Multiple elastic scattering of electrons in condensed matter

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• 作者：A. Jablonski ; ^{ajablonski@ichf.edu.pl}
• 关键词：Elastic electron scattering on atoms ; Electron transport in condensed matter ; Elastic electron backscattering from surfaces ; Electron inelastic mean free path
• 刊名：Computer Physics Communications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：210
• 期：Complete
• 页码：92-102
• 全文大小：2843 K
• 卷排序：210

文摘

Since the 1940s, much attention has been devoted to the problem of accurate theoretical description of electron transport in condensed matter. The needed information for describing different aspects of the electron transport is the angular distribution of electron directions after multiple elastic collisions. This distribution can be expanded into a series of Legendre polynomials with coefficients, AlAl. In the present work, a database of these coefficients for all elements up to uranium (Z=92) and a dense grid of electron energies varying from 50 to 5000 eV has been created. The database makes possible the following applications: (i) accurate interpolation of coefficients AlAl for any element and any energy from the above range, (ii) fast calculations of the differential and total elastic-scattering cross sections, (iii) determination of the angular distribution of directions after multiple collisions, (iv) calculations of the probability of elastic backscattering from solids, and (v) calculations of the calibration curves for determination of the inelastic mean free paths of electrons. The last two applications provide data with comparable accuracy to Monte Carlo simulations, yet the running time is decreased by several orders of magnitude. All of the above applications are implemented in the Fortran program MULTI_SCATT. Numerous illustrative runs of this program are described. Despite a relatively large volume of the database of coefficients AlAl, the program MULTI_SCATT can be readily run on personal computers.Program summaryProgram title: MULTI_SCATTProgram Files doi:http://dx.doi.org/10.17632/cvt9yz9gj8.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: Typically, elastic electron backscattering probabilities are estimated from results of Monte Carlo simulations of electron trajectories in a solid. This approach, although very convenient for a programmer, has major drawbacks: (i) the solid acceptance angles of analyzers are rather small, thus a large number of electron trajectories must be generated to obtain reasonable statistics; (ii) large running times are needed to reach an acceptable precision; (iii) results are always burdened with a statistical error. In the program MULTI_SCATT, an analytical formalism is implemented which leads to an accuracy comparable with Monte Carlo simulations, however it is faster by several orders of magnitude.Solution method:   The program MULTI_SCATT requires presence of the database containing the coefficients AlAl for the series expansion of the differential elastic-scattering cross section. This database has a large volume of 96.2 Mb; a major computer effort is needed to complete these. This database has a large volume of 96.2 Mb; a major computer effort is needed to complete these calculations, and it has been decided to avoid this stage by providing the precalculated AlAl data. To obtain the AlAl value for a selected energy, the data from the database are interpolated in logarithmic coordinates. The elastic electron scattering probability is calculated from a slowly convergent series. Acceleration of the series convergence was found to be effective in the considered range of electron energies. Consequently, the running times of the program MULTI_SCATT are considerably decreased.Additional comments:(i) Restrictions: The program MULTI_SCATT can be used for electron energies between 100 and 5000 eV. Calculations of the elastic-backscattering probability and the calibration curve are performed under the assumption of normal incidence of the primary beam. This program is designed for applications in elemental solids, however, it can also be generalized to compounds (see text).(ii) Unusual features: To facilitate the use of the program MULTI_SCATT, several additional databases of parameters needed for the calculations are implemented using the DATA statement: inelastic mean free paths for different classes of materials, number of electrons in the valence band, atomic masses, densities, etc.Running time: Depending on the option selected, the running time is typically of the order of tens of milliseconds. For example, for tungsten and an electron energy of 500 eV, the running time varies from 12.8 milliseconds (angular distribution of electron directions after two collisions) to 60.2 milliseconds (calibration curve).