Collective Behavior of Polar Impurities in Ionic Crystals. II

The purpose of this work is to construct an improved theory to account for the low-temperature dielectric properties of KCl crystals containing substitutional (OH)^- impurities. The statistical mechanics of a pair of dipoles is discussed in order to derive a generalized Langevin function giving the mean moment of such a pair along the direction of an applied field. A hybrid approximation is then developed, in which the interaction of a typical pair of (OH)^- dipoles is treated explicitly while the remaining interactions are treated by means of a modified Onsager-Pirenne approximation. The resulting equations predict a second-order phase transition at kTcNm²0.53, where N is the concentration and m the dipole moment of the (OH)^- ions. The predicted spontaneous polarization increases continuously below Tc to reach a saturation value of P₀ᵐᵃˣ0.45Nm. A thermodynamic analysis indicates that the asymptotic behavior of the model as T→0 is essentially in agreement with the third law, and also enables one to show that the critical temperature should be independent of the applied field.