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Journal of Mathematical Physics

On the Pair Distribution Function of a Hard Sphere Bose System

Leopoldo S. Garcı́a-Colı́n, Jean Peretti

Published Mar 1, 1960• 6 citations
Publisher

Abstract

The pair distribution function for a quantum Bose gas is expressed as a power series in terms of the fugacity, the coefficients of which are temperature dependent. For the hard sphere case, these coefficients have been evaluated to the first order in a/λ, (a being the scattering length and λ the thermal wavelength) by using torons with two fixed points, or alternatively U cluster functions. The result gives the first order correction to the ideal gas formula of London and Placzek introduced by the interactions between the particles.

DOI
10.1063/1.1703648

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The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.

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Physical Review•Feb 1, 1957•186 citations
The pair distribution function for a quantum Bose gas is expressed as a power series in terms of the fugacity, the coefficients of which are temperature dependent. For the hard sphere case, these coefficients have been evaluated to the first order in a/λ, (a being the scattering length and λ the thermal wavelength) by using torons with two fixed points, or alternatively U cluster functions. The result gives the first order correction to the ideal gas formula of London and Placzek introduced by the interactions between the particles.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where nnn is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
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.
It is the purpose of this article to give an analysis of the one- and two-particle distribution functions of the ideal Bose gas and the hard sphere Bose gas. The distribution functions of the ideal Bose gas are expanded in two forms which are useful for the cases of r ≫λTT and r ≪λT, respectively. The analysis for the hard sphere gas are performed in the chain approximation or the Bogolyubov approximation. Asymptotic forms are given for the various limiting cases separately, where the cases are classified by the relative magnitudes of r/Rc, λT/Rc and unity.
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.