I

Monte Carlo Studies of Electron

and Photon Transport at

& Energies up to 1 000 MeV

Stephen M. Seltzer

and g

Martin J. Berger

Radiation Physics Division

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

July 1 978

Prepared for Office of Naval Research

I ^Arlington, Virginia 22217

100 I .U56 i 73 -153^

I

' --- —- ^

I

Wslfional Biiirea!! of StandtHf^O aRY 1'

NBSIR 78-1534

\ t * MONTE CARLO STUDIES OF ELECTRON

AND PHOTON TRANSPORT AT

ENERGIES UP TO 1000 MeV

Stephen M. Seltzer

and

Martin J. Berger

Radiation Physics Division

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

July 1978

Prepared for Office of Naval Research

Arlington, Virginia 22217

' ' 'Irp - __i_

U.S. DEPARTMENT OF COMMERCE. Juanita M. Krepa, Sacrettry

Dr. Sidney Harman, Under Sscretery

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernaat Ambler, Director

MONTE CARLO STUDIES OF ELECTRON AND PHOTON TRANSPORT

AT ENERGIES UP TO 1000 MeV

Stephen M. Seltzer

and

Martin J. Berger

Radiation Physics Division

Center for Radiation Research

National Bureau of Standards

Washington, DC 20234

This report describes calculations of electron-photon showers initiated

by electron beams with energies from 50 to 1000 MeV. Results obtained with

the Monte Carlo program ETRAN are presented for diverse problems including:

(1) the spatial distribution of energy deposited in water, air and lead

targets; (2) the escape of scattered electrons and secondary bremsstrahlung

from air and lead targets; (3) energy-loss straggling of primary electrons

in air; (4) depth-dependent electron spectra (differential tracklength

distributions) in air.
1. INTRODUCTION

This report describes calculations of electron-photon showers initiated

by electron beams with energies from 50 to 1000 MeV in water, air and lead

media. The calculations were done with the Monte Carlo computer program ETRAN.

The ETRAN program was originally developed^ solely as a tool for solving

2 electron transport problems, to be applied at energies up to a few MeV. The

treatment of electron transport in ETRAN is based on a segment model in which

the multiple-scattering deflections and energy-losses in successive track

segments are sampled from pertinent probability distributions given by

analytical theories. The ETRAN program was later extended to include the

propagation of secondary bremsstrahlung photons, and was applied in the

energy region from a few MeV to 100 MeV where the bremsstrahlung photons

3 4 5 are making an increasingly significant contribution to energy transport. ’ ’

The transport of photons is treated in ETRAN by simulating photoelectric

absorption, Compton scattering and pair production in rather direct analogy

to the physical processes.

In the present work the application of the ETRAN code has been extended

to the energy region from 100 to 1000 MeV, where energy transport by photons

is the dominant phenomenon and energy transport by electrons is a significant

but relatively minor factor.

Because of its origin as an electron transport program, ETRAN treats

energy losses and angular deflections in successive track segments in rather

detailed fashion. Energy losses are sampled according to the Landau^ energyloss straggling distribution, instead of being evaluated in the continuousslowing-down approximation. Angular multiple-scattering deflections are

sampled from the Goudsmi t-Saunderson^ distribution, and the latter is

evaluated with the use of an elastic scattering cross section based on a

2
o combination of the Mott cross section with a screening correction due to

Q

Molifere. The Mott cross section is an improvement over the Rutherford cross

section, taking into account relativistic and spin effects according to the

Dirac equation.

There is a considerable literature on electron-photon shower calculations

at high energies. Th early work was largely analytical and is summarized, for

example, in the book by Rossi. Later work has been done mainly with use of the

11-18

Monte Carlo method, usually with a track-segment model for electron transport.

The track segment models used in high-energy shower calculations differ from

those in ETRAN in two respects; (a) the sampling of angular deflections from a

Gaussian diffusion approximation, or from the Moli^re theory based on the

Rutherford cross section; (b) the use of the continuous-slowing-down approximation rather than Landau's straggling theory to take into account the cumulative

effect of small energy transfers to atoms. Additional simplifications have

sometimes been employed to reduce the computational effort involved in following

the large numbers of secondary electrons in a shower. The various Monte Carlo

programs are similar to ETRAN in that they carefully simulate events resulting

in sudden large energy transfers to knock-on electrons or bremsstrahlung photons

whose subsequent histories are also followed.

The Goudsmi t-Saunderson multiple scattering distribution used in ETRAN

has the form of a Legendre series, which - while formally exact - converges

quite slowly at high energies. The use of the more approximate Moli^re distribution, in which this Legendre series is already implicitly summed, avoids this

convergence difficulty. However, it appears difficult to treat the energy

history of electrons at very high energies adequately while at the same time

choosing track segments long enough so that the available multiple-scattering

theories are fully valid. In a refined model one might want to sample from a

pi ural -scattering distribution valid for any number of collisions, but such a

3
refinement is probably unwarranted in the context of a high-energy shower

calculation. We also note that the use of the Landau straggling distribution

in ETRAN complicates the program and lengthens the numerical computations

compared to the use of the continuous-slowing-down approximation.

Finally, it should be pointed out that the refinement of the multiplescattering distributions used to simulate electron tracks is only one of the

factors determining the length of the entire Monte Carlo calculation. Another,

perhaps more important, factor is the number of segments into which each track

is divided. The choice of the lengths of successive segments should ideally

depend on the electron energy, and should be arranged so as to obtain a faithful model of the electron transport, while at the same time minimizing the

number of segments.

