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Physical Review C

New Integrodifferential Equation and Integral Constraint for the Correlation Function Between Two Nucleons in Nuclear Matter

M. E. Grypeos

Published Jan 1, 1973• 4 citations
Publisher

Abstract

A nonlinear integrodifferential equation for the correlation function between two nucleons in nuclear matter is obtained. This equation results by writing the (trial) expression of the energy per particle in nuclear matter in a suitable form and applying subsequently the variational principle. The investigation of the behavior of the equation for large internucleon distances ${r}_{12}$ leads to a new integral constraint on the correlation function.

DOI
10.1103/physrevc.7.95

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A nonlinear integrodifferential equation for the correlation function between two nucleons in nuclear matter is obtained. This equation results by writing the (trial) expression of the energy per particle in nuclear matter in a suitable form and applying subsequently the variational principle. The investigation of the behavior of the equation for large internucleon distances r12{r}_{12}r12​ leads to a new integral constraint on the correlation function.
Calculations of the energy for two different nuclear matter models are carried out using Jastrow-Slater wave functions and an integral equation developed previously by Chakravarty and Woo. Even though the same variational parameters are used, our results differ significantly with those of Ceperley, Chester, and Kalos who employed a Monte Carlo technique. The discrepancy, mainly in the potential energy, is traced to short-range behavior of the pair correlation function. No satisfactory explanation can be offered at this time. Our integral equation is then subjected to a formal analysis for a weakly interacting Bose gas. It is shown to perform at least as well for boson systems as the Bogoliubov-Born-Green-Kirkwood-Yvon equation.NUCLEAR STRUCTURE Many-body variational calculation applied to nuclear matter model.
The hypotheticai system of nucleons known as nuclear matter provides a very interesting field of study with relevance to both nuclear physics and astrophysics. The task of finding nucleon-nucleon potentials that explain the empirical properties of nuclear matter, as well as the two-body data, has been hampered by the lack of an adequate many-body theory for treating the complicated forces involved. Realistic nucleon-nucleon potentials have strong spin, isospin, tensor and spin-orbit components which induce correspondingly strong correlations in nuclear matter. In this thesis we present a variational theory designed to cope with these complicated correlations. To place the theory in perspective we give an overview of several different many-body theories and compare their predictions in the somewhat simpler systems of the 4He and 3He liquids, and for a simple central potential model of nuclear matter. The variational theory is then developed, starting with the choice of wave function. A generalized Jastrow product of two-body correlation functions containing central, spin, isospin, tensor, and spin-orbit operators is used. The correlation function is generated by a series of two-body Schrodinger equations with boundary conditions which require the wave function to heal at a distance d. The wave function is parameterized by d and by the magnitudes of the non-central correlations. Expectation values with variational wave functions of the Jastrow type may be studied conveniently with generalized Mayer diagrams. Diagram rules are given and a general diagrammatic cluster expansion is derived. It is an expansion in powers of commutators of the non-central operators, and in their absence reduces to the usual Mayer cluster expansion. Evaluation of diagrams in the cluster expansion is complicated by the presence of the non-commuting operators. Some simple rules and useful methods for calculating the contribution due to the central, spin, isospin, and tensor operators are given. Spin-orbit operators are neglected beyond this point, but the approach used is extendible to them. Large classes of diagrams contributing to an expectation value can be summed by means of integral equations. The development of single chain, hypernetted chain, and Fermi hypernetted chain (FHNC) equations for central correlations is reviewed. The chain summation methods are extended to sum single operator chains (SOC) for the non-central operators. Methods are also given for treating the leading commutator corrections in the expansi0n by simple vertex factors. The energy expectation value for potentials with six operators are evaluated using the FHNC/SOC functions, with an exact treatment of the commutators involved. The energy is found to have a minimum with respect to variations in all parameters. Results of calculations with model potentials based on the Reid, Bethe-Johnson, Hamada-Johnston, and Gammel-Thaler potentials are reported. A crude estimate of the effect of the neglected spin-orbit potentials on the nuclear matter binding energy and saturation density indicates that it could be significant. A correct treatment of the spin-orbit potentials is the next major step that should be taken.
The self-consistent Green’s function method within the ladder approximation provides a microscopic description of correlated nuclear systems which properly treats the nucleon-nucleon correlations induced by the short-range and tensor components present in any realistic interaction. These correlations produce a sizable depletion of low momenta below the Fermi surface as well as the occupation of high momenta in the nuclear ground state. A few representative results for nuclear matter are presented to illustrate the present progress in the application of this method to nuclear systems.
A trial wavefunction is constructed for nuclear matter in terms of the reaction matrix. The trial function includes only two-particle excitations and the resulting energy expectation value for two-nucleon forces with a state-independent repulsive core of Yukawa shape approaches in the thermodynamic limit the energy given by the lowest-order Brueckner theory with standard dispersion. Evaluation of the energy expectation value is carried out in a systematic way by use of an identity that holds between the Hamiltonian and reaction matrix.
The equation of state of symmetric nuclear matter is addressed starting both from a realistic interaction derived from nucleon-nucleon scattering processes and from a low-momentum effective potential. The approach is based on finite temperature Green’s functions. The internal energy per particle is estimated from the summation of diagrams and through the Galitskii-Koltun’s sum rule. At present, there is not a unique and reliable theory of nuclear interactions. From the study of nucleon-nucleon scattering processes different models for the basic interaction between nucleons have been developed, but difficulties arise when they are used in calculations which aim to reproduce the properties of bound nuclei. These models all reproduce the experimental N-N phase shifts up to the threshold for pion production with high accuracy, but differ in the short-wavelength region, where less experimental constraints are available. In deriving the properties of dense nuclear matter, if we start from a bare N-N potential we have to take into account the complex correlations induced
The conventional description of nuclear structure relies heavily on the nuclear mass density function ρ(r) which captures the distribution of nucleons within the nucleus. Recent theoretical developments suggest that this view is incomplete because it neglects the spatial distribution of quantum informational content that arises from many–body correlations and entanglement among nucleons. In this paper we introduce the Nuclear Information Density Spectrum (NIDS), a formalism that assigns to each point inside a nucleus an information density function I(r). This function quantifies the amount of quantum information encoded at each spatial coordinate and is defined through the von Neumann entropy of reduced density matrices of local subsystems. By integrating I(r) over the nuclear volume we recover the total quantum informational content, while the shape of I(r) reflects the distribution of correlation strength throughout the nucleus. We derive explicit expressions for I(r) in terms of the two–body correlation function g2(r1,r2) and outline a strategy to extract I(r) from high–energy elastic and inelastic scattering experiments. The NIDS formalism provides a new lens to view the internal structure of nuclei, potentially redefining shell closures, deformation, and collective modes. Moreover, it offers a quantitative framework to assess the role of quantum correlations in nuclear binding, with implications for the interpretation of exotic nuclei and the equation of state of dense matter. The proposal that the nucleus contains an intrinsic information density invites fresh experimental investigations that could lead to a Nobel–level paradigm shift in nuclear physics.