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Journal of Physics G Nuclear and Particle Physics

Many-body approximations in the presence of proton-neutron pairing

Stuart Pittel, Jan Dobeš

Published Jan 1, 1999• 3 citations
Publisher

Abstract

The applicability of the generalized BCS approximation in the presence of proton-neutron pairing is studied in the context of two algebraic models. The analysis suggests that approximations based solely on pair correlations cannot describe the ground states of these models. Full symmetry restoration or inclusion of four-body -like correlations is required.

DOI
10.1088/0954-3899/25/4/015

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The applicability of the generalized BCS approximation in the presence of proton-neutron pairing is studied in the context of two algebraic models. The analysis suggests that approximations based solely on pair correlations cannot describe the ground states of these models. Full symmetry restoration or inclusion of four-body -like correlations is required.
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 mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two-, and four-body operators within a basis of particle-number-projected BCS states are analytically expressed in terms of a single function Q(N)Q(N)Q(N) depending on the number of particles, NNN. The function Q(N)Q(N)Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any iteration step. An illustrative example is numerically considered in a restricted single-particle space. Some specific features of the standard BCS, the projection after variation approach, and the variation after projection formalism are pointed out.
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, NNN. The function Q(N) is calculated using a recursion in NNN in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after variation approach as well as for the variation after projection formalism, are pointed out.
It is pointed out that the ground state of nnn neutrons and nnn protons in a single-jjj shell, interacting through an isoscalar (T=0T=0T=0) pairing force, is not paired, J=0J=0J=0, but rather spin aligned, J=nJ=nJ=n. This observation is explained in the context of a model of isoscalar PPP (J=1J=1J=1) pairs, which is mapped onto a system of ppp bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in jjjjjj coupling.
The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.