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Progress of Theoretical Physics

Analysis of Collective-Noncollective Couplings in a Degenerate Many j-Shell Model

H. Aiba, K. Matsuyanagi

Published Mar 1, 1989• 2 citations
Publisher

Abstract

The selfconsistent-collective-coordinate method is applied to anharmonic quadrupole vibrations in a degenerate many j-shell model with the pairing-plus-quadrupole force. By comparing the result obtained by this method with that obtained by the boson-expansion method, it is shown that 1) the selfconsistent-collective-coordinate method yields, after a suitable canonical quantization, the result that agrees to order (j + 1/2)-1 with that obtained by the boson-expansion method, in the limit that the time-dependent Hartree-Bogoliubov space is truncated to the multiphonon space built out of the 2+ phonons, and 2) the couplings between the RPA 2+ phonons and the other modes (the pairing rotations and the noncollective modes) considerably modify the fourth-order boson Hamiltonian obtained by the conventional boson-expansion method.

DOI
10.1143/ptp.81.690

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Non-Unitary Realization of the Selfconsistent Collective-Coordinate Method

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Recently we reformulated the selfconsistent collective•coordinate (See) method of Marumori, Maskawa, Sakata and Kuriyama.In this reformulation, biunitary forms of state vectors are used and the resultant representation corresponds to a c-number image of the Dyson-type boson expansion theory.This non-unitary version of the see method is rederived from a general point of view in order to clarify the relation to the original unitary version.Moreover, it is shown that the expansion technique to solve the basic equations is as applicable to this new version as to the unitary one, so that applications to realistic problems are easily done. § 1. IntroductionRecently it has bec;:ome increasingly important to determine the collective subspace selfconsistently for understanding the anharmonicity or non-linearity of the large amplitude nuclear collective motion.The selfconsistent collective-coordinate (See) method,*) originally proposed by Marumori, Maskawa, Sakata and Kuriyama,l) is promising for this purpose because it can properly take into account the effect of coupling to non-collective degrees of freedom.It is known that, if the coupling effect or the dynamical anharmonicity effect is neglected, this theory turns to the wellknown boson expansion theory for only the collective phonons under a suitable quantization procedure.Then, it enables us to construct a new type of "dynamical boson expansion theory" which incorporates the dynamical anharmonicity as well as the kinematical one originating from the Pauli principle.This new dynamical boson theory is expected to provide us a powerful method for investigating the nuclear structure problems full-microscopically.The original version of the see method corresponds, roughly speaking, to the Holstein-Primakoff type boson theory.6),7)Quite recently we have shown in Ref. 2) (from now on referred to as 1) that there exists another version corresponding to the Dyson-type boson theory; namely, a "non-unitary realization" of the see method, so to speak.Although both the types of boson theory are strictly equivalent to each other, the Dyson theory is more convenient for realistic applications, since a method of hermitian treatment of the theory is established.3 ) Therefore the newly proposed Dyson version of the see method can be superior for the purpose of investigating the dynamical coupling effect, because the kinematical anharmonicity is evaluated without any approximation such as truncation of the expansion.In other words, the "zero-th order collective subspace" is constructed without any ambiguities.*) The see method itself has a wide range of applicabilities, e.g., to heavy ion reactions, spontaneous fissions and rotational motions.lO)However, we are mainly concerned, in this paper, with an application to the boson description of low-lying collective states.

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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.]

