Appl. Radiat. ht. Vol. 42, No. IO, pp. 905-916, ht. J. Radial. Appl. Insrrum. Port A Printed in Great Britain

0883-2889/91 $3.00 + 0.00 Pergamon Press plc

1991

Differences in the Multiple Scattering of Positrons and Electrons MARTIN National

Institute

of Standards

J. BERGER*

and Technology,

Gaithersburg,

MD 20899, U.S.A.

(Received 26 March 1991) After a brief review of differences between the scattering cross sections for electrons and positrons, results are presented on the transmission and reflection of these particles by foils. These results were obtained with the Monte Carlo code ETBAN, which takes into account positron-electron differences of stopping powers and elastic-scattering cross sections. The principal output of the calculations consists of number and energy transmission and reflection coefficients for electrons and positrons. A few results are also shown pertaining to differences of energy spectra and angular distributions of positrons and electrons reflected from thick foils. The calculated transmission coefficients for positrons are greater, and the reflection coefficients smaller, than those for electrons of the same energy. These results are in qualitative agreement with experimental data, but the quantitative agreement is not very close.

Introduction The penetration of charged particles through matter is determined by the outcomes of countless Coulomb collisions with atoms and atomic electrons. It is a useful schematization to classify these collisions as either soft or hard. Soft collisions, with impact parameters large compared to atomic dimensions, are extremely numerous, and give rise to small energy losses and deflections. Hard collisions, with small impact parameters, are much less frequent, and result in large energy losses and deflections. Both types of collisions make essential contributions to the transport process. For example, a particle can have its direction reversed and be reflected from a foil solely due to the cumulative effect of many small deflections, or also due to the effect of one or a few deflections through large angles. The cross sections for soft collisions of positrons and electrons with atoms and atomic electrons are practically the same. There are considerable positronelectron differences for hard collisions, and these give rise, in turn, to small but non-negligible differences in penetration and diffusion. Experimental evidence for such differences has been available for a long time. Seliger (1952) found that the backscattering of electrons from a j2P beta-particle source exceeds that of positrons from a 22Na source. In a later experiment Seliger (1955) observed that the transmission of

*Contractor.

This work

was supported

50SBN9C5555 with the National and Technology.

Institute

by Contract of Standards

905

monoenergetic positrons through foils is greater, in general, than that of electrons of the same energy. However, for electrons with energies below 400 keV in aluminum he obtained a small reverse effect. Takhar (1966) and Ram et al. (1981) reported the transmission through foils to be greater for positrons than for electrons in experiments with beta-sources with roughly comparable endpoint energies. Patrick and Rupall (1971) also found this to be the case for lead foils, but observed the reverse effect for aluminum foils. Rohrlich and Carlson (1954) took into account the differences between inelastic scattering cross sections and derived energy-loss straggling distributions for positrons and electrons that are modifications of the Landau distribution (Landau, 1944). They also calculated positron-electron differences for the average cosine of the multiple-scattering angle, and for the penetration depth at which the particles lose their memory of their initial direction. Hansen and Ingerslev-Jensen (1982) used the theory of Lewis (1950) to calculate differences between the transport of electrons and positrons in a homogeneous unbounded medium. They obtained mean penetration depths which are greater for positrons than for electrons at all energies in lead, and at energies above 100 keV in aluminum. Using simplified transport models, Batra and Sehgal (1981) and Batra and Singh (1989) estimated differences between the practical ranges of electrons and positrons, and Singh and Batra (1987) applied similar methods to obtain differences between attenuation coefficients for positrons and electron from beta sources. Transport calculations by Rogers

906

MARTIN

(1984a,b) with the EGS Monte Carlo code took into account positron-electron differences in stopping power, and also the annihilation in flight of positrons, but disregarded differences in elastic scattering Rogers (1984a) calculated depth-close curves with in tissue phantoms, and found the peak values of these curves to be lower by several percent for positron beams than for electron beams of the same energy. In calculations of the response function of germanium detectors Rogers (1984b) found considerable differences between the photofractions (probabilities of complete absorption of the incident energy) for positrons and electrons. The present paper begins with a brief survey of positron-electron differences in respect to inelastic and elastic scattering cross sections, and then describes transport calculations for positrons and electrons, with emphasis on reflection and transmission by foils. These results were obtained with the Monte Carlo transport code ETRAN (Berger, 1963, 1988; Seltzer, 1988) which takes into account positron-electron differences in regard to stopping power and elastic scattering.

Differences

Between Scattering Cross Sections for Positrons and Electrons

Inelustic stuttering

The cross section for the scattering of a positron by a free electron (Bhabha, 1935) can be compared with the cross section for the scattering of an electron by a free electron (Moller, 1932). Both cross sections take into account spin and relativistic effects. The Moller cross section also includes exchange effects, and the Bhabha cross section has a corresponding term related to the possibility of positron annihilation [see Jauch and Rohrlich (1955)]. Ratios of the Bhabha to the Msller cross section are plotted in Fig. 1, as a function of the fraction W/T, where T is the initial energy and W is the energy loss. A family of curves is shown for different values of T. The BHABHA/

MBLLER

2.2 20

0.0 1 MEV

I

1.9 1.6

/

10-2

10-l

,

0.1

1

J. BERGER

J

10-Z

10-l Energy

Fig. 2. Ratio

100

( MeV

10’

1

of collision stopping powers that for electrons.

for positrons

Bhabha-Mraller cross section ratios approach unity for small values of W/T. For large W/T they exceed unity for T > 200 keV, and fall below unity for T < 200 keV. Stopping powers

and ranges

The collision stopping power, i.e. the energy loss per unit path length due to collisions that result in atomic ionization and excitation, is the sum of two parts: (1) the contribution from soft collisions, which is the same for electrons and positrons and is given by Bethe’s stopping-power theory; (2) the contribution from hard collisions, which is evaluated as an energy-weighted integral over the Merller cross section for electrons, or over the Bhabha cross section for positrons [see, e.g Uehling (1954), or ICRU (1984)]. For positrons the integration includes all possible energy losses W < T. For electrons the integration only includes losses W < T/2, because the faster of the two electrons involved in the collision is by definition considered to be the primary electron whose energy loss is being calculated. Positron-electron ratios of stopping powers, obtained from the 1.1

1

W/T

of electrons and positrons by free plotted is the ratio of the Bhabha (positron-electron) cross section to the Moller (electronelectron) cross section, as a function of the fractional energy loss, W/r, for various values of the incident energy T. Fig.

