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Physical Review C

Two-Body Interactions: The Method of Correlated Quasiparticles. I.

Richard R. Chasman

Published Jan 1, 1972• 12 citations

Abstract

A method is proposed for determining the ground state of the Hamiltonian $H=\ensuremath{\Sigma}\stackrel{}{k}{\ensuremath{\epsilon}}_{k}{N}_{k}\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{\mathrm{ijkl}}{V}_{\mathrm{ijkl}}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}$when the residual interaction is such that $〈{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}〉={〈{N}_{i}{N}_{j}(1\ensuremath{-}{N}_{k})(1\ensuremath{-}{N}_{l})〉}^{\frac{1}{2}}{〈{N}_{k}{N}_{l}(1\ensuremath{-}{N}_{i})(1\ensuremath{-}{N}_{j})〉}^{\frac{1}{2}}$is a good approximation. The method is applied to pairing interactions involving one type of nucleon as well as to the generalized pairing problem containing neutron-neutron, neutron-proton, and proton-proton interactions.

DOI: 10.1103/physrevc.5.29

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A method is proposed for determining the ground state of the Hamiltonian

H=\ensuremath{\Sigma}\stackrel{}{k}{\ensuremath{\epsilon}}_{k}{N}_{k}\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{\mathrm{ijkl}}{V}_{\mathrm{ijkl}}{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}

when the residual interaction is such that

〈{a}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{k}{a}_{l}〉={〈{N}_{i}{N}_{j}(1\ensuremath{-}{N}_{k})(1\ensuremath{-}{N}_{l})〉}^{\frac{1}{2}}{〈{N}_{k}{N}_{l}(1\ensuremath{-}{N}_{i})(1\ensuremath{-}{N}_{j})〉}^{\frac{1}{2}}

is a good approximation. The method is applied to pairing interactions involving one type of nucleon as well as to the generalized pairing problem containing neutron-neutron, neutron-proton, and proton-proton interactions.

The new method presented by B. Bremond and J. G. Valatin is generalized which at-tache ~ four body correlation to the Bogoliubov transformation. The generalization carried in this paper is similar to that from the Bogoliubov transformation to the generalized Bogoliubov one. We find a criterion of whether a trial ground state vector has an essential feature of four body correlation or it is like that of the generalized Bogoliubov transformation. This method is applied to a system with a charge independent pairing interaction. There appear two. different kinds of single quasiparticle excitation energy due to proton-neutron interaction.

The method of coherent correlated pairs is applied with a realistic effective Hamiltonian to the four-nucleon system. Three methods of solution are compared for ease of implementation and convergence rates. A Monte Carlo method yields the best upper bound within the coherent correlated pair space. The full variational treatment of the four-nucleon coherent correlated pair problem yields equations which have poor stability properties. An ansatz variational method proves to be rapidly convergent and yields results very close to those of the Monte Carlo method. The results for binding energy and the one body density distribution are compared with Hartree-Fock results using the same effective Hamiltonian in the same model space. As in previous soluble model applications the coherent correlated pair method provides a superior upper bound to the ground state energy.

It is pointed out that the ground state of

n

neutrons and

n

protons in a single-

j

shell, interacting through an isoscalar (

T=0

) pairing force, is not paired,

J=0

, but rather spin aligned,

J=n

. This observation is explained in the context of a model of isoscalar

P

(

J=1

) pairs, which is mapped onto a system of

p

bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in

jj

coupling.

Neutron-proton pairing correlations are studied within the context of two solvable models, one based on the algebra SO(5) and the other on the algebra SO(8). In both of these models, particles interact in

L=0

pairs only. Boson-mapping techniques are applied to these models and shown to provide a convenient methodological tool both for solving such problems and for gaining useful insight into general features of pairing. We first focus on the SO(5) model, which involves generalized

T=1

pairing. Neither boson mean-field methods nor fermion-pair approximations are able to describe in detail neutron-proton pairing in this model. The analysis suggests, however, that the boson Hamiltonian obtained from a mapping of the fermion Hamiltonian contains a pairing force between bosons, pointing to the importance of boson-boson (or equivalently four-fermion) correlations with isospin

T=0

and spin

S=0

. These correlations are investigated by carrying out a second boson mapping. Closed forms for the fermion wave functions are given in terms of the fermion-pair operators. Similar techniques are applied---albeit in less detail---to the SO(8) model, involving a competition between

T=1

and

T=0

pairing. Conclusions similar to those of the SO(5) analysis are reached regarding the importance of four-particle correlations in systems involving neutron-proton pairing.