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.本研究旨在设计一款面向凝聚态物质光电子输运理论模型的钱德拉塞卡函数（Chandrasekhar function，即H函数）计算算法。已有研究表明，针对非保守各向同性散射场景，仅需反照率w最大值达0.85时的H函数取值即可。本研究对多款H函数计算算法开展分析，以筛选出可实现14位小数精度的参数取值区间。结果显示，双指数规则（double-exponential rule）算法通用性最强，可在跨越10个数量级的参数范围内输出高精度H函数值。但在参数值较小的区间，采用近似解析表达式的算法运行时长更短；而在本研究涉及的最高反照率区间，斯蒂布斯-韦尔算法（Stibbs and Weir, 1959）的表现更优。基于上述结论，本研究构建了一种混合算法，并通过被积函数含H函数的积分计算对其进行验证——这类积分是光电子输运形式体系中所需的核心计算项。针对固体发射光电子流或光电子平均逃逸深度的计算任务，即便需多次执行此类积分，整体运行时长通常仍远低于1秒，且可达到13位及以上的期望精度。