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Progress of Theoretical Physics

A New Method for Microscopic Description of the So-Called "Many-Phonon" States in Spherical Even-Even Nuclei. I

S. Iwasaki, Fumihiko Sakata, Kosuke Takada

Published Apr 1, 1977• 18 citations
Publisher

Abstract

A new method for systematic treatment of the anharmonicity effect in the many-phonon states in fermion space is proposed. The most important part of the method is a recursion formula for the norm of many-phonon states in fermion space. This recursion formula is applied to the calculation of every matrix element of an arbitrary operator. A successive approximation procedure to calculate the recursion formula is also proposed. This new method is rewritten so as to be suitable for the case of collective-phonon truncation of fermion space. The recursion formula in this case is expressed in a form expanded with respect to to what extent the anharmonicity effect due to the Pauli principle is taken into account. It is shown that the cutoff factor appearing in the boson representation of the SU(6) algebra is obtained by the lowest order of the successive approximation procedure for this expansion and the values of the cutoff factor estimated in this approximation show good fit to those obtained from the exact solution of the many-quasi-particle states which were previously calculated by the present authors. This implies that the new method proposed in this paper gives good results even in the lowest order approximation.

DOI
10.1143/ptp.57.1289

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Physical Review C•Apr 1, 1996•9 citations
A new method for systematic treatment of the anharmonicity effect in the many-phonon states in fermion space is proposed. The most important part of the method is a recursion formula for the norm of many-phonon states in fermion space. This recursion formula is applied to the calculation of every matrix element of an arbitrary operator. A successive approximation procedure to calculate the recursion formula is also proposed. This new method is rewritten so as to be suitable for the case of collective-phonon truncation of fermion space. The recursion formula in this case is expressed in a form expanded with respect to to what extent the anharmonicity effect due to the Pauli principle is taken into account. It is shown that the cutoff factor appearing in the boson representation of the SU(6) algebra is obtained by the lowest order of the successive approximation procedure for this expansion and the values of the cutoff factor estimated in this approximation show good fit to those obtained from the exact solution of the many-quasi-particle states which were previously calculated by the present authors. This implies that the new method proposed in this paper gives good results even in the lowest order approximation.
A new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system is developed, with the purpose of analysing the “anharmonic effects” on the collective oscillations in spherical even nuclei from the standpoint of the microscopic theory of the collective excitations. In the “harmonic approximation”, a pair of bound quasi-particles is replaced by the “phonon” as an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effect due to the residual interactions omitted under the “harmonic approximation” and the effect of the Pauli-principle among the quasi-particles composing the different pairs. According to our theory, the two effects which cause “anharmonic effects” can be evaluated correctly, in principle. It is not our purpose to go into detailed quantitative calculations, but rather to develop the basic idea.
We propose a new boson expansion theory that does not premise the closed-algebra approximation, indispensable for formulation until now, as an extension of the conventional practical boson expansion methods that have tried to elucidate nuclear collective motion, which is a method that allows the closed-algebra approximation not to be used or to be used appropriately, enables us to obtain the boson expansion easier, and reproduces the fermion subspace onto the boson subspace more faithfully than the conventional practical methods. The two-phonon norm matrix composed of all the phonon excitation modes is investigated in detail, which reveals the mechanism, essential for the boson expansion methods, of how we should construct the fermion subspace to be mapped from the whole fermion space. The conventional practical boson expansion methods have applied the closed-algebra approximation improperly to strengthen the effect of the Pauli principle inappropriately, which should be replaced by those that do not use that approximation or use it properly.
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 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.
The introduction of an adequate auxiliary space allows us to replace the problem of the treatment of coupled phonons, which is very complicated and voluminous, with the problem of the treatment of usual matrices. As applications of this approach, the formulation of a Wick's theorem for coupled phonons and recursion formulas which can be used to calculate the overlap matrix elements of multiphonon states built with n coupled phonons of different types and collectivities are given. \textcopyright{} 1996 The American Physical Society.