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PUBLICATIONS

NBS

AlllOb MOflOAb

NBSIR 82-2572

Status of Electron Transport

Cross Sections

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

September 1 982

Prepared for: Office of Naval Research

Arlington, Virginia 22217

Space Science Data Center

NASA Goddard Space Flight Center

Greenbelt, Maryland 20771

Office of Health and Environmental Research

Department of Energy

jyj Washington, DC 20545

c.2

'»AT10*AL. BURKAU

OF »TAMTARJ»a LIBRARY SEP 2 0 1982

NBSIR 82-2572

STATUS OF ELECTRON TRANSPORT

CROSS SECTIONS

S. M. Seltzer and M. J. Berger

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

September 1 982

Prepared for: Office of Naval Research

Arlington, Virginia 22217

Space Science Data Center

NASA Goddard Space Flight Center

Greenbelt, Maryland 20771

Office of Health and Environmental Research

Department of Energy

Washington, DC 20545

(A Q*

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
u r ju

'

l. . ^

.. w >

.• - H
Status of Electron Transport Cross Sections

*+

Stephen M. Seltzer and Martin J. Berger

National Bureau of Standards

Washington, D.C. 20234

This report describes recent developments and improvements pertaining

to cross sections for electron-photon transport calculations. The topics

discussed include: (1) electron stopping power (mean excitation energies,

density-effect correction); (2) bremsstrahlung production by electrons

(radiative stopping power, spectrum of emitted photons); (3) elastic

scattering of electrons by atoms; (4) electron-impact ionization of atoms.

Key words: bremsstrahlung; cross sections; elastic scattering; electronimpact ionization; electrons; photons; stopping power;

transport.

Summary of a paper presented at the Annual Meeting of the American Nuclear Society,

June 7-11, 1982, Los Angeles, California.

TThis work was supported by the Office of Naval Research, by the National Space

Science Data Center at the NASA Goddard Space Flight Center, and by the Office of

Health and Environmental Research of the Department of Energy.

Status of Electron Transport Cross Sections

*+

Stephen M. Seltzer and Martin J. Berger

National Bureau of Standards

Washington, D.C. 20234

This report describes recent developments and improvements pertaining

to cross sections for electron-photon transport calculations. The topics

discussed include: (1) electron stopping power (mean excitation energies,

density-effect correction); (2) bremsstrahlung production by electrons

(radiative stopping power, spectrum of emitted photons); (3) elastic

scattering of electrons by atoms; (4) electron-impact ionization of atoms.

Key words: bremsstrahl ung ; cross sections; elastic scattering; electronimpact ionization; electrons; photons; stopping power;

transport.

Summary of a paper presented at the Annual Meeting of the American Nuclear Society,

June 7-11, 1982, Los Angeles, California.

+This work was supported by the Office of Naval Research, by the National Space

Science Data Center at the NASA Goddard Space Flight Center, and by the Office of

Health and Environmental Research of the Department of Energy.
1. INTRODUCTION

In this paper we highlight certain improvements to the cross-section data base

for various Monte Carlo computer codes such as ETRAN [1],"* EGS [2], SANDYL [3],

TIGER [4], ACCEPT [5], etc., which are used to calculate the transport of electrons

and associated bremsstrahlung at energies from ~ 10 keV up to 1 GeV. The discussion

will be in terms of the cross-section information needed by the ETRAN code [1], as supplied by the program DATAPAC [6], but is also applicable to the SANDYL, TIGER, and

ACCEPT codes which are decendants of ETRAN.

2. ELECTRON STOPPING POWER

2.1. Mean excitation energies

We have completed a critical review of the mean excitation energy I, the key

quantity in the Bethe theory of the collision stopping power. Based on current

information derived from stopping-power and range data, oscillator-strength distributions for gases, and dielectric-response functions for condensed media, I-values were selected for 43 elemental substances and 54 compounds. I-values for other

elements were obtained by interpolation with respect to atomic number (see table 1 and fig. 1). I-values for other compounds were obtained through the application of

the Bragg additivity rule, using I-values for atomic constituents modified to take

into account chemical binding and phase effects (see table 2). Further details can

be found in references [7-9].

2.2. Density-effect correction

The density-effect correction, the other non-trivial quantity entering into the

Bethe formula, was also re-examined. For this correction, DATAPAC uses the algorithm

of Sternheimer and Peierls [10], which is a universal fit to previously calculated

results. The use of this fit involves some sacrifice in accuracy, and in recent

work [9,11] we have gone back to the numerical evaluation of the density effect using

Sternheimer ' s [12] model in conjunction with updated values of mean excitation

energies and atomic binding energies. We have found that our numerical results for

the density effect are in good agreement with those calculated by Inokuti and Smith

[13] for aluminum and by Ashley [14] for water with semi -empirical dielectric-response

functions. The changes in the collision stopping power that result from using the new numerical results instead of the Sternheimer-Peierls algorithm are shown in

figure 2 for selected materials.

