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.
"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."
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 ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function
g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The
n-particle distribution functions
p(n) are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the
g(r) function for
n=3and4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using
p(3) function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function
U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for
U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of
U(r),
p(3),
p(4), and the ground-state energy are computed using the Gaussian approximation for
g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of
p(n) in terms of
g(r).
Abstract The main task of microscopic many-body theory is to provide an understanding and an explanation of properties of macroscopic systems from no other information than the properties of the underlying Hamiltonian, the particle statistics, and the macroscopic geometry of the system. Variational wave functions have been a very successful technique for examining the ground state and dynamic properties of dense quantum fluids. Specifically the “optimized (Fermi)-hypernetted chain” ((F)HNC-EL) hierarchy of approximations reproduces, for simple interactions like electrons and quantum fluids, the most basic features of a self-bound many-body system, namely binding, saturation, and spinodal decomposition. The relationships between Green’s functions based perturbation theory and variational wave functions have been clarified in much detail in the past. In the language of Feynman diagrams, a self-consistent summation of ring- and ladder-diagrams is the minimum requirement for a satisfactory microscopic description of these basic features. Realistic nucleon–nucleon interactions pose further problems: The most difficult aspect of the nuclear many-body problem is caused by the form of the microscopic nucleon–nucleon interaction which depends at least on the spin, isospin, the relative orientation and angular momentum of the interacting particles. Simple variational methods as used for electrons and quantum fluids are inadequate. Correlation functions that have the same structure as the interactions lead to so-called “commutator diagrams” that have, so far, been mostly neglected. We have shown in the past that these corrections can be very important if the two-body interactions are very different in different spin- or isospin channels. To deal with the problem of realistic nuclear interactions we have combined techniques of the Jastrow–Feenberg variational method and the local parquet-diagram theory. In the language of diagrammatic perturbation theory, “commutator diagrams” can be identified with non-parquet diagrams. We examine the physical processes described by these terms and include the relevant diagrams in a way that is suggested by the Jastrow–Feenberg approach. We show that the corrections from non-parquet contributions are, at short distances, larger than all other many-body effects. We examine here neutron matter as a prototype of systems with state-dependent interactions. Calculations are carried out for neutrons interacting via the so-called
v8 v 8 version of four popular interactions. We determine the structure and effective interactions and apply the method to the calculation of the energetics, structure and dynamic properties such as the single-particle self-energy and the dynamic response functions as well as BCS pairing in both singlet and triplet states. We find that many-body correlations lead to a strong reduction of the spin-orbit interaction, and, therefore, to a suppression of the
3P2 3 P 2 and
3P2 3 P 2 -
3F2 3 F 2 gaps. We also find pairing in
3P0 3 P 0 states; the strength of the pairing gap depends sensitively on the potential model employed.
A nonperturbative derivation of the Brueckner approximation to the ground-state energy of nuclear matter (with standard choice of hole and particle energies) is given within the framework of the method of correlated basis functions. The structure of the factor-cluster (FIY) expansion of the expectation value of the Hamiltonian with respect to a trial function incorporating short-range correlations in a general fashion, is analyzed for a uniform extended medium of fermions. With a special choice of correlated wave function, the Brueckner approximation is extracted from the FIY expansion by selective summation of two-body combination terms to all orders in the smallness parameter $\ensuremath{\kappa}$ associated, in this case, with both Brueckner-Bethe-Goldstone and FIY energy expansions. A reexamination of the numerical comparison of conventional Brueckner and Jastrow methods carried out earlier for two simple models of nuclear matter leads to the conclusion that it is inadvisable, at least in a Jastrow calculation satisfying the average Pauli condition, to incorporate the $O(\ensuremath{\kappa})$ contribution of the analog of the dispersion correction of reaction-matrix theory without at the same time including the $O(\ensuremath{\kappa})$ contribution of the analog of the Bethe-Faddeev term. For the uniform extended medium for the Jastrow choice of wave function, there is a formal cancellation of these two three-body contributions, the scale of this cancellation being governed by the size of $\ensuremath{\kappa}$. In the numerical example considered, $\ensuremath{\kappa}$ is about 0.22 and the cancellation goes something like +14 MeV-15 MeV=-1 MeV. These findings, when juxtaposed with the standard Brueckner results for the simple models in question, suggest that a departure from the standard prescription of hole and particle energies in reaction-matrix theory may be necessary when $\ensuremath{\kappa}$ is sizeable-as in liquid
3He and the high-density neutron gas, as well as equilibrium nuclear matter described with some potentials.
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.