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Physical Review

Influence of the Exclusion Principle on the Scattering Theory of Nonlocal, Separable Interactions

C. B. Duke

Published Oct 12, 1964• 2 citations
Publisher

Abstract

For a nonlocal, separable interaction which describes low-energy nucleon-nucleon scattering, the scattering-theory solution to the Brueckner-Bethe-Goldstone (BBG) equation is obtained analytically. The associated scattering amplitudes are evaluated for all momenta of the interacting pair. The off-energy-shell amplitudes are defined by analytic continuation to complex values of the relative momentum of the pair. The positions of singularities in the amplitudes are investigated for various values of both the two-body kinematical parameters and the density of the passive BBG Fermi gas. A comparison of the amplitudes calculated with and without incorporating the effects of the exclusion principle indicates that even for repulsive interactions the latter may differ from the former by as much as 50% for Fermi-gas densities as low as $\frac{1}{500}$ of the observed nuclear density.

DOI
10.1103/physrev.136.b49

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Physical Review C•May 1, 1973•19 citations
For a nonlocal, separable interaction which describes low-energy nucleon-nucleon scattering, the scattering-theory solution to the Brueckner-Bethe-Goldstone (BBG) equation is obtained analytically. The associated scattering amplitudes are evaluated for all momenta of the interacting pair. The off-energy-shell amplitudes are defined by analytic continuation to complex values of the relative momentum of the pair. The positions of singularities in the amplitudes are investigated for various values of both the two-body kinematical parameters and the density of the passive BBG Fermi gas. A comparison of the amplitudes calculated with and without incorporating the effects of the exclusion principle indicates that even for repulsive interactions the latter may differ from the former by as much as 50% for Fermi-gas densities as low as 1500\frac{1}{500}5001​ of the observed nuclear density.
It is observed that unlike the versions which rely on auxiliary potentials designed to cancel diagrams exactly, the continuous version of the Brueckner-Bethe-Goldstone theory can be used to calculate the properties and in particular the ground state energy of repulsive many body systems. The only proviso amounts to a trivial requirement on the effective mass at the Fermi momentum. These results are related to those concerning the predictions of the corresponding versions of Green's function theory for the analytical properties of the mass operator. The case of attractive many body systems is also considered.
Abstract This chapter discusses the significance of two body interactions to many-body properties, beginning with a schematic model showing the relevance of short-range correlations beyond those of the Fermi gas. These correlations are studied within Brueckner theory and the help of the Bethe-Goldstone equation, mainly for nuclear matter proper but extensions to finite temperature are presented. Effective forces of Skyrme type are derived through an expansion of the G-matrix to small relative momenta. This scheme hinges essentially on the Brueckner-Hartree-Fock approximation, for which results for binding energy and equilibrium density are presented. An alternative variational method based on Jastrow functions is sketched, which demonstrates the necessity for considering effective three body forces. The concept of the nuclear equation of state is presented and parameterized by Skyrme forces with density dependent parameters. Finally, a semi-classical transport theory is presented which accounts for statistical collisions of the nucleons.
For a system of fermions with a three-body contact interaction the second-order contributions to the energy per particle $\ensuremath \bar E(k_f)$ are calculated exactly. The three-particle scattering amplitude in the medium is derived in closed analytical form from the corresponding two-loop rescattering diagram. We compare the (genuine) second-order three-body contribution to $\ensuremath \bar E(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependent effective two-body interaction, and find that the latter term dominates. The results of the present study are of interest for nuclear many-body calculations where chiral three-nucleon forces are treated beyond leading order via a density-dependent effective two-body interaction.
The momentum density, n(k) of interacting many-body Fermionic systems is studied (for k>kF) using examples of several well-known two-body interaction models. It is shown that n(k) can be approximated by a zero-range model for momenta k less than about 0.1/re, where re the effective range. If the scattering length is large and one includes the effects of a fixed value of re≠0, n(k) is almost universal for momenta k up to about 2/re. However, n(k) can not be approximated by a zero-range model for momenta k greater than about 1/(are2)1/3, where a is the scattering length, and if one wishes to maintain a sum rule that relates the energy of a two component Fermi-gas to an integral involving the density. We also show that the short separation distance, s, behavior of the pair density varies as s6.
We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.
A nonperturbative derivation of the Brueckner approximation to the ground-state energy of nuclear matter (with standard choice of hole and particle energies) is given within the framework of the method of correlated basis functions. The structure of the factor-cluster (FIY) expansion of the expectation value of the Hamiltonian with respect to a trial function incorporating short-range correlations in a general fashion, is analyzed for a uniform extended medium of fermions. With a special choice of correlated wave function, the Brueckner approximation is extracted from the FIY expansion by selective summation of two-body combination terms to all orders in the smallness parameter $\ensuremath{\kappa}$ associated, in this case, with both Brueckner-Bethe-Goldstone and FIY energy expansions. A reexamination of the numerical comparison of conventional Brueckner and Jastrow methods carried out earlier for two simple models of nuclear matter leads to the conclusion that it is inadvisable, at least in a Jastrow calculation satisfying the average Pauli condition, to incorporate the $O(\ensuremath{\kappa})$ contribution of the analog of the dispersion correction of reaction-matrix theory without at the same time including the $O(\ensuremath{\kappa})$ contribution of the analog of the Bethe-Faddeev term. For the uniform extended medium for the Jastrow choice of wave function, there is a formal cancellation of these two three-body contributions, the scale of this cancellation being governed by the size of $\ensuremath{\kappa}$. In the numerical example considered, $\ensuremath{\kappa}$ is about 0.22 and the cancellation goes something like +14 MeV-15 MeV=-1 MeV. These findings, when juxtaposed with the standard Brueckner results for the simple models in question, suggest that a departure from the standard prescription of hole and particle energies in reaction-matrix theory may be necessary when $\ensuremath{\kappa}$ is sizeable-as in liquid 3He^{3}\mathrm{He}3He and the high-density neutron gas, as well as equilibrium nuclear matter described with some potentials.