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Physical Review

Low-Temperature Behavior of a Dilute Bose System of Hard Spheres. I. Equilibrium Properties

T. D. Lee, Chongming Yang

Published Dec 1, 1958• 193 citations
Publisher

Abstract

By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of $a$, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.

DOI
10.1103/physrev.112.1419

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Journal of Mathematical Physics•Dec 1, 1974•1 citation
By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of aaa, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided (a\ensuremathλ)\ensuremath≪1(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1(\ensuremathλa​)\ensuremath≪1 and \ensuremathρa3\ensuremath≪1\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1\ensuremathρa3\ensuremath≪1, where \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the thermal wavelength and \ensuremathρ\ensuremath{\rho}\ensuremathρ is the particle density.
We obtain thermodynamical functions for a dilute Bose gas of hard spheres by using the efffective Hamiltonian which is obtained by a quasi-linear transformation proposed previously and may be recognized as an improvement of the model Hamiltonian of Lee and Yang which is not correct about the terms of second order in the diameter of hard spheres when it is applied to the problem of the phase transition. The Bose condensation occurs at a temperature below which the degenerate phase of the quasi-particles appears. We evaluate the amount of the gap of the specific heat and that of the compressibility at the transition point.
The method developed in the previous paper for the treatment of the quantum-mechanical NNN-body hard-sphere problem is applied to a calculation of the grand partition function of an imperfect Bose gas with hard-sphere interactions. The grand partition function is calculated to second order in an expansion in powers of $\frac{a}{\ensuremath{\lambda}}$, where aaa is the hard-sphere diameter and $\ensuremath{\lambda}$ the thermal wavelength. The approximate equation of state for the gas phase is thereby obtained by calculating all the virial coefficients to order ${(\frac{a}{\ensuremath{\lambda}})}^{2}$.The first-order energy levels obtained in the previous paper embody some interesting physical properties. A fictitious system with exactly such energy levels is considered. The partition function for such a system can be calculated exactly and the exact equation of state obtained. It is shown that there is a phase transition, which more closely resembles an ordinary gas-liquid transition than the Bose-Einstein condensation.
The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.
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)
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.