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Physical Review

Ground-State Energy of Neutron Gas

Charles T. P. Wang

Published Jan 20, 1969• 5 citations
Publisher

Abstract

In the calculation of the energy per neutron in the zeroth cluster and first perturbative order at different neutron densities, a complete set of correlated basic functions has been used to represent the ground-state wave function for a system of neutron gas. Three potentials, all fitting two-nucleon data, were used. They are the Hamada-Johnston (HJ), Breit, and Brueckner-Gammel-Thaler (BGT) potentials. The results show that the energy decreases monotonically, as neutron density decreases and that the energy values at low densities are nearly the same for all three potentials. But at high densities, the BGT result departs quite markedly from those for HJ and Breit potentials. The results also indicate that a bound state due to the nuclear $n\ensuremath{-}n$ force alone does not exist for a system of neutron gas.

DOI
10.1103/physrev.177.1452

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In the calculation of the energy per neutron in the zeroth cluster and first perturbative order at different neutron densities, a complete set of correlated basic functions has been used to represent the ground-state wave function for a system of neutron gas. Three potentials, all fitting two-nucleon data, were used. They are the Hamada-Johnston (HJ), Breit, and Brueckner-Gammel-Thaler (BGT) potentials. The results show that the energy decreases monotonically, as neutron density decreases and that the energy values at low densities are nearly the same for all three potentials. But at high densities, the BGT result departs quite markedly from those for HJ and Breit potentials. The results also indicate that a bound state due to the nuclear n\ensuremath−nn\ensuremath{-}nn\ensuremath−n force alone does not exist for a system of neutron gas.
We assume that the neutron-neutron potential is well-behaved and velocity-dependent. We can then apply perturbation theory to find the energy per particle of a neutron gas, in the range of Fermi wave numbers 0.5<kf<20.5<{k}_{f}<20.5<kf​<2 ${\mathrm{f}}^{\ensuremath{-}1}$. The energy through first order is found in closed form, or by a single numerical integration. We use two different velocity-dependent potentials adjusted to fit observed nucleon-nucleon 1S^{1}S1S and 1D^{1}D1D phase shifts. In the range of densities 0.5<kf<10.5<{k}_{f}<10.5<kf​<1 ${\mathrm{f}}^{\ensuremath{-}1}$, our two potentials give nearly the same energy/particle (within 0.5 Mev); our values tend to run an Mev below values found by Brueckner et al., for the Gammel-Thaler potential. Wider divergences appear at higher densities. Our values, and Brueckner's are higher than those found by Salpeter by a semiempirical approach. A crude estimate of the second-order energy for our potentials indicates that perturbation theory converges rapidly in the density range considered. Our results suggest that at moderately low densities the energy/particle in a many-body system is insensitive to the shape or nonlocal character of the assumed two-body potential.
We review the properties of neutron matter in the low-density regime. In particular, we revise its ground state energy and the superfluid neutron pairing gap and analyze their evolution from the weak to the strong coupling regime. The calculations of the energy and the pairing gap are performed, respectively, within the Brueckner–Hartree–Fock (BHF) approach of nuclear matter and the Bardeen–Cooper–Schrieffer (BCS) theory using the chiral nucleon-nucleon interaction of Entem and Machleidt at N3LO and the Argonne V18 phenomenological potential. Results for the energy are also shown for a simple Gaussian potential with a strength and range adjusted to reproduce the1S0neutron-neutron scattering length and effective range. Our results are compared with those of quantum Monte Carlo (QMC) calculations for neutron matter and cold atoms. The Tan contact parameter in neutron matter is also calculated, finding a reasonable agreement with experimental data from ultra-cold atoms only at very low densities. We find that low-density neutron matter exhibits a behavior close to that of a Fermi gas at the unitary limit, although, this limit is actually never reached. We also review the properties (energy, effective mass, and quasiparticle residue) of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons already studied by the author in a recent work where it was shown that these properties are very close to those of an attractive Fermi polaron in the unitary limit.
We present calculations of the energy per particle of pure neutron and symmetric nuclear matter with simplified Argonne nucleon-nucleon potentials for different many-body theories. We compare critically the Brueckner-Hartree-Fock results to other formalisms, such as the Brueckner-Bethe-Goldstone expansion up to third order, self-consistent Green's functions, auxiliary field diffusion Monte Carlo, and Fermi hypernetted chain. We evaluate the importance of spin-orbit and tensor correlations in the equation of state and find these to be important in a wide range of densities.
Uniform neutron matter is approximated by a cubical box containing a finite number of neutrons, with periodic boundary conditions. We report variational and Green's function Monte Carlo calculations of the ground state of fourteen neutrons in a periodic box using the Argonne ${v8}^{\ensuremath{'}}$ two-nucleon interaction at densities up to one and half times the nuclear matter density. The effects of the finite box size are estimated using variational wave functions together with cluster expansion and chain summation techniques. They are small at subnuclear densities. We discuss the expansion of the energy of low-density neutron gas in powers of its Fermi momentum. This expansion is strongly modified by the large nn\mathrm{nn}nn scattering length, and does not begin with the Fermi-gas kinetic energy, as assumed in both Skyrme and relativistic mean field theories. The leading term of neutron gas energy is approximately half the Fermi-gas kinetic energy. The quantum Monte Carlo results are also used to calibrate the accuracy of variational calculations employing Fermi hypernetted and single operator chain summation methods to study nucleon matter over a larger density range, with more realistic Hamiltonians including three-nucleon interactions.
The Fermi-hypernetted-chain method for quantum many-body calculations is studied and investigated in some detail by calculations of the ground-state energy for neutron matter. The calculations are done for five different two-body potentials and results are compared with other theoretical results. Our Fermi-hypernetted-chain results are rather close to results obtained by lowest-order constrained variation calculations, and the results are quite dependent on the chosen potential, especially at high densities.
The ground state energy of the Bose particle system at low densities is investigated by using the method which is based on summing up the terms of the conventional perturbation series giving rise to the lowest order with respect to the density. The summation is shown to be greatly simplified if the scattering matrix is introduced, and a simple example is demonstrated for the repulsive square well potential by calculating a few terms of the series first directly and secondly with the aid of scattering matrix. The final result is shown to be valid both for weak and strong interactions and the calculation is carried out analytically for some types of potential function. The connection of the present method with the pseudopotential method and with the Brueckner theory is investigated and some problems associated with the attractive interactions are discussed.