Dipole Correlations in a Random Dilute Ferroelectric

We consider the statistical mechanics of a set of randomly distributed dipole impurities in a nondipolar medium. The impurity concentration is assumed to be sufficiently low that no long-range order exists even near $T=0$. Two cases are treated, one in which the impurity dipoles may have only two orientations, and another in which they may have six orientations along the $〈100〉$ directions. The partition function is evaluated by using an approximation to the dipole interaction and an expansion in a power series of the impurity concentration. The result is then averaged over the dipole orientations, and the free energy is obtained by averaging the logarithm of the partition function over all spatial configurations. The dipole correlation function is obtained for the case T→0. It is found in a certain approximation that the dipoles are strongly correlated to each other if they are within a "correlation distance" from each other but approximately uncorrelated otherwise. The probability distribution of the random internal dipolar field is discussed. If we assume the static (dc) dipole moment to be twice the value reported at optical frequencies, the model gives the ferroelectric susceptibility of OH^- ions in KCl, in very good agreement with experiment over the whole temperature and concentration range reported by K\"anzig et al. The low-temperature specific heat predicted from an Ising model is linear in temperature and independent of the impurity concentration.