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

2
2.3. Conclusions

(i) The analysis of experimental data confirms the theoretical prediction [15]

that, due to atomic shell effects, I/Z is an irregular function of Z. (ii) The

Bragg additivity rule, with a simple assignment of I-values for atomic constituents

that takes into account chemical binding and phase effects, provides an accurate

representation of the I-values for over 50 compounds and is expected to be a reliable method for predicting I-values for other materials, (iii) The combined

effect of using the new I-values and density-effect corrections results in changes of

no more than 3-4%, more typically 1-2%, in the collision stopping power as compared

to our older values from DATAPAC.

3.1. Radiation stopping power

As discussed in reference [16], DATAPAC makes use of a combination of BetheHeitler Born-approximation formulas, with empirical corrections, at energies below

50 MeV, and above 50 MeV uses the exact high-energy theory of Davies, Bethe, Maximon,

and Olsen [17], which includes a Coulomb correction. A significant improvement in

the calculation of the cross section for bremsstrahlung in the field of the screened

nucleus was made by Pratt et at. [18], who solved the Dirac equation numerically for

a static, screened Coulomb potential, and evaluated the bremsstrahlung matrix

elements numerically from the wave functions. These authors give the cross section,

differential in emitted photon energy, for atomic numbers Z from 2 to 92 and for

electron kinetic energies from 1 keV to 2 MeV.

For the calculation of electron radiative stopping powers [9,19], we have

bridged the gap between 2 and 50 MeV by interpolating between the results of

Pratt et at. and the results from the high-energy theory. Two additional improvements

were also made: (a) the high-energy results were re-evaluated using screening

functions based on Hartree-Fock rather than Thomas-Fermi form factors; (b) the

contribution from bremsstrahlung produced in the field of the atomic electrons was evaluated on the basis of Haug's [20] cross section. The results can be conveniently

summarized in terms of the scaled radiative energy-loss functions

3. BREMSSTRAHLUNG PRODUCTION BY ELECTRONS

dk (electron-nucleus)

and

3
kZ

do

(e) (d,T)

4> (e)

rad

(T)

1 1

T+mc

2 dk (electron-electron) , dk

where a is the fine structure constant (1/137.03604), r

0

is the classical

electron radius (2.817938 x 10”^ 3 cm), me

2 is the electron rest mass (0.5110034 MeV),

Z is the atomic number, T is the electron kinetic energy, ^ 1S the bremsstrahlung

production cross section differential in emitted photon energy k, either in the

field of the atomic nucleus (denoted by superscript n) or in the field of the atomic

electrons (denoted by superscript e), and T' = mc^T/[T + 2nrc - (T(T + 2mc J)'1

] is the maximum possible energy of photons emitted in an electron-electron collision.

The radiative stopping power can then be written as

1

P = — ar

2

(T+mc

2

) A

e rad

i <4

ed(

T > z &> »

23 -1 where N is Avagadro's number (6.022045 x 10 mol ) and A is the atomic

a / \ weight of the material. Figure 3 gives results for from 1 keV to 10 GeV in

selected materials. The ratio ^a^^rad^ * Prevlous ^y assumed equal to unity

in DATAPAC, is shown in figure 4 for hydrogen, carbon, and gold. At high energies,

this ratio is greater than unity due to the differences in screening and Coulomb

effects between the electron-electron and the electron-nucleus systems. At low

energies, the ratio tends to vanish due to the lack of a dipole moment for the

electron-electron system.

3.2. Spectrum of emitted photons

A similar synthesis for the cross section differential in emitted photon energy

is in progress. A further refinement is included to account for the non-zero value

of the electron-nucleus cross section at the high-frequency limit (tip of the

bremsstrahlung spectrum, k = T). This is accomplished (a) by interpolating between

the tip values predicted by Jabbur and Pratt [21] for extremely high energies and

those given by Pratt et al. [18] for energies below 2 MeV, as illustrated in figure 5;

and (b) by applying a Coulomb correction, shown in figure 6, which combines the

results of Davies, Bethe, Maximon, and Olsen [17] with the Elwert [22] factor so as to insure that the cross section goes smoothly into the tip value. Figures 7a and b

give the cross sections at 50 MeV for carbon and gold, and are typical of the highenergy (T > 50 MeV) results; figures 7c and d, for carbon and gold at 50 keV,

illustrate the results for T < 2 MeV. The accuracy of the interpolated results in

4
the gap region from 2 to 50 MeV is confirmed by the comparisons, shown in figures 8a

and b, with the results available from the exploratory, exact numerical calculations

of Tseng and Pratt [23], for A£ and U at 5 and 10 MeV.

