THE STATISTICAL MECHANICS OF INTERACTING TUNNELING STATES

The low temperature properties of amorphous, or "glassy" materials are believed to arise from tunneling units which experience a random local potential. An assumption of a constant density of excitation energies explains the experimentally observed anomalous properties of amorphous materials. However, one knows neither what tunnels in the amorphous material nor the nature of the tunneling state. In order to shed further light on this problem, we study the statistical mechanics of some well known tunneling units distributed in alkali halide crystals. One such tunneling unit, for example is for a Li('+) ion dissolved in KCl and which is found experimentally to have eight orientations. In order to simulate the glass problem, we study the properties of one dimensional chain of multi-orientational tunneling dipoles. We relate the partition function (PF) of the multiorientational dipoles to the PF of an Ising model in a transverse field, a problem which has already been solved previously. The derived solutions apply to a one-dimensional chain of atoms with a nearest neighbor interaction. For low concentrations of tunneling units it is sufficient to consider only the interacting pairs of tunneling dipoles instead of the infinite chain. We also generalized our results and show that the PF of two-dimensional system of multiorientational dipoles can be related to the exact solution of the two-dimensional Ising model. Finally we prove that the low temperature specific heat of pairs of quadrupoles have a constant density of low energy excitations. It is therefore expected that low concentrations of pairs of quadrupoles should have a similar behavior to "glassy" materials.