Dipole interactions among polar defects: A self-consistent theory with application to OH−impurities in KCl

Abstract

We consider a set of dipole impurities randomly distributed in a nonpolar medium where the only interaction between the impurities is the dipole-dipole interaction. The dipoles are assumed to be oriented in the six equivalent (1,0,0) directions found experimentally to exist when O${\mathrm{H}}^{-}$ impurities are dissolved in KC1 crystals. Effects arising from tunneling between the six equivalent directions are neglected. We set up the expression for the random molecular electric field vector $\overrightarrow{E}$ at a particular impurity site and derive self-consistently the probability distribution of $\overrightarrow{E}$ for all temperatures and (sufficiently low) impurity concentrations. The thermodynamic properties of the system are then obtained by integrating the thermodynamic variable of a single dipole in a fixed vector field $\overrightarrow{E}$ over the distribution of all fields. Evaluating the thermodynamic properties arising from the dipole-dipole interaction we show that the dielectric susceptibility scales with the ratio of the temperature $T$ to the impurity concentation $c$. Similarly the specific heat per impurity scales with $\frac{T}{c}$. Even though the experimentally measured $T^{\frac{3}{2}}{c}^{-\frac{1}{2}}$ dependence of the specific heat is consistent with our "scaling" requirements, indicating that the dipole-dipole interaction is involved in the excess low-temperature specific heat, our theory gives that the specific heat is linear in $T$ and independent of $c$ for very low temperatures. Thus, the dipole-dipole interaction alone, using a molecular-field approximation, does not explain the experimentally observed low-$T$ specific heat. A detailed study of the temperature dependence, the concentration dependence, and the scaling properties of the specific heat and the dielectric susceptibility are discussed and compared with experiment.