In the present exploratory calculations, we were mainly interested in

seeing how well the ETRAN code works at very high energies, and in obtaining

immediate answers to certain problems. We therefore have made no effort to

optimize the choice of track segments (which would require a good deal of

numerical experimentation) but instead have made simple choices taken over

from earlier work at lower energies. For example, when sampling the showers

in water generated by an electron with an initial energy of 1 GeV, we have

chosen segments such that the average energy loss per segment was approximately

11% of the electron energy at the beginning of the segment, and have further

subdivided each segment into two equal sub-segments for the purpose of sampling

multiple-scattering angular deflections. It appears to us now that the segment

sizes were somewhat too large at high energies (near the source energy) and

too small for the numerous secondary electrons with low energies.

The results to be presented in the following Sections of this Report fall

into five categories: (1) spatial distribution of energy deposition (absorbed

dose) in water and air, as a function of depth in the medium and radial distance

4
from the beam axis; (2) the backscattering of energy, in the form of secondary

bremsstrahlung photons, from a semi-infinite air medium; (3) energy-loss

straggling of electrons in air (including a comparison of the Monte Carlo

results with a semi-analytical calculation); (4) evaluation of electron flux

spectra as a function of depth in the medium, for primary as well as secondary

electrons; (5) interaction of a high-energy electron beam with a lead target,

including the spatial distribution of energy deposited in the target and the

escape of radiation energy (transmission, reflection) from the target, for

both perpendicular and oblique incidence of the beam onto the target.

2. SPATIAL DISTRIBUTION OF ENERGY DEPOSITION IN WATER AND AIR

A monoenergetic, narrowly collimated ( point-monodi recti onal ) beam of

electrons was assumed to irradiate a semi-infinite medium, and energy

deposition was calculated as a function of the depth z in the medium and of

the radial distance p from the beam axis. Actually the target was assumed

to be a cylinder centered on the direction of the incident beam, with a

radius of 70 g/cm and a length of 460 g/cm . The leakage of energy through

the lateral surface and the transmission face of the cylinder amounted to

only a fraction of a percent of the incident energy.

Electrons in the shower were accurately followed (according to the

detailed Monte Carlo model) down to an energy of 2 MeV, and the small amount

of energy deposition along their residual tracks below that energy was

estimated approximately. Photons were followed down to an energy of 10 keV.

The contribution to the total amount of energy deposited in the target by

photons with energies less than 10 keV was insignificant.

Results for the depth dose, i.e. the energy deposition per unit depth

(integrated over all radial distances) are given in Fig. 1 for a water medium

and various electron-beam energies:

5
EnBrgy (MbV) 1000 500 250 125 60

Size of Monte Carlo sample 1000 2000 4000 8000 10000

The size of the sample refers to the number of primary electrons. The total

number of electrons in the cascade was many times larger. At 1000 MeV it is

19

possible to make comparison with the experimental data of Crannell et al.,

and with the results obtained by Nelson^*^ with the Monte Carlo program EGS.^^

The agreement with experiment is quite good and is well within the combined

limits of theoretical and experimental error. The agreement with the EGS

results is close for most depths; however, for very large depths the EGS results

are somewhat lower than the ETRAN and experimental results. This discrepancy

is possibly due to the fact that the cylindrical target assumed in Nelson's

calculation had a radius of only 24 g/cm (resulting in a significant lateral

escape of energy from the target). Beyond the shower maximum the depth-dose

curves, on a semi-logarithmic plot, are almost linear, indicating that the

transport process is controlled by the approximately exponential attenuation

of the secondary bremsstrahlung. In the region up to the shower maximum the

transition curves are governed by the penetration of the primary and secondary

knock-on electrons. The energy deposition at the surface of the medium is

smaller than that which one would predict using the stopping power (mean

collision loss), due to the forward transport of energy by energetic knock-on

electrons.

A comparison of depth-dose curves for a 500-MeV beam in water and in air

is shown in Fig. 2. The transition curve for air is shifted towards smaller

depths compared to that for water, because the mass stopping power is larger

in air than in water for electrons of the same energy. This difference of

stopping power is the result of the polarization of a dense medium by the

electron beam (density effect correction).

6
Fig. 3a shows the two-dimensional distribution of energy deposition in

water from a 1000-MeV beam, as a function of the depth z and radial distance p

from the axis of the point-monodirectional incident electron beam. The

results shown are averaged over the indicated depth intervals. Also shown

for comparison are experimental results of Crannell et al.,^^ and Monte Carlo

(EGS) results of Nelson. In Nelson's work, the dependence of the energydeposition distribution on the choice of track segment sizes was explored,

and the final results reflect a choice which gives very good agreement with

Crannell 's experimental data. The agreement among the ETRAN, EGS and experimental results is generally to within 20% or better, which we consider

satisfactory in view of the expected theoretical and experimental uncertainties.

Fig. 3b compares two-dimensional energy-deposition distributions in water and

in air from a 500-MeV beam. The differences are relatively small, generally

no greater than 20%. The differences are similar in magnitude to those for

the corresponding depth-dose curves in Fig. 2, and have their origin in the

differences in the density-effect correction to the stopping power for water

and air. However, the statistical fluctuations of the Monte Carlo results

obscure the situation.

Tables 1 to 4 give the two-dimensional energy-deposition distributions

in water for 1000-, 500- , 250- , and 125-MeV beams. The results are presented

in terms of the percentage of the incident energy that is deposited in

various depth intervals and radial intervals.

