The properties of a hard-sphere, Bose–Einstein gas are investigated numerically in the t-matrix, ladder approximation, using both the coordinate-space treatment of Brueckner and Sawada, as modified by Parry and ter Haar, and the pseudopotential method of Liu and Wong. For each treatment, the form of the excitation spectrum and the thermodynamic properties of the gas at low temperatures are compared with the corresponding properties of liquid helium. The pseudopotential method gives better agreement for a reasonable choice of the radius of the "hard-core" interaction. This result is not surprising since the actual potential corresponding to the pseudopotential of Liu and Wong is intermediate between the "soft-core" potential of Brueckner and Sawada and a true "hard-core" potential.
A method developed by the authors for treating Bose-Einstein particles at low temperature is specialized to the case of hard sphere interactions. It is shown that the integral equations determining the energy can in this case be replaced by an integral involving the unknown excitation spectrum. The energy is first determined as a power series in ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and
a the hard-sphere radius. This solution is shown to diverge badly at high density. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low density and to a good approximation at high density. The excitation spectrum of the system is shown to have a peculiar nonmonotonic relation between energy and momentum very similar to that deduced by Landau from the low-temperature properties of
He4.
The thermodynamics of a free Bose gas with effective temperature scale
T~ and hard-sphere Bose gas with the
T~ scale are studied.
T~ arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction
Vosc, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential
VHe between
4He atoms. The repulsive part of
VHe is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the
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
a0,a0∗ (originally rejected by Bogoliubov). Theoretical heat capacity
CV(T) exhibits good agreement, below 2.1 K, with the experimental heat capacity graph observed in
4He at saturated vapour pressure. The phase transition to the low-temperature phase, with a Bose-Einstein condensate, occurs in the HSET at $T_λ=$2.17 K, and is accompanied, in the modified HSET version, by a singularity of
CV(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
4He, but improve those of the free Bose gas.
The thermodynamics of a free Bose gas with effective temperature scale [Formula: see text] and hard-sphere Bose gas with the [Formula: see text] scale are studied. [Formula: see text] arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction V osc , which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential V He between 4 He atoms. The repulsive part of V He is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the [Formula: see text] temperature scale (HSET), and particle mass and density equal to those of 4 He , is investigated, first, by the Bogoliubov–Huang method and next by an improved version of this method, which describes He II in terms of dressed bosons and takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators a0, [Formula: see text] (originally rejected by Bogoliubov). Theoretical heat capacity CV(T) exhibits good agreement, below 1.9 K, with the experimental heat capacity graph observed in 4 He at saturated vapour pressure. The phase transition to the He II phase, occurs in the HSET at Tλ = 2.17 K, and is accompanied, in the modified HSET version, by a singularity of CV(T). The fraction of atoms in the momentum condensate at 0 K equals 8.86% and agrees with other theoretical estimates for He II. The fraction of normal fluid falls to 8.37% at 0 K which exceeds the value 0% found in He II.
A hard-sphere (HS) Bose gas in a trap is investigated at finite temperatures in the weakly interacting regime and its thermodynamic properties are evaluated using the static fluctuation approximation. The energies are calculated with a second-quantized many-body Hamiltonian and a harmonic oscillator wave function. The specific heat capacity, internal energy, pressure, entropy, and the Bose–Einstein occupation number of the system are determined as functions of temperature and for various values of interaction strength and number of particles. It is found that the number of particles plays a more profound role in the determination of the thermodynamic properties of the system than the HS diameter characterizing the interaction, that the critical temperature drops with the increase of the repulsion between the bosons, and that the fluctuations in the energy are much smaller than the energy itself in the weakly interacting regime.