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TRANSMISSION AND REFLECTION

OF ELECTRONS BY ALUMINUM FOILS

MARTIN J. BERGER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

\~e.cUv\icai ^Ttote 187

ISSUED APRIL 1, 1963

TRANSMISSION AND REFLECTION

OF ELECTRONS BY ALUMINUM FOILS

Martin J. Berger

This work was supported in part by the National

Aeronautics and Space Administration,

under Contract No. R-80.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 15 cents

Transmission and Reflection of Electrons by

Aluminum Foils

Martin J. Berger

Electron transmission and reflection coefficients for aluminum foils, pertaining both to

number and energy, are given for source energies

between 0.125 and 2.0 Mev, source obliquities

ranging from perpendicular to grazing incidence,

and foil thicknesses up to one half of the electron mean range. The results were obtained by a Monte Carlo calculation.

1. Introduction

In the course of an investigation of the Monte Carlo method in

application to the transport of fast charged particles, the author has

had occasion to compute reflection and transmission coefficients for

electrons incident on aluminum foils. These results were obtained in

preliminary model studies; they disregard energy-loss straggling, and

take into account only the deviation of the electron path from a straight line due to multiple Coulomb scattering.

For foils of small and intermediate thicknesses this approximation

is actually quite good. Nevertheless, on account of the preliminary

nature of the results, only some of them have been included for publication in [l]. It has been pointed out to the author that there is some interest in transmission and reflection data in connection with the

shielding of space-vehicles against electrons encountered in the Van

Allen belt. There is scarcity of such data, experimental or calculated,

and the coverage of the preliminary results is greater than that

available from other sources, and includes not only number - but also

energy - reflection and transmission coefficients, and their dependence

on direction of the incident electron beam as well as on its energy . Accordingly, these results are presented here, with the proviso that

they are subject to future revision. Further work of this type,

Figures in brackets indicate literature references at the end of

this paper.
including the effect of energy-loss straggling and the associated

bremsstrahlung emitted by the electrons, is currently in progress at

the National Bureau of Standards.

2. Calculation

A beam of monoenergetic electrons, with energy E , is assumed to

be incident in direction 6 on a plane-parallel foil of aluminum of

thickness z. The angle of incidence, G , is the angle between the

direction of incidence and the normal to the foil. Thus, the value

6 = 0° corresponds to perpendicular incidence, and the value O = 90

to the limiting case of grazing incidence. The azimuth of the direction of incidence does not enter into the calculation.

The following quantities are computed:

1. Number reflection coefficient,

number of electrons reflected

N number of electrons incident

2. Number transmission coefficient,

number of electrons transmitted

N

" number of electrons incident

3. Energy reflection coefficient,

r - energy reflected

E energy incident

4. Energy transmission coefficient,

_ energy transmitted

E energy incident
The calculation of these quantities is achieved by simulating the

actual physical multiple scattering process by an (artificially con- structed) random walk, each step of which takes into account the com- bined effect or many successive Coulomb scatterings. The transition

probabilities for this random walk are obtained from the Bethe theory

of electron stopping power, and the Molie"re theory of multiple scattering. A detailed description of the procedures used can be found in [l].'

3. Results

It is useful to present reflection and transmission coefficients as functions of the "reduced foil thickness" z/r , where r is the mean

range of an electron of energy E , i.e., the average rectified pathlength it would travel in an unbounded medium from the time it has

energy E

Q until it comes to a stop. The reason for the introduction of

the reduced thickness is a scaling law, suggested by theoretical con- siderations and verifiable empirically, which states that in first approximation, particularly for source energies E

Q below 1 Mev, transmission and reflection coefficients are nearly independent of E > provided the foil thickness is expressed in units of z/r . In other words,

the energy dependence of reflection and transmission is largely taken

into account by the energy dependence of r

Q

. This facilitates interpolation of the Monte Carlo results with respect to E .

Table 1 contains values of the mean range at the energies treated

in this report. Actually, two sets are shown: the first set does not

take into account energy loss by bremsstrahlung, and corresponds to the

actual input data used in the Monte Carlo calculation; the second set

includes the effects of bremsstrahlung, and should preferably be used

to convert actual to reduced foil thickness. Actually, there is a significant difference only at the highest source energy listed, 2 Mev.

