Andreev-Lifshitz Theory Applied to Normal Solids under Pressure*

On letting the superfluid density go to zero, the Andreev-Lifshitz hydrodynamic theory of supersolids becomes applicable to an ordinary solid.\footnote{A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP 29, 1107 (1969).} Under applied pressure $P_{a}$, needed to produce solid He$^3$ and He$^4$ or to be of geophysical relevance, the system has both an elastic stress $\lambda_{ik}$ and an internal pressure $P$, with $P\delta_{ik}=P_{a}\delta_{ik}+\lambda_{ik}$ in equilibrium. $P$ may be thought of as being due to a vacancy fluid. For $P_a$ small compared to the bulk modulus, Maxwell relations give $P\sim P_{a}^{2}$. The dynamical equations lead to three sets of propagating elastic modes (longitudinal and transverse sound) and two diffusive modes (one largely of entropy density and one largely of vacancy density -- or, more generally, defect density), all of which we study for non-zero $P_{a}$.\footnote{M. R. Sears and W. M. Saslow, Phys. Rev. B 82, 134304 (2010).} The vacancy diffusion mode has diffusion constant $D_{L}\sim P_{a}^{2}$, and is diffusive because its associated internal pressure fluctuation $P'$ nearly cancels its lattice stress fluctuation $\lambda'_{ik}$. This mode permits the system to respond differently to transducers with different surface treatments. We specifically have in mind solid $^4$He, which requires $P_a \sim 25$ bars to solidify; however, the results should apply to any solid under pressure.