3.3. Conclusions

The use of the cross sections of Pratt et al ., together with interpolation

linking the low- and high-energy theories, results in a significant improvement over

the cross-section package and empirical correction factors used in DATAPAC. The

procedures outlined above will provide the basis for a planned comprehensive

tabulation of bremsstrahlung cross sections for electron energies greater than 2 MeV.

4. ELECTRON ELASTIC SCATTERING

4.1. Discussion of cross sections

The accurate evaluation of the elastic scattering cross section requires - in

principle - exact phase shift calculations for the solution of the Dirac equation for

the electron in a screened Coulomb potential. In practice, an approximation is - in

most cases - quite adequate in which the cross section is calculated as the product

of two factors: (a) the unscreened Mott [24] cross section which includes spin and

relativistic effects; and (b) a screening correction term.

We have considered two methods of calculating the screening factor. The first

method, introduced by Spencer [25] and used in DATAPAC, consists of using the factor

2 2

(1 - cose) /(I - cose + 2n) where n is an energy- and Z-dependent screening

parameter derived by Molifere [26] in a calculation for the Thomas-Fermi atom. The

second method consists of using the impulse approximation in which the screening

factor is taken to be [1 - F (q , Z ) ] , where F(q,Z) is a Hartree-Fock atomic form

factor for momentum transfer q and atomic number Z. At low energies the factorization of the elastic scattering cross section becomes

inaccurate (see, e.g. , Zeitler and Olsen [27]), and it becomes necessary to carry out

a full phase-shift analysis. The most comprehensive calculations of this kind are those of Riley et al. [28], who give results at 9 energies from 1 to 256 keV and for

selected elements from Z = 2 to 92, and who also give a convenient approximation for

Z = 1 - 94 which facilitates the incorporation of their cross sections into DATAPAC

(see also Haggmark et al. [29]).

We have investigated the validity of the factorization in terms of the transport

(or momentum transfer) cross section a-j = a(l - cose ) , where a is the total cross section (integrated over all angles) and cose is the mean cosine of the scattering

angle. As pointed out by Moli&re [26] and by Bethe [30], the transport cross section

5
has the dominant influence on the multiple elastic scattering process. In figure 9 we compare the transport cross sections of Riley et al. with those from the MolifcreMott approximation and those from the impulse approximation.

4.2. Conclusions

(i) Comparisons with the exact results of Riley et al. indicate that the

Moli£re-Mott approximation used in DATAPAC begins to break down, in terms of the

transport cross section, at electron energies below ~ 128 keV in Au, ~ 64 keV in

Cu, ~ 16 keV in A£, and ~ 4 keV in Be. (ii) To the extent that transport calculations depend only on the transport cross section, the data of Riley et al. and/or the

results from the Molifcre-Mott approximation provide sufficient data at all energies.

However, it would be desirable to extend the exact calculations to energies somewhat

higher than 256 keV for the high-Z elements, (iii) The incorporation of the exact

cross sections into the multiple scattering algorithm in DATAPAC could be greatly

simplified by retaining the Molifere-Mott approximation, but adjusting the screening

parameter n so as to obtain the correct value of the transport cross section a-j.

The accuracy of this approximation remains to be investigated.

5. ELECTRON-IMPACT IONIZATION

5.1. Discussion of cross sections

As an energy-loss mechanism, this process is included in the collision stopping

power. Knowledge of ionization cross section is needed when including in a transport

calculation the emission of x rays subsequent to ionizations of inner shells.

Numerous calculations of this cross section, done in various approximations, have

been reported. In DATAPAC, we have used the approximate formula of Kolbenstvedt [31]

for K-shell ionization, which was derived by the Weizsacker-Williams method. Recently,

the ab initio calculations of Scofield [32] have become available for the K and

L shells, for Z > 18, and for T > 50 keV. We have applied the Weizsacker-Williams

method for all shells, going beyond Kolbenstvedt through the use of more detailed

theoretical and experimental photoionization cross sections for soft collisions and

through the use of binary-encounter theory for hard collisions. In figures 10 and 11 we compare results from Scofield and from our Weizsacker-Williams calculations with

experimental data [33-51] for Au, Ni, and A&.

6
5.2. Conclusions

(i) The results obtained by our application of the Weizsctcker-Williams method

agree quite well with experiment and with Scofield's rigorous results at energies

above about 20 times the ionization threshold energy, (ii) At energies below ~ 20

times the ionization threshold energy, the Weizsacker-Williams results appear to be

more consistent with experimental data than Scofield's results. ( i i i

)

The WeizsackerWilliams method, which requires a modest computational effort, can be used to extend

coverage to any energy, atom, and shell of interest (for example, for Z < 18 not

covered by Scofield). It is a useful alternative to semi -empirical methods for

estimating the electron-impact ionization cross section, such as the formula of

Lotz [52].

REFERENCES

[1] M. J. Berger and S. M. Seltzer, National Bureau of Standards Report 9837 (1968);

see also Oak Ridge National Laboratory Publ . ORNL-RSIC Report CCC-107 (1968).

