Statistical Mechanics of an Impure Crystal at Low Temperatures

A simple model of a regular crystal containing a dilute concentration of noninteracting interstitial impurities is analyzed with particular attention to thermal properties at low temperatures. Tunneling produces a lifting of the degeneracy of the spatial configurations of the impurities among the interstitial sites, and the consequent impurity band energy structure is studied under various conditions of temperature and impurity concentration. At relatively high temperatures the tunneling band entropy $S$ approaches the conventional value corresponding to static impurities in random solution. At lower temperatures, $S$ decreases and the heat capacity rises as $T^{-2\mathrm{}}$, reaching a peak near the tunneling temperature $T_t\simeq \frac{\hslash }{k\tau }$, where $\tau$ is the tunneling time between adjacent sites. At temperatures below $T_t$, thermodynamic properties approach those of ideal gases. For sufficiently low $T$, the impurities undergo Fermi or Bose condensation. The general dependence of $S$ on $T$ and $T_t$ suggests that it may in principle be used for refrigeration; the ultimate temperatures possible in an "ideal" substance being on the order of ${10}^{-12\mathrm{}}$ deg.