I.

Scattering

electrons. The quantity

to

1o-2

10-l

Energy

100

( MeV)

Fig. 3. Ratio of the radiative stopping power for positrons to that for electrons.

Multiple scattering of positrons and electrons

907

the theory of Mott (1929) for an unscreened potential.

Coulomb

Positron annihilation in fright

10-z

10-l Energy

100

10'

(Me%)

Fig. 4. Ratio of the CSDA range of positrons to that of electrons. tables in ICRU (1984), are illustrated in Fig. 2 for water and lead. It can be seen that the stopping powers for positrons are greater than those for electrons above 200 keV, and smaller below 200 keV, and that the differences greater for lead than for water. The radiative stopping power, that is, the energy loss per unit path length due to the emission of bremsstahlung quanta, is smaller for positrons than for electrons at all energies (ICRU, 1984; Kim et al., 1986). Positron-electron ratios of the radiative stopping powers are shown in Fig. 3 for water and lead. The CSDA range, that is the mean rectified path length calculated in the continuous-slowing-down approximation, is obtained by integrating the reciprocal of the total stopping power (collision plus radiative) with respect to energy. Positron-electron ratios of CSDA ranges are shown in Fig. 4 for water and lead. Elastic scattering Cross sections for the elastic scattering of positrons and electrons were calculated in the present work with a computer code of Riley (Riley, 1974; Riley et al., 1975). This code implements the partial-wave expansion method and evaluates phase shifts for scattering in a static screened Coulomb potential, combining the numerical solution of the Dirac equation with use of the WKB approximation. The screened Coulomb potentials were obtained from electron-density distributions calculated with a relativistic Dirac-Fock wave-function code of Desclaux (1975). Positron-electron ratios of elastic-scattering cross sections, as functions of the deflection angle, are plotted in Fig. 5(a) for carbon and in Fig. 5(b) for gold, at energies of 1024, 256, 64 and 16 keV. The cross sections for positrons are always smaller than those for electrons. The differences are greatest for deflection angles near 90”, are smaller than 10% in carbon, but become as great as 80% in gold. In order to indicate the importance of screening, Figs S(a) and (b) show cross section ratios that were obtained from

Small differences between the transport of positrons and electrons can arise because of the possibility that positrons will be annihilated before they are slowed down to rest. The most important mode of annihilation-in-flight involves the emission of two quanta. The cross section for this process has been given by Bethe (1935). The other mode which must be included in transport calculations involves the emission of a single quantum, and arises from the interaction of the positron with strongly bound electrons (mainly in the K shell). The annihilation cross section for interactions with K-shell electrons is proportional to the fifth power of the atomic number and is important only for heavy atoms. In the present work, cross sections calculated by Tseng and Pratt (1973) were used for single-quantum annihilation involving K-shell electrons. With guidance provided by the calculations of Broda and Johnson (1972) for the K and L shells, the K-shell cross sections of Tseng and Pratt were increased by 20% for lead, and by 13% for silver to include, approximately, the interaction with L-shell electrons. As an example, Fig. 6 shows the probabilities that positrons with an initial energy of 1 MeV, slowing down in carbon or lead, will survive down to a lower energy T without being annihilated. These survival probabilities can be seen to differ from unity by only a few percent.

Features of the ETRAN

Code

Condensed-random walk model The Monte Carlo model used in the ETRAN code avoids the sampling of individual Coulomb collisions. Instead, the electron or positron tracks to be simulated are divided into segments, and the angular deflections and energy losses in successive segment are sampled from appropriate multiple-scattering distributions. The number of track segments is typically no greater than a few hundred, and thus is an order of magnitude smaller than the number of individual collisions, which leads to a considerable saving of computing time. Goudsmit-Saunderson

distribution

Angular deflections in track segments are sampled in the ETRAN code from the distribution given by the multiple-scattering theory of Goudsmit and Saunderson (1940). This distribution is calculated as a Legendre expansion whose coefficients depend on the elastic scattering cross section. GoudsmitSaunderson distributions for positron and electrons multiply-scattered in tungsten are compared in Figs 7(a) and (b) at 1000 and 10 keV. As expected on the basis of the cross section differences, the multiplescattering distributions for positrons are greater than

908

MARTIN

J.

BERGER

16 keV

0

120

60

180

,

Deflection

60

0 angle

120

180

(deg)

(b) 1.0 0.9

1024

keV

0.6 0.7 0.6 0.5 0.4 03 02 0.1

t

0

I

I

60

120

0 ; cz

1.0 0.9 0.6 0.7 06 0.5 0.4 a3 0.2 0.1 0

60

120

160

Deflection

0

angle

180

(deg)

Fig. 5. Positron-electron ratios of the elastic-scattering cross sections. Solid curve were calculated with, and dashed curves without screening effects. (a) Elastic scattering by carbon atoms; (b) elastic scattering by gold atoms.

Multiple Initial

positron

energy

:

1

scattering

of positrons

and electrons

MeV

2 = 74,

909

To = 1000

k&J,

s=O.O8

g/Cm2

10’

‘OO.OI

(a) --__

Electrons

- - -

Positrons

-.

100 r 0

-

r L

G ‘\

16’

‘.

,,zf

‘..__

r

0

10“

lo-2

Energy

20

40

10°

60

80

Deflection

100

angle

140

160

160

ldeg)

( MeV) Z = 74,

6. Probability that a positron with an initial kinetic energy of I MeV will be slowed down to various energies without having been annihilated in flight.

those for electrons at small angles, and smaller at the cases large angles, with the cross-over-in plotted-occurring near 60”.