[2] R. L. Ford and W. R. Nelson, Stanford Linear Accelerator Center Report SLAC-210,

UC-32 (1978).

[3] H. M. Colbert, Sandia Laboratories Report SLL-74-0012 (1973).

[4] J. A. Halbleib and W. H. Vandevender, Sandia Laboratories Report SLA-73-1 026

(1974).

[5] J. A. Halbleib, Sandia Laboratories Report SAND79-0415 (1979).

[6] M. J. Berger and S. M. Seltzer, National Bureau of Standards Report 9836 (1968);

also Oak Ridge National Laboratory Publ. ORNL-RSIC Report CCC-107 (1968).

[7] S. M. Seltzer and M. J. Berger, Inti. J. of Applied Radiation and Isotopes, in

press.

[8] M. J. Berger and S. M. Seltzer, Proc. of the Seminar on Charge States and

Dynamic Screening of Swift Ions in Solids, Honolulu, Hawaii, Jan. 25-29 (1982),

to be published.

[9] M. J. Berger and S. M. Seltzer, National Bureau of Standards Report

NBSIR 82-2550 (1982).

[10] R. M. Sternheimer and R. F. Peierls, Phys. Rev. B 3_ , 3681 (1971 ).

[11] R. M. Sternheimer, S. M. Seltzer, and M. J. Berger, submitted to Phys. Rev.

[12] R. M. Sternheimer, Phys. Rev. 88, 851 (1952).

7
[13] M. Inokuti and D. Y. Smith, Phys. Rev. B 25, 61 (1982).

[14] J. C. Ashley, Rad. Res. 89, 32 (1982).

[15] W. K. Chu and D. Powers, Phys. Lett. 40A, 23 (1972).

[16] M. J. Berger and S. M. Seltzer, Phys. Rev. C 2 , 621 (1970).

[17] H. Davies, H. A. Bethe and L. C. Maximon, Phys. Rev. 9^, 788 (1954);

H. Olsen, Phys. Rev. 99, 1335 (1955).

[18] R. H. Pratt, H. K. Tseng, C. M. Lee, L. Kissel, C. MacCallum, and M. Riley,

Atomic Data and Nuclear Data Tables 20^, 175 (1977); errata in 26, 477 (1981 ); see also revised data in L. Kissel, C. MacCallum, and R. H. Pratt,~5andia National

Laboratories Report SAND81 -1 337 (1981).

[19] S. M. Seltzer and M. J. Berger, Inti. J. of Applied Radiation and Isotopes, In

press.

[20] E. Haug, Z. f. Naturforsch. 30a , 1099 (1975).

[21] R. J. Jabbur and R. H. Pratt, Phys. Rev. 129^, 184 (1963); Phys. Rev. 1_32» B1090

(1964).

[22] G. Elwert, Ann. Physik. 34, 178 (1939).

[23] H. K. Tseng and R. H. Pratt, Phys. Rev. A 19^, 1525 (1979).

[24] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford

University Press, London), 2nd edition (1949J.

[25] L. V. Spencer, National Bureau of Standards Monograph 1 (1959).

[26] G. Molifere, Z. Naturforsch. 3a_, 78 (1948).

[27] E. Zeitler and H. Olsen, Phys. Rev. 136 , A1546 (1964).

[28] M. E. Riley, C. J. MacCallum, and F. Biggs, Atomic Data and Nuclear Data Tables

1 5., 443 (1975).

[29] L. G. Haggmark, C. J. McCallum, and M. E. Riley, Trans. Am. Nucl. Soc. T9, 471

(1974).

[30] H. A. Bethe, Phys. Rev. 89, 1256 (1953).

[31] H. Kolbenstvedt, J. Appl . Phys. 38, 4785 (1967).

[32] J. H. Scofield, Phys. Rev. A 18, 963 (1978).

[33] L. M. Middleman, R. L. Ford, and R. Hofstader, Phys. Rev. A 2 , 1429 (1970).

[34] K. Ishii, M. Kamiya, K. Sera, S. Morita, H. Tawara, M. Oyamada, and T. C. Chu,

Phys. Rev. A 15., 906 (1977).

[35] H. H. Hoffmann, H. Genz, W. Low, and A. Richter, Phys. Lett. 65A , 304 (1978).

[36] G. R. Dangerfield and B. M. Spicer, J. Phys. B 8, 1744 (1975).

8
[37] K. H. Berkner, S. N. Kaplan, and R. V. Pyle, Bull. Am. Phys. Soc. 15^, 786 (1970).

[38] D. H. Rester and W. E. Dance, Phys. Rev. 152 , 1 (1966).

[39] H. Hansen and A. Flammersfeld, Nucl . Phys. 79_

, 135 (1966).

[40] J. W. Motz and R. C. Placious, Phys. Rev. 136 , A662 (1964).

