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Physical Review C

Number-conserving random phase approximation with analytically integrated matrix elements

M. Kyotoku, K.W. Schmid, Frank Grümmer, Amand Faessler

Published Jan 1, 1990• 11 citations
Publisher

Abstract

In the present paper a number conserving random phase approximation is derived as a special case of the recently developed random phase approximation in general symmetry projected quasiparticle mean fields. All the occurring integrals induced by the number projection are performed analytically after writing the various overlap and energy matrices in the random phase approximation equation as polynomials in the gauge angle. In the limit of a large number of particles the well-known pairing vibration matrix elements are recovered. We also present a new analytically number projected variational equation for the number conserving pairing problem.

DOI
10.1103/physrevc.41.284

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In the present paper a number conserving random phase approximation is derived as a special case of the recently developed random phase approximation in general symmetry projected quasiparticle mean fields. All the occurring integrals induced by the number projection are performed analytically after writing the various overlap and energy matrices in the random phase approximation equation as polynomials in the gauge angle. In the limit of a large number of particles the well-known pairing vibration matrix elements are recovered. We also present a new analytically number projected variational equation for the number conserving pairing problem.
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.
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.
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 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.