To = 10

keV.

distributions

Energy losses in track segments are sampled from the Landau distribution (Landau, 1944) with a binding correction from Blunck and Leisegang (1950). In the Landau theory no limit is placed on the energy losses in individual collisions, so that the mean value of the Landau distribution is in effect unbounded. As discussed by Seltzer (1988), it is necessary to truncate the distribution in order to obtain a finite mean value consistent with the stopping power. This truncation procedure allows one to take into account positronelectron stopping power differences. In Landau’s theory, large energy transfers in hard collisions are treated according to the Rutherford cross section. Rohrlich and Carlson (1954) developed a correction to the Landau distribution which involves the replacement of the Rutherford cross section by the Msller cross section for electrons or the Bhabha cross section for positrons. The Rohrlich-Carlson correction changes the central part of the Landau distribution very little. For example, in calculations of penetration through lead with the ETRAN code the lengths of the track segments in lead used in the ETRAN code, the most probable energy loss in each track segment is changed by only a fraction of a percent, and the full width at halfmaximum of the energy-loss distribution is reduced by less than 1.5%. The complications that would result from the use of the Rohrlich-Carlson correction therefore did not seem to be worth the trouble, and the uncorrected Landau distribution was used.

s= 0.0008

g/cm’

1

10’

-.

(b)

-

Electrons

- - -

PoJltrons

6 ‘_

S

Energy-loss

120

B

. .._ _-_

lo-’ =

0

20

40

60

Deflection

60 awe

100

e

120 (deg

140

--__

160

160

1

Fig. 7. Goudsmit-Saunderson angular multiple-scattering distributions for electrons and positrons in tungsten. (a) Initial energy 1000 keV, path length 0.08 g/cm2, final energy 909.7 keV. (b) Initial energy 10 keV, path length 0.00008 g/cm2, final energy 9.26 keV.

taneously. Thus it is not yet possible to treat primary positrons and secondary knock-on electrons in one calculation. The simulation of positron annihilation in flight is not yet included in ETRAN. In the present work it was considered accurate enough to apply an annihilation correction to the output from ETRAN. Let S(T,, T) denote the energy spectrum of transmitted or reflected positrons where T, is the initial energy and T the spectral energy. Let P(T,,, T) be the probability that the positron will survive down to energy T (as illustrated, for example, in Fig. 6). The spectrum corrected for annihilation in flight can be aproximated by the product S(T,,, T)P(T,, T), and corrected transmission or reflection coefficients are obtained by integrating this product over all spectral energies.

Transmission and Reflection of Positrons and Electrons by Foils

Unfinished business

Description

In regard to the treatment of positrons, the ETRAN code is still incomplete in some respects. The code uses cross sections either for electrons or for positrons, but not for both types of particles simul-

The results in this Section pertain mainly to number reflection and transmission coefficients (average fractions of the number of incident particles that are reflected or transmitted) and energy reflection and

of the calculation

910

MARTIN J. BERGER Table

I.

Transmission

and reflectionof electrons and positrons by aluminum

foils.Results are for perpendicular incidence

Fraction of particles: Transmitted

Fraction of energy:

Reflected

Transmitted

Reflected

T

.1

CkeV)

(n/cmZ)

