The Case of a Single J State

Within a manifold of angular momentum J composed of f electron wavefunctions the most general operator equivalent potential with cubic point symmetry may be expressed as 

Here the reference axis is taken along the fourfold axis. 

Leask and Wolf (24) in their classical paper provided a detailed treatment of the ions for J values ranging from 2 to 8. For computational convenience the above expression may be rewritten as 

For the reference axis lying along the 110 and 111) directions, die crystal field Hamiltonian may be expressed as (79) 

Segal and Wallace (25) reported detailed calculations involving the effect of hexagonal crystal field in rare earth ions. Their Hamiltonian is of the form 

---