Theory of third sound and stability of thin ³He -⁴He superfluid films

In this paper, we summarize the results of recent studies of third sound in thin, superfluid ³He -⁴He mixture films and the relation of the third sound spectrum to the question of the films' thermodynamic stability. We have considered films on several representative substrates: Nuclepore, glass, Li and Na. Our approach utilizes the variational, hypernetted chain/Euler-Lagrange (HNC-EL) theory as applied to inhomogeneous boson systems to calculate chemical potentials for both the ⁴He superfluid film and the physisorbed ³He. Numerical density derivatives of the chemical potentials lead to the sought-after third sound speeds. On all substrates, the third sound speeds show a series of oscillations as a function of film coverage that is driven by the layered structure of the ⁴He film. We find that the effect on the third sound response of adding a small amount of ³He to the ⁴He film can depend sensitively on the particular ⁴He film coverage. The third sound speed can either increase or decrease. In fact, in some regimes, the added ³He destabilizes the film and can drive "layering transitions" leading to quite complicated geometric structures of the film in which the outermost layer consists of phase-separated regimes of ³He and ⁴He. Finally, we examine the range of applicability of the usual film-averaged hydrodynamic description. We find that at least up to film thicknesses of six liquid layers, there is no regime in which this hydrodynamic description is applicable.