[41] D. V. Davis, V. D. Mistry, and C. A. Quarles, Phys. Lett. 35A, 169 (1972).

[42] S. I. Salem and L. D. Moreland, Phys. Lett. 37A, 161 (1971).

[43] M. Green, Ph.D. Dissertation, Univ. of Cambridge, Cambridge, England (1962);

data obtained from reference 42.

[44] V. K. Park, M. T. Smith, and W. Scholz, Phys. Rev. A 1_2, 1358 (1975).

[45] W. Hink and A. Ziegler, Z. Physik 226, 222 (1969).

[46] H. Genz, C. Brendel , P. Eschwey, U. Kuhn, W. Low, A. Richter, P. Seserko, and

R. Sauerwein, Institute fur Kernphysik, Technische Hochschule Darmstadt,

Report IKDA 81/16 (August, 1981).

[47] A. Li-Scholz, R. Col 1 ^ , I. L. Preiss, and W. Scholz, Phys. Rev. A 7_, 1957 (1973).

[48] S. A. H. Seif el Nasr, D. BerSnyi, and Gy. Bibok, Z. Physik 267_, 169 (1974).

[49] L. T. Pockman, D. L. Webster, P. Kirkpatrick, and K. Harwort, Phys. Rev. 71_,

330 (1947).

[50] A. E. Smick and P. Kirkpatrick, Phys. Rev. 67_, 153 (1945).

[51] J. Jessenberger and W. Hink, Z. Physik A 275 , 331 (1975).

[52] W. Lotz, Z. Physik 232^, 101 (1970).

[53] N. E. Holden, Pure and Appl . Chem. 5J_, 405 (1979).

[54] J. F. Ziegler, Handbook of Stopping Cross Sections for Energetic Ions in all

Elements , Vol . 5, Pergamon Press, New York (1980).

[55] J. H. Hubbell, W. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and

R. J. Howerton, J. Phys. Chem. Ref. Data 4_, 471 (1975); errata in 6_, 615 (1977).

[56] J. H. Hubbell and I. J. 0verbd, J. Phys. Chem. Ref. Data 8, 69 (1979).

[57] J. A. Wheeler and W. E. Lamb, Phys. Rev. 55, 858 (1939); errata in 101, 1836

(1956).

9
Table 1 . Values of the

otherwise, themean excitation

values are for

energy I for

the substance

the elements. Unless noted

in the condensed phase.

Z Element Symbol A, a g/mol ]l, b

eV

1 hydrogen H 1.0079 19.2

21.8

± 0.4 molecular gas

± 1 .6 liquid

2 helium He 4.00260 41.8 ± 0.8 gas

3 1 ithium Li 6.941 40 ± 5

4 beryllium Be 9.01218 63.7 ± 3.0

5 boron B 10.81 76.0 ± 8.0

6 carbon C 12.011 78.0 ± 7.0 graphite

7 nitrogen N 14.0067 82.0 ± 2.0 molecular gas

8 oxygen 0 15.9994 95.0 ± 2.0 molecular gas

9 fl uorine F 18.998403 (115 ± 10) gas

10 neon Ne 20.179 137 ± 4 gas

11 sodium Na 22.98977 (149 ± 10)

12 magnesium Mg 24.305 (156 ± 10)

13 aluminum Ail 26.98154 166 ± 2

14 silicon Si 28.0855 173 ± 3

15 phosphorus P 30.97376 (173 ± 15)

16 sulfur S 32.06 (180 ± 15)

17 chlorine Cl 35.453 (174 ± 15) gas

18 argon Ar 39.948 188 ± 10 gas

19 potassium K 39.0983 (190 ± 15)

20 calcium Ca 40.08 191 ± 8

21 scandium Sc 44.9559 216 ± 8

22 titanium Ti 47.88 233 ± 5

23 vanadium V 50.9415 245 ± 7

24 chromium Cr 51.996 257 ± 10

25 manganese Mn 54.9380 272 ± 10

26 iron Fe 55.847 286 ± 9

27 cobalt Co 58.9332 297 ± 9

28 nickel Ni 58.69 311 ± 10

29 copper Cu 63.546 322 ± 10

30 zinc Zn 65.38 330 ± 10

31 gal 1 ium Ga 69.72 (334 ± 20)

32 germanium Ge 72.59 350 ± 11 33 arsenic As 74.9216 (347 ± 25)

