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Physical Review C

Study of Approximations in the Nuclear Pairing-Force Problem

Mordechai Bishari, I. Unna, A. Mann

Published Apr 1, 1971• 13 citations
Publisher

Abstract

The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.

DOI
10.1103/physrevc.3.1715

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The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.
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 new technique is developed in this paper for solving Hamiltonians of the form $H=\ensuremath{\Sigma}\stackrel{}{k>0}{\ensuremath{\epsilon}}_{k}({{a}_{k}}^{+}{a}_{k}+{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}k})\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{k,l>0}G(k,l){{a}_{k}}^{+}{{a}_{\ensuremath{-}k}}^{+}{a}_{\ensuremath{-}l}{a}_{l},$ where G(k,l)G(k, l)G(k,l) may have the forms: G(k,l)=GG(k, l)=GG(k,l)=G (independent of kkk and lll), G(k,l)=GkGlG(k, l)={G}_{k}{G}_{l}G(k,l)=Gk​Gl​, or G(k,l)=G(k,l)G(k, l)=G(k, l)G(k,l)=G(k,l) (nonseparable in kkk and lll). The eigenvalues obtained with this technique are compared with the exact solutions of Kerman, Lawson, and Macfarlane. The agreement with the exact solutions is fairly good.
The BCS and/or Hartree-Fock-Bogoliubov theories are extended by treating the effect of four quasiparticle states perturbatively. The approach is tested on the pairing Hamiltonian, showing that it combines the advantage of standard perturbation theory, valid at low pairing strength, and of nonperturbative approaches breaking the particle number, valid at higher pairing strength. Including the restoration of particle number further improves the description of pairing correlation. In the presented test, the agreement between the exact solution and the combined perturbative plus projection is almost perfect. The proposed method scales friendly when the number of particles increases and provides a simple alternative to other more complicated approaches.
Exact eigenstates of the pairing-force Hamiltonian have been calculated for 2N2N2N particles in 2N2N2N equispaced, doubly degenerate single-particle levels for N=4,8,12,and16N=4, 8, 12, \mathrm{and} 16N=4,8,12,and16. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of NNN than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.