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arXiv:0904.2830 · Physical Review C

New exact solutions of the standard pairing model for well-deformed nuclei

Feng Pan, Mingxia Xie, Xin Guan, L. R. Dai, Jerry Paul Draayer

Published Oct 9, 2009• 21 citations

Abstract

A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a $k$-particle Hilbert subspace can be mapped onto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Feng Pan, V. G. Gueorguiev, and J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004)] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis shows that the standard pairing model with the first step approximation displays similar pair structures of the first few exact low-lying states of the model, which, therefore, provides a link between the two models.

DOI: 10.1103/physrevc.80.044306

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A new step-by-step diagonalization procedure for evaluating exact solutions of the nuclear deformed mean-field plus pairing interaction model is proposed via a simple Bethe ansatz in each step from which the eigenvalues and corresponding eigenstates can be obtained progressively. This new approach draws upon an observation that the original one- plus two-body problem in a

k

-particle Hilbert subspace can be mapped onto a one-body grand hard-core boson picture that can be solved step by step with a simple Bethe ansatz known from earlier work. Based on this new procedure, it is further shown that the extended pairing model for deformed nuclei [Feng Pan, V. G. Gueorguiev, and J. P. Draayer, Phys. Rev. Lett. 92, 112503 (2004)] is similar to the standard pairing model with the first step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is taken into consideration. Our analysis shows that the standard pairing model with the first step approximation displays similar pair structures of the first few exact low-lying states of the model, which, therefore, provides a link between the two models.

Exact solution of the nuclear mean-field plus separable pairing model is reexamined. New auxiliary constraints for solving the Bethe ansatz equations of the model are proposed. By using these auxiliary constraints, the Bethe ansatz form of eigenvectors of the mean-field plus separable pairing Hamiltonian with nondegenerate single-particle energies and nondegenerate separable pairing strengths purposed previously is verified. Since the solutions of the model with one- and two-orbit cases are known, verification of the solutions for these two special cases is made. To demonstrate structure and features of the solution, the model with three orbits in the

ds

shell is taken as a nontrivial example, of which two-pair results and the ground state of the three-pair case are provided explicitly. Since the number of equations involved increases with the number of orbits and pairs, to solve these equations for a large number of orbits and pairs seems still difficult.

The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.

The present contribution discusses a connection between the exact Bethe Ansatz eigenstates of the reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and the multi-phonon states of the Tamm-Dancoff Approximation (TDA). The connection is made on the algebraic level, by means of a deformed quasi-spin algebra with a bosonic Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. Each exact Bethe Ansatz eigenstate is mapped on a unique TDA multi-phonon state, shedding light on the physics behind the Bethe Ansatz structure of the exact wave function. The procedure is illustrated with a model describing neutron pairing in 56 Fe.

An exact boson mapping of the deformed mean-field plus equal strength pairing Hamiltonian is considered. In the mapping, fermion pair operators are mapped exactly to the corresponding bosons. The mapping occurs at the level of the Richardson-Gaudin equations. The image of the mapping results is a Bose-Hubbard model with level-dependent hopping. Although the resultant Bose-Hubbard Hamiltonian is non-Hermitian, a part of spectrum of the Bose-Hubbard Hamiltonian with U/t = 1 determined by the corresponding Richardson-Gaudin (Bethe ansatz) equations corresponds exactly to the whole spectrum of the pairing Hamiltonian. © 2009 IOP Publishing Ltd.

A general procedure, based on the Bethe ansatz, is proposed for finding algebraic solutions for low-lying

J=0

states of

2k

nucleons interacting with one another through a

T=1

charge-independent pairing interaction. The results provided by Richardson are shown to be valid for up to two pairs,

k<~2;

expressions are given here for up to three pairs,

k<~3.

The results show that a set of highly nonlinear equations must be solved for

k>~3.