Microscopic and hydrodynamic theory of superfluidity in periodic solids

The microscopic theory of fourth sound and of the superfluid fraction for perfect one-component periodic solids has been derived. It is applicable to finite temperatures and is restricted to the case of well-defined excitations. One finds that the superfluid fraction is a tensor $\frac{{\rho }_{\alpha \beta }^s}{{\rho }_0}$ and that the fourth-sound velocity $C_4$ is a tensor ${\left(C_4^2\right)}_{\alpha \beta }={\left(\frac{\partial {\rho }_0}{\partial {\mu }_0}\right)}^{-1\mathrm{}}{\rho }_{\alpha \beta }^s$, where ${\mu }_0$ and ${\rho }_0$ are the spatially averaged values of the chemical potential (per unit mass) and of the number density. In addition, the exact nonlinearized hydrodynamics is derived, and for fourth sound is found to give agreement with the microscopic theory. Because the superfluid velocity for a periodic solid cannot be generated by a Galilean transformation, we find that elastic waves are loaded by the average mass density of the system. This is in contrast to the result of Andreev and Lifshitz, which involves only the superfluid fraction. Therefore one cannot look to (hydrodynamic) elastic waves for an obvious signature of superfluidity. A study of the effect of a transducer indicates that fourth sound will be generated to a non-negligible extent only when the crtystal is imperfect (i.e., it has vacancies, interstitials, or impurities). On the other hand, a heater might be an effective generator of fourth sound, provided that the mean free path for umklapp processes is sufficiently small. In the limit of zero crystallinity the theory shows that second sound, rather than fourth sound, occurs. Detection of superflow by rotation experiments is also considered. It is pointed out that, because the superfluid velocity is not Galilean, two-fluid counterflow does not occur. Hence, it appears that rapid angular acceleration or deceleration would be the best technique for bringing the superfluid into rotation.