3. BACKSCATTERING OF PHOTONS FROM AN AIR MEDIUM

When a semi-infinite air medium is irradiated with high-energy electron

beams, the transport of energy is predominantly in the forward direction. The

very small amount of backscattered energy is nevertheless of interest for

radiological safety considerations. We have investigated this problem for

the case of monoenergetic , point-monodirectional beams with energies up to

7
1000 MeV, and have studied the emergence of radiation energy from a semiinfinite medium as a function of the distance from the electron beam.

We have found in our calculations that the backscattered radiation energy

escaping from the medium consists almost entirely of photons, and that only

0.09% of the energy from a 100-MeV and 0.014% of the energy from a 1000-MeV

beam is backscattered. Such results cannot readily be obtained merely by the

direct-analogue simulation of particle trajectories, because excessively large

numbers of tracks would have to be sampled. A collision density method has

been found effective in dealing with this problem. When generating photon

Monte Carlo histories, we focussed attention on the collision density, i.e.

the set of all events in which a photon was born or scattered. For each event

we then computed (a) the probability that it would emerge from the medium

without undergoing a further scattering, and (b) the radial distance between

the point of emergence and the beam axis.

An average over the collision density then provides the desired backscattering results. Calculations were done for the following cases:

Beam energy (MeV) 1000 500 200 100 50

No. of primary electrons

in Monte Carlo sample 1000 2000 2000 5000 5000

The number of collision points which contributed to the backscattering results

amounted to several hundred thousand.

In Fig. 4a, pertaining to a 500-MeV beam, the calculated' dose shown is

that which would be recorded by an air-equivalent detector placed at the

entrance plane of the air medium. The dose is given as a function of the

distance p from the detector to the beam axis. It has been found that this

radial dependence, for all beam energies considered, can be fitted by the

empirical one-parameter formula

8
D(p) = 8.1 X 10 ^ Pq + e

where D(p) is in MeV/g(air) and the parameter p^ is in meters. The dependence

of p^ on the beam energy is shown in Fig. 4b. The number of backscattered

2 photons, and the amount of backscattered photon energy per cm are given in

Fig. 4c, both as functions of the radial distance p. Also shown is the mean

energy of the backscattered photon, which is independent of radial position.

In fact, the mean energy for all beam energies and radial positions was found

to be ~ 150 keV.

4. ENERGY-LOSS STRAGGLING IN AIR

The ETRAN program has also been used to calculate the energy-loss straggling

in air, i.e. the statistical distribution of the energy losses which primary

electrons suffer in the course of penetrating a layer of air. In this work,

attention was focussed on the distribution in energy of the transmitted

electrons, and not on the energy deposition in the air layer, so that secondary

electrons could be disregarded. The calculations were done for initial

electron energies of 100 and 1000 MeV and for air layers with thicknesses up

2 2

to 90 g/cm . For layers with thicknesses up to 10 g/cm , the differences (due

to multiple scattering) between the actual pathlengths traveled and the layer

thickness z is negligibly small, and it is possible also to make semianalytical calculations of the straggling distribution which can be compared

with the Monte Carlo results.

The energy-loss straggling results from the combined effects of (a) energy

losses to atoms and secondary electrons in ionization and excitation events, and

C (b) the emission of bremsstrahlung quanta. Landau has treated straggling

including only effect (a), using the Rutherford cross section. We have

extended his treatment to include also effect (b), using the Bethe-Hei tier

bremsstrahlung cross section (high-energy approximation, complete screening,

formula 3BS(a) in the review article of Koch and Motz ). In setting up the

9
22

solution, we have followed the formalism of Vavilov, who - in contrast to

Landau - imposed an upper limit on the energy transfer in a single collision.

In our case, the limit was set equal to one half the kinetic energy in

ionizing collisions, and the total kinetic energy in bremsstrahlung events.

These restrictions affect the results very little. In any case, our solution,

just as those of Landau and Vavilov, is valid only when large energy losses

are very rare, and when the average net energy loss over the pathlength

considered is small compared to the initial electron energy.

The resulting energy-loss straggling formula is given in the Appendix,

and involves a numerical integral over a rather complicated integrand. This

integrand is obtained by analytically inverting the Laplace transform of the

straggling distribution. We have found the required numerical integration to

be difficult, because the integrand is highly oscillatory. Perhaps it would

be more efficient to evaluate the inverse of the Laplace transform of the

straggling distribution numerically with use of Fast Fourier Transform techniques.

For different air target thicknesses, the results of the Monte Carlo

calculations of energy-loss straggling (based on 20,000 histories of primary

electrons for each source energy) are compared in Figs. 5a-d with the

predictions of the formula given in the Appendix. This formula appears to

fit the Monte Carlo data quite well. We find that the value of the most

probable energy loss is very little affected by bremsstrahlung losses, and

could therefore be predicted by using the Landau theory alone.

The results of the Monte Carlo calculations for air layers of greater

thickness are summarized in Fig. 6, which shows the fractional transmission

of electrons with energy losses less than through a layer of thickness.

10
5. DIFFERENTIAL TRACKLENGTH DISTRIBUTIONS FOR ELECTRONS IN AIR

A complete description of electron-photon showers consists of the specification of electron and photon tracklength distributions that are functions of

energy, direction and position. Such data can be obtained from the ETRAN

program. By way of example, we present here electron tracklength distributions,

that are differential in energy, as functions of the depth in the medium. These

results are integrals over all directions and radial positions of electrons.