Tables 2, 3, 4- and 5 contain reflection and transmission co- efficients for electron number and energy. These are raw results, as they come off the computer, and could be smoothed and somewhat improved

by cross-plotting them against source energy and obliquity, and against

foil thickness. Three significant figures are given, but the last is

not really significant and has been left in to facilitate cross-plotting

and comparisons.

Reflection coefficients are listed only for z/rQ ^ 0.3 because a further increase of foil thickness does not further increase the amount

of reflection. Transmission coefficients are listed only for

z/r ^ 0.5> because for much greater thicknesses the neglect of energy

* r In the terminology of I 1 !* the calculations presented here are based

on Model {i, PL(l6, 96), EC, AM, DLT}.
loss straggling would begin to have serious consequences. The transmission coefficients plotted against z/r have an approximately straight

slope between z/r - 0.3 and 0.5. A linear extrapolation of this portion

of the curve yields a lower limit for transmission, and the intercept,

at zero-transmission, is the so-called extrapolated electron range.

For each source energy, the results for all values of 6Q and z/r

were obtained in one run of n = 1000 Monte Carlo case histories. The

standard deviations of the number reflection and transmission co- efficients are

a(RN

) = ^

N

(l - R

N

)/n

a(T

N

) = J[Jl - T

N

)/n

The standard deviations of the energy transmission and reflection

coefficients have not been estimated directly. Experience with gammaray Monte Carlo problems indicates that their fractional standard

deviations (ratio of standard deviation to the coefficient itself) is

approximately 2-3 times greater than those for the number transmission

and reflection coefficients.

It can be seen from tables 2-5 that the coefficients, particularly

those for transmission, are rather independent of the value of E , at

least up to 1 Mev, as has been predicted. For many purposes it is

reasonable, therefore, to take coefficients averaged over a set of source energies, such as those given in table 6. The ratios Tg/Tj^ and

RE

/Rjg, also shown there, represent the ratios of the mean energy of the

transmitted (reflected) electrons to the incident energy.

Table 7 compares number transmission coefficients for perpendicular

incidence from this paper with results obtained by Perkins [2]. The

calculation of Perkins is based on the same general principles as the

present work, but differs in procedural details, and includes the effects of energy-loss straggling. The mean ranges assumed by Perkins

(1.20 g/cm^ at 2 Mev, 0.545 g/cm^ at 1 Mev) differ somewhat from those

used by us. In the comparison, his transmission curves, which he

plotted against z/r , were first converted to an absolute thickness

scale and then to a z/r scale using our values for r . For E = 2 Mev

the agreement is quite good, and for E

Q = 1 Mev it is fair. The transmission coefficients of Perkins are generally somewhat higher, which
at least in part can be ascribed to our disregard of energy-loss

straggling. Perkins gives data at 2 Mev which indicate that for

z/rQ ~ 0.5 the inclusion of energy-loss straggling may raise the transmission coefficient by about 3-4%. According to a private communication from him, he imposed a certain upper limit on the magnitude of the

multiple scattering deflections in his calculation, which, he estimates,

may have decreased his reflection coefficients from 10-20$. This of course is accompanied by a corresponding increase of his transmission

coefficient and contributes to the discrepancy between his and our results.

In table 8, a comparison is made with experimental transmission

data of Agu, Burdett and Matsukawa [3]> the agreement being quite close.

Further comparisons, as well as indications of the effect of the

systematic errors due to the assumed Monte Carlo model of the calculation, will be found in |~l]. Finally, it should be pointed out that the

error of the energy transmission and reflection coefficients, due to

the neglect of energy loss straggling, is likely to be larger than the

corresponding error for the number transmission and reflection co- efficients.

4-. References

[l] M. J. Berger, Monte Carlo Calculations of the Penetration and

Diffusion of Fast Charged Particles, to be published in "Methods

in Computational Physics", B. Alder, S. Fernbach and M. Rotenberg,

Vol. I, Academic Press, New York, N.Y. (1963).

[2] J. F. Perkins, Monte Carlo Calculation of Transport of Fast Electrons, Phys. Rev. 126 , l7bl (1962).

