The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.
The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.
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.
A theory is formulated to derive an effective interaction in nuclei in the framework of the unitarymodel-operator approach.A unitary operator is introduced to describe two-body correlations, and it is determined from the condition that the effective interaction should be decoupled between the lowmomentum model space and the high-momentum space.The unitary-transformed Hamiltonian is represented in a cluster expansion form.The one-and two-body cluster terms are mainly discussed.The theory is applied to the calculation of the energies of the ground-and one-body states with the Reid soft core potential.A large gain of the energy has been obtained for the ground state in comparison with the result of the usual G-matrix theory.As for the one-body energies obtained, agreement with the experimental values is fairly good.Although the discrepancies with the experiments still remain at the 1O~ 15 % level, the present approach shows to have some desirable features compared with the nuclear many-body theories proposed so far. § 1. IntroductionA fundamental problem in nuclear physics is to clarify the properties of nuclei in terms of the interaction among their constituent particles.Although there have been many theories or approaches, we have not yet reached the goal of reproducing successfully the experimental data.The investigation of nuclear properties, in the usual nonrelativistic approach, proceeds in two steps: The first one is to determine the interaction among nucleons, and the second is to solve the many-body SchrOdinger equation.Concerning the nucleon-nucleon force, it seems that the phenomenological potential, such as the Reid potential, would be incomplete and one should wait until quark dynamics, probably quantum chromodynamics (QCD), solves finally the problem o~ nuclear force.It is of no doubt that in order to fit the experiments we need more accurate knowledge on the nuclear force including the threebody force.However, if we want to reach the goal, we need also a good many-body theory.Regardless of the accuracy of the nuclear force, we shall not be able to solve the problem unless we have a way of constructing an accurate solution for a many-body system.We believe that many-body theory should share the responsibility for the discrepancy between theory and experiment.It will be sure that nuclear many-body theory is still to be improved.Among many theories for nuclear many-body problem, the most widely applied method is, needless to say, the linked-diagram perturbation theory.This approach may also be referred to as the folded-diagram theory based on the G-matrix interaction especially due 'to Brandow 1l and Kuo et al. 2l An outstanding problem in this approach is the convergence of the perturbation expansion.Since Schucan and Weidenmliller 3 ) pointed out the divergence of the perturbation expansion due to the presence of intruder states, this difficulty has long been a serious problem.Present status of this problem is
A closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.
The formalism based on correlated basis functions and the cluster-expansion technique has been recently employed to derive an effective interaction from a realistic nuclear Hamiltonian. To gauge the reliability of this scheme, we perform a systematic comparison between the results of its application to the Fermi hard-sphere system and the predictions obtained from low-density expansions, as well as from other many-body techniques. The analysis of a variety of properties, including the ground-state energy, the effective mass, and the momentum distribution, shows that the effective-interaction approach is quite accurate, thus suggesting that it may be employed to achieve a consistent description of the structure and dynamics of nuclear matter in the density region relevant to astrophysical applications.
A series of lectures on nuclear matter is presented. The work done at Utrecht in the area of the general theory of systems of N Fermi particles, application of the theory to nuclear matter, and a critical survey of the works of Brueckner are discussed. An introduction and history, the Hamiltonian, a general study of the resolvent by the method of perturbations, matrix element of the resolvent R/sub z/ for the ground state, study of the ground state, wave function of the ground state, excited states, singular forces, case of low densities, the Brueckner theory, numerical results for E/sub 0/, and unresolved problems are considered. (J.S.R.)
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."