1. 1 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, Indiana August 8-12, 1999

2. 2 Abstract We have generated the data on the number transmission coefficient  TN of electrons residual energy T r 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  TN is given.

3. 3 Introduction Definitions of the quantities treated –Number transmission coefficient  TN : the ratio of the number of electrons transmitted by a slab absorber to the number of incident electrons –Residual energy T r 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. 4 Introduction (continued) Motivations –Better empirical formulas for  TN and T r are necessary for improving the semiempirical depth–dose code EDMULT. –An empirical formula for  TN 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  TN be made?”

5. 5 Introduction (continued) Related previous work –Monte Carlo calculations of  TN Normal incidence: Seltzer & Berger, NIM 119, 157 (1974) (ETRAN) Oblique incidence: Watts & Burrell, NASA TN D-6385 (1971) (for Al; ETRAN) –Empirical formulas for  TN 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 T r Normal incidence for light materials only –Empirical formulas for T r Simple linear relation for low-Z materials (normal incidence) Approximate expressions used in depth– dose algorithms (normal incidence)

6. 6 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

7. 7 Method (continued) Empirical formula for  TN under normal incidence –Determine extrapolated ranges r ex from Monte Carlo transmission curves –Express r ex /r 0 by an analytic expression (r 0 : CSDA range; use NIST database values) –Compare fits to two types of function and select the better one Rao type Ebert type

8. 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. 9 Transmission curves: Normal incidence (continued) –r ex : Comparison with r ex from charge-deposition distributions in semi-infinite medium Appreciable differences: only for low energy electrons incident on the highest Z absorbers.

10. 10 Transmission curves: Normal incidence (continued) –The reason for the differences in r ex

11. 11 Transmission curves: Normal incidence (continued) –Analytic expression for r ex /r 0 The same functional form as used by Tabata et al., [NIM B 119, 463 (1996)] for fitting r ex /r 0 from charge-deposition distributions.

12. 12 Transmission curves: Normal incidence (continued) –Analytic expression for r ex /r 0 (cont.)

13. 13 Transmission curves: Normal incidence (continued) –Empirical formula for  TN 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. 14 Transmission curves: Normal incidence (continued) –Empirical formula for  TN (cont.) · Coefficient  : values and expression The coefficient  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.

15. 15 Transmission curves: Normal incidence (continued) –Empirical formula for  TN (cont.) · Analytic expression for  · Why does  become smaller again at high energies?

16. 16 Transmission curves: Normal incidence (continued) – Empirical formula for  TN (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. 17 Transmission curves: Dependence on angle of incidence,  – Comparison of PENELOPE results with previous data: Watts & Burrell (1971) by ETRAN, Knock-on electrons included

18. 18 Transmission curves: Dependence on  (continued) –Empirical Formula Extension to include the dependence on 

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

20. 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. 21 Residual energy: At different angles of incidence – PENELOPE results

22. 22 Concluding Remarks Comprehensive data sets on  TN and T r have been generated by PENELOPE according to the strict definitions of these parameters. Interesting trends have been found for  TN and r ex. A general empirical formula for  TN, which includes the dependence on , has been obtained. A similar formula for T r is going to be studied.