Results for the case of a 500-MeV electron beam in air are shown in

Fig. 7 which illustrates the evolution of the electron energy spectrum with

increasing depth. Close to the entrance face of the medium, the spectrum

consists almost entirely of primary electrons with energies close to the

incident beam energy. At larger depths, the primary electron component is

increasingly attenuated and a buildup of a secondary component occurs.

Eventually the secondary component of the spectrum is completely predominant

and has a shape that is rather independent of depth for spectral energies

below ~ 20 MeV. These results are based on a sample of 2000 electron-photon

cascades.

6. INTERACTION OF AN ELECTRON BEAM WITH A LEAD TARGET

In this Section, we consider the interaction of a 500-MeV electron beam

with a target of high atomic number. The beam is assumed to be incident on

the flat endface of a lead cylinder. The diameter of the target was kept at

30 radiation lengths, and the length of the cylinder was varied up to a value

* of 20 radiation lengths. As shown in Fig. 8a, the point-monodirectional beam

was assumed to be incident at the center of the entrance face either perpendicularly or obliquely at an angle 0^ = 70° with respect to the cylinder axis.

* 2 The radiation length in lead was taken to be 6.5 g/cm .

11
Fig. 8a shows the depth-dose curves in a lead target for perpendicular and

oblique incidence, obtained from a Monte Carlo sample of 500 electron-photon

showers. In Fig. 8b the same data are replotted as function of the slant

distance measured along the incident-beam direction. In this representation the

dependence of the depth-dose curves on the initial beam obliquity 6^ is greatly

reduced. Fig. 8c shows the radial distribution of deposited energy in the lead

target, at a depth of 3-4 radiation lengths (where the depth-dose peaks). The

results in Fig. 8c are for perpendicular incidence.

We have also calculated data on the escape of energy from irradiated lead

cylinders through the entrance face (reflection), front face (transmission) and

through the lateral surface of the cylinder. Fig. 9a shows the dependence of

the energy escape on the length of the cylinder, for the cases of perpendicular

incidence and oblique incidence. The energy leaving the cylinder by reflection

or through the lateral surface is carried almost entirely by photons. For

escape through the front face, this is the case also, except for cylinders

thin enough so that the primary electrons have a significant chance to be

transmi tted.

Figs. 9b, c and d show the energy spectra of the electrons and photons

transmitted through the front face and reflected from the entrance face of

cylinders with diameters of 30 radiation lengths and lengths of 1 , 5 and

20 radiation lengths. These results are for perpendicular incidence of a

500-MeV electron beam. The photon spectra include, in addition to bremsstrahlung, annihilation quanta and characteristic x-rays. The corresponding

angular distributions of the escaping radiations are shown in Figs. 9e, f and g.

12
APPENDIX; ELECTRON ENERGY-LOSS STRAGGLING FORMULA

Let P(A)dA be the probability that an electron after traversing a pathlength s has suffered a net energy loss between A and A + dA. The probability

density P(A) can be shown to be

P(A) = (2e‘^^/7TT) e'^^^cos(yX + Kf2)dy,

where

a = 1 - (x/K)[n'(iin 2 + y) - t(n' - nt/2)],

fi = -cos y - y Si(y) - (x/K){n'[^n y

- Ci(2y)] + t[b sin 2y/(2y)

+ rit(l - cos 2y)/(2y)^]},

f 2 = y[iin y

- Ci(y)] + sin y - (x/k){ti ' Si (2y ) - t[{n' - b) cos 2y/(2y)

- nt sin 2y/(2y)^]},

A = I(A - A^) - K{£n 2 - Y - [T"/8 - y(2T + p)£n 2]/(T + y)^}.

Other symbols have the following meanings:

K = 2^T,

C

= 2Tirg2N^(Z/A)(M/Bys,

X

= 4arYN (2 + l)(Z/A)s,

c a

n = £n(183 Z‘‘^y,

n' = (4n + l/3)/3,

t = T/(T + u),

b = ri' - nt,

^c ^dx^col^’

Y

= Euler's constant = 0.577216...,

N^ = Avagadro's number = 6.022 x 10^^ mol

13
r = classical electron radius = 2.818 x 10 cm,

e

u

= electron rest mass = 0.511 MeV,

a = 1/137,

Si is the sine integral,

Ci is the cosine integral,

T = incident electron kinetic energy,

3

= electron velocity in units of light velocity,

s = pathlength, g/cm^

,

Z = atomic number,

A = atomic weight,

(^)coi collision-loss stopping power, MeV cm^/g.

14
Table

1.

Percentage

of

energy

from

1000-MeV

point-monodirectional

electron

beam

that

is

deposited

in

the

indicated

depth

and

radial

intervals

in

a

water

medium.

O) o

o

CD

OC

cLlJ QQ

U

CD

fO>

s-O)

+-> c

CL

O) Q

O O^ CM CM

o o CD O r— CM

O O CM CO

o o OO CM

o o CO 00

o o

^ CD

O O oo 'd"

