On the role of certain rotational invariants in crystal-field theory

We examine a perturbation scheme in which the effect of the crystal-field Hamiltonian in mixing free-ion states is represented in second order by an effective operator acting only on the free-ion state under consideration. The shifts in the centers of gravity of the free-ion levels caused by J mixing is derived in terms of three parameters s2, s4, s6, which are quadratic rotational invariants. The second moment of the Stark levels corresponding to a particular free-ion state is described by these same parameters in the absence of J mixing. Using Ho3+ as an example, we show how these parameters can be extracted from experimental data in a simple manner and how they can be used to correct the data for J-mixing effects.