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Canadian Journal of Physics

Liquid helium and the properties of a Bose–Einstein gas. I. A numerical study of hard-core models

D. F. Goble, Lynn E. H. Trainor

Published Apr 1, 1968• 11 citations
Publisher

Abstract

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.

DOI
10.1139/p68-107

Related papers

Liquid helium and the properties of a Bose–Einstein gas. III. The density dependence of hard-core models

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We have used the results of a previous paper by Goble and Trainor to compute the density dependence of the hard-core Bose–Einstein gas in the t-matrix ladder approximation, utilizing the coordinate-space method of Brueckner and Sawada as modified by Parry and ter Haar, and the pseudopotential treatment of the hard-core boundary condition presented by Liu and Wong. Various thermodynamic parameters of these model systems are compared with the properties of liquid helium four. The disagreements which are found are shown to be primarily related to differences in the magnitudes of the Landau parameters.

Canadian Journal of Physics•Jun 1, 1970•1 citation

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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)

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On a qualitative basis we have examined the identification of the properties of a hard-core Bose–Einstein (BE) gas with those of liquid helium four below the lambda temperature. It is suggested that the use of the generalized condensation of Girardeau together with the inclusion of "three-body ladders" from the perturbation series may improve the agreement between the hard-core BE model system and the physical helium four system to such an extent that reasonable quantitative agreement may be attained.

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Bose-Einstein Gas with Repulsive Interactions: Hard Spheres at High Density

K. A. Brueckner, K. Sawada

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 ${\mathrm{He}}^{4}$.

Physical Review•Jun 15, 1957•95 citations
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.
We have used the results of a previous paper by Goble and Trainor to compute the density dependence of the hard-core Bose–Einstein gas in the t-matrix ladder approximation, utilizing the coordinate-space method of Brueckner and Sawada as modified by Parry and ter Haar, and the pseudopotential treatment of the hard-core boundary condition presented by Liu and Wong. Various thermodynamic parameters of these model systems are compared with the properties of liquid helium four. The disagreements which are found are shown to be primarily related to differences in the magnitudes of the Landau parameters.
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)
On a qualitative basis we have examined the identification of the properties of a hard-core Bose–Einstein (BE) gas with those of liquid helium four below the lambda temperature. It is suggested that the use of the generalized condensation of Girardeau together with the inclusion of "three-body ladders" from the perturbation series may improve the agreement between the hard-core BE model system and the physical helium four system to such an extent that reasonable quantitative agreement may be attained.
As a model for liquid helium II, we study the Bose–Einstein gas with a two-body interaction potential of the form ~δ(r − a), where r is the interparticle separation. Excitation spectra for various values of a are calculated using the Brueckner–Sawada approach based on the concept of the T matrix. However, unlike the work of Brueckner and Sawada and many other related works that followed, we take into account the dependence of the T matrix on the center-of-mass momentum of the interacting particles. Excitation spectra calculated with and without this dependence, using two different expressions for the two-particle propagator, indicate the validity of the Brueckner and Sawada center-of-mass approximation for physically interesting values of a. The difference between the two spectra is found to increase as a increases.
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 aaa 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{\mathrm{He}}^{4}He4.