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Physical Review

Neutron Gas

J. S. Levinger, L. M. Simmons

Published Nov 1, 1961• 53 citations
Publisher

Abstract

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 $0.5<{k}_{f}<2$ ${\mathrm{f}}^{\ensuremath{-}1}$. 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 $^{1}S$ and $^{1}D$ phase shifts. In the range of densities $0.5<{k}_{f}<1$ ${\mathrm{f}}^{\ensuremath{-}1}$, 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.

DOI
10.1103/physrev.124.916

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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 $n\ensuremath{-}n$ force alone does not exist for a system of neutron gas.
I study the properties of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons. In particular, I analyze its energy (${E}_{\ensuremath{\downarrow}}$), effective mass (${m}_{\ensuremath{\downarrow}}^{*}$), and quasiparticle residue (${Z}_{\ensuremath{\downarrow}}$). Results are compared with those of state-of-the-art quantum Monte Carlo calculations of the attractive Fermi polaron realized in ultracold atomic gases experiments, and with those of previous studies of the neutron polaron. Calculations are performed within the Brueckner-Hartree-Fock approach using the chiral two-body nucleon-nucleon interaction of Entem and Machleidt at N3LO{\mathrm{N}}^{3}\mathrm{LO}N3LO with a 500 MeV cut-off and the Argonne V18 phenomenological potential. Only contributions from the 1S0{}^{1}{S}_{0}1S0​ partial wave, which is the dominant one in the low-density region considered, are included. Contributions from three-nucleon forces are expected to be irrelevant at these densities and, therefore, are neglected in the calculation. It is shown that for Fermi momenta between $\ensuremath{\approx}0.25$ and $\ensuremath{\approx}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ the energy, effective mass, and quasiparticle residue of the impurity vary only slightly, respectively, in the ranges $\ensuremath{-}0.604\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.635\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.300\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.085\phantom{\rule{0.16em}{0ex}}m$, and $0.741<{Z}_{\ensuremath{\downarrow}}<0.836$ in the case of the chiral interaction, and $\ensuremath{-}0.621\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.643\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.310\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.089\phantom{\rule{0.16em}{0ex}}m$, and $0.739<{Z}_{\ensuremath{\downarrow}}<0.832$ when using the Argonne V18 potential. These results are compatible with those derived from ultracold atoms and show that a spin-down neutron impurity in a free Fermi gas of spin-up neutrons with a Fermi momentum in the range $0.25\ensuremath{\lesssim}{k}_{F}\ensuremath{\lesssim}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ exhibits properties very similar 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.