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Physical Review C

Approximate projection of physical states in the Dyson boson description of nuclear collective motion

Ching‐Teh Li

Published Jun 1, 1984• 15 citations
Publisher

Abstract

A new approximate projection method is proposed to treat the spurious state problem in the Dyson boson description of nuclear collective motion. In this method, successive orders of approximate projection are used to construct the various physical states in the Dyson boson representation of the shell model. Even in the lowest orders of approximation, the approximately projected boson states obtained turn out to be quite accurate in producing physical quantities. Furthermore, this approximate projection method is applicable not only to vibrational or transitional nuclei but also to rotational nuclei.

DOI
10.1103/physrevc.29.2309

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Progress of Theoretical Physics•May 1, 1987•9 citations
A new approximate projection method is proposed to treat the spurious state problem in the Dyson boson description of nuclear collective motion. In this method, successive orders of approximate projection are used to construct the various physical states in the Dyson boson representation of the shell model. Even in the lowest orders of approximation, the approximately projected boson states obtained turn out to be quite accurate in producing physical quantities. Furthermore, this approximate projection method is applicable not only to vibrational or transitional nuclei but also to rotational nuclei.
In order to truncate a high-dimensional Hilbert space for a many-nucleon shell model to a small collective subspace, we carry out a Dyson-type boson mapping and select a small number of collective bosons. In this way, we can divide the Dyson boson Hamiltonian into collective and non-collective parts; the latter includes the renormalization of the coupling effects between the collective and non-collective degrees of freedom. The boson Hamiltonian thus obtained is written in terms of the selected collective bosons only. Diagonalizing this boson Hamiltonian and comparing the numerical results with those obtained from exact shell-model calculations for some sd- and pf-shell nuclei, we find that our method gives a good approximation of the exact shell model. It is shown that the coupling effect between the collective and non-collective degrees of freedom plays an essential role. This method is also applied to some heavier pfg-shell nuclei for which it is unfeasible to carry out exact shell-model calculation because of the very large number of dimensions of the Hilbert space. Our numerical results are in very good agreement with experiment.
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)
Based on the correspondence of projection to an eigenvalue problem via the underlying group structure, we generalize the projection algorithm proposed by Morrison et al. This algorithm is eligible for large-scale computations, because we can avoid the accumulation of errors. As an example, we apply it to the spurious c.m. motion in a multi-ℏω\hbarωℏω shell-model calculation. In comparison with Lawson's method, the present method is exact and nevertheless does not need longer CPU time. It is pointed out that the exact treatment of the spurious c.m. motion is crucial to some physical quantities.
In phenomenological descriptions of nuclear vibrations a close connection is implied or expressed between the most-collective transition operator and the harmonic-oscillator (boson) variables which are introduced to describe the collective motion. With the help of a widely used example - the monopole vibration model of Meshkov, Lipkin, and Glick - it is shown how this relationship suggests, for the microscopic case, a new and flexible class of boson representations. As shown, these are capable of yielding accurate descriptions of the system not only in the extremes of weak and strong coupling, but in the usually more inaccessible intermediate regions as well.
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.