Physical Review C•Feb 1, 1978•4 citations
The selfconsistent-collective-coordinate method is applied to anharmonic quadrupole vibrations in a degenerate many j-shell model with the pairing-plus-quadrupole force. By comparing the result obtained by this method with that obtained by the boson-expansion method, it is shown that 1) the selfconsistent-collective-coordinate method yields, after a suitable canonical quantization, the result that agrees to order (j + 1/2)-1 with that obtained by the boson-expansion method, in the limit that the time-dependent Hartree-Bogoliubov space is truncated to the multiphonon space built out of the 2+ phonons, and 2) the couplings between the RPA 2+ phonons and the other modes (the pairing rotations and the noncollective modes) considerably modify the fourth-order boson Hamiltonian obtained by the conventional boson-expansion method.
We investigate the validity of the canonical quantization of the collective Hamiltonian derived by the selfconsistent-collective-coordinate method. Special attention is paid to the boundary condition for the (η*, η) expansion, which fixes the canonical coordinate system to carry out the quantization procedure. We remove the ambiguity of the boundary condition by requiring that the boson representations obtained by the canonical quantization agree, to the order 1/Ω, with those derived by the modified-Marumori boson-expansion method in a certain limit. Numerical examples for an exactly solvable model show that our quantization procedure provides us with remarkably accurate results.
An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in nuclei with pairing correlations on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a manner similar to that of the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.
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.
The boson expansion method was developed with the purpose of continuously following even nuclear systems near their ground states, from the region of slightly anharmonic vibrations through the transitional region formed by a nonsharp phase transition to a region of permanent distortion characterized by the occurrence of rotational spectra. However, as the boson quantum was defined in the extreme vibrational limit, it was not possible to obtain quantitatively correct spectra in the transitional or deformed region. Believing that the major cause for the lack of success was the disregard of the self- consistent change of the basic degree of freedom, a simple variational method to be used with the boson expansion technique is presented. Its ability to cover the vibration-to-rotation transition is demonstrated by comparison with an exact solution for a four-level model with pairing force. Further the coverage of a quadrupole phase transition is shown for a system interacting via pairing-Plus- quadrupole forces. A successful calculation of static quadrupole moments of Cd and Te isotopes is presented. Finally the approach in general situations with more complicated forces is discussed. (auth)
Recently we reformulated the selfconsistent collective•coordinate (See) method of Marumori, Maskawa, Sakata and Kuriyama.In this reformulation, biunitary forms of state vectors are used and the resultant representation corresponds to a c-number image of the Dyson-type boson expansion theory.This non-unitary version of the see method is rederived from a general point of view in order to clarify the relation to the original unitary version.Moreover, it is shown that the expansion technique to solve the basic equations is as applicable to this new version as to the unitary one, so that applications to realistic problems are easily done. § 1. IntroductionRecently it has bec;:ome increasingly important to determine the collective subspace selfconsistently for understanding the anharmonicity or non-linearity of the large amplitude nuclear collective motion.The selfconsistent collective-coordinate (See) method,*) originally proposed by Marumori, Maskawa, Sakata and Kuriyama,l) is promising for this purpose because it can properly take into account the effect of coupling to non-collective degrees of freedom.It is known that, if the coupling effect or the dynamical anharmonicity effect is neglected, this theory turns to the wellknown boson expansion theory for only the collective phonons under a suitable quantization procedure.Then, it enables us to construct a new type of "dynamical boson expansion theory" which incorporates the dynamical anharmonicity as well as the kinematical one originating from the Pauli principle.This new dynamical boson theory is expected to provide us a powerful method for investigating the nuclear structure problems full-microscopically.The original version of the see method corresponds, roughly speaking, to the Holstein-Primakoff type boson theory.6),7)Quite recently we have shown in Ref. 2) (from now on referred to as 1) that there exists another version corresponding to the Dyson-type boson theory; namely, a "non-unitary realization" of the see method, so to speak.Although both the types of boson theory are strictly equivalent to each other, the Dyson theory is more convenient for realistic applications, since a method of hermitian treatment of the theory is established.3 ) Therefore the newly proposed Dyson version of the see method can be superior for the purpose of investigating the dynamical coupling effect, because the kinematical anharmonicity is evaluated without any approximation such as truncation of the expansion.In other words, the "zero-th order collective subspace" is constructed without any ambiguities.*) The see method itself has a wide range of applicabilities, e.g., to heavy ion reactions, spontaneous fissions and rotational motions.lO)However, we are mainly concerned, in this paper, with an application to the boson description of low-lying collective states.
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.]