[3] B. N. C. Agu, T. Burdett and E. Matsukawa, Transmission of Electrons through Aluminum Foils, Proc. Phys. Soc. London, 71 , 201

(1958).
Table 1. Mean Range of Electrons in Aluminum

E

(Mev)

r , g/cm

Bremsstrahlung

included

Bremsstrahlung

not included

2.0

1.0

0.5

0.25

0.125

1.214

0.5501

0.2246

0.08165

0.02690

1.237

0.556

0.2258

0.08196

0.02706

Table 2. Number Transmission Coefficient, T

N

E e z/r

(Mev) (Degrees)

' 0.1 0.2 0.3 0.4 0.5

2.0 0.988 0.962 0.847 0.689 0.469

45 0.922 0.803 0.637 0.459 0.276

60 0.804 0.648 0.483 0.340 0.161

75 0.600 0.458 0.328 0.186 0.091

90 0.237 0.178 0.106 0.064 0.026

1.0 0.983 0.925 0.785 0.608 0.373

45 0.881 0.758 0.582 0.417 0.218

60 0.766 0.608 0.445 0.283 0.145

75 0.573 0.429 0.299 0.166 0.065

90 0.233 0.170 0.114 0.067 0.021

0.5 0.972 0.890 0.751 0.559 0.361

45 0.877 0.747 0.582 0.400 0.202

60 0.768 0.619 0.451 0.270 0.129

75 0.564 0.436 0.305 0.165 0.075

90 0.254 0.197 0.114 0.069 0.023

0.25 0.977 0.894 0.738 0.538 0.306

45 0.863 0.721 0.539 0.349 0.189

60 0.733 0.575 0.408 0.249 0.122

75 0.544 0.393 0.268 0.149 0.061

90 0.237 0.167 0.109 0.064 0.021

0.125 0.967 0.880 0.731 0.531 0.321

45 0.861 0.703 0.543 0.346 0.176

60 0.744 0.570 0.414 0.242 0.113

75 0.546 O.404 0.260 0.151 0.063

90 0.230

_ . ..

0.168 0.116 0.059 0.023
Table 3. Energy Transmission Coefficient, T

b

E(Mev)

e (Degrees)

