Abstract
The elastic electron back-scattering is a problem that is important for many theoretical and experimental techniques, especially in the determination of the inelastic mean free paths. This effect arises when a monoenergetic electron beam bombards a solid target and some of the electrons are scattered without energy loss.The description of the flow can be written as an integral equation and may be solved by Monte Carlo methods.
In this paper we investigate the possibility of improving the convergence of the Monte Carlo algorithm by using scrambled low-discrepancy sequences. We demonstrate how by taking advantage of the smoothness of the differential elastic-scattering cross-section a significant decrease of the error is achieved. We show how the contribution of the first few collisions to the result can be evaluated by an appropriate integration method instead of direct simulation, which further increases the accuracy of our computations without increase of the computational time. In order to facilitate these techniques, we use spline approximation of the elastic cross-section, which is more accurate than the widely used tables of Jablonski.
Supported by the Ministry of Education and Science of Bulgaria under contracts NSF I-1201/02 and I-1405/04.
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References
Atanassov, E.: A New Efficient Algorithm for Generating the Sobol Sequences with Scrambling. In: Dimov, I.T., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) NMA 2002. LNCS, vol. 2542, pp. 83–90. Springer, Heidelberg (2003)
Atanassov, E., Dimov, I.: A new optimal Monte Carlo method for calculating integrals of smooth functions. Monte Carlo Methods and Applications 5(2), 149–167 (1999)
Atanassov, E., Dimov, I., Dubus, A.: A new weighted Monte Carlo algorithm for elastic electron backscattering from surfaces. Math. and Comp. in Sim. 62(3-6), 297–305 (2003)
Atanassov, E.I., Dimov, I.T., Durchova, M.K.: A New Quasi-Monte Carlo Algorithm for Numerical Integration of Smooth Functions. In: Lirkov, I., Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2003. LNCS, vol. 2907, pp. 128–135. Springer, Heidelberg (2004)
Atanassov, E.I., Durchova, M.K.: Generating and Testing the Modified Halton Sequences. In: Dimov, I.T., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) NMA 2002. LNCS, vol. 2542, pp. 91–98. Springer, Heidelberg (2003)
Bachvalov, N.S.: On the approximate computation of multiple integrals. Vestnik Moscow State University, Ser. Mat., Mech. 4, 3–18 (1959)
Bachvalov, N.S.: Average Estimation of the Remainder Term of Quadrature Formulas. USSR Comput. Math. and Math. Phys. 1(1), 64–77 (1961)
Dimov, I.T., Atanassov, E.I., Durchova, M.K.: An improved Monte Carlo Algorithm for Elastic Electron Backscattering from Surfaces. In: Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2001. LNCS, vol. 2179, pp. 141–148. Springer, Heidelberg (2001)
Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes on Mathematics, vol. 1651. Springer, Berlin (1997)
Dubus, A., Jablonski, A., Tougaard, S.: Evaluation of theoretical models for elastic electron backscattering from surfaces. Prog. Surf. Sci. 63, 135–175 (2000)
Jablonski, A.: NIST Electron Elastic Scattering Cross-Section Database, Version 3.0, http://www.nist.gov/srd
NIST Electron Inelastic-Mean-Free-Path Database: Version 1.1, http://www.nist.gov/srd/nist71.htm
Jablonski, A.: Elastic Scattering and Quantification in AES and XPS. Surf. Interf. Anal. 14, 659 (1989)
Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. John Wiley & Sons, New York (1974)
Owen, A.B.: Scrambled Net Variance for Integrals of Smooth Functions. Annals of Statistics 25(4), 1541–1562
Standard E673, Annual Book of the ASTM Standards, American Society for Testing and Materials, West Conshohocken, PA, vol. 3.06 (1998)
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Atanassov, E.I., Durchova, M.K. (2005). A Quasi-Monte Carlo Method for an Elastic Electron Back-Scattering Problem. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_14
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DOI: https://doi.org/10.1007/978-3-540-31852-1_14
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