E

P

E

P

E

P

E

P

PIE

960

0.040

0.990

0.990

1.000

0.010

0.008

0.825

0.923

0.924

1.002

0.008

0.006

0.824

0.080

0.956

0.958

1.002

0.039

0.033

0.842

0.809

0.816

1.008

0.024

0.020

0.848

O.I20

0.881

0.887

1.007

0.075

0.067

0.894

0.662

0.674

I.017

0.039

0.034

0.891

0.160

0.769

0.784

1.020

0.093

0.084

0.905

0.507

0.523

1.032

0.045

0.040

0.898

0.200

0.636

0.656

I.031

0.096

0.088

0.908

0.363

0.380

1.046

0.045

0.041

0.900

0.240

0.485

0.511

1.052

0.097

0.088

0.908

0.239

0.255

1.066

0.045

0.041

0.900

0.280

0.338

0.360

1.067

0.097

0.088

0.908

0.142

0.154

1.084

0.045

0.041

0.900

0.320

0.205

0.225

1.098

0.097

0.088

0.908

0.073

0.082

I.111

0.045

0.041

0.900

0.360

0.103

0.117

I.134

0.097

0.088

0.908

0.031

0.036

I.151

0.045

0.041

0.900

0.008

0.989

0.990

I.001

0.011

0.009

0.842

0.942

0.943

I.001

0.009

0.008

0.839

336

250

I59

P/E

P/E

P/E

0.016

0.956

0.960

1.005

0.042

0.036

0.849

0.853

0.859

1.006

0.030

0.025

0.847

0.024

0.895

0.904

I.010

0.083

0.072

0.872

0.736

0.74s

I.013

0.052

0.045

0.867

0.032

0.808

0.820

I.014

0.113

0.100

0.879

0.606

0.618

I.019

0.065

0.057

0.874

0.040

0.703

0.716

I.019

0.126

0.112

0.884

0.478

0.490

1.025

0.069

0.061

0.879

0.048

0.586

0.599

1.023

0.130

0.115

0.883

0.358

0.370

I.031

0.070

0.062

0.879

0.056

0.460

0.474

I.031

0.130

0.115

0.883

0.253

0.263

1.042

0.070

0.062

0.879

0.064

0.340

0.354

I.041

I.130

0.115

0.883

0.168

0.176

1.049

0.070

0.062

0.879

0.072

0.231

0.243

I.051

0.130

0.115

0.883

0.102

0.108

I.061

0.070

0.062

0.879

0.008

0.974

0.977

1.003

0.026

0.022

0.852

0.901

0.904

1.003

0.020

0.017

0.850

0.016

0.889

0.895

1.007

0.086

0.076

0.883

0.732

0.739

I.010

0.054

0.047

0.876

0.024

0.741

0.751

I.015

0.123

0.110

0.894

0.529

0.538

I.018

0.070

0.062

0.887

0.032

0.560

0.570

1.018

0.129

0.116

0.899

0.341

0.350

1.025

0.072

0.064

0.891

0.040

0.368

0.378

1.029

0.129

0.116

0.899

0.190

0.197

1.035

0.072

0.064

0.891

0.048

0.200

0.208

1.038

0.129

0.116

0.899

0.087

0.091

1.044

0.072

0.064

0.891

0.056

0.083

0.087

1.050

0.129

0.116

0.899

0.030

0.032

1.060

0.072

0.064

0.891

0.064

0.024

0.025

1.067

0.129

0.116

0.899

0.007

0.008

1.080

0.072

0.084

0.891

0.072

0.003

0.004

1.297

0.129

0.116

0.899

0.001

0.001

I.291

0.072

0.064

0.891

0.004

0.963

0.966

1.004

0.037

0.032

0.881

0.888

0.890

1.002

0.029

0.025

0.876

0.008

0.863

0.872

I.010

0.105

0.091

0.872

0.706

0.712

1.009

0.068

0.059

0.870

0.012

0.706

0.715

I.013

0.138

0.121

0.878

0.501

0.507

I.013

0.082

0.072

0.874

0.016

0.517

0.523

I.013

0.143

0.126

0.880

0.314

0.319

I.016

0.084

0.073

0.876

0.020

0.327

0.329

1.005

0.143

0.126

0.880

0.169

0.172

I.013

0.084

0.073

0.876

0.024

0.167

0.170

I.021

0.143

0.126

0.880

0.073

0.075

1.032

0.084

0.073

0.876

0.028

0.063

0.066

1.048

0.143

0.126

0.880

0.023

0.025

1.058

0.084

0.073

0.876

0.032

0.015

0.017

I.115

0.143

0.126

0.880

0.005

0.005

1.122

0.084

0.073

0.876

0.036

0.002

0.002

I.142

0.143

0.126

0.880

0.001

0.001

I.132

0.084

0.073

0.876

*Tis the energy of the incident particles.and x is the foilthickness. Transmission and reflectioncoefficientsfor electrons are indicated by (E),

and

those for positrons by (P)

transmission coefficients (average fractions of the incident energy that is reflected or transmitted). For a given energy and direction of the incident particle, the reflection and transmission coefficients for all foil thicknesses were obtained from samples of 100,000 Monte Carlo histories of electrons and positrons. The histories were followed until the particle energies fell below 1 keV, or until the residual ranges were so short that the particles could no longer reach the nearest foil boundary. For each case treated, the same sequence of pseudo-random numbers was used twice, once for electrons and then for positrons. The calculations included only primary positrons and electrons, and the histories of secondary electrons from ionizing collisions were not followed. The number of histories sampled was sufficiently large so that the statistical errors (relative standard deviations) of the reflection and transmission coefficients are generally on the order of 1% or smaller, except for transmission through very thick foils. Of greater importance than the statistical errors are the systematic errors due to the approximations in the Monte Carlo model and due to the

uncertainties of the input cross sections. Numerous comparisons with experimental results [see, e.g. Berger (1988)] have established the reliability of predictions made with the ETRAN code. However, one cannot exclude the possibility of sytematic errors up to perhaps 3% for reflection and transmission coefficients, especially at energies below a few hundred keV. The positron-electrons ratios of these coefficients are likely to be more accurate than the coefficients themselves, because of the cancellation of errors. Results for monoenergetic particle beums Transmission and reflection coefficients for positrons and electrons incident perpendicularly onto foils of various thicknesses are compared in Table 1 for aluminum foils, and in Table 2 for lead foils. The results are given for initial particle energies of 960, 336, 250 and 159 keV. The positron-electron differences in regard to transmission are greater for lead than for aluminum, and show systematic trends consistent with the fact that the elastic scattering cross sections at large scattering angles are smaller for positrons than for electrons. At all energies, the

Multiple

scattering

of positrons

and electrons

911

Table 2. Transmission and reflection of electrons and positrons by leadfoils. Resultsare forperpendicular incidence Fractionof particles: Transmitted T

Fractions of energy:

Reflected

Transmitted

Reflected

x

OW

(g/cm’)

