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.
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.
A recently developed formalism for Helium II is generalized by introducing a 2-body interaction of spheres with diameter depending on the momentum exchanged between two atoms in an interaction process. A larger class of atomic collisions is also admitted. These modifications allow to account for some details of the interatomic potential V He (r) between two 4 He atoms, which were previously disregarded, and to improve the theoretical graphs of Helium II momentum distribution and normal fluid fraction.
Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)
The properties of Bose-Einstein-condensed systems of hard spheres at high density are studied by taking into account higher-order corrections beyond the Hartree-Fock-Bogoliubov approximation. Based on a diagrammatic technique of the perturbation expansion starting from the Bogoliubov approximation, we show that the effective interactions separate into two distinct parts: the screened and condensate-mediated interactions. The latter is described by the exchange of virtual quasiparticles between noncondensate bosons via the Bose-Einstein condensate and is attractive for small energy transfer. We calculate the condensation temperature, excitation spectrum, and condensate density in the whole region of the gas parameter. It is shown that the condensate temperature decreases compared to that of an ideal Bose gas due to hard-sphere interactions. We find that the condensate-mediated attraction between noncondensate bosons leads to an enhancement of the anomalous self-energy and, as a consequence, to the emergence of a roton minimum in the excitation spectrum and to the strong suppression of the condensate fraction in the region of high density.
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.