Cooperative Dynamic Jahn-Teller Effect. I. Molecular Field Treatment of Spinels

Abstract

A theory of cubic-to-tetragonal transformations occurring in spinels containing octahedrally coordinated ($B$ site) $d^9$ and $d^4$ transition-metal ions in high concentration is developed. The problem is treated in the spirit of the molecular field approximation. For each Jahn-Teller (JT) susceptible ion in the unit cell, a Schrödinger equation is solved, in which the variables of the octahedra (the occupancies of the degenerate states or the distortion from cubic symmetry) appear as temperature-dependent unknown averages. These are then determined self-consistently, utilizing the solution of the Schrödinger equation. The present treatment introduces two improvements on the previous theories given by Wojtowicz and Kanamori. The first is to take into account the dynamic JT effect, thus covering a much wider experimental range, including moderate or even weak anisotropic (nonlinear) JT couplings. The other innovation is the inclusion of the excited vibronic states. Five characteristic regions of solution are distinguished; (a) the static limit, to which previously studied cases belong; (b) the three-state region, which differs from the previous one in that it allows the splitting between the lowest doublet and singlet to become comparable to the molecular field energy; (c) the general regime: the anisotropic energy, the kinetic energy, and the molecular field are of the same order of magnitude (all the excited vibronic states must be taken into account); (d) the strong molecular field limit; (e) the Ising-model limit of weak molecular field, in which case the ground vibronic doublet alone needs to be considered. Thermodynamic quantities such as the specific heat and entropy of transition, as well as bulk distortion at different temperatures and concentrations, are computed and compared with those obtained from previous theories. The phase transition is of first order, but in regimes (d) and (e) the transition entropy is very small. The first-order transition temperature reduces, as a result of introducing dynamic effects, to about half the value obtained by Wojtowicz. A digest of the experimental data is given in an Appendix.