E

P

PIE

E

P

PIE

E

PIE

E

P

960

0.0200 0.0400 0.0600 0.0800 0.1000 0.1400 0.1600 0.1800 0.2000

0.921 0.796 0.683 0.591 0.503 0.419 0.339 0.265 0.202 0.150

0.970 0.890 0.801 0.716 0.636 0.560 0.483 0.409 0.340 0.275

1.053 I.119 I.173 1.212 1.264 1.338 1.425 1.542 1.684 1.838

0.079 0.203 0.309 0.382 0.429 0.454 0.466 0.471 0.473 0.473

0.028 0.106 0.189 0.261 0.314 0.347 0.368 0.378 0.383 0.385

0.361 0.522 0.612 0.683 0.733 0.765 0.789 0.803 0.810 0.813

0.893 0.736 0.596 0.481 0.382 0.297 0.224 0.165 0.118 0.083

0.944 0.832 0.712 0.601 0.502 0.415 0.337 0.270 0.211 0.162

1.057 I.130 I.195 1.249 I.314 1.399 1.503 1.632 1.790 1.960

0.073 0.178 0.257 0.304 0.330 0.343 0.348 0.350 0.351 0.351

0.026 0.091 0.154 0.201 0.232 0.249 0.258 0.262 0.264 0.265

0.357 0.511 0.597 0.659 0.701 0.725 0.741 0.749 0.753 0.754

336

0.0075 0.0150 0.0225 0.0300 0.0375 0.0450 0.0525 0.0600 0.0675 0.0750

0.824 0.645 0.509 0.387 0.279 0.186 0.115 0.067 0.037 0.018

0.909 0.763 0.638 0.520 0.411 0.309 0.221 0.151 0.096 0.058

I.104 I.182 1.253 1.343 1.476 1.658 1.922 2.258 2.633 3.211

0.176 0.351 0.455 0.502 0.517 0.522 0.522 0.523 0.523 0.523

0.090 0.232 0.337 0.398 0.425 0.435 0.437 0.438 0.438 0.438

0.509 0.662 0.740 0.793 0.821 0.833 0.837 0.837 0.838 0.838

0.789 0.575 0.414 0.287 0.189 0.116 0.067 0.036 0.018 0.009

0.875 0.691 0.534 0.401 0.292 0.204 0.135 0.086 0.051 0.029

I.109 1.201 1.291 1.395 1.547 1.749 2.015 2.366 2.778 3.352

0.163 0.304 0.374 0.401 0.409 0.410 0.411 0.411 0.411 0.411

0.082 0.197 0.270 0.306 0.319 0.323 0.324 0.325 0.325 0.325

0.503 0.650 0.721 0.762 0.781 0.788 0.790 0.790 0.790 0.790

250

0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0505

0.825 0.649 0.510 0.383 0.272 0.180 0.112 0.063 0.034 0.016

0.908 0.765 0.639 0.518 0.404 0.299 0.211 0.142 0.091 0.051

1.100 I.180 1.252 I.351 1.488 1.659 1.893 2.245 2.704 3.227

0.175 0.347 0.451 0.499 0.514 0.518 0.519 0.519 0.519 0.519

0.091 0.230 0.332 0.391 0.417 0.426 0.428 0.429 0.429 0.429

0.524 0.663 0.737 0.784 0.811 0.822 0.826 0.826 0.827 0.827

0.792 0.580 0.418 0.285 0.186 0.114 0.066 0.035 0.017 0.008

0.874 0.694 0.537 0.401 0.289 0.199 0.131 0.082 0.050 0.026

1.196 1.287 1.406 I.555 1.748 1.996 2.367 2.844 3.381

I.104

0.162 0.302 0.372 0.400 0.408 0.409 0.409 0.409 0.409 0.409

0.084 0.196 0.267 0.302 0.316 0.319 0.320 0.321 0.321 0.321

0.517 0.650 0.718 0.756 0.774 0.781 0.783 0.783 0.783 0.783

0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 0.0225 0.0250

0.805 0.624 0.488 0.368 0.262 0.176 0.109 0.063 0.035 0.018

0.891 0.739 0.611 0.493 0.382 0.283 0.201 0.135 0.086 0.052

I.106 1.185 1.253 1.340 1.456 I.612 I.851 2.136 2.429 2.974

0.195 0.371 0.474 0.520 0.535 0.539 0.540 0.540 0.540 0.540

0.109 0.256 0.359 0.416 0.440 0.449 0.451 0.452 0.452 0.452

0.558 0.689 0.757 0.800 0.821 0.832 0.836 0.837 0.837 0.837

0.772 0.558 0.399 0.276 0.181 0.113 0.065 0.036 0.019 0.009

0.857 0.669 0.512 0.382 0.275 0.190 0.126 0.079 0.048 0.027

I.110 I.199 1.284 1.385 1.520 1.686 1.933 2.227 2.563 3.102

0.181 0.326 0.397 0.423 0.431 0.433 0.433 0.433 0.433 0.433

0.100 0.220 0.292 0.326 0.339 0.343 0.344 0.345 0.345 0.345

0.552 0.676 0.737 0.771 0.786 0.793 0.795 0.795 0.795 0.795

0.1200

159

P

PIE

*T istheenemy of theincident uarticles. and x isthe foilthickness. Transmission and reflection coefficients forelectrons are indicated by (E),and chosefor positrons by (P).’

transmission coefficients for positrons are greater than those for electrons, and the differences increase with the foil thickness. Conversely, the reflection coefficients for positrons are always smaller than those for electrons, with the differences becoming larger as the thickness of the foil decreases. It is of interest to scale the transmission data for positrons and electrons in Tables 1 and 2, by expressing foil thicknesses in units of the CSDA ranges for positrons and electrons. Thereby the influence of stopping-power and range differences on the transmission curves is to a large extent eliminated, and the residual positron-electron differences are mainly due to differences in elastic scattering. Scaled transmission curves are shown in Fig. 8(a) for aluminum, and in Fig. 8(b) for lead. The CSDA ranges used for the scaling were obtained by interpolation in the range tables in (ICRU, 1984) and are given in Table 3. Seliger (1955) measured the transmission of positrons and electrons through foils with the same

experimental arrangement. Comparisons with some of his results are shown in Table 4. For lead foils, the agreement is fairly close at energies of 960, 336 and 250 keV, but poor at 159 keV. For aluminum foils the agreement is never close. At 159,250 and 336 keV the measured transmission coefficients for positrons are smaller than those for electrons, whereas the opposite effect is predicted by the calculations. It is difficult to conceive of changes in the calculations that would remove these discrepancies. It should be mentioned in this connection that number transmission coefficients for electrons passing through aluminum foils, as calculated with the ETRAN code, have been found to be in rather good agreement with many experimental results at energies from 1 MeV down to 50 keV (Berger, 1988). Results for isotropic sources Table 5 gives calculated results on the reflection of positrons and electrons from isotropic monoenergetic sources with energies from 2000 to 10 keV. These

MARTIN

912

J. BERGER

336

90

ksV

80 70 60 50 40 30

I

I

I

0.15

0.30

0.45

I

I= go80

-

70

-

60

-

5040

-

30

-

20

-

10

-

0

0

0.60

Fail thickness

0.15

0.30

0.45

0.60

(x/r,,)

(b)

90

960

kBV

80 70 60 50 40 30 2 -

5

20 10 0

.-3 E

r

e

c

100 90

$59

keV

80 70 60 50 40 30 20

‘,’ 0

t 0.06

0.12

0.18

0.24

Fail

thickness

I

0

0.06

( x/r0

I 0.12

I 0.18

0.24

1

Fig. 8. Transmission of through foils. Solid curves arc for electrons, and dashed curves for positrons. Particles are incident perpendicularly onto the foils. Foil thickness are expressed in units of the CSDA range for electrons or positrons, x/rO. (a) Transmission through aluminum foils; (b) transmission through lead foils.

