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arXiv:1006.5825

A refined Bogoliubov-Huang approach to helium II thermodynamics

Jan F. Maćkowiak, Dawid Borycki

Published Jun 30, 2010• 0 citations
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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_λ=$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.

DOI
10.48550/arxiv.1006.5825

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The thermodynamics of a free Bose gas with effective temperature scale T~\tilde{T}T~ and hard-sphere Bose gas with the T~\tilde{T}T~ scale are studied. T~\tilde{T}T~ arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction VoscV_\textrm{osc}Vosc​, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential VHeV_\textrm{He}VHe​ between 4He^{4}\textrm{He}4He atoms. The repulsive part of VHeV_\textrm{He}VHe​ is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the T~\tilde{T}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∗a_0,\,\,a^*_0a0​,a0∗​ (originally rejected by Bogoliubov). Theoretical heat capacity CV(T)C_V(T)CV​(T) exhibits good agreement, below 2.1 K, with the experimental heat capacity graph observed in 4He^{4}\textrm{He}4He at saturated vapour pressure. The phase transition to the low-temperature phase, with a Bose-Einstein condensate, occurs in the HSET at Tλ=T_λ=Tλ​=2.17 K, and is accompanied, in the modified HSET version, by a singularity of CV(T)C_V(T)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^{4}\textrm{He}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.
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.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.