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Physical Review C

Ground state energy of a hard-sphere Fermi fluid

George Allen Baker Jr., L. P. Benofy, Mauricio Fortes, MANUEL DE LLANO

Published Aug 1, 1986• 7 citations
Publisher

Abstract

A new method of computing the ground state energy of a Fermi fluid is developed and applied to the hard-sphere Fermi fluids of two and four components. We find the results to be highly accurate for low densities. In this paper we compute and report the perturbation series for the ground state energy for the soft, repulsive, square-well problem at a number of densities for a four-component Fermi system as well as the ladder approximation energies for a hard-sphere potential at the same densities.

DOI
10.1103/physrevc.34.678

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Physical Review A•Nov 1, 1987•11 citations
A new method of computing the ground state energy of a Fermi fluid is developed and applied to the hard-sphere Fermi fluids of two and four components. We find the results to be highly accurate for low densities. In this paper we compute and report the perturbation series for the ground state energy for the soft, repulsive, square-well problem at a number of densities for a four-component Fermi system as well as the ladder approximation energies for a hard-sphere potential at the same densities.
We evaluate the fourth-order Hugenholtz diagrams from the Rayleigh-Schr\"odinger many-body perturbation series for a repulsive, square-well potential. We consider a system containing four different equal-mass identically interacting particle species, similar to nuclear matter when only the repulsive core is considered. We use the results of these calculations to extend the L expansion of the ground-state energy of a Fermi fluid by one more term. From this extension, we are able to provide accurate values of the ground-state energy over a density range which includes that for infinite nuclear matter.
Based on the well-known low-density expansion for the ground-state energy per particle of a many-fermion system interacting pairwise via a central potential we develop a perturbation scheme in which the zero-order state is that of the purely-repulsive-spheres fluid. For the hard-core plus square-well pair-interaction case we have deduced, for two-species fermion matter, perturbation formulas up to sixth order in the attractive coupling parameter. We explicitly report results up to fourth order here. Pad\'e-approximant analyses in the density are carried out in each order with the intent of providing the basis necessary for extensive equation-of-state calculations of various many-fermion systems. In particular, for zero order (the fermion hard-sphere fluid) we predict an uncertainty-principle divergence in the energy, and hence pressure, corresponding to the random close packing of spheres, at a density of about 0.5 times the recently obtained value for boson hard spheres and about 0.2 times the well-known empirical value for classical hard spheres.
I am looking at the old problem of the ground-state energy of quantum many-Fermion systems. My tools will be a version of perturbation theory. It is plain that the ordinary, low-density rearrangement can not be used directly to compute the saturation energy and density of a system for which the two-body potential has a strong repulsive core and an attractive portion beyond the core, because the path of continuation from low-density to saturation density passes through the two-phase (vacuum, liquid) region. Likewise it is difficult to supply an expansion directly in lambda for fixed rho (density) because of the difficulty of the strong repulsive core and the tediousness of the calculations. My approach here is to expand the energy in a double series in k/sub F/, the Fermi momentum, and lambda, the strength of the attractive part of the potential. The plan is then to sum up, by the Pade method or a generalization of it, the k/sub F/ expansions of the coefficients of each power of lambda into a function of k/sub F/. The coefficient of lambda/sup 0/ is just the repulsive-core energy itself.
The London ground-state energy formula as a function of number density for a system of identical boson hard spheres, corrected for the reduced mass of a pair of particles in a "sphere-of-influence" picture, and generalized to fermion hard-sphere systems with two and four intrinsic degrees of freedom, has a double-pole at the ultimate regular (or periodic, e.g., face-centered-cubic) close-packing density usually associated with a crystalline branch. Improved fluid branches are constructed based upon exact, field-theoretic perturbation-theory low-density expansions for many-boson and many-fermion systems, extrapolated to intermediate densities via Padé and other approximants, but whose ultimate density is irregular or random closest close-packing as suggested in studies of a classical system of hard spheres. Results show substantially improved agreement with the best available Green-function Monte Carlo and diffusion Monte Carlo simulations for bosons, as well as with ladder, variational Fermi hypernetted chain, and so-called L -expansion data for two-component fermions.
In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
A perturbation scheme based not on the ideal gas but on a system of purely repulsive cores is applied to a typical fully interacting fermion gas. This is ``neutron matter'' interacting via (a) the repulsive ``Bethe homework-problem'' potential, (b) a hard-core--plus--square-well potential, and (c) the Baker-Hind-Kahane modification of the latter, suitable for describing a more accurate two-nucleon potential. Pad\'e extrapolation techniques and generalizations thereof are employed to represent both the density dependence as well as the attractive coupling dependence of the perturbation expansion. Equations of state are constructed and compared with Jastrow--Monte Carlo calculations as well as expectations based on semiempirical mass formulas. Excellent agreement is found with the latter.