Improved electron transport mechanics in the PENELOPEMonte-Carlo modelA.F. Bielajewa,*, F. SalvatbaDepartment of Nuclear Engineering and Radiological Sciences, University of Michigan, 1906 Cooley Building, 2355 Bonisteel Boulevard,Ann Arbor, MI 48109-2104, Michigan, USAbFacultat de Fõsica (ECM), Universitat de Barcelona, Societat Catalana de Fõsica, Diagonal 647, 08028 Barcelona, SpainReceived 17 March 2000AbstractWe describe a new model of electron transport mechanics, the method by which an electron is transported geo-metrically in an in®nite medium as a function of pathlength, s, the accumulated elastic multiple-scattering angularde¯ection characterized by Hs, the polar scattering angle, and U, a random azimuthal angle. This model requires onlyone sample of the multiple-scattering angle yet it reproduces exactly the following spatial moments and space±angularcorrelations: hzi, hx sin H cos Ui, hy sin H sin Ui, hz cos Hi, hx2i, hy2i and hz2i. Moreover, the distributions associated withthese moments exhibit a good improvement over the PENELOPE transport mechanics model when compared self-consistently with the results of analog simulations. When we split the transport step into two steps with equal path-length, we observe excellent agreement with the distributions, indicating that the algorithm nearly matches higher ordermoments when employed in this way. The equations described herein are relatively inexpensive to employ in an iterativeMonte-Carlo code. We have employed the new model to demonstrate the usefulness of the new mechanics for severalexamples that span the dynamic range of application. Ó 2001 Elsevier Science B.V. All rights reserved.PACS: 02.50Ng; 13.60Fz; 25.30Bf; 34.80BmKeywords: Monte-Carlo simulation; Condensed history; Elastic scattering; Multiple-scattering; Coulomb scattering1. IntroductionOne of the most challenging problems in theMonte-Carlo simulation of high-energy electron(and positron) transport is the generation of spa-tial displacements of the particle. In each step ofthe simulation, the electron is moved a certainpathlength, s, through the medium. The angularde¯ection after this pathlength is determined bythe polar multiple-scattering angle, Hs, and theazimuthal angle U, which is distributed uniformlyon 0; 2p. For a given elastic cross-section, thetheory of Goudsmit and Saunderson [1,2] providesthe multiple-elastic scattering distribution fromwhich Hs can be sampled. The diculty comesNuclear Instruments and Methods in Physics Research B 173 (2001) 332±343www.elsevier.nl/locate/nimb*Corresponding author. Tel.: +1-734-7646364; fax: +1-734-7634540.E-mail address: [email protected] (A.F. Bielajew).0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.PII: S 0 168-583X(00)00363-3from the fact that the space displacement x; y; zat the end of the step, although strongly correlatedwith the angular de¯ection, is not known. Formalsolutions of the transport equation [3,4] provideclosed expressions only for the moments of thespace displacements and space±angle correlations.Since only a few of these moments can be evalu-ated and employed in a practical transport scheme,there is not enough information to characterizex; y; z unambiguously. The prescription that re-lates x; y; z to Hs, U and s will be called the``electron transport mechanics'' [4].The PENELOPE Monte-Carlo code system [5±7], a general purpose coupled ec Monte-Carlocode, employs a ``random hinge'' electron trans-port mechanics' scheme that can be summarized asfollows:x=s r sin Hscos U;y=s r sin Hssin U;z=s 1 ÿ rr cos Hs;1where r is a random number sampled uniformly on0; 1, and the pathlength, s, for which the multiple-scattering angle Hs is calculated and interpretedas the total curved pathlength that the electrontravels through the medium.Although this scheme is an ansatz, it produceshigh quality results as indicated through compli-ance [4] with Lewis' moments [3]. The Lewis mo-ments studied in the previous work werehziZs0ds0eÿg1s0;hzli13Zs0ds0eÿg1sÿs01 2eÿg2s0;hx sin H cos U y sin H sin Ui23Zs0ds0eÿg1sÿs01 ÿeÿg2s0;2hz2i13Zs0ds0Zs00ds00eÿg1s0ÿs001 2eÿg2s00;hx2 y2i23Zs0ds0Zs00ds00eÿg1s0ÿs001 ÿ 2eÿg2s00;where the gs are moments of the single-scatteringcross-section rl with Legendre polynomials,g` 2pNAqAZs0ds0Z1ÿ1dlrl1 ÿP`l; 3in which NAis Avogadro's number, A is atomicweight (we assume single-element medium) and q isthe density of the medium. Here, the distance s isexpressed as a unit of length and the integrationvariable l is the cosine of the polar scattering angle.Since the scattering model we are considering isazimuthally symmetric, hx sin H cos Ui hy sin Hsin Ui, hx2ihy2i, and are combined in Eq. (2).The angular distribution after a pathlength, s,isgiven byf l; sX1l0l12eÿsg`P`l; 4and we also havehlieÿg1s; hl2i1 2eÿg2s3: 5Here, we have ignored the energy dependence ofthe single-scattering cross-section, which allowsfor greater analytic development. The aboveequations may be expressed in energy-dependentform, for example, employing the continuousslowing down approximation (CSDA), wherebythe integrals over pathlength, s, in Eq. (2) are re-placed by integrals over energy, and pathlength andenergy are related through a stopping-power rela-tionship. We leave this, or similar, adaptations tofuture work. Having ignored energy loss, the inte-grals in Eq. (2) may be performed with the resulthzis1ÿeÿnn;hzlis13n1ÿeÿn2eÿnÿeÿcncÿ1;hxsinHcosUy sinHsinUis23n1eÿcnÿceÿncÿ1;hz2is223cn2cneÿnÿc ÿ22eÿcnÿceÿncÿ1;hx2y2is243cn2cnÿc1ÿeÿcnÿc2eÿncÿ1;6where n sg1and c g2=g1which spans the rangefrom 0 (backward scattering) to 3 (high energy,A.F. Bielajew, F. Salvat / Nucl. Instr. and Meth. in Phys. Res. B 173 (2001) 332±343 333forward directed) and has the value unity for iso-tropic scattering.Generally, much of the range of applicationinvolves small values of n which suggests an ex-pansion of the moments in a Taylor series in n.These results have been stated elsewhere [4], butwe include them here for completeness and for thenext order in n for later use. To
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