Thermodynamics of an incommensurate quantum crystal

We present a simple theory of the thermodynamics of an incommensurate quantum solid. The ground state of the solid is assumed to be an incommensurate crystal, with quantum zero-point vacancies and interstitials and thus a non-integer number of atoms per unit cell. We show that the low temperature variation of the net vacancy concentration should be as $T^4$, and that the first correction to the specific heat due to this varies as $T^7$; these are quite consistent with experiments on solid $^4$He. We also make some observations about the recent experimental reports of ``supersolidity'' in solid $^4$He that motivate a renewed interest in quantum crystals.