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Transmission Coefficients and Residual Energies of

Electrons: PENELOPE Results and Empirical

Formulas

Tatsuo Tabata and Vadim Moskvin*

Osaka Prefecture University and IDEA *Indiana University School of Medicine

Third International Workshop on Electron and Photon Transport Theory Applied to Radiation Dose Calculation

Hyatt Regency Hotel, Indianapolis, IndianaAugust 8-12, 1999

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Abstract We have generated the data on

• the number transmission coefficient ηTN of electrons

• residual energy Tr of transmitted electrons

by the PENELOPE Monte Carlo code, and have compared them with those in the literature.

The primary electrons were assumed to be incident • with energies 0.1–50 MeV• on absorbers of atomic numbers 4-92 • at different angles.

Improvement of empirical formulas given previously for these parameters is in progress by using the data obtained.• A general formula for ηTN is given.

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Introduction

• Definitions of the quantities treated– Number transmission coefficient ηTN: the ratio of the number of electrons transmitted by a slab absorber to the number of incident electrons

– Residual energy Tr of transmitted electrons: the ratio of the total energy of electrons transmitted by a slab absorber to the number of transmitted electrons

Note: Knock-on electrons are included in most experimental and MC results as “transmitted electrons,” but not in the present work.

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Introduction (continued)

• Motivations – Better empirical formulas for ηTN

and Tr are necessary for improving the semiempirical depth–dose code EDMULT.

– An empirical formula for ηTN is useful for simple evaluation of“the average depth of electron penetration” (Ref. Moskvin)

– Bichsel’s comment on Berger’s talk at 2nd I WEPT lead us to a question: “How accurate can an empirical formula for ηTN be made?”

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Introduction (continued)

• Related previous work – Monte Carlo calculations of ηTN

• Normal incidence: Seltzer & Berger, NIM 119, 157 (1974) (ETRAN)

• Oblique incidence: Watts & Burrell, NASA TN D-6385 (1971) (for Al; ETRAN)

– Empirical formulas for ηTN • Normal incidence: Tabata et al., NIM 127,

429 (1975) & papers cited there• Oblique incidence: Tabata et al., NIM 136,

533 (1976) (for Al only)

– Monte Carlo calculation of Tr• Normal incidence for light materials only

– Empirical formulas for Tr• Simple linear relation for low-Z materials

(normal incidence)• Approximate expressions used in depth–

dose algorithms (normal incidence)

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• Generation of Data– Monte Carlo (MC) code used

PENELOPE (Ref. Fernández-Valea)– Present treatment

• Included generation of SE and photons

• The above not traced for scoring• Used “the method of full

trajectories” (Ref. Moskvin)– Incident energies

0.1–50 MeV– Absorber materials

Be, C, Al, Cu, Ag, Au, U– Angles of incidence

0–80 deg at 10-deg step, 89.9 deg

Method

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Method (continued)

• Empirical formula for ηTN under normal incidence– Determine extrapolated ranges rex

from Monte Carlo transmission curves

– Express rex /r0 by an analytic expression (r0 : CSDA range; use NIST database values)

– Compare fits to two types of function and select the better one• Rao type• Ebert type

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Results• Transmission curves: Normal

incidence– MC results compared with

experimental data

Experiment: Harder and Poschet, Phys. Lett. 24B, 519 (1967); insensitive to secondary electrons only when incident on the detector with the primary

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• Transmission curves: Normal incidence (continued)– rex: Comparison with rex from

charge-deposition distributions in semi-infinite medium

Appreciable differences: only for low energy electrons incident on the highest Z absorbers.

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• Transmission curves: Normal incidence (continued)– The reason for the differences in

rex

rex =xm +A ym

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• Transmission curves: Normal incidence (continued)– Analytic expression for rex/r0

rexr0

= 1a1 +a2 1+a3 τ0

a4 +a5τ0a6( )

a1 =0.2946Z0.2740

a2 =18.4Z−3.457+1.377lnZ

a3 =6.59Z−2.414+1.094lnZ

a4 =0.05242Z0.3709

a5 =0.958Z−2.02+1.099lnZ

a6 =0.2808Z0.2042

τ0 =incident energy in units of the rest energy of the electron

The same functional form as used by Tabata et al.,[NIM B 119, 463 (1996)] for fitting rex/r0 from charge-deposition distributions.

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• Transmission curves: Normal incidence (continued)– Analytic expression for rex/r0 (cont.)

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• Transmission curves: Normal incidence (continued)– Empirical formula for ηTN

• Rao type [Rao, NIM 44, 155 (1966)]

• Ebert type [Ebert, Lauzon & Lent, Phys. Rev. 183, 422 (1969)]

• The average of weighted rms relative deviations of fits to a total of 63 transmission curves– Rao Type: 3.4%– Ebert type: 2.4%; adopted

ηTN z( )= 1+exp−s( )1+exp s+2( ) z rex( )−s[ ]

ηTN z( )=exp−α z rex( )β[ ]α = 1−1β( )1−β

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• Transmission curves: Normal incidence (continued)– Empirical formula for ηTN (cont.)

· Coefficient β: values and expression

The coefficient β takes on a maximum at an energy 15–30 MeV.To avoid complication of the functional form, we have considered an expression applicable up to 20 MeV.

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• Transmission curves: Normal incidence (continued)– Empirical formula for ηTN (cont.)

· Analytic expression for β

· Why does β become smaller again at high energies?

β =1+b1 + b21+ b3 τ0( )b4

b1 =4.89Z−0.338−0.075lnZ

b2 =5.73Z−0.41+0.17lnZ

b3 =22.6Z−0.92+0.163lnZ

b4 =1.65Z−0.13+0.15lnZ

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• Transmission curves: Normal incidence (continued)– Empirical formula for ηTN (cont.)

• Comparison with MC results

Some systematic deviations indicate that the functional form is not flexible enough, but the formula is moderately good as a whole.

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• Transmission curves: Dependence on angle of incidence, θ– Comparison of PENELOPE results

with previous data:

Watts & Burrell (1971) by ETRAN,Knock-on electrons included

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• Transmission curves: Dependence on θ (continued)– Empirical Formula

• Extension to include the dependence on θ

rex θ( ) =rex 0( )cosc1θβ θ( )=β 0( )cosc2θc1 =c1 θ,τ0,Z( ) : an expression with 10 adjustable coefficientsc2 =c2 θ,τ0,Z( ) : an expression with 12 adjustable coefficients

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• Transmission curves: Dependence on θ (continued)– Empirical Formula (cont.)

• Comparison with MC results

Larger errors at larger angles Tolerable errors up to 30 or 40 deg

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• Residual energy: Normal incidence– PENELOPE results

• Comparison with an approximate expression [used in the depth–dose algorithm by Tabata et al., Radiat. Phys. Chem. 53, 205 (1998)]

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• Residual energy: At different angles of incidence– PENELOPE results

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Concluding Remarks• Comprehensive data sets on ηTN

and Tr have been generated by PENELOPE according to the strict definitions of these parameters.

• Interesting trends have been found for ηTN and rex.

• A general empirical formula for ηTN, which includes the dependence on θ, has been obtained.

• A similar formula for Tr is going to be studied.