Multiple scattering of positrons and electrons Table 3. Ranges ofelectrons (E) and positrons (P). calculated continuous-slowina-down aooroximation Aluminum Energy (W) 960 336 250 159

E range 527.1 128.2 82.17 40.22

P (mg/cm2) 531.6 125.6 79.71 38.43

Table

in the

5. Reflection of electrons and positrons from semi-infinite aluminum, silver and lead targets Fraction of particles reflected

Lead P/E 1.007 0.980 0.970 0.955

P (m&m*)

E range 748.6 195.8 128.0 64.64

P/E

E

P

P/E

E

P

PIE

2000 1000 500 200 100 50 20 IO

0.337 0.369 0.388 0.404 0.408 0.410 0.399 0.391

0.331 0.359 0.377 0.390 0.392 0.391 0.371 0.353

0.980 0.970 0.971 0.967 0.960 0.953 0.930 0.902

0.232 0.266 0.290 0.312 0.318 0.321 0.310 0.305

0.228 0.259 0.281 0.301 0.304 0.305 0.288 0.275

0.984 0.973 0.972 0.967 0.959 0.951 0.929 0.902

Ag

2000 1000 500 200 100 50 20 IO

0.506 0.550 0.577 0.598 0.602 0.593 0.563 0.522

0.476 0.519 0.543 0.562 0.562 0.547 0.503 0.447

0.940 0.943 0.940 0.939 0.934 0.923 0.893 0.856

0.387 0.439 0.474 0.502 0.508 0.501 0.470 0.433

0.360 0.409 0.439 0.465 0.468 0.454 0.412 0.365

0.929 0.93 I 0.928 0.927 0.921 0.907 0.877 0.844

Pb

2000 1000 500 200 100 50 20 IO

0.590 0.637 0.663 0.682 0.680 0.666 0.618 0.567

0.535 0.583 0.626 0.626 0.621 0.600 0.537 0.468

0.906 0.915 0.944 0.918 0.913 0.901 0.869 0.826

0.477 0.533 0.567 0.593 0.593 0.578 0.525 0.473

0.420 0.476 0.507 0.533 0.530 0.509 0.445 0.385

0.882 0.893 0.893 0.899 0.894 0.880 0.848 0.814

(k;)

0.997 0.980 0.953

(1) by interpolating in Table 5 to the mean energy of the beta spectra for “Na and 32P; (2) by integrating the results in Table 5 with respect to energy over the beta spectra.

Fraction of energy reflected

Al

I .052

787.3 195.2 125.4 61.62

results are for sources placed next to thick targets of aluminum, silver and lead. At all energies and for all foil materials, the reflection coefficients for positrons are smaller than those for electrons, with the differences increasing with the atomic number of the foil. Seliger (1952) measured the reflection of positrons from a 22Na source, and the reflection of electrons from a “P source, using the same experimental arrangement. Bisi and Braicovich (1964) also measured the reflection of positrons from a 22Na source. Corresponding theoretical reflection coefficients were obtained in two ways:

‘Results are for point-isotropic sources that are located at the txget boundary and emit particles of energy T. Reflection coefficients for electrons are indicated by (E) and those for positrons by(P).

sections, or refinements of the Monte Carlo model, would change the calculated results enough to bring them into agreement with the experimental results. The interpretation of the experimental positronelectron ratios of reflection coefficients may be obscured by the uncertainties of the measured reflection

The two procedures gave results that agreed within 1% or better. As shown in Table 6, the agreement between the calculated and measured reflection coefficients is rather poor. Just as in the case of transmission, it appears unlikely that improvements of the cross Table 4. Comparison

913

of calculated transmission coefficients for aluminum exmrimental result of Seliger (1955) Transmission

and lead with

coefficients: Measured

Calculated Z

T (keV)

x (mg/cm’)

E

P

P/E

E

P

PIE

13

I59

8 I6 24 I6 32 48 24 48 72 120 240 360

0.863 0.517 0.167 0.889 0.560 0.200 0.895 0.586 0.231 0.881 0.485 0.103

0.872 0.523 0.170 0.895 0.570 0.208 0.904 0.599 0.243 0.887 0.511 0.117

1.010 I.013 I.021 I.007 I.018 I.038 1.010 I.023 I.051 1.007 1.052 I.134

0.89 0.58 0.23 0.90 0.60 0.36 0.90 0.64 0.3 I 0.86 0.49 0.13

0.83 0.50 0.18 0.90 0.55 0.27 0.89 0.59 0.26 0.92 0.65 0.33

0.93 0.86 0.78 I .oo 0.92 0.75 0.99 0.92 0.84 I .07 I .32 2.53

5 IO I5 IO 20 30 I5 30 45 60 120 I80

0.624 0.368 0.176 0.649 0.383 0.180 0.645 0.387 0.186 0.683 0.419 0.202

0.739 0.493 0.283 0.765 0.518 0.299 0.763 0.520 0.309 0.801 0.560 0.340

I.185 1.340 I.612 I.180 I.351 1.659 I.182 I.343 I.658 I.173 I.338 1.684

0.57 0.29 0.13 0.60 0.32 0.14 0.62 0.34 0.17 0.68 0.41 0.23

0.62 0.35 0.18 0.72 0.45 0.24 0.75 0.48 0.29 0.86 0.65 0.42

I .08 I.21 I .38 I .20 I.41 I.71 I.21 I.41 1.71 1.26 1.59 I .83

250

336

960

82

159

250

336

960

‘Results pertain to electrons (E) and positrons perpendicularly onto foils of thickness x.

(P) with energy

T that are incident

MARTINJ. BERGER

914 Table 6. Calculated

and measured number reflection coefficients for electrons and beta-particle sources located at the boundary of thick

positrons from isotropic

foil targets Reflection

coefficient Silver

Aluminum

Lead

(a) Electrons from“P beta spectrum; endpoint mergy 1710.4 keV; merug~ energy 494.9 keV 0.414 0.573 0.655 Expt, Seliger (1955)

Calculated (b/W~~;~

0.565 0.652 0.380 from “Nu beia spec~um; endpoint energy 545.5 ke V; me-rage energy 0.446

0.289

Expt, Seliger (1955) Expt, Bisi and Eraicovitch (1964) Calculated (c)Ratio q/reflection coeficientfor positronsfrom from “P source

0.519

0.407 0.537 0.214 0.556 0.617 0.388 “No sauce to rhor /or elecrrons Ratio Aluminum

0.698 I.021

Seliger (I 955) Calculated

coefficients. From a review of such measurements by Thummel and Krivan (1964) one gets the impression that these uncertainties can easily be as large as 10%. Another complication arises from the circumstance 90

a5 -

(O) 694.9 - keV electrons

60 -

-

75 -

- - - 215.5 - keV powtrons

70 65 60 55 -

45 -___-___’ 40 1 / 0 10 20

I 1 30

Angle

70 I 65-

I 40

1 50

of lncldence

(deg

1 80

90

1 1

(b)

60 -

-

694.9 - keV electrons

55 50 45 40 35 c 30 25 20 15 10 5-

- - -

215.5 - ksV positrons

0

I 70

I 60

9. Directional

1 20

1 30

I 40

I 50

I 60

I 70

I 60

90

t deg 1

Acceptance

angle

dependence

of the reflection

of 694.9 keV

electrons and 215.5 keV positrons from a thick lead target. The energies of the incident particles are equal to the mean energies of the jzP and ‘*Na beta spectra, respectively. (a) Reflection of monodirectional beams, as a function of the direction of incidence. Note that an angle of 0” corresponds to perpendicular incidence. (b) Reflection of particles from an isotropic source next to a thick target, as a function of the acceptance angle of the cone of acceptance. Note that 90” acceptance angle corresponds to a 271 solid angle.