10
34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

Element Symbol k,a

g/mol \,

b

eV

selenium Se 78.96 (348 + 30)

bromine Br 79.904 (343

(357

+

+ 30) gas

30) condensed

krypton Kr 83.80 352 + 25 gas

rubidium Rb 85.4678 (363 + 30)

strontium Sr 87.62 (366 + 30)

yttrium Y 88.9059 (379 + 30)

zirconium Zr 91.22 393 + 15

niobi urn Nb 92.9064 417 + 15

molybdenum Mo 95.94 424 + 15

technetium 98Tc 97.907 (428 + 35)

ruthenium Ru 101.07 (441 + 35)

rhodium Rh 102.9055 449 + 20

palladium Pd 106.42 470 + 20

si 1 ver Ag 107.868 470 + 10

cadmium Cd 112.41 469 + 20

i ndium In 114.82 488 + 20

tin Sn 118.69 488 + 15

antimony Sb 121.75 (487 + 40)

tellurium Te 127.60 (485 + 40)

iodine I 126.9045 (474

(491

+

+ 40) gas

40) condensed

xenon Xe 131.29 482 + 30 gas

cesium Cs 132.9054 (488 + 40)

barium Ba 137.33 (491 + 40)

lanthanum La 138.9055 (501 + 40)

cerium Ce 140.12 (523 + 40)

praseodymium Pr 140.9077 (535 + 45)

neodymium Nd 144.24 (546 + 45)

promethium 145

Pm 144.913 (560 + 45)

samarium Sm 150.36 (574 + 45)

europium Eu 151.96 (580 + 45)

gadolinium Gd 157.25 591 + 50

terbium Tb 158.9254 (614 + 55)

dysprosium Dy 162.50 (628 + 55)

holmium Ho 164.9304 (650 + 60)

erbium Er 167.26 (658 + 60)

thulium Tm 168.9342 (674 + 60)

11
z Element Symbol A,a

q/mol l

f

l?

«2V

70 ytterbium Yb 173.04 (684 + 65)

71 1 utetium Lu 174.967 (694 + 65)

72 hafnium Hf 178.49 (705 + 65)

73 tantalum Ta 180.9479 718 + 30

74 tungsten W 183.85 727 + 30

75 rhenium Re 186.207 (736 + 70)

76 osmium 0s 190.2 (746 + 70)

77 iridium Ir 192.22 757 + 30

78 platinum Pt 195.08 790 + 30

79 gold Au 196.9665 790 + 30

80 mercury Hg 200.59 (800 + 75)

81 thallium U 204.383 (810 + 75)

82 lead Pb 207.2 823 + 30

83 bismuth Bi 208.9804 (823 + 80)

84 polonium 209

Po 208.982 (830 + 80)

85 astatine ^At 209.987 (825 + 80)

86 radon 222

Rn 222.018 (794 + 80) gas

87 francium 223

Fr 223.020 (827 + 80)

88 radium Ra 226.0254 (826 + 80)

89 actinium Ac 227.0278 (841 + 80)

90 thorium Th 232.0381 (847 + 80)

91 protactinium Pa 231.0359 (878 + 80)

92 uranium U 238.0289 890 + 30

93 neptunium Np 237.0482 (902 + 80)

94 plutonium 239

Pu 239.052 (921 + 85)

95 americium 243

Am 243.061 (934 + 85)

96 curium 247Cm 247.070 (939 + 85)

97 berkel ium

277

Bk 247.070 (952 + 85)

98 californium 251

Cf 251.080 (966 + 90)

99 einsteinium 252Es 252.083 (980 + 90)

100 fermium 257Fm 257.095 (994 + 90)

aThe atomic weights A are those recommended by the Commission on Atomic

Weights of the International Union of Pure and Applied Chemistry’ (Holden [53])

The values are for naturally occuring

isotope is indicated.

isotopic mixtures, unless a particular

^Values in parentheses are estimated by interpolation of l/l vs . z,» or by

extrapolation for Z > 92.

12
Table 2. Mean excitation energies adopted in the present work for atomic

constituents of compounds.

GASES

Constituent I(eV)

H 19.2

C 70

N 82

0 97

LIQUIDS AND SOLIDS

Constituent I(eV)

H 19.2

C 81

N 82

0 106

F 112

a iso

Others 1.13 x I, where I is

the I-value for the element

in the condensed phase given

in Table 1

.

13
ATOMIC NUMBER Z

Fig. 1. Ratio of the mean excitation energy I to the atomic number Z for

elements. Points indicated as experimental are the result of a critical

analysis of experimental data; interpolated results are based in part on the theoretical results of Chu and Powers [15] for gases or of Ziegler [54]

for solids.

14
Fig. 2. Percent change in collision stopping power (^j resulting from the use of the density-effect correction from a new, more accurate, numerical

evaluation rather than that calculated according to the algorithm of

Sternheimer and Peierls [10]. The solid curves, given as a function of

electron kinetic energy T, are from the present work evaluated according

to Sternheimer 1 s [12] model; the broken curves were derived from experimental information on the dielectric-response function by Inokuti and

Smith [13] for aluminum and by Ashley [14] for liquid water. The sharp corners in the curves are due to the approximate cut-off procedure in the

Sternheimer-Peierls algorithm.

