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Physical Review C

Algebraic approach to vibrational collective motion in a model with both collective and noncollective degrees of freedom

Moyez G. Vassanji, Abraham Klein, C.H. Dasso

Published Feb 1, 1978• 4 citations
Publisher

Abstract

An algebraic approach, within a framework using equations of motion, is applied to the study of vibrational collective motion in exactly soluble shell models with symmetry R(5) and R(5) \ifmmode\times\else\texttimes\fi{} R(5). The latter exhibits two phonon branches, permitting the study of the effect of the noncollective on the collective branch. The results of this method, including the splitting of the two-phonon degeneracy, agree satisfactorily with exact results for a range of parameters well into the transition region between vibrational and rotational motion.[NUCLEAR STRUCTURE Approximate many-body methods applied to a solvable model. Vibrational spectrum. Equations of motion method.]

DOI
10.1103/physrevc.17.755

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An algebraic approach, within a framework using equations of motion, is applied to the study of vibrational collective motion in exactly soluble shell models with symmetry R(5) and R(5) \ifmmode\times\else\texttimes\fi{} R(5). The latter exhibits two phonon branches, permitting the study of the effect of the noncollective on the collective branch. The results of this method, including the splitting of the two-phonon degeneracy, agree satisfactorily with exact results for a range of parameters well into the transition region between vibrational and rotational motion.[NUCLEAR STRUCTURE Approximate many-body methods applied to a solvable model. Vibrational spectrum. Equations of motion method.]
The program of the algebraic approach to nuclear collective motion is summarized. In essence, it should permit an accurate solution of the many-body problem within a subspace of the many-particle space if the conditions defining collective motion are present. The formulation, containing both dynamical and kinematical aspects, leads to coupled nonlinear algebraic equations for the matrix elements between collective states of fermion pair operators. An explicit set of equations is given for the problem of NNN (even) particles confined to a single-jjj shell and described by the conventional pairing plus quadrupole-quadrupole interaction Hamiltonian. This formulation is designed to be nearly exact in the pairing-dominant (seniority) limit and to be accurate enough (based on plausibility arguments) to describe the transition from spherical to deformed shape. That this is so is verified for configurations, the largest being jN=(212)6{j}^{N}={(\frac{21}{2})}^{6}jN=(221​)6, for which exact solutions are available. The present method can be viewed as an angular momentum conserving alternative to the description of the transition based on the Hartree-Bogoliubov-BCS method. Qualitative and detailed comparisons of the two methods are carried out. In particular, it is shown from approximate analytic results that the two theories predict the transition to occur for almost identical values of the parameters. Over-all agreement with exact results is much better for the algebraic method, however, confirming its status as an approximate method of variation after projection.
Microscopic investigation of collective models is illustrated with the study of the multilevel pairing force model. The coherent-state description of pairing excitations through the variation of the pairing field and the classification of generators into collective and noncollective modes are discussed. Numerical calculations are performed for collective pairing rotations and vibrations built on the ground state. The coupling to the noncollective state is studied. Exact results from the Richardson method are given for comparison.
Investigates the question of whether or not the vibrational spectrum of a two-dimensional interacting boson model can be transformed into the rotational one via quantum deformation of the group structure. It turns out that although the spectrum of the q-deformed model shows one or another of the rotational-like features at different values of the deformation parameter, nevertheless, all the requirements of the rotational behavior cannot be obtained simultaneously. This result indicates that the quantum algebraic treatment cannot give an analytical solution to the eigenvalue problem of the interacting boson models of nuclear and molecular physics in the whole range of the physically relevant breaking of dynamical symmetries.
This is the second of a series of papers aimed at exploiting a generalized self-consistent approach to the study of collective excited states in finite many-particle systems. It contains: (i) A restatement of the equations of the method with further discussion of the basic approximations involved; (ii) a study, through the medium of the exactly soluble problem of uniform translational motion, of the inherent limitations in accuracy of the method when applied to the "strong-coupling" limit for which it was designed. A mode of approximation which can be made self-consistent is indicated. (iii) This approximation is applied to the problem of rotational motion in a plane. In the further limit in which one assumes the spectrum of the rigid rotator, one regains the results of the self-consistent cranking model. In contrast to previous work, the possibility of removing this limitation is indicated, but remains to be carried through.