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Physical Review

Nuclear Many-Body Problem with Nonuniform Density. I. Theory

Daniel S. Koltun, L. Wilets

Published Jan 15, 1963• 8 citations
Publisher

Abstract

The many-body problem for nuclear matter is considered under the constraint that the density of the system vary periodically in space; this is equivalent to the problem of an external periodic potential. The method of Martin, Schwinger, and Puff is employed to derive, through second order in the amplitude of the density ripple, a set of tractable equations for the energy of the system. No new approximations are needed beyond those already assumed for the uniform density case. The model exhibits some interesting complications of the finite nuclear problem, and yet possesses sufficient simplicity for solubility. In the long-wavelength limit, an expression obtains for the explicit dependence of the energy on the density gradient; this relationship is useful in a study of the nuclear surface problem. Numerical calculations are planned.

DOI
10.1103/physrev.129.880

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The Martin-Schwinger-Puff theory for a many-fermion system is applied to a calculation of the appropriate density propagator. The real part of this function, the system's polarizability, is a measure for the lowest order change in energy due to a harmonic density ripple enforced on the infinite system. The imaginary part, the linear response function, yields the inelastic cross section for the scattering of a weak external agent. The linear response contains additional information on excited states and, in particular, on collective states, excited by an external field. The formalism necessitates the calculation of the self-energy of a particle and from it the optical model for nucleon scattering can be derived. The relation of the theory to perturbative and other nonperturbative approaches is discussed and a comparison is made with a calculation of the polarization energy of nuclear matter by Koltun and Wilets.

Physical Review•Jan 15, 1963•6 citations
The many-body problem for nuclear matter is considered under the constraint that the density of the system vary periodically in space; this is equivalent to the problem of an external periodic potential. The method of Martin, Schwinger, and Puff is employed to derive, through second order in the amplitude of the density ripple, a set of tractable equations for the energy of the system. No new approximations are needed beyond those already assumed for the uniform density case. The model exhibits some interesting complications of the finite nuclear problem, and yet possesses sufficient simplicity for solubility. In the long-wavelength limit, an expression obtains for the explicit dependence of the energy on the density gradient; this relationship is useful in a study of the nuclear surface problem. Numerical calculations are planned.
For one decade, several fields in physics as well microscopic as macroscopic benefit from the computational particle-models (astrophysics, electronics, fluids mechanics...). In particular, the nuclear matter offers an interesting challenge as many body problem, owing to the quantal nature of its components and the complexity of the in-medium interaction. Using a model derived from semi-classical Vlasov equation and the projection of the Wigner function on a Gaussian coherent states basis (pseudo-particles), static and dynamical properties of nuclear matter are studied, featuring the growing of bulk instabilities in dilute matter. Using different zero and finite range effective interactions, the effect of the model parameters upon the relation total energy - density - temperature and surface energy of the pseudo-particles fluid is pointed out. The dynamical feature is first based upon a model of the 2-body Uehling-Ulhenbeck collisionnal term. A study of the relaxation of a nucleonic system is performed. At last, the pseudo-particle model is used in order to extract time scale for the growing of density fluctuations. This process is supposed to be a possible way to clusterization during heavy nuclei collisions.
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.
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 Martin-Schwinger-Puff theory for a many-fermion system is applied to a calculation of the appropriate density propagator. The real part of this function, the system's polarizability, is a measure for the lowest order change in energy due to a harmonic density ripple enforced on the infinite system. The imaginary part, the linear response function, yields the inelastic cross section for the scattering of a weak external agent. The linear response contains additional information on excited states and, in particular, on collective states, excited by an external field. The formalism necessitates the calculation of the self-energy of a particle and from it the optical model for nucleon scattering can be derived. The relation of the theory to perturbative and other nonperturbative approaches is discussed and a comparison is made with a calculation of the polarization energy of nuclear matter by Koltun and Wilets.