General expressions for Stevens and Racah operator equivalents

TL;DR

This paper clarifies various definitions and expressions of crystal field operators, connecting Stevens and Racah formalisms, and provides explicit computational methods and code reviews for these operators in quantum systems.

Contribution

It introduces explicit formulas for crystal-field operators in terms of angular momentum quantum numbers and operators, clarifying their relation to Stevens and Racah operators.

Findings

Unified expressions for crystal-field operators

Connections established between Stevens and Racah formalisms

Review of computational codes using different conventions

Abstract

Several definitions of the crystal field have been used over time and their variety has lead to many misunderstandings, in both theoretical and experimental literature. Two categories of definitions can be mentioned, the first being the operators equivalents introduced by Stevens in 1952 and the second being the crystal-field operators, introduced by different authors from 1962 and expanded on the Racah spherical tensors. This paper aims at providing some clarification in this field. We first make a review of several expressions introduced in various references to compute crystal-field operators and we describe connections between them. Then, we introduce an explicit way to compute crystal-field operators, in terms of angular momentum quantum numbers $j$ and $m$ as well as in terms of $J^2$ and $J_z$ operators. We eventually give some connections between the Stevens operators equivalents and the crystal-field operators, and make usage of the coefficients of fractional parentage for the expression of the crystal-field operators for the many-body states. Various computational codes, using different crystal-field conventions, are also reviewed.