General expressions for Stevens and Racah operator equivalents

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.