z/r

0.1 0.2 0.3 0.4 0.5

2.0 0.889 0.749 0.546 0.351 0.183

45 0.780 0.556 0.360 0.205 0.094

60 0.638 0.412 0.245 0.133 0.051

75 0.437 0.260 0.148 0.067 0.025

90 0.165 0.096 0.048 0.022 0.008

1.0 0.892 0.727 0.515 0.321 0.154

45 0.757 0.538 0.343 0.193 0.080-

60 0.622 0.407 0.238 0.121 0.048

75 0.434 0.259 0.146 0.064 0.019

90 0.169 0.099 0.053 0.023 0.006

0.5 0.890 0.711 0.507 0.313 0.157

45 0.765 0.546 0.357 0.200 0.081

60 0.632 0.422 0.257 0.128 0.048

75 0.442 0.280 0.161 0.070 0.024

90 0.191 0.119 0.057 0.027 0.007

0.25 0.902 0.730 0.511 0.301 0.143

45 0.757 0.535 0.338 0.181 0.079

60 0.616 0.402 0.237 0.120 0.047

75 0.432 0.258 0.144 0.067 0.022

90 0.183 0.106 0.059 0.027 0.007

0.125 0.897 0.728 0.521 0.318 0.158

45 0.763 0.537 0.354 0.191 0.081

60 0.630 0.409 0.253 0.127 0.050

75 0.439 0.274 0.152 0.075 0.025

90 0.181 0.113 0.064 0.029 0.009
Table 4» Number Reflection Coefficient, R

N

E e z/r

(Mev) (Degrees) 0.3

0.1 0.2 or larger

2.0 0.008 0.015 0.023

45 0.051 0.078 0.083

60 0.130 0.159 0.161

75 0.293 0.313 0.314

90 0.711 0.719 0.720

1.0 0.017 0.061 0.090

45 0.118 0.198 0.220

60 0.231 0.318 0.330

15 0.423 0.489 0.494

90 0.766 0.792 0.796

0.5 0.028 0.087 0.108

45 0.122 0.209 0.229

60 0.230 0.304 0.316

75 0.430 0.485 0.494

90 0.745 0.774 0.782

0.25 0.023 0.069 0.097

45 0.133 0.216 0.237

60 0.264 0.349 0.360

75 0.451 0.523 0.531

90 0.763 0.796 0.798

0.125 0.033 0.093 0.121

45 0.135 0.228 0.235

60 0.253 0.339 0.347

75 0.448 0.514 0.518

90 0.765 0.785 0.788
Table 5. Energy Reflection Coefficient, R

E e z/r

(Mev) (Degrees) 0.3

0.1 0.2 or larger

2.0 0.008 0.015 0.023

45 0.051 0.078 0.083

60 0.130 0.159 0.162

75 0.293 0.312 0.314

90 0.711 0.719 0.721

1.0 0.013 0.035 0.044

45 0.083 0.118 0.124

60 0.168 0.204 0.208

75 0.337 0.364 0.365

90 0.714 0.724 0.725

0.5 0.021 0.050 0.057

45 0.091 0.132 0.1-38

60 0.174 0.209 0.212

75 0.354 0.378 0.380

90 0.702 0.714 0.715

0.25 0.018 0.043 0.053

45 0.099 0.140 0.147

60 0.204 0.245 0.249

75 0.381 0.413 0.415

90 0.720 0.735 0.735

0.125 0.026 0.061 0.072

45 0.105 0.154 0.157

60 0.201 0.254 0.248

75 0.380 0.414 O.416

90 0.725 0.733 0.734
Table 6. Average Transmission and Reflection Coefficients,

Obtained by Averaging over the Results for Source

Energies E

Q - 1.0, 0.5, 0.25 and 0.125 Mev

zAo

9o

(Degrees) % TE teAn RN Re RE/RN

0.1 0.975 0.895 0.92 0.025 0.018 0.72

45 0.871 0.760 0.87 0.127 0.095 0.75

60 0.753 0.625 0.83 0.245 0.187 0.76

75 0.557 0.436 0.78 0.43b 0.363 0.83

90 0.239 0.181 0.76 0.760 0.715 0.94

0.2 0.897 0.724 0.81 0.078 0.047 0.60

45 0.732 0.539 0.74 0.213 0.136 0.64

60 0.593 0.410 0.69 0.328 0.226 0.69

75 0.416 0.268 0.64 0.503 0.392 0.78

90 0.176 0.111 0.63 0.787 0.727 0.92

0.3 0.751 0.513 0.68 0.104 0.057 0.55

45 0.562 0.348 0.62 0.230 0.142 0.62

60 0.4-30 0.246 0.57 0.339 0.229 0.68

75 0.283 0.151 0.53 0.509 0.394 0.77

90 0.113 0.058 0.51 0.791 0.727 0.92

0.4- 0.559 0.313 0.56

4-5 0.378 0.191 0.51

60 0.261 0.124 0.47

75 0.158 0.069 0.44

90 0.065 0.027 0.42

0.5 0.34-0 0.153 0.45

4-5 0.196 0.080 0.41

60 0.127 0.048 0.38

75 0.066 0.023 0.35

90 0.022 0.007 0.32

10
Table 7. Number Transmission Coefficients, T»i, for

Perpendicular Incidence. Comparison with

the Results of J. F. Perkins, [2]

Eo

(Mev) zA This Report Perkins

2.0 0.2 0.962 ± 0.006 0.96

0.3 0.84-7 ± 0.015 0.86

0.4- 0.689 ± 0.015 0.68

0.5 0.4-69 ± 0.016 0.4-8

1.0 0.2 0.925 ± 0.008 0.94

0.3 0.785 ± 0.013 0.83

0.4 0.601 ± 0.015 0.64

0.5 0.373 ± 0.015 0.42

Table 8. Number Transmission Coefficients, T^, for Perpendicular

Incidence. Comparison with the Results of Agu, Burdett

and Matsukawa, [3]. The Experimental Results Represent

an Average for Various Source Energies between .25 and

.75 Mev. The Calculated Results are from Table 6.

Agu et al

z/r

o Experiment Calculation

0.1 0.97 0.975 ± 0.003

0.2 0.88 0.897 ± 0.005

0.3 0.74 0.751 ± 0.007

0.4 0.54 0.559 ± 0.008

0.5 0.34 0.340 ± 0.008

11

ft U. S. GOVERNMENT PRINTING OKF1CE : 1963 O - 678864

DEPARTMENT OF COMMERCE

Luther H. 11 >ry FIONAL BUREAU OF STANDARDS

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