Pressure and density of vacancies in solid $^4$He

TL;DR

This paper investigates how hard-core repulsion influences the quantum statistical behavior of vacancies in solid helium-4, revealing a transition from Bose-like to Fermi-like behavior with increasing temperature.

Contribution

It introduces a virial expansion approach to model vacancy interactions in solid helium-4 and calculates the second virial coefficient to analyze their statistical behavior.

Findings

Vacancies behave as ideal Bose particles at low temperatures.

Hard-core interactions cause vacancies to exhibit more fermion-like behavior at higher temperatures.

The virial expansion provides a quantitative framework for vacancy interactions.

Abstract

Crystals of $^4$He contain vacancies that move around by a quantum mechanical hopping process. The density and pressure of these vacancies can be experimentally studied. The accuracy of the experiments is high enough to detect the effect of the Bose statistics of the vacancies. In this paper we examine the effect of the hard-core repulsion between the vacancies, which should also have a measurable effect on their behaviour. We set up a virial expansion for a lattice gas of hard-core particles, and calculate the second virial coefficient. It turns out that the vacancies behave as ideal Bose particles at low temperatures, but that the hard-core interaction makes them behave more and more like fermions as the temperature increases.