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Physical Review A

Fluid-Solid Phase Transition of a Hard-Sphere Bose System

Jean-Pierre Hansen, D. Levesque, Daniel Schiff

Published Feb 1, 1971• 59 citations
Publisher

Abstract

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/${\mathrm{\ensuremath{\sigma}}}^{3}$.

DOI
10.1103/physreva.3.776

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The ground-state energy of a system of hard-sphere bosons of diameter \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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 \ensuremathρfluid=0.23\ifmmode±\else\textpm\fi0.02{\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02\ensuremathρfluid​=0.23\ifmmode±\else\textpm\fi0.02 particles/\ensuremathσ3{\mathrm{\ensuremath{\sigma}}}^{3}\ensuremathσ3 and \ensuremathρsolid=0.25\ifmmode±\else\textpm\fi0.02{\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02\ensuremathρsolid​=0.25\ifmmode±\else\textpm\fi0.02 particles/\ensuremathσ3{\mathrm{\ensuremath{\sigma}}}^{3}\ensuremathσ3.
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.
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.
Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at rs=139{r}_{s}=139rs​=139, where rs{r}_{s}rs​ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at rs=2{r}_{s}=2rs​=2, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part v(r>d)v(r>d)v(r>d) and a constant part $\ensuremath{\lambda}$ at r<dr<dr<d of the bare interaction are assumed to contribute to the average field. The correlation function f(r)f(r)f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance ddd is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of ddd in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present f(r)f(r)f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.
By a Monte Carlo variational calculation, it is shown that the description of the ground-state properties of a hard-sphere Bose system is improved if the Jastrow wave function contains a correlation structure at distances intermediate between the first and the second shell of neighbors. We relate these correlations to the zero-point motion of rotons. We predict that the oscillations of the structure factor S(k)S(k)S(k) and of the radial distribution function g(r)g(r)g(r) increase at finite temperatures. Such intermediate-distance correlations improve the description of the solid phase by a Jastrow wave function, but the description is still not satisfactory compared to localized wave functions. Backflow associated with rotons induces explicit three-particle correlations in the wave function, and a new form for such correlations is proposed for variational calculation for 4He^{4}\mathrm{He}4He.