Theory of Phenomenological Coefficients in Solid-State Diffusion. III. Limiting Law Conductance for Pure Ionic Crystals

Abstract

The cycle diagrams of the previously derived cluster expansions are studied in detail for pure ionic crystals containing Schottky defects. The chain diagram approximation for the equilibrium pair correlation function is employed throughout. Some diagrams are then zero and the remainder yield, in the continuum limit, the Onsager limiting law for the relaxation effect for the conductance at zero frequency. The coefficient of the correction of lowest order in κa to this limiting law which arises when the cycle diagrams are evaluated allowing for the discreteness of the lattice is determined for the NaCl structure. The predictions of this correction are compared with exact numerical evaluations of the cycle diagrams. The corrections from the time-dependent correlation terms are well approximated by the analytic result and are very close to those predicted by the Pitts ion-size correction for electrolyte solutions. However, the remaining corrections from the time-independent correlation terms are but poorly represented by the analytic result and tend to cancel the other corrections. It is concluded that the major approximation in the use of the Onsager limiting law is the use of the chain diagram approximation for the pair correlation function (equivalent to the Debye–Hückel linearized result in the continuum limit) and this is probably relatively more important than effects arising from the discreteness of the lattice or corrections arising from the dynamics of vacancy pairs, triplets, etc. Further numerical study of this point is required. The generalization of the results to Frenkel defects is considered briefly but without detailed results.