Bose-Einstein condensation and thermodynamic functions of 4He film

A monolayer of ⁴He atoms is treated as a system of hard-sphere bosons in a thin film geometry, with a finite thickness. The method of pseudopotential is used to calculate first the energy spectrum, and then the Helmholtz free energy and other thermodynamic functions of the system. It is found that Bose-Einstein condensation exists below a definite temperature. Much like a liquid-gas transition, the boson system displays a high temperature normal phase, a low temperature condensed superfluid phase and coexistence region. In the present treatment, the minimum momentum associated with the finite thickness of monolayer is used as a parameter. We find that the transition temperature is linearly proportional to the density of the ⁴He film. After performing double-tangent construction of the Helmholtz free energy curve we find for the specific heat a rounded peak at the transition temperature, in agreement with recent experiments. The ratio of the superfluid density at the transition point to the transition temperature is found to be essentially a constant.