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 3 He -4 He 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 4 He superfluid film and the physisorbed 3 He . 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 4 He film. We find that the effect on the third sound response of adding a small amount of 3 He to the 4 He film can depend sensitively on the particular 4 He film coverage. The third sound speed can either increase or decrease. In fact, in some regimes, the added 3 He 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 3 He and 4 He . 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.

A MATHEMATICAL VERIFICATION OF THE EXISTENCE OF STRINGS OF CIRCULATION IN SUPERFLUID FILMS ON POROUS MEDIA

A superfluid film flowing along the walls of a porous material can be modeled by a harmonic differential on a punctured Riemann surface satisfying integrality conditions on its periods and residues. In this paper, we use classical Riemann surface theory to mathematically investigate the observed physical phenomenon of the appearance of pairs of vortices in the fluid and the resulting patterns of circulation. We show that the existence of strings of circulation in the superfluid depends on a quantized divisibility condition between the number of vortex–antivortex pairs and the circulation of the fluid.