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Progress of Theoretical Physics

Modification of Quasi-Particle Theory in Spherical Nuclei. III: Couplings between Pairing Vibration and Quadrupole Vibration

M. Hasegawa, Nobuo Kanesaki

Published Sep 1, 1980• 4 citations
Publisher

Abstract

The couplings between pairing vibration and quadrupole vibration are investigated by the method presented in our preceding paper which deals with the pairing and quadrupole correlations on the number conserving basis. The results are compared with the conventional quasi-particle treatment. It is clarified that our method and the quasi-particle treatment give different couplings. The difference is evaluated in numerical examples of a two-level model.

DOI
10.1143/ptp.64.897

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The couplings between pairing vibration and quadrupole vibration are investigated by the method presented in our preceding paper which deals with the pairing and quadrupole correlations on the number conserving basis. The results are compared with the conventional quasi-particle treatment. It is clarified that our method and the quasi-particle treatment give different couplings. The difference is evaluated in numerical examples of a two-level model.
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 order to go beyond the conventional BCS plus RPA approach for the description of pairing correlation, we have developed a possible canonical theory. In this theory, we can exactly get rid of the well-known difficulty of the particle-number non-conservation in the BCS approximation. We give, in this paper, its approximate form in order to clarify the relation with the conventional approach. We show that our theory in its approximate form can reproduce the same result as that obtained by the BCS plus RPA approach. The equivalence between the approach developed by Suzuki et al. and the present one is also shown, within the RPA order at least. We give some perspective for the description of higher-order non-linear effect, such as the coupling between the pairing rotation and pairing vibration, in conclusion.