A Lattice Model of Liquid Helium, II

Abstract

The partition function of the lattice model of liquid helium introduced in the preceding paper is calculated by making use of the Kikuchi's approximation. Setting two parameters, the effective mass m=1/7m0 (mo is the mass of He atom), and the lattice spacing d=3.1A, an excellent agreement of the density dependence of A-temperature with experiment is obtained. The general trend of other thermodynamical quantities calculated in this paper is in good accordance with the observation. In the preceding paper/> hereafter referred to as I, we have introduced Hamiltonian of the lattice model of liquid helium and proved its equivalence to that of a spin system with anisotropic exchange coupling. The A-transition was shown there to correspond to the spin system, and making use of the molecular field approximation, we could obtain right dependence of the A-temperature on density. Phonons in liquid helium were proved to correspond to spin waves in the spin system. All the properties of our lattice model, however, were derived by translating the results obtained for the spin system, under certain assumptions which were not necessarily without question open to criticism. For example, in the molecular field approximation we assumed the existence of the long range order of magnetization in the 'ICY plane, but there is some difficulty in defining the long range order because of the fact that the magneti­ zation in the x y plane is not constant of motion owing to the anisotropic exchange coupling. Furthermore, the crude approximation adopted in I fails to afford not only quantitative conclusions on specific heat and pressure, etc., but also qualitative explanation for the last of the three questions proposed at the begining of I. Namely as to the question why liquid helium has negative thermal expansion coefficient just below the J.-point, the previous treatment seems unsatisfactory for obtaining a correct solution. To remedy these points, we shall employ in this paper a rather direct method of approximation without referring to the spin system for calculating the partition function of the lattice model, and try to understand qualitatively the thermodynamical behavior of