Perturbational approach to glasslike low-energy excitations in interacting tunneling dipoles and quadrupoles

Second- and third-order peturbation theory is used to calculate the low-energy spectrum of pairs of interacting tunneling dipoles (TD’s) and tunneling quadrupoles (TQ’s). The TD’s and/or TQ’s are assumed to be in an n-orientational potential well, determined by the local environment; hence assume n directions of orientation. For each of the orientations, the TD’s and/or TQ’s are allowed to tunnel only to their nearest-neighbor potential wells. We derive equations which relate the energy eigenvalues of a pair of interacting TD’s and/or TQ’s to the energies of the noninteracting tunneling Hamiltonian. We show that the pair has low-energy excitations provided the interaction potential is sufficiently greater than the tunneling matrix element. In particular we show that dilute concentrations of strongly interacting TQ’s randomly distributed in a nondipolar crystalline host give low-energy excitations at low temperature T. Our calculation predicts that the very low T thermal properties of very small concentrations of eight-orientational TQ’s (${\mathrm{CN}}^{\mathrm{-}}$ in KBr) are different than those of TD’s and other TQ’s dissolved in alkali halides.