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 interstitial 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 T4 and that the first correction to the specific heat due to this varies as T7; 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.
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