Theory of Zeeman effect for rare earth ions in crystal field with D2 symmetry and application to rare earth garnets

Abstract

Rare earth ions in solids frequently exhibit sharp-line optical spectra. In this paper a part of the theory of the Zeeman effect for the lines is developed, first for a single rare earth ion in D2 symmetry, and secondly for rare earth ions at different inequivalent sites in garnet crystals. The results for the case in which the crystal field and Zeeman splittings are comparable have application for magnetic garnets such as the rare earth iron garnets, while the results for the case where the Zeeman splittings are treated as a perturbation on the crystal field states have application for nonmagnetic garnets such as the rare earth aluminum or gallium garnets. In both cases it is found that when the magnetic field is restricted to the xy plane of the local site coordinate, there exists an antiunitary symmetry operation which simplifies the energy level and transition probability calculations, and which leads to the assignment of a quantum number ζ = ± 1 to the states. There are shown to be two types of states: Type A, for which the assignment of ζ value is not sensitive to the choice of coordinate axes, and Type B states for which it is. It is also shown, in the case of rare earth ions with an odd number of 4f electrons, that the Types of the states are determined by the relative signs of the principal values of the g factors. Furthermore, it is found that the dependence of line intensity on magnetic field direction for A→A and B→B transitions is clearly different from that of A→B and B→A transitions. Finally, and most important, it is shown that studies of the energies and intensities of the Zeeman lines as a function of magnetic field direction in suitable crystal planes can be used to distinguish between spectra arising from different inequivalent rare earth ion sites in garnets. This result has immediate importance for experimental study of site selectivity problems in garnets, and, when combined with the other conclusions of this work, leads to definite suggestions for experimental determinations of at least some of the details of the states.