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Physical Review C

Neutron matter calculations by the Fermi-hypernetted-chain method

Børge Arntsen, Erlend Østgaard

Published Jul 1, 1984• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.30.335

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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.
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.
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.
The half-diagonal two-body density matrix ${\ensuremath{\rho}}_{2h}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{{1}^{\ensuremath{'}}})$ is a key quantity in theoretical descriptions of nucleon knockout reactions at intermediate energies, due to its central role in treatments of the propagation and final-state interactions of ejected nucleons in the nuclear medium. This quantity is calculated for a simple model of the ground state of infinite symmetrical nuclear matter, based on a Jastrow-correlated wave function and a Fermi hypernetted-chain analysis. The dependence of ${\ensuremath{\rho}}_{2h}$ on the variables ${r}_{{11}^{\ensuremath{'}}}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}|,$ ${r}_{12}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}|,$ and the angle $\ensuremath{\theta}$ between ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}$ and ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$ is investigated in some detail. Significant departures from ideal Fermi gas behavior in certain domains reflect the importance of short-range correlations and their interplay with statistical correlations. The quality of approximations to ${\ensuremath{\rho}}_{2h}$ proposed by Gersch et al., Rinat, and Silver is assessed by comparison with the Fermi hypernetted-chain results.
We apply ideas of the parquet-diagram and optimized Fermi-hypernetted chain methods to determine the short-range structure of the pair wave function in neutron matter and compare these with Bethe-Goldstone results and those of low-order variational calculations. It is shown that the induced interaction, describing the exchange of density, spin, and tensor fluctuations, has a profound influence on the short-ranged structure of the pair wave function and, hence, on effective interactions in neutron matter.
In this paper, we calculate some properties of polarized neutron matter using the lowest order constrained variational method with the AV18{\mathrm{AV}}_{18}AV18​ potential and employing a microscopic point of view. A comparison is also made between our results and those of other many-body techniques.
The nucleonic matter problem is investigated by comparing the lowest order constrained variational ( LOCV ) method with the Fermi hypernetted chain ( FHNC ) theory, emphasizing the role of the LOCV correlation functions. In this way, the central correlation functions are used in the LOCV formalism, for the Bethe homework problem. It is shown that the LOCV computations reasonably agree with those of FHNC . Moreover, the FHNC calculations are performed with the LOCV correlation functions. It is found that, assuming the LOCV or the parametrized correlation functions, the FHNC computations do not change significantly. So, one may conclude that the mentioned consistencies refer to the choice of the LOCV correlation functions. Because, the contribution of the many-body cluster terms can be ignored, if the LOCV correlation functions satisfy the normalization constraint. Then, using the AV 18 interaction, the operator-dependent ( OD ) correlation functions are employed in the LOCV calculations. Note that the LOCV OD correlation functions are obtained by averaging over the states. It turns out that the overall behaviour of the LOCV OD correlation functions are similar to those of FHNC . Although, due to the many-body effects which are considered in the FHNC calculations, the LOCV results fairly differ from those of FHNC . Finally, it is worth mentioning that, unlike the recent FHNC calculations, the spin-orbit-dependent correlation functions are included in the LOCV approach.