Distribution of the total angular momentum in relativistic configurations

TL;DR

This paper analyzes the distribution of total angular momentum in relativistic atomic configurations using cumulants and generating functions, providing new formulas and series expansions for practical calculations.

Contribution

It introduces a novel approach using cumulants and generating functions to study angular momentum distributions in relativistic configurations, including recurrence relations and series expansions.

Findings

Efficient recurrence relations for the generating function of J distribution.

Representation of the distribution by a Gram-Charlier-like series.

Approximate formulas for transition counts in spin-orbit split arrays.

Abstract

This paper is devoted to the analysis of the distribution of the total angular momentum in a relativistic configuration. Using cumulants and generating function formalism this analysis can be reduced to the study of individual subshells with $N$ equivalent electrons of momentum $j$. An expression as a nth-derivative is provided for the generating function of the $J$ distribution and efficient recurrence relations are established. It is shown that this distribution may be represented by a Gram-Charlier-like series which is derived from the corresponding series for the magnetic quantum number distribution. The numerical efficiency of this expansion is fair when the configuration consists of several subshells, while the accuracy is less good when only one subshell is involved. An analytical expression is given for the odd-order momenta while the even-order ones are expressed as a series which provides an acceptable accuracy though being not convergent. Such expressions may be used to obtain approximate values for the number of transitions in a spin-orbit split array: it is shown that the approximation is often efficient when few terms are kept, while some complex cases require to include a large number of terms.