Thermodynamics of an incommensurate quantum crystal

We present a 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 at low temperature T, the 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 helium-4. We also make some observations about the recent experimental reports of "supersolidity" in solid helium-4 that motivate a renewed interest in quantum crystals.