The Chandrasekhar function for modeling photoelectron transport in solids

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, w, 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.