Lead

0.778 0.984

0.792 0.946

that the reflection coefficients for isotropic sources are rather sensitive to the condition of the surface of the target foil. As has been shown by Kanter (1957) one can significantly decrease reflection coefficients by slightly roughening the surface of the foil. The surface roughness affects particularly those particles which travel in grazing directions, almost parallel to the surface, either when entering or when leaving the foil. It may be the case that the assumption of a perfectly smooth foil surface in the calculation is not sufficiently realistic for comparisons with experiments. Results of two other calculations are shown in Fig. 9 that pertain to the reflection from lead of 215.5 keV positrons and 694.9 keV electrons (mean energies of the 32P and **Na beta spectra). Figure 9(a) shows the dependence of the reflection coefficient on the obliquity of the direction of incidence of monodirectional beams. The biggest difference between positrons and electrons (14%) occurs for normal incidence, and is smaller than the difference in the experiment of Seliger with an isotropic source. Figure 9(b) shows the reduction of the reflection coefficient for an isotropic source that occurs if only particles are included that emerge from the foil within a certain acceptance cone extending from 0% (perpendicular direction of emergence) to a maximum angle smaller than 90 Energy

10

Silver

spectra

and angular

distributions

Energy spectra of the particles emerging from the foils were obtained in all of the cases, and angular distributions in some of the cases treated. Some illustrative results will now be shown. Figures 10(a) and (b) compare energy spectra of positrons and electrons reflected from a thick lead target, for 500 keV beams incident perpendicularly and for an isotropic 500 keV source placed next to the target. These spectra include all particles regardless of their direction of emergence from the target. The positron spectra are lower than the electron spectra at energies close to the source energy, but this difference tends to vanish at lower spectral energies. Figures I I(a) and (b), for the same conditions, compare the angular

Multiple Pb,

500

keV,

scattering

of positrons

and electrons Pb,

0 deg

915 500

keV,

normal

incidence

7.”

;>

3.5

g -

3.0 t

0

o.ie7 $ 0.16 -

(a)

_

-

Electrons

---

Positrons

LA1 0.05

_0.10

0.15

0.25

500

keV,

0.30

” 0.35

0.40

0.14

-

3 .o

0.12

-

1 0.45

0

0.50

-

Electrons

---

Positrons

, 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.6

0.9

,.O

( MeV 1

Energy

Pb,

11

’ 0.20

-

(a)

Isotropic

source

Pb,

500

keV,

Isotropic

scarce

0.16, _

20-

7 > ;

la-

-

14-

s .0 r

lo-

x

8-

P z 2

(b) -

Electrons

_

---

Positrons

;

16-

:,

0.14-

(b)

0.12-

12-

Electrons

- - -

Positrons

64-

2 0

0.05

0.10

0.15

0.20 Energy

0.25

030

( MeV

0.35

0.40

0.45

0.50

1

Fig. 10. Energy spectra of 500 keV electrons and positrons reflected from a thick lead target. (a) Spectra for the case of perpendicular incidence; (b) spectra for an isotropic source placed next to the target.

distributions

of the

reflected

electrons,

integrated

over all spectral energies. The differences between the angular distributions of positrons and electrons are greatest for perpendicular emergence from the target.

Concluding

Remarks

The transport calculations with the ETRAN code provide a plausible and consistent picture of the differences in the reflection and transmission of positrons and electrons by foils. The agreement with experimental results is generally not very close. The origin of the discrepancies is probably at least as much experimental as theoretical. Additional experiments would be interesting, especially measurements with monoenergetic particles which are easier to interpret than those with beta-particle sources. On the theoretical side there are possibilities of improving the ETRAN code in a number of respects. The elastic scattering cross sections used here are actually for atomic gases, and more realism could be achieved by including solid-state effects. However, such improvements would be largely confined to small deflections from soft collisions [see, e.g. Berger and Wang (1988)], and are not

0

0.1

I 0.2

I 0.3

I 0.4

I 0.5

I 0.6

I

I

I

07

0.6

0.9

Fig. II. Angular distribution of 500 keV electrons and positrons reflected from a thick lead target. Note that cos 0 = I .O corresponds to emergence in a direction perpendicular to the target, and cos 0 = 0.0 to grazing emergence. (a) Angular distribution for the case of perpendicular incidence. (b) Angular distribution for an isotropic source placed next to the target.

expected to change positron4ectron transport differences very much. The treatment of energy-loss straggling in track segments could be improved at low energies through the replacement of the Landau distribution by a more accurate energyloss distributions based on detailed, realistic scattering cross sections. This would improve the accuracy of the spectra of particles emerging from the foil, but would have only a minor effect on reflection and transmission coefficients. Finally there is the possibility, suggested by the experiments of Brunger and Menz (1965) and Hilgner and Kessler (1965) at energies below 100 keV, that quantum-mechanical interference effects (diffraction) due to the presence of polycrystalline regions in foils modify the penetration of electrons or positrons. It remains to be seen how such phenomena can be incorporated into a Monte Carlo electron transport model. Acknowledgements-The author would like to dedicate this paper to his colleagues in the former Radiation Theory Section of the National Bureau of Standards for the friendship and inspiration he received from them for many years. He would particularly like to acknowledge his debt to Ugo Fano and Lew Spencer, who taught him the

MARTIN J. BERGER

916

fundamentals of radiation transport theory. Their ideas and results provided the foundations for much of the author’s work. The author would also like to express his gratitude to Steve Seltzer whose collaboration for more then twenty five years was essential for the development of the ETRAN code and for its many applications, including the present paper.

