The quadrupolar term in the equivalent crystal-field Hamiltonians for various central-ion point symmetries

Abstract

The method of the determination of the principal axes system (PAS) orientation with respect to the initial reference frame for any quadrupolar component of the crystal-field Hamiltonian is presented. The developed method is based on the extreme points of the axial crystal-field parameter map, B20(?, ?), where both the partial derivatives ?B20/?? and ?B20/?? simultaneously vanish, and ?, ? stand for the two Euler angles with respect to the initial reference frame. In general, there are three such points corresponding to a maximum, minimum and saddle point on the map. These particular points fix the orthogonal PAS in which three equivalent, two-parameter (B20, B22), orthorhombic-like parameterizations coexist according to the three options of the z-axis. Hence, the standardization of parameterizations becomes simplified and lies in the particular choice between the three forms, conventionally that with the maximal |B20|.