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Physical Review C

Nucleon-number-conserving description of pairing correlations in terms of occupation probability amplitudes

M. Hasegawa, S. Tazaki

Published Apr 1, 1987• 14 citations
Publisher

Abstract

A practical number-conserving method of treating pairing correlations is proposed. The theory is formulated in a form having clear correspondence with quasiparticle theory and, moreover, it works well even in the weak pairing limit. Various matrix elements are approximately expressed in terms of the occupation probabilities containing the blocking effects within the number-conserving framework. The exact occupation probabilities can be used in the case of a constant pairing force. The present method is able to take the place of previous fixed-number BCS methods and is suitable for treating residual correlations further. Application to the cranking model is discussed.

DOI
10.1103/physrevc.35.1508

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A practical number-conserving method of treating pairing correlations is proposed. The theory is formulated in a form having clear correspondence with quasiparticle theory and, moreover, it works well even in the weak pairing limit. Various matrix elements are approximately expressed in terms of the occupation probabilities containing the blocking effects within the number-conserving framework. The exact occupation probabilities can be used in the case of a constant pairing force. The present method is able to take the place of previous fixed-number BCS methods and is suitable for treating residual correlations further. Application to the cranking model is discussed.
A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.
In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.
Two advances are reviewed in the application of pairing-force theory in the nuclear shell model. The first exploits a discovery that a wide range of two-nucleon interactions conserve seniority as a good quantum number. Asa consequence, the eigenstates of a hamiltonian with such an interaction belong to irreducible representations of a compact unitary-symplectic group. This makes it possible to extend the simply-solvable models with J = 0 pairing forces to a much richer set of models and still obtain states uniquely classified by their seniority and angular momentum quantum numbers. Moreover, many of the low-lying energy levels of such models can be obtained algebraically; in technical terms, the models are in some cases completely solvable and in other cases partially solvable by algebraic methods. The second advance exploits the discovery that, in a coherent state representation, states of good nucleon number can be projected analytically from BCS vacuum and excited quasiparticle states. This makes it possible to perform calculations in a number-projected BCS basis without losing much of the advantage of working in a quasiparticle scheme.
A method for projecting the Bardeen-Cooper-Schrieffer states with respect to the particle number is suggested. This method does not involve the Fowler-Darwin integrals which are usually applied for this purpose. Energy matrix elements of an arbitrary single-particle operator and overlap integrals are calculation in the quasi- particle representation and are expressed analytically in terms of the u and v parameters. The volume of calculations according to the method suggested increases linearly with the size of the system in the case of factorizable pairing matrix elements (the weaker the particle number fluctuations, the smaller the quantity of calculations). The method is illustrated by some model systems. In the paper the limiting case of weak correlations is considered and comparison with perturbation theory is carried out.
An equations-of-motion method is proposed for treating neutron-proton pairing correlations conserving both nucleon number NNN and isospin TTT. This method makes it possible to find approximate solutions to the charge-independent pairing Hamiltonian by a step-by-step procedure. It is shown that for the degenerate model the exact results are obtained.NUCLEAR STRUCTURE Charge-independent treatment of pairing correlations.