A variational theory of nuclear matter
Robert B. Wiringa
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.
•Jan 1, 1978•0 citations