On the Superfluid Fraction and the Hydrodynamics of Supersolids

This review largely considers the author’s extensions of two foundation works in supersolids: Andreev and Lifshitz’s hydrodynamics, and Leggett’s Non-Classical Rotational Inertia (NCRI) with estimates of the T =0 upper limit. Relative to the case of a perfect lattice, the Andreev and Lifshitz equations contain an additional degree of freedom, which permits a liquid-like internal pressure P that is distinct from the lattice stress (or ‘solid-like pressure’). This is particularly relevant to 4 He, which requires an applied pressure P a to solidify; use of a Maxwell relation relating P and strain yields that ; we estimate that near the melting pressure the liquid-like pressure in equilibrium is about 1/4 of the solid-like pressure. This new freedom also permits vacancy diffusion, which we have studied for both ordinary solids and supersolids. In both cases, for the vacancy diffusion mode the liquid-like pressure and the lattice stress cancel. Further, since at T =0 the supersolid fraction f s is less than unity and the excitation part of the normal fraction is zero, we argue that there must be an additional source of “normal” mass, to which we attribute a velocity that in principle is distinct from the lattice velocity associated with elasticity. Relative to NCRI we have made numerous estimates of the upper limit for the superfluid fraction f s ; we find f s values on the order of 0.2 for realistic models of the atomic density. Correlation effects in the solid cause the superfluid velocity of one particle to depend on correlations with the positions of other particles, and this leads to a more complex theory for the flow pattern and for the upper limit on f s .