A refined Bogoliubov-Huang approach to helium II thermodynamics

Abstract

The thermodynamics of a free Bose gas with effective temperature scale $\tilde{T}$ and hard-sphere Bose gas with the $\tilde{T}$ scale are studied. $\tilde{T}$ arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction $V_\textrm{osc}$, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential $V_\textrm{He}$ between $^{4}\textrm{He}$ atoms. The repulsive part of $V_\textrm{He}$ is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the $\tilde{T}$ temperature scale (HSET), is investigated, first, by the Bogoliubov-Huang method and next by a modified version of this method, which takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators $a_0,\,\,a^*_0$ (originally rejected by Bogoliubov). Theoretical heat capacity $C_V(T)$ exhibits good agreement, below 2.1 K, with the experimental heat capacity graph observed in $^{4}\textrm{He}$ at saturated vapour pressure. The phase transition to the low-temperature phase, with a Bose-Einstein condensate, occurs in the HSET at $T_\lambda=$2.17 K, and is accompanied, in the modified HSET version, by a singularity of $C_V(T)$. Other thermal properties of HSET, such as the momentum distribution function, the fraction of atoms in the momentum condensate and normal fluid density, agree qualitatively with those of $^{4}\textrm{He}$, but improve those of the free Bose gas.