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Nuclear Matter Considered as a Fermi Liquid; LA MATIERE NUCLEAIRE CONSIDEREE COMME LIQUIDE DE FERMI

Leo Van Hove

Published Feb 1, 1958• 0 citations
OpenAlex

Abstract

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.)

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Physical Review•Mar 2, 1964•7 citations
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.)
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 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.
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.
Recent advances in variational and perturbative theories are surveyed which offer genuine promise that nuclear matter will soon become a viable tool for investigating nuclear interactions. The basic elements of the hypernetted chain expansion for Jastrow variational functions are briefly reviewed, and comparisons of variational and perturbative results for a series of increasingly complicated systems are presented. Prospects for investigating realistic forces are assessed and the unresolved, open problems are summarized.
This article is an account of the present position of theories of nuclear matter initiated some years ago mainly by Jastrow and Brueckner. Section 1 introduces the basic ideas. Section 2 discusses the variational approach. Section 3 develops methods involving partial summations in perturbation theory. The difficulties confronting the latter approach, and their partial resolution by ideas from superconductivity theory, are analysed in § 4. Much of the formalism is relevant also in the theory of individual nuclei (see for example the review by Eden 1959) and in fields other than nuclear physics (see for example The Many Body Problem, Les Houches, 1958). However, in the selection of material our criterion has been relevance for the ground state of nuclear matter.
Perturbation theoretic equations for several properties of the zero-temperature many-fermion normal system are rederived from general considerations. Though none of the equations are new, some of the derivations are, and taken together they form a brief and simple summary of many-fermion relations and a basis for further investigation of the basic perturbation description of the many-fermion normal system: the Brueckner-Goldstone expansion. A "change-of-parameter" technique is then developed and employed to investigate the possible use of the true momentum density instead of the unperturbed Fermi distribution in the Brueckner-Goldstone expansion and in the Brueckner KKK-matrix approximation. The result is a simpler, approximate perturbation series for the interaction energy, whose accuracy for nuclear matter is estimated to be approximately \ifmmode\pm\else\textpm\fi{}2 MeV. The new approximation is exact to the fourth order.