IONIC SIZE EFFECTS IN VALENCE TRANSITIONS

Valence changes in rare-earth compounds are accompanied by significant changes in the rare-earth ionic radii. We consider the effects of such ionic size variation on the thermodynamics of phase changes in these systems and on the effective lifetime of individual integral valence states. It is now well established that many compounds of rare-earth elements from the middle or the ends of the lanthanide series exhibit intermediate valence in certain regions of pressure-temperature-alloying space [I], and that varying these parameters can lead to transitions, of varying degrees of sharpness, from integral to non-integral valence and, in appropriate cases, from semiconducting to metallic character. It seems clear that the trigger for these transitions is electronic in origin, a delocalized (band) fn-'(sd)' state becoming degenerate with a localized fn ionic state, but equally it seems likely that in most cases positive feedback for the transition is provided by lattice contraction, driven by the radical reduction in ionic size in changing from f" to fn-'(sd)' (for example the ionic radius changes by almost 20 % in going from Sm+ + to Sm+++) [2]: Coulombic interactions between delocalized electrons and the holes they leave behind also accelerate the transition. A reasonably complete theoretical analysis of intermediate valence systems seems out of the question at present and all of the theoretical work to date has inevitably involved drastic approximation. For example, the electronic properties have been considered in a quasi-particle manner as effective single particle f or (sd) excitations above an fn-' core, despite the fact that the difference of the ionic radii of the f* and fn-' ions shows that, even in Hartree-Fock approximation, the self-consistently determined single-particle orbitals making up these ionic states are different in the two cases. The effect of the lattice energy on the phase stability has been considered both empirically [3, 41, employing a simple compressibility expression, with [3] and without [4] interpolation for the dependence of the equilibrium volume on valence, and with two opposite extreme microscopic models [5, 61 ; in one of these extreme models the lattice energy was calculated as though each rare-earth ion had the average f-electron occupation and ionic radius [5] ; in thd other, each ion was taken as being either fn or fn-' in the proportions required by the overall valence, but randomly mixed and interacting elastically (but not just harmonically) [6]. It should be emphasized that the latter can be a well-defined procedure for calculating the energy without implying that the actual ions do always have a particular pure valence. One of the aims of this paper is to consider the factors which determine which of the above models, which we shall refer to henceforth as the static and adiabatic approximations, is the more appropriate. Our second objective is to consider the related effects of electron-lattice interaction on the effective lifetime of the pure ionic states ; with one exception [7] this has been ignored in the literature. For simplicity we shall restrict ourselves here to a zero-temperature analysis although extension will be