Angular momentum distribution in a relativistic configuration: Magnetic quantum number analysis

TL;DR

This paper analyzes the distribution of the magnetic quantum number in relativistic electron subshells using cumulants, generating functions, and recurrence relations, providing efficient computational methods and exploring approximation series.

Contribution

It introduces analytical expressions for the distribution's generating function and cumulants, along with recurrence relations, applicable to any relativistic configuration.

Findings

Efficient recurrence relations for magnetic quantum number distribution.

Analytical expressions for cumulants and generating functions.

Series expansions like Gram-Charlier and Edgeworth have limited convergence but useful approximations.

Abstract

This paper is devoted to the analysis of the distribution of the total magnetic quantum number $M$ in a relativistic subshell with $N$ equivalent electrons of momentum $j$. This distribution is analyzed through its cumulants and through their generating function, for which an analytical expression is provided. This function also allows us to get the values of the cumulants at any order. Such values are useful to obtain the moments at various orders. Since the cumulants of the distinct subshells are additive this study directly applies to any relativistic configuration. Recursion relations on the generating function are given. It is shown that the generating function of the magnetic quantum number distribution may be expressed as a n-th derivative of a polynomial. This leads to recurrence relations for this distribution which are very efficient even in the case of large $j$ or $N$. The magnetic quantum number distribution is numerically studied using the Gram-Charlier and Edgeworth expansions. The inclusion of high-order terms may improve the accuracy of the Gram-Charlier representation for instance when a small and a large angular momenta coexist in the same configuration. However such series does not exhibit convergence when high orders are considered and the account for the first two terms often provides a fair approximation of the magnetic quantum number distribution. The Edgeworth series offers an interesting alternative though this expansion is also divergent and of asymptotic nature.