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Physical Review

Quantum Hard-Sphere Gas. II

Marshall Luban

Published May 17, 1965• 6 citations
Publisher

Abstract

Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at $T=0$\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian $\mathcal{H}$ defined for all relative distances of the $N$ particles. The terms of $\mathcal{H}$ corresponding to the forward motion of a particle through the system lead to the notion of an effective mass ${m}^{*}$ for each particle in excess of its actual mass $m$, with $\frac{{m}^{*}}{m}$ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and $a$ are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches $\frac{{p}^{2}}{2{m}^{*}}$. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.

DOI
10.1103/physrev.138.a1033

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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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Physical Review•Dec 1, 1958•193 citations
Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at T=0T=0T=0\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian H\mathcal{H}H defined for all relative distances of the NNN particles. The terms of H\mathcal{H}H corresponding to the forward motion of a particle through the system lead to the notion of an effective mass m∗{m}^{*}m∗ for each particle in excess of its actual mass mmm, with m∗m\frac{{m}^{*}}{m}mm∗​ given by a monotonically increasing function of \ensuremathρa3\ensuremath{\rho}{a}^{3}\ensuremathρa3, where \ensuremathρ\ensuremath{\rho}\ensuremathρ and aaa are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta p\ensuremath≳\ensuremathℏap\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}p\ensuremath≳a\ensuremathℏ​, possesses further extrema at higher momenta, and asymptotically approaches p22m∗\frac{{p}^{2}}{2{m}^{*}}2m∗p2​. The pair distribution function vanishes in the low-density limit for particle separations r\ensuremath≤ar\ensuremath{\le}ar\ensuremath≤a.
We pose the problem of finding an operator, defined for all relative distances of two particles, which plays the role of a Hamiltonian such that the subsequent eigenvalue problem has as its only solutions precisely the eigenfunctions for two hard spheres, each of core diameter aaa. This operator is explicitly constructed. Its crucial aspect is that, despite its being defined for all relative distances r12{r}_{12}r12​, its eigenfunctions serve as a complete set for expressing a two-particle wave function only in the restricted interval ${r}_{12}\ensuremath{\ge}a$. Subsequent papers of this series will be devoted to the application of these results to the problem of quantum hard-sphere gases.
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.
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)
We investigate the dynamic structure factor of a system of Bose particles at zero temperature using quantum Monte Carlo methods. Interactions are modeled using a hard-sphere potential of size aaa and simulations are performed for values of the gas parameter na3n{a}^{3}na3 ranging from the dilute regime up to densities nnn where the thermodynamically stable phase is a solid. With increasing density, we observe a crossover of the dispersion of elementary excitations from a Bogoliubov-type spectrum to a phonon-maxon-roton curve and the emergence of a broad multiphonon contribution accompanying the single-quasiparticle peak. In particular, for na3=0.2138n{a}^{3}=0.2138na3=0.2138, which corresponds to superfluid 4{}^{4}4He at equilibrium density, the extracted spectrum turns out to be in good agreement with the experimental energy-momentum dispersion relation in the roton region and for higher momenta. The behavior of the spectral function at the same density in the stable solid and metastable gas phase above the freezing point is also discussed.
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.
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 $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.