15
Fig. 3. Radiative energy-loss function <J>^ for bremsstrahlung in the field of

the atomic nucleus. Points below 2 MeV are from the calculations of

Pratt et al. [18]; points above 50 MeV were calculated using Bethe-Heitler

theory with Hartree-Fock form-factor [55-56] screening corrections and the

Coulomb correction of Davies, Bethe, Maximon, and Olsen [17]. Curves are from a least-squares fit to the theoretical points.

16
1.2

Fig. 4. Ratio of the radiative energy-loss functions and which

represent the mean energy losses resulting from bremsstrahlung emission in

the field of the atomic electrons, and the field of the atomic nucleus,

(e) respectively. The results for

<|>J; ^ are based on Haug's [20] theory of

electron-electron bremsstrahlung with screening corrections evaluated

according to Wheeler and Lamb [57] using Hartree-Fock incoherent scattering

functions [55]. The total radiative stopping power is proportional to

z2 *rad + Z »rad •

17
kd<r

Fig. 5. High-frequency limit (tip) of the electron-nucleus differential bremsstrah2 2 lung cross section. The scaled tip value, ($ /Z )k(da/dk) for k = T, is

plotted as a function of the variable /l-g 2 /g

2

, where 6 is the incident

electron velocity divided by the speed of light. The corresponding initial

electron kinetic energy is given on the upper scale. The squares are the

results of the theory of Jabbur and Pratt [21] for 8=1; the circles

below 2 MeV are from the data of Pratt et al. [18]. The curves, nearly

straight lines when plotted in terms of these variables, are least-squares

fits to the theoretical points.

18
Fig. 6. Contribution of the Coulomb correction to the high-energy electron-nucleus

bremsstrahl ung cross section differential in emitted photon energy. The

percent contribution to the final cross section is given as a function of

the final kinetic energy T

2 of the electron after the emission of a bremsstrahl ung photon. The results are nearly the same for all initial

electron kinetic energies > 50 MeV.

19
Fig. 7. Bremsstrahlung production cross section, differential in emitted photon energy. Shown is the scaled cross section B

2 k(do/dk) va. the ratio k/T of the emitted photon energy to the initial electron kinetic energy. The solid curve is the electron-nucleus cross section

divided by Z

2

, and the dashed curve is the electron-electron cross section divided by Z, where Z is the atomic number of the target.

a. Carbon (Z = 6), T = 50 MeV. For T = 50 MeV, the electron-nucleus results are from a synthesis of Bethe-Heitler

theory, screening corrections evaluated in the high-energy approximation using HartreeFock atomic form factors [55,56], and a Coulomb correction combining the theoretical

results of Davies, Bethe, Maximon, and Olsen [17] and of Elwert [22], modified to go smoothly into the high-frequency limit given by the theory of Jabbur and Pratt [21].

The electron-electron results are from the theory of Haug [20], with screening

corrections evaluated in the high-energy approximation using Hartree-Fock incoherent

scattering factors [55].

20
Fig. 7. Bremsstrahl ung production cross section, differential in emitted photon energy. Shown is

the scaled cross section B

2 k(da/dk) vs. the ratio k/T of the emitted photon energy to the initial electron kinetic energy. The solid curve is the electron-nucleus cross section

divided by Z

2

, and the dashed curve is the electron-electron cross section divided by Z,

where Z is the atomic number of the target.

b. Gold (Z = 79), T = 50 MeV. For T = 50 MeV, the electron-nucleus results are from a synthesis of Bethe-Heitler

theory, screening corrections evaluated in the high-energy approximation using HartreeFock atomic form factors [55,56], and a Coulomb correction combining the theoretical

results of Davies, Bethe, Maximon, and Olsen [17] and of Elwert [22], modified to go

smoothly into the high-frequency limit given by the theory of Jabbur and Pratt [21].

The electron-electron results are from the theory of Haug [20], with screening

corrections evaluated in the high-energy approximation using Hartree-Fock incoherent

scattering factors [55].

21
12

.aE

•» JC JO

8

b c —

OJCQ. -

Z 6

T * 50 keV

.electron - nucleus (x l/Z 2

)

V-v

\electron- electron (xl/Z )

0.2 0.4 0.6

K/T

0.8 1.0

Fig. 7. Bremsstrahlung production cross section, differential in emitted photon energy. Shown is

the scaled cross section 3

2 k(da/dk) ys. the ratio k/T of the emitted photon energy to the initial electron kinetic energy. The solid curve is the electron-nucleus cross section

divided by Z

2

, and the dashed curve is the electron-electron cross section divided by Z,

where Z is the atomic number of the target.

c. Carbon (Z = 6), T = 50 keV.

For T = 50 keV, the electron-nucleus results are from the calculations of Pratt et al.

[18]. The electron-electron results are from the theory of Haug [20]; screening

corrections which would somewhat modify the rather steep portion of the curves near

k = 0 have been neglected.