O O CM cn

o o r— CM

CO O

CM CO

O CM

CO

fO Csi > E

s- o

•o <U

fO +-> C71 Ctl

ns

CO CO LO C3^ CM CTi 00 cn o r— o CO o CO CO cn <— o o o o o o O o ,

o o o o o

00 CO cn CTv o CM CO 1— CM CO CM o CM oO <T> o CO CM CO cn CM LO CM o o o o O r— r- o o o o O

o cn o CO un 00 CO 1— CO cn

CO CO cn 00 CO LO 00 o o LO CO r“ o o o o CM 1

CM r— o o o o

LO00 ,60 cnoLT) cn CO

00

r*~ CO

cn 00CM o r- CM CM CO CM CO CM ,

o o o O o

cn o CO r— o LO 00 LO 00 LO LO

cn CX3 LO CO CO r— o o <— o

o CM P~ r— o o o o o o o

1-^ r— CM CO cn CO LO O CO LO CO CD CO <— cn CO LO CO CM CM o o o o o

CM CM r- o o O o o o o o o

,65 ,33 CD

CD oo

LO

CO

CO LOo

COo

CMo

fmmm o o

CMo

r~" o

,

r— o CO o o o o o o o o o o

00 CO cn LO CO CO CM CM ,

o r~ o CM CM — o o o o o o o o o

CM o o o o o o o o o o o o o

00 CO o LO CD CM CM , o o o r— o CM o O o o o o o o o o o

CM o o o O o o o o o o o o o

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

o o o o o o o o o o o o o o

CO CM o o o o o o o o o o o o LO o o o o o o o o o o o o o

o o o o o o o o o o o o o <~>

CO o o o o o o o o o o o o o CO o o o o o o o o o o o o o

o o o o <d o o o o o o o o o

LO LO O r— r— CMCO^CDOOCMCDO o o 1— 1— CM OJ 1^

I I I I I I I I I I I I I I cn cn OOl— 1— CMC0"5d-C000CMC0O o t— CM oo
Table

2.

Percentage

of

energy

from

500-MeV

point-monodirectional

electron

beam

that

is

deposited

in

the

indicated

depth

and

radial

intervals

ina

water

medium.

OJ o

lT> oq;

<=c UJ

CQ

C\J CT> CO U3 CM o o o o o o r— r— CO CO CO CsJ uo CM oC\J CNJ o o o o o o o o o o O O o O

UD o o CT> CO UD CM o o o o r— r— ro cn n— CO LT) CO >x> CM

UD o « • 1 C\J o o o o o o o o 1— o O O o o , CO CO O CM CM CM CTl LD CT o o CM CO CO CO CO LD 1 o LD CO LD 1

CM CD • . o o O o o o P“~ ' CM ' o o o o

un cn CM CO CO cn CTl

O CO CM CTl o o CO CD CTi r— o LD CTl o LD CM CO r—

CO CM o o O ' CM CM CM CM '

OO o o

CNEo CT) LD CO =d" cn CM LD CO LD o o cn CM CO CM CM CTL CM o 1— o

CD CD CO o O

— CM CM CM >—

O o o o o o

CO

fO cr>O ID CO CT) CTl ID CM LD CO CO > O o r— CM CO CM CO ID LD CO o o o o S- CD

O) ,— CM CM CM CO r—

O o o o o o o -t-j c

»—<

CD CO COO o CO r— CO CM r— CM 1—

jC o O r— CO LD

O CTl CO CO r— o o o o o o +-> CO • CL r— r— 1 O o o o o o o o o o

O) Q

O CM CM 1 CT LD in CM r— ,CM fmmm o O CM r~ CT CO t— r-“ o o o o o o o CM CO

CM CM o o O o o o o o o o o o

CO LD CO CM LD r“ o o O CO CM 1— o o o o o o o o o r— CM • CO o O o o o o o o o o o o o

CD o CM o o o o o o o o o LD O CT 1— o o o o o o o o o o o o r— • ' o o o o o o o o o o o o o O CO o o o o o o o o o o o CM LD o o o o o o o o o o o o o o o o o o o o o o 'o o o o

CT o o o o o o o o o o o o O CM LD o o o o o o o o o o o o o

CO o o o o o o o o o o o o o

LD LD

t/^ • • O 1— 1 oo CD LD CO CM LD o o o r— fD <N CM CM

fO > E

•r- S_

u ! 1 1 1 1 1 1 1 1 1 1 1 1 1 -a O)

HD +J CT LO

D

q; c • • H— o o 1 r— CM CD LD CO CM LD o o 1— «

CM CM

T /r
Table

3.

Percentage

of

energy

from

250-MeV

point-monodirectional

electron

beam

that

is

deposited

in

the

indicated

depth

and

radial

intervals

ina

water

medium.

OJ omCM

>-C3

cxl

cC

LU

CQ Eu

cn

to>

s-O)

-(-> D.

0) Q , CM CO CT> cn CM CM 00 00 o o o O o o o o r— CO CM CM CM oCM CM o o o o o o o o O o o o o o

CO CO CO CO <n CO CO CO

OO o o o o CO LO CO LO CM

COO ( CM o o o o o o o o o o o o o o

LO 00 CM CO o 00 LO cn 1^ o 00 CO CO o O o o (—. (O 00 LO CO CO CM

CM CO1 o o o CD o o o o ( O o o o o

CM CO CM CO CO o CM cn cn LO CM 00 o o f— CO LO r— r— CO r— LO CO LO fmm. CO CMr— o o o O r— •“ CM CM fmmm o o o o o o CO CM p~ cn 00 CD LO 00 00 o o CM CO LO LO LO CO CO LO CM r— o CO 00 o o o o r*~ r— CM O o o o o

CTi CD CT> LO CO cn CO CO CO LO o o LO CM CO cn 00 CO o CO r— o o CO o ' (— CO CM CO ' O o o o o

LO 00 CO CO CO CM LO CM o o LO 00 CM cn CM o o o o o oo o r— r— r-“ CM ' o o o o o o o o

