The Chandrasekhar function for modeling photoelectron transport in solids

Abstract

The objective of this study was to design an algorithm for calculating the Chandrasekhar function (H-function) dedicated for theoretical models of photoelectron transport in condensed matter. It has been shown that only the H-function values for non-conservative isotropic scattering are needed with the largest albedo, ω, values reaching 0.85. Different algorithms for calculating the H-function were analyzed to identify values of arguments for which an accuracy of 14 decimals is reached. It turned out that the most universal approach was an algorithm implementing the double-exponential rule which provided accurate values of the H-function for arguments varying over 10 orders of magnitude. However, the execution time was found to be shorter for algorithms implementing approximate analytical expressions in the region of small argument values, and the Stibbs and Weir algorithm (Stibbs and Weir, 1959) in the region of largest albedo values considered here. Based on these results, a mixed algorithm was created, and tested in calculations of integrals with integrands containing the H-function that are needed in the formalism for photoelectron transport. The execution time of calculations of the photoelectron current emitted from a solid or the photoelectron mean escape depth usually was very short, well below 1 s despite the fact that several such integrals were calculated, and the desired and attained accuracy was 13 decimals or better. Program summaryProgram title: H_FUNProgram Files doi: http://dx.doi.org/10.17632/xf4k25cnpk.1Licensing provisions: GNU General Public License 3 (GPL)Programming language: Fortran 90Nature of problem: Analytical formalism of photoelectron transport requires access to a program providing accurate values of the Chandrasekhar H-function within a possibly short execution time. However, the theory is based on non-conservative scattering; thus the albedo values associated with photoelectron transport vary in a limited range of values, not exceeding ω=0.85. Furthermore, the H-function in the range of small arguments is expressed by analytical formulas that can be implemented in programs providing the H-function with high speed and high accuracy. It would be of interest to develop an algorithm for calculating the H-function with the following features: (i) optimized with respect to execution time, (ii) providing the results with accuracy of 14 decimals, and (iii) taking into account the limited range of albedo values.Solution method: It has been proved that the H-function for both arguments smaller than about 10−5–10−4 are obtained with the desired accuracy from analytical formulas. In the remaining region of arguments, algorithms involving the integral representation need to be used. In recent reports, the integration method based on the double-exponential rule was recommended as an accurate and fast approach. Similarly, an algorithm implementing the Stibbs and Weir theory was also recently indicated as a very effective source of the H-function. In the present work, the execution times for the above algorithms were compared for different pairs of arguments. Finally, separate domains were ascribed to tested algorithms in such a way that the execution times were always the shortest. Based on these results, a program HFUNELE dedicated to calculations of photoelectron transport was developed.Additional comments:(i) Restrictions: The program HFUNELE optimized with respect to speed can be used for the directional variable, μ, varying in the range from zero to unity, while the allowed albedo, ω, range is from zero to 0.85, i.e., the range needed for applications in photoelectron transport theory.(ii) Unusual features: The enclosed test program demonstrates the fact that the typical integrals over the integrands containing the H-function reach or even exceed an accuracy of 13 decimals. Furthermore, application of the program HFUNELE is illustrated in exemplary calculations related to photoelectron transport theory (determination of the photoelectron signal intensity or the photoelectron mean escape depth). To facilitate the performance of the test program, several databases of parameters needed for the calculations are implemented using the DATA statement (electron inelastic mean free paths, albedo values for elements, atomic weights and densities), or included as accompanying text files: BAND_MG.TXT and BAND_AL.TXT (atomic subshells, subshell photoionization cross sections, electron binding energies, and asymmetry parameters).