Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields

Abstract

The calculation of energy levels of magnetic ions in crystalline electric fields is often the cause of considerable confusion. This confusion largely arises, not through the fundamental theoretical principles which are now well established, but from the large number of different and often not fully defined notations used, occasional errors, and the fact that one author seldom gives an example of calculation from start to finish. This chapter contains no original contribution to the problem, but it is hoped that by illustrating how the energy levels may be calculated on the basis of a simple Point-charge ionic model of the crystal lattice, the connection between the various forms of crystal-field Hamiltonians will be clarified. Particular reference is made to fields of cubic symmetry. If the crystalline electric field effects are taken as a perturbation on the appropriate free-ion wave functions and energy levels, the problem becomes that of finding the perturbing Hamiltonian and its matrix elements. The energy levels in the crystal field can then be found from standard perturbation theory. The simple point-charge model used to calculate the Hamiltonian is known to possess several weaknesses. It neglects the finite extent of charges on the ions, the overlap of the magnetic ions' wave functions with those of neighboring ions, and the complex effects of “screening” of the magnetic electrons by the outer electron shells of the magnetic ion. However, it serves as a first approximation to illustrate the principles involved, and may be used to calculate ratios of terms of the same degree in the Hamiltonian for lattice sites of high symmetry, since these ratios are independent of the model used and are determined solely by the symmetry.