Theory of the dilute interacting quadrupole glass: Four-orientational tunneling quadrupoles

The free energy F for dilute, interacting, four-orientational tunneling quadrupoles (TQ’s) randomly distributed in a nonpolar medium is derived for low fractional TQ concentrations c. F is developed into a virial expansion in c and is evaluated exactly up to terms in ${\mathit{c}}^3$. From the microscopic free energy we obtain the low-energy excitation spectrum for dilute interacting four-orientational TQ’s from fundamental considerations and show that the low-energy excitations arise from strongly interacting TQ’s. The derivations give (a) a constant density of excitation energies for low energies, (b) a specific heat linear in T for low temperature T, and (c) a strong broadening of the Schottky specific heat even for very low TQ concentrations. Properties (a) and (b) are similar to those observed in glasses. The predicted specific heat is compared with experiments on dilute ${\mathrm{CN}}^{\mathrm{-}}$ dissolved in KBr.