References Batra R. K. and Sehgal M. L. (1981) Range of electrons and positrons in matter. Phys. Rev. B23, 4448. Batra R. K and Sineh B. (1989) Difference in ranges of positrons and eleccons in rare-earth metals. Physy Rec. B39, 2692. Berger M. J. (1963) Monte Carlo calculations of the penetration and diffusion of fast charged particles. Methods in Computational Physics, Vol. I (Eds Alder B.. Fernbach S. and Rotenberg M.), p. 135. Academic Press, New York. Berger M. J. (1988) ETRAN-Experimental benchmarks. Monte Carlo Transport of Electrons and Photons (Eds Jenkins T. W., Nelson W. R. and Rindi A.), p. 183. Plenum Press, New York. Berger M. J. and Wang R. (1988) Multiple-scattering angular deflections and energy-loss straggling. Monte Car/o Transoort of Electrons and Phorons (Eds Jenkins T. W.. Nelson W.” R. and Rindi A.), p. 21. Plenum Press, New York. Bethe H. (1935) On the annihilation radiation of positrons. Proc. R. Sot. (London) 150, 129. Bethe H. A., Rose M. E. and Smith L. P. (1938) The multiple scattering of electrons. Proc. Am. Phil. Sot. 78, 573. Bhabha H. J. (1936) The scattering of positrons by electrons with exchange on Dirac’s theory of the positron. Proc. R. Sot. (London) A154, 195. Bisi A. and Braicovich L. (1964) Backscattering of positrons. Nucl. Phys. 58, 171. Blunck 0. and Leisegang S. (1950) Zum Energieverlust schneller Elektronen in dunnen Schichten. Z. Phys. 128, 500. Broda K. W. and Johnson W. R. (1972) Single-quantum annihilation of positrons by screened K- and L-shell electrons. Phvs. Reo. A6, 1693. Brunger W. anh Menz W. (1965) Wirkungquerschnitte fiir elastiche und inelastische Elektronenstreuung an amorphen C- und Ge-Schichten. 2. f. Phys. 184, 271. Ddsclaux J. P. (1975) A multiconfiguration relativistic Dirac-Fock program. Compul. Phys. Comm. 9, 31. Goudsmit S. and Saunderson J. L. (1940) Multiple scattering of electrons. Phys. Rer. 57, 24. Hansen H. E. and Ingerslev-Jensen U. (1928) Penetration of fast electrons and positrons. J. Phys. D: Appl. Phys. 16, 1353. Hilgner W. and Kessler J. (1965) Mehrfachstreuung mittelschneller Elektronen in Folien. Z Phys. 187, 119. ICRU (1984) Report 37, Stopping Powers for Electrons and Posirrons, International Commission on Radiation Units and Measurements, Bethesda, MD.

Jauch J. M. and Rohrlich F. (1955) The Theory of Photons and Electrons. Addison-Wesley, Cambridge, Mass. Kanter H. (1957) Riickstreuung von Elektronen im Energiebereich von IO bis 100 keV. Ann. Phys. 20, 144. Kim L., Pratt R. H., Seltzer S. M. and Berger M. J. (1986) Ratio of positron to electron bremsstrahlung energy loss: an approximate scaline law. Phvs. Rec. A33. 3002. Landai‘L. (1944) On tie energy’ loss of fast' particle by ionization. JI. Phvs. USSR 8. 201. Lewis H. W. (19;O) Multi& scattering in an infinite medium. Phys. Rev. 78, 52%. MBller C. (1932) Zur Theorie des Durchgangs schneller Elektronen durch Materie. Ann. Phys. 14, 568. Mott N. F. (1929) The scattering of fast electrons by atomic nuclei. Proc. R. Sot. A124, 426. Patrick J. R. and Rupall A. S. (1971) Transmission of low energy positrons through thin metallic foils. PI?ys. Lett. 35A, 235. Ram N., Sundara Rao I. S. and Mehta M. K. (1981) Relative transmission of 0.324- and 0.544-keV positrons and electrons in Be, Al, Cu, Ag and Pb. Phys. Ret>. A23, 1202. Riley M. E. (1974) Reiaritlistic, Elastic Eleclron Scalfering ,from Atoms at Energies Greater lhan I keV. Sandia Laboratories Report SLA-74-0107. Riley M. E., MacCallum C. J. and Biggs F. (1975) Theoretical electron-atom elastic scattering cross sections. At. Data Nucl. Data Tables 15, 201. Rogers D. W. G. (1984a) Fluence to dose equivalent conversion factors calculated with EGS3 for electrons from 100 keV to 20 GeV and photons from I I GeV to 20 keV. Health Phys. 46, 89 I. Rogers D. W. G. (1984b) Low energy electron transport with EGS. Nucl. Instr. Merh. 227, 535. Rohrlich F. and Carlson B. C. (1954) Positron+lectron differences in energy loss and multiple scattering. Phys. Rev. 93, 38. Seliger H. H. (1952) The backscattering of positrons and electrons. Phys. Reo. 88, 408. Seliger H. H. (1955) Transmission of positrons and electrons. Phys. Reo. 100, 1029. Seltzer S. M. (1988) An overview of ETRAN Monte Carlo methods. Monte Carlo Transport of Electrons and Phorons (Eds Jenkins T. W., Nelson W. R. and Rindi A.). p. 153. Plenum Press, New York. Singh B. and Batra R. K. (1987) A method for calculating mass attenuation coefficients for beta particles. Appl. Radial. Isot. 38, 1027. Takhar P. S. (1966) Direct comparison of the penetration of solids by positrons and electrons. Phys. Lerr. 23, 219. Thummel H. W. and Krivan V. (1964) Untersuchungen zur Energieabhangigkeit der Ruckstreufaktoren von Betastrahlung mit einem Szintillationszahler. Nuc/eonik 6, 379. Tseng H. K. and Pratt R. H. (1973) Comments on screening and shell ratio effects in single-quantum pair annihilation. Phys. Rec. A7, 1423. Uehling E. A. (1954) Penetration of heavy charged particles in matter. Ann. Reo. Nurl. Sri. 4, 3 15.