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Transmission Coefficients and Residual Energies of Electrons: PENELOPE Monte Carlo Results

This study presents new data on the number transmission coefficient (ηTN) of electrons and the residual energy (Tr) of transmitted electrons, generated using the PENELOPE Monte Carlo code. We examined primary electrons with energies ranging from 0.1 to 50 MeV incident on absorbers with atomic numbers 4-92 at various angles. By comparing our findings with existing literature, we aim to refine empirical formulas for these parameters, facilitating improved depth-dose calculations in radiation therapy. Detailed insights into methods and results are explored in this comprehensive review.

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Transmission Coefficients and Residual Energies of Electrons: PENELOPE Monte Carlo Results

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  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.

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