We consider the problem of very dilute concentrations of interacting tunneling dipoles or quadrupoles (TD’s or TQ’s) randomly distributed in a host matrix. We assume that the TD’s and TQ’s can have n equivalent directions of orientation determined by the minima in the local potential. Each TD and TQ is only allowed to tunnel to its nearest-neighbor potential wells. Starting from a microscopic Hamiltonian, we use the special properties of ‘‘circulant’’ matrices to obtain the exact energy eigenvalues for the following interacting pairs: (a) TD’s or TQ’s with three, four, and six orientations in a plane (tunneling clock model); (b) TD’s or TQ’s in three dimensions with four, six, and eight orientations. For each of the cases considered, we obtain low-energy excitations from strongly interacting tunneling units. For very low concentrations and a random distribution of TD’s or TQ’s in the medium, we use a virial expansion of the free energy to obtain the density of states and the heat capacity. For an ${\mathit{r}}^{\mathrm{-}3}$ interaction our results are in good agreement with the experimentally measured heat capacity for 340 ppm of ${\mathrm{CN}}^{\mathrm{-}}$ dissolved in KBr and dilute concentrations of ${\mathrm{Li}}^{+}$ in KCl. We find that the experimentally observed broadening of the Schottky specific heat arises from strongly interacting tunneling units.

• Received 13 January 1993

DOI:https://doi.org/10.1103/PhysRevB.47.11097