CM o CO cn o 00 CO CO CM CO o o CO r— CM CO r— o o o o o o

CM CO ' CO CM 1 o o o o o o o o o

CM CO CO CO o CM cn CO CO CM CM o o r— CO 00 CO CO 1— o o o o o o o o

CM

LO CM o o o o o o o o o o o o o LO LO r~ o o o o o LO o 00 CM o o o o o o o o o o o

f—

CO o o o o o o o o o o o o o

CM o o o o o o o o o o

CM LO CM o O o o o o o o o o o o o

CM o O o o o o o o o o o o o

CO CM

O o o o o o o o o o o o o CM o O o o o o o o o o o o o

-- o O o o o o o o o o o o o

LO LO

(/) • • o r— CM CO CO 00 CM CO o o o

(O CM 1— CM CM

(T3 > E

i. (_) 1 1 1 1 1 1 1 1 1 1 s 1 1 1 "O (U

(T3 +J cr LO LO

Qc: c • • 1^-1 o o r— r— CM CO CD 00 CM CO o o 1

•

CM CM

17
Table

4.

Percentage

of

energy

from

125-MeV

point-monodirectional

electron

beam

that

is

deposited

in

the

indicated

depth

and

radial

intervals

ina

water

medium.

O)

lO

C\J

>-

C3

Cd

UJ

<c

UJ

CQ Eo

<n>

s-O)

Q.

cu Q

O o o 1

CM CO VO LO CM cr> cn CM o o O o o o o o o o p— r— CD CM CM oC\J CM O o o o o o o o o o o o o o

CM CO 00 r-^ CO o LO LO o O o o o o o o CM r— CM CM CO CM O ' C\J CO o o o o o o o o o o CD o o

CO LO VO LO LO CM CD 00 VO o o o o o o r— r— CO CO LO CO CO LO CM

C\J VO ' ' o o o o o o o o o o o o o o

VO oo VO cr> CO VO CO CO CO o LO o o o 1— 1

r— CO 00 00 CM 00 VO VO CM

00 CM* CO o o o o o o o 1 o o o o o

00 00 o CO VO 00 CO VO o cn CO CM o o o r~ CM CM LO LO (Ti LO CO CO r—

VO 00 o o o o o o CD o o o o CD o o

VO cr> CM CM CO VO LO CO 00 o o CM LO VO VO VO VO CO 00 VO CM CM o VO o o o o r— ' CM 1 1 o o o CD o

CT> VO LO CT> 00 LO CM cn LO 00 CM o o CM o CM VO CO o CO CO CM CD o o o CO o o f— CM CM CO ) o o CD o o o

oo LO CM CO 00 o 00 cn o LO CO LO CM o o cr> CM 00 CT> 1— CM r— o o o o C\J oo

CO CM CM CM CO CM o o o o CD o o

CM CT^ 00 o LO cn o CM CM CO o o (Ti VO CO CM o o o o o o r— CM

LO CO ' 1 o o o o o CD o CD o

ro CM o CTt CO CM o o un o 00 CM 1

o o o o o o o o <D o r“~ VO 1 o o o o o o o o o o CD o

LO CM o o o o o o o CD o OsJ LD LO o o o o o o o o o o CD o o CD o o o o o o o Q CD CD o

CO CO o o o o o o o o o CD o o CM CTi o o o o o o o o o o o CD o

CM o o o o o o o o o o o CD o

LD LO

Ln • • r— o r— (

CM CO VO 00 CM VO o CD o r— to CM r— CM CM

fO > Eu 1 1 1 1 1 1 1 1 1 1 1 1 t 1 T3 cu

fO 4-> cr. LO LO

Dd sz • • o o r— 1

CM CO >^1- VO 00 CM VO o o r— CM CM

18
FIGURE CAPTIONS

Fig. 1. Energy deposition per unit depth in a water target irradiated by

electron beams with energies T between 60 and 1000 MeV. Results are normalized to one incident electron.

The histograms (and the curves in the insert) were calculated with

the Monte Carlo code ETRAN. For T = 1000 MeV, the circles (o) and

triangles (A) are replicate experimental data of Crannell et al [19];

the crosses (x) are results obtained by Nelson [20] with the Monte

Carlo code EGS [17] and have been plotted at the midpoints of his

histogram bins.

Fig. 2. Comparison of calculated energy deposition per unit depth in water

and air at 500 MeV.

Fig. 3. Radial dependence of energy deposition per unit volume from pointmonodirectional electron beam. The electrons are incident along the

z-axis, and the radial dependence is given as a function of the

distance p from the z-axis. The results are normalized to one incident electron and are averaged over various depth intervals.

Some histograms are shifted along the abscissa, with the p=0 position

indicated.

a) 1000-MeV electron beam, water target. Solid histograms are calculated with ETRAN code, crosses (x) are calculated with the

EGS code [20], and the dotted histograms are experimental results

of Crannel et al [19].

b) 500-MeV electron beam, comparison of energy deposition in water

and in air targets. Results were obtained with ETRAN code,

assuming a density of 0.001205 g/cm^ for air. The results for

air are shown in scaled form and must be multiplied by the cube

of the density to obtain the energy deposition in MeV/cm^.

Fig. 4. Backscattering of secondary bremsstrahlung photons from an air target

irradiated by a point-monodirectional 500-MeV electron beam incident

perpendicularly. The results are shown as functions of the distance

from the beam axis, and are normalized to one incident electron.

a) Absorbed dose delivered to a small air-equivalent detector at the

surface of the target. The dotted curve is a fit by a formula

given in Fig. 4b. To convert from MeV/g per incident electron to

units of rads/coulomb, the ordinate scale should be multiplied by

10 ''.

number of photons per cm^ reflected from target.

