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Progress of Theoretical Physics

On the New Kinematical Constraints of the Pair Operators in the Algebraic Approach to the Theory of Collective Motion

M. Yamamura

Published Aug 1, 1974• 8 citations
Publisher

Abstract

New kinematical constraints of the pair operators in the algebraic approach to the theory of collective motion are proposed. These relations are the natural extension, to the operator forms, of the well-known constraints of the generalized density matrix in the Hartree-Bogoliubov theory. On the basis of the new constraints, a microscopic theory of rotational motion given by Belyaev and Zelevinsky is reconstructed.

DOI
10.1143/ptp.52.538

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New kinematical constraints of the pair operators in the algebraic approach to the theory of collective motion are proposed. These relations are the natural extension, to the operator forms, of the well-known constraints of the generalized density matrix in the Hartree-Bogoliubov theory. On the basis of the new constraints, a microscopic theory of rotational motion given by Belyaev and Zelevinsky is reconstructed.
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.
A microscopic theory of collective and independent-particle motion in many-fermion system is developed in the framework of the classical theory. The basic idea is an extension of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.