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Canadian Journal of Physics

Liquid Helium and the Properties of a Bose–Einstein Gas. IV. The Effect of Various Approximations

D. F. Goble

Published Dec 15, 1971• 1 citation
Publisher

Abstract

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.

DOI
10.1139/p71-370

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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)

•Sep 26, 1960•0 citations
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.
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.
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.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
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)