Jpi amount of photon energy (MeV per cm^) reflected

from target

E = niean energy of reflected photons, in MeV

b) Formula fitted to the Monte Carlo data for the absorbed dose

from reflected photons.

19
Fig. 5. Energ>'-loss straggling of electrons in air. P(A) is the probability

that the electrons will suffer an energy loss between A and A+dA

when traversing a plane-parallel target of thickness z. The area under each distribution shown is equal to the number transmission

coefficient.

Monte Carlo results: Histograms are for an incident electron energy

T = 500 MeV. Points are for T = 1000 MeV

aHd are plotted at midpoints of^histogram bins.

Analytical results (for T = 500 MeV): Dotted curves are from

Landau's theory of collision-loss straggling.

Solid curves are from an extension of Landau's

theory that includes fluctuations due to

bremsstrahlung emission; dashed curve is an extrapolation.

a)

b)

c)

d)

Target thickness

Target thickness

Target thickness

Target thickness

z=2 g/cm^

z=5 g/cm^

z=10 g/cm^

z=20 g/cm^

Fig. 6. Integral energy-loss straggling curves in air derived from the Monte

Carlo results for P(A) at 500 MeV. QlA^iax) fraction of the

incident electrons transmitted through a layer of thickness z with an energy loss A < A^qxFig. 7. Doubly differential electron tracklength distribution per unit spectral

energy and depth. Results were calculated with the ETRAN code for a 500-MeV beam of electrons incident perpendicularly on a semi-infinite

air medium. The distributions are normalized to one incident electron

and are averaged over the indicated depth intervals. The distributions

include primary electrons as well as all secondary electrons from

ionizing collisions and photon interactions.

Fig. 8. Spatial distribution of energy deposited in a cylindrical lead target

by a 500-MeV electron beam. As indicated in the diagram, the beam is

incident at the center of the entrance face of the cylinder, at an angle Go with respect to the cylinder axis.

Energy deposition has been computed with the ETRAN code as a function

of the depth z in the cylinder and the radial distance p from the

axis. These distances are expressed in units of the radiation length

X[^ (= 6.5 g/cm^ Pb)

.

a) Energy deposition per unit depth, for perpendicular incidence

(00=0°) and oblique incidence (0o=7O°).

b) Same data as in Fig. 8a, plotted as function of the scaled

depth z/(X^cos0o) measured along the direction of incidence.

c) Energy deposition per unit mass as a function of the radial

distance p. Results shown are for 0o=O°, and are averaged

over the depth interval from 3 to 4 radiation lengths.

20
Fig. 9. Transmission and reflection of energy by a cylindrical lead target

irradiated with a 500-MeV electron beam. The beam is incident at the

center of the entrance face, at an angle 6o with respect to the axis.

The escape of energy from the target has been computed with the ETRAN

code as a function of the target thickness z (distance from entrance

to front face), for a fixed diameter of 195 g/cm^ (equal to 30 Xp»

where is the radiation length).

a) The solid curves show the fraction of the incident energy that

emerges from the various parts of the cylinder surface in the

form of electrons and bremsstrahlung photons. The dotted curves

give the transmission in the form of photons only.

b) Spectra of radiations escaping from target. The quantity

plotted is the number per unit spectral energy E of photons

or electrons transmitted through the front surface or reflected

through the entrance surface. Results are normalized to one incident electron.

The quantity Pn is the area under the spike associated with the

escape of annihilation quanta. Target thickness is 1 radiation

length.

c) Similar to b), but for a target thickness of 5 radiation lengths.

d) Similar to b), but for a target thickness of 20 radiation lengths.

Electron spectra are very small and are not shown. Spectrum of

reflected photons (not shown) is practically the same as that in

c).

e) Angular distribution of radiations escaping from target through

the front face (transmission) or the entrance face (reflection).

The quantity given is the intensity, i.e. the amount of energy

per unit solid angle, carried by electrons and photons. The

results are normalized to one incident electron.

The exit angle 6 is defined such that both for transmission and

reflection the values of 6 go from 0° (perpendicular emergence)

to 90°. Target thickness is 1 radiation length.

f) Similar to e) but for a target thickness of 5 radiation lengths.

g) Similar to e) but for a target thickness of 20 radiation lengths.

21
[I

'

i

%lt^®

;

/..> < .! ^Lf*:-'' , -T "2-r 9f#'' ? ' 3

^it^

' ' 'i^fj

mJ

;i!-

4-^, ^

l; ;V ,,vV

s*'
c

DEPTH

z.g/cm*
ENERGY

DEPOSITION

PER

UNIT

DEPTH,

MeV

cm^/g
§

8

iQ

CO n

bO

•HPu

_ O

m

ir>

g/cm2
in

o

in

CO

o

CO

U)

O!

o

CM

in

in

o

Jig

.

