On the Problem of Unsaturated Helium Films

Abstract

A theoretical explanation is given for the lowering of the transition temperature, and the increase of the entropy, with decreasing film thickness as met with in unsaturated helium films. It is shown that in the ideal gas approximation the boundary conditions obeyed by the eigenfunctions assume rather an important role. There is, however, no quantitative agreement between the experimental and the theoretical results even when the dependence of the film density on the thickness of the film is taken into account. On the other hand, if one considers the non-ideal gas approximation for the liquid and takes into account (i) the slow variation of the parameters J and p. with the film thickness, and (ii) the contributions arising from the presence of surface tension waves on the liquid surface, the experimental data are explained in a good quantitative manner. In this case, however, the boundary conditions are found to become rather unimportant. Experimental investigations into the properties of unsaturated helium films!) have shown that the transition from helium I to helium II takes place at a tem­ perature lower than that for the bulk and this transition temperature decreases monotonically with the decreasing film thickness. Further, the entropy of such films is found to be larger than the bulk value. Whereas the properties of liquid helium in bulk have been investigated rather in detail, the behaviour of thin films is comparatively less studied. Particularly, the monotonic variation of the film properties with its thickness has remained almost unexplained. Recently it was suggested by the authors2) that such a behaviour of the thin films could be understood if one enumerated the eigenfunctions in the bounded continuum more exactly than is ordinarily done (by taking into account the surface term also). In the present communication we have investigated this problem in greater detail. In Sections 2 and 3 we have studied the ideal gas problem by (i) carrying out the summation over states as such, so as to avoid the divergence dif­ ficulty met with in reference 2), and (ii) taking into account the variation of the specific volume of the fluid with film thickness. We find that if one tries to understand the experimental data by assuming the ideal gas approximation for liquid helium, the boundary conditions assume rather an important role, i.e., one is obliged to assume the boundary condition 0cp/on=O. In the next section we use the non-ideal Bose gas spectrum characterised by an energy gap J and an effective mass /1. It is found that the experimental data can be quantitatively un,~