The Theory of Liquid Helium

Abstract

The free energy, $F$, of an assembly of interacting helium atoms is expressed in terms of the trace of the density operator, $\exp (-\beta H)$, where $\beta =\frac{1}{\mathrm{kT}}$, $H$ is the Hamiltonian and $T$ the temperature. The Hamiltonian is written as $H_0+gW$, where $H_0$ is the part corresponding to the kinetic energy and $\mathrm{gW}$ the potential energy. The resulting expression for $F$ is expanded in powers of $g$, the first two terms of the series being calculated explicitly. The first term, which is independent of $g$, gives the free energy of the London theory, and leads to all the usual results. In particular the transition at the lambda temperature, $T_{\lambda }$, is of the third order. The second term in the expansion of $F$ raises the transition to one of the second order; all the second-order derivatives of this term are discontinuous at $T_{\lambda }$, while $T_{\lambda }$ and $\frac{\partial T_{\lambda }}{\partial \rho }$ are the same as in the London theory. Numerical values are obtained for the specific heat and the discontinuity in the specific heat. These are compared with experiment and it is found that there is an improvement as compared with previous theories.