3b
s/A^M ‘ssvR iim Noiiisojaa

P,

METERS
METERS

50 100 200 500 1000

To. MeV

Fig. 4b
MEAN ENERGY, MeV

7 ^ o o

u

'dao •H

P,

METERS
P(A

),

MeV

Fig. 5a
10

10

10

10

T

1—TT"

» 1 t i

c Fig. 5b
P(A),

MeV

Fig. 5c
P(A),

MeV

-1

Fig. 5d

c
z, g/cm2

Fig. 6
§

%

§

8

i A*M 'Hldaa lINn Uld H10N31 MOW! 1VllNiU3ddia

00 •H

0

M

100

200

500

2

6

10

20

B0

100

200

600

2

6

10

20

T>««V

T,

M«V

T,

M«V
ENERGY

DEPOSITION

PER

UNIT

DEPTH,

D(z),

MeVcm^/g

z, g/cm2

n

u

'1C

h

z/Xr

Fig. 8a
ENERGY

DEPOSITION

PER

UNIT

SLANT

DEPTH

cos

Sg

D(z),

MeV

cm^/g

2/ COS g/cm^

Fig. 8b
ENERGY

DEPOSITION

PER

UNIT

MASS,

MeV/g

Fig. 8c
,

g/cm^

z,

g/cm^

60

80

100

120

20

40

60

80

100

120 00

<o

C*4

fT‘ O

f“

00

CDo

CO a^

00

•iH

oc

Z/Xo

2/X
ANNIHILATION

QUANTA

P.=

0.20

L-A9IAI 'IAinH103dS

Fig.

9b
>

Q)

LU

Fig.

9c
Fig.

9d
INTENSITY,

MeV/sr

Fig. 9e
Fig. 9f
INTENSITY,

MeV/sr

6 , DEGREES

fig. 9g
REFERENCES

[

1 ]

[ 2 ]

[3]

[4]

[5]

[ 6 ]

[7]

[ 8 ]

[9]

[ 10 ]

[ 11 ]

[ 12 ]

[13]

[14]

[15]

M. J. Berger, Methods in Computational Physics , B. Adler, S. Fernback

and M. Rotenberg, Eds., Vol . I, p. 135, Academic Press, New York (1963).

S. M. Seltzer, Transmission of Electrons Through Foils, National Bureau of

Standards Report NBSIR 74-457 (1974); S. M. Seltzer and M. J. Berger,

Nucl. Instr. Meth. 119 , 157 (1974).

M. J. Berger and S. M. Seltzer, Computer Code 107, Radiation Shielding

Information Center, Oak Ridge National Laboratory (1968).

M. J. Berger and S. M. Seltzer, Phys. Rev. C 621 (1970).

M. J. Berger and S. M. Seltzer, Ann. New York Acad. Sci . 1 61 , 8 (1969).

L. Landau, J. Phys. USSR 8, 201 (1944).

S. Goudsmit and G. J. L. Saunderson, Phys. Rev. 24 (1940).

N. F. Mott, Proc. Roy. Soc. A 124 , 425 (1929).

G. Molifere, Z. Naturforsch. 78 (1948).

B. Rossi, High-Energy Particles, Prentice-Hall, New York (1952).

J. C. Butcher and H. Messel , Nucl. Phys. 15 (1960); D. F. Crawford

and H. Messel, Phys. Rev. 128 , 2352 (1962); H. Messel, A. D. Smirnov,

A. A. Valfolomeev, D. F. Crawford and J. C. Butcher, Nucl. Phys. 1

(1962); H. Messel and D. F. Crawford, Electron-Photon Shower Distribution

Function Tables, Pergamon Press, New York (1970).

C. D. Zerby and H. S. Moran, J. Appl . Phys. M, 2445 (1963).

H. H. Nagel and C. H. Schlier, Z. f. Phys. 174 , 464 (1963); H. Nagel,

Z. Phys. 1%, 319 (1965).

U. Vblkel, Deutsches El ektron-Synchrotron Report DESY 65/6, July 1965;

DESY 67/16, May 1967.

R. G. Alsmiller and H. S. Moran, Nucl. Sci. Eng. 131 (1969).
[16] E. Longo and I. Sestili, Nucl . Instr. Meth0. 1 28 , 283 (1975).

[17] R, L. Ford and W. R. Nelson, EGS/PEGS, Computer Programs for Monte Carlo

Simulation of Electromagnetic Cascade Showers, in preparation.

[18] K. Shin, Y. Hayashida and T. Nakamura, Nuc. Instr. Meth. 1 51 , 271 (1978);

T. Nakamura, H. Hirayama and K. Shin, Nucl. Instr. Meth. 1 51 , 277 (1978).

[19] C. J. Crannell, H. Crannell, R. R. Whitney and H. D. Zeman, Phys. Rev. 184 ,

426 (1969).

[20] W. R. Nelson, Stanford Linear Accelerator Center, private communication,

January 1978.

[21] H. W. Koch and J. W. Motz, Rev. Mod. Phys. 920 (1959).

[22] P. V. Vavilov, Zh. Exper. Teor. Fiz. 320 (1957); JETP 5, 749 (1957).

c
U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUm.lCATION OR REPORT NO.

NBSIR 78-1534

2. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Monte Carlo Studies of Electron and Photon Transport

at Energies up to 1000 MeV.

S. Publication Date

4. Peiforming Organization Code

7. AUTHOR(S)

Stephen M. Seltzer and Martin J. Berger

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 5331439

11. Contract /Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Office of Naval Research

Arlington, VA 22217

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 2(X)-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report describes calculations of electron-photon showers initiated by

electron beams with energies from 50 to 1000 MeV. Results obtained with the

Monte Carlo program ETRAN are presented for diverse problems including:

(1) the spatial distribution of energy deposited in water, air and lead

targets; (2) the eacape of scattered electrons and secondary bremsstrahlung

from air and lead targets; (3) energy-loss straggling of primary electrons

in air; (4) depth-dependent electron spectra (differential tracklength

distributions) in air.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only die first letter of the first key word unless a proper name; separated by semicolons)

Electron; electron photon showers; energy deposition; monte carlo method; photon;

straggling; transport theory

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