  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.
