Method of Treating Zeeman Splittings of Paramagnetic Ions in Crystalline Fields

In this paper, we present an alternative method to the "spin Hamiltonian" for treating the behavior of a paramagnetic ion under the combined influence of the host crystal and an applied magnetic field. This method has the advantage of being applicable to all strengths of couplings between the paramagnetic ion and the surrounding crystal. It is applicable to free-ion $S$ states in the same way that it applies to all other states. It gives all of the constants one needs to describe the energy level structure as a function of field as restricted by symmetry and the properties of the levels considered. The method is more general and is expected to be also more accurate than the conventional spin Hamiltonian formalism. It is illustrated by the application to the problem of the $\left(d^5\right){}_{}{}^6{}_{}{}^{}S$ level in a cubic crystalline environment.