTES

16. ABSl RAC 1 (A 200-\vord or 7e.si- (actual sun)m<iry of most si^iificant inforwotion. If Jocur.icii t includes j .<itf.nificanC biblio^,Taphy or Uteralurc survey, mention it here.) The transmission of electrons through foils has been studied by a Monte Carlo

method. Cases involving electrons \N?ith energies from 50 keV to 1 MeV normally

incident on beryllium, mylar, aluminum, and titanium foils are considered. Good

agreement with experimental results has been found for quantities such as the number

transmission, the energy and angular distribution of the emergent electrons, and the

spatial distribution of energy deposited in the foil.

A comprehensive set of results has been generated for 100, 150, 200, 300 and

400-keV electrons incident on beryllium, mylar, aluminum, and titanium foils that

are commonly used as vacuum windows in conjunction with low energy electron ac- celerators. Quantities given are the electron number and energy transmission and

reflection, the energy absorbed, and the energy and angular distribution of the

transmitted electrons. It is shown that much of the results can be presented in a scaled form which reduces the explicit dependence on, and facilitates the interpolation x-7ith respect to, the incident energy.

17. KP.Y WORDS (six to twelve entries; alphubelical order; capitalize only the first letter of the first key wonl ur.le^s a prop-, r name; separated by semicolons) Angular distribution; electrons; energy spectra; reflection;

transmission; transport calculation

18. AVAILABILITY [jc' Unl iiiiitcd 19. SHCURITV CLASS 21.' NO. OF PAC,'k<.

( LUIS KL:POiri')

83 u

[_

' For Olficial Oi.stribut luii. Do Noi Rcicaso ro NTIS UNCL ASSIFH J) r Order From Sup. of Doc, U.S. Go veriimc-nt Printing Office 20. Slit l!Ri 1 ^ CLASS 22. Price

Wa.shinpton, D.C. 2()i02, SD Cai. No. CI', ( 1 lils pac;f)

Or.'.fr !• rom National Technical Information Set vice (NTIS)

Spr:n.i;fiol.l, Vir,:;ini.i 221 "SI UNCLA.SSIMliD

LI sc Of.is<- 1^ c t CC''. ;Nr 74