22
0.4

k/T

Fig. 7. Bremsstrahlung production cross section, differential in emitted photon energy. Shown is

the scaled cross section 8

2 k(do/dk) vs. the ratio k/T of the emitted photon energy to the initial electron kinetic energy. The solid curve is the electron-nucleus cross section

divided by Z

2

, and the dashed curve is the electron-electron cross section divided by Z, where Z is the atomic number of the target.

d. Gold (Z = 79), T = 50 keV.

For T = 50 keV, the electron-nucleus results are from the calculations of Pratt et al.

[18]. The electron-electron results are from the theory of Haug [20]; screening

corrections which would somewhat modify the rather steep portion of the curves near

k = 0 have been neglected.

23
Fig. 8. Comparison of interpolated differential electron-nucleus bremsstrahlung

cross sections with the results of exact calculations by Tseng and Pratt

[23]. Shown is the scaled cross section (8 /Z )k(do/dk) plotted as a function of the ratio k/T, where 8 is the ratio of the initial electron

velocity to the speed of light, Z is the atomic number of the target, k is the emitted photon energy, and T is the initial electron kinetic

energy. The curves are from interpolation between the high-energy results

(T > 50 MeV) and the results of Pratt et al. [18] (T < 2 MeV). The

points are from the pilot calculations of Tseng and Pratt [23]; the error

bars are drawn to indicate a ± 5% uncertainty.

a. AJl and U, T = 5 MeV.

24
kdcr/dk,

mb

Fig. 8. Comparison of interpolated differential electron-nucleus bremsstrahlung

cross sections with the results of exact calculations by Tseng and Pratt

[23]. Shown is the scaled cross section (6 /Z )k(da/dk) plotted as a function of the ratio k/T, where 8 is the ratio of the initial electron

velocity to the speed of light, Z is the atomic number of the target, k

is the emitted photon energy, and T is the initial electron kinetic

energy. The curves are from interpolation between the high-energy results

(T > 50 MeV) and the results of Pratt et al. [18] (T < 2 MeV). The

points are from the pilot calculations of Tseng and Pratt [23]; the error

bars are drawn to indicate a ± 5% uncertainty.

b. AS, and U, T = 10 MeV.

25
Fig. 9. Transport cross sections for elastic scattering, at electron energies

2 below 1 MeV. The quantity given is 6 a-|(T), where 6 is the ratio of

the electron velocity to the speed of light, and

(T) = 2trf [da(0,T)/df2](l - cos0)sin0d0. The curves are results based

on the factorization of da/dft (Mott cross section x screening correction).

The solid curves are for a screening correction calculated with a screening

parameter according to Molifere [26]. The dashed curves are with a screening correction in terms of a form factor, (1 - F) . The points are from the results of the phase-shift calculations of Riley et al. [28] for

a static, screened Coulomb potential.

26
ELECTRON

IMPACT

IONIZATION

CROSS

SECTION.

Fig. 10. Comparison of theoretical electron impact ionization cross sections for K and L shells in gold with experimental results. The solid curves are from

our Wei zsacker-Wi 1 1 iams calculations, and the broken curves are from

Scofield's [32] Born-approximation calculations. The points are from

various measurements [33-44].

27
ELECTRON

IMPACT

IONIZATION

CROSS

SECTION,

Fig. 11. K-shell electron impact ionization cross sections in aluminum and nickle.

The solid curves are from our Weizsacker-Williams calculations; the broken curve is from Scofield's [32] Born-approximation calculations (Scofield

did not calculate the cross section for A£). The points are from various

measurements [34,45-51].

28
NBS-114A (REV. 2 -80 ) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

PUBLICATION OR

REPORT NO.

NBSIR 82-2572

2. Performing Organ. Report No, 3. Publication Date

September 1982

4. TITLE AND SUBTITLE

STATUS OF ELECTRON TRANSPORT CROSS SECTIONS

5. AUTHOR(S)

S. M. Seltzer and M. J. Berger

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Repo r t & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Office of Naval Research

Arlington, Virginia 22217

Space Science Data Center

NASA Goddard Space Flight Center

Greenbelt, Maryland 20771

Office of Health and

Environmental Research

Department of Energy

Washington, D.C. 205'

10. SUPPLEMENTARY NOTES

PJ Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This report describes recent developments and improvements pertaining

to cross sections for electron-photon transport calculations. The topics

discussed include: (1) electron stopping power (mean excitation energies,

density-effect correction); (2) bremsstrahlung production by electrons

(radiative stopping power, spectrum of emitted photons); (3) elastic

scattering of electrons by atoms; (4) electron-impact ionization of atoms.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) bremsstrahlung; cross sections; elastic scattering; electron-impact ionization;

electrons; photons; stopping power; transport.

13. AVAILABILITY

PP Unlimited

pj For Official Distribution. Do Not Release to NTIS

PH Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[~y] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

30

15. Price

$7.50

US COMM- DC 6043-P80