Tatsuo Tabata and Vadim Moskvin * Osaka Prefecture University and IDEA

1. Third International Workshop on Electron and Photon Transport Theory Applied to Radiation Dose Calculation Hyatt Regency Hotel, Indianapolis, Indiana August 8-12, 1999 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

2. Abstract • We have generated the data on • the number transmission coefficientηTNof 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 forηTNis given.

3. Introduction • Definitions of the quantities treated • Number transmission coefficientηTN: 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.

4. Introduction (continued) • Motivations • Better empirical formulas forηTNand Tr are necessary for improving the semiempirical depth–dose code EDMULT. • An empirical formula forηTNis 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 forηTNbe made?”

5. Introduction (continued) • Related previous work • Monte Carlo calculations ofηTN • Normal incidence: Seltzer & Berger, NIM 119, 157 (1974) (ETRAN) • Oblique incidence: Watts & Burrell, NASA TN D-6385 (1971) (for Al; ETRAN) • Empirical formulas forηTN • 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)

6. Method • 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 energies0.1–50 MeV • Absorber materialsBe, C, Al, Cu, Ag, Au, U • Angles of incidence0–80 deg at 10-deg step, 89.9 deg

7. Method (continued) • Empirical formula forηTNunder 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

8. 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

9. 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.

10. Transmission curves: Normal incidence (continued) • The reason for the differences in rex

11. Transmission curves: Normal incidence (continued) • Analytic expression for rex/r0 The same functional form as used by Tabata et al., [NIM B 119, 463 (1996)] for fitting rex/r0 from charge-deposition distributions.

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

13. Transmission curves: Normal incidence (continued) • Empirical formula forηTN • 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

14. Transmission curves: Normal incidence (continued) • Empirical formula forηTN(cont.) • Coefficientβ: values and expression The coefficientβtakes 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.

15. Transmission curves: Normal incidence (continued) • Empirical formula forηTN(cont.) • Analytic expression forβ • Why doesβbecome smaller again at high energies?

16. Transmission curves: Normal incidence (continued) • Empirical formula forηTN(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.

17. Transmission curves: Dependence on angle of incidence,θ • Comparison of PENELOPE results with previous data: Watts & Burrell (1971) by ETRAN, Knock-on electrons included

18. Transmission curves: Dependence onθ(continued) • Empirical Formula • Extension to include the dependence onθ

19. Transmission curves: Dependence onθ(continued) • Empirical Formula (cont.) • Comparison with MC results • Larger errors at larger angles • Tolerable errors up to 30 or 40 deg

20. 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)]

21. Residual energy: At different angles of incidence • PENELOPE results

22. Concluding Remarks • Comprehensive data sets onηTNand Tr have been generated by PENELOPE according to the strict definitions of these parameters. • Interesting trends have been found forηTNand rex. • A general empirical formula forηTN, which includes the dependence onθ, has been obtained. • A similar formula for Tr is going to be studied.