An exactly solvable spherical mean-field plus extended monopole pairing model
Introduction
It is well known that the monopole pairing interaction is one of the important residual interactions in a nuclear mean-field theory [1], and is the key to elucidate ground state and low-energy spectroscopic properties of nuclei, such as binding energies, odd–even effects, single-particle occupancies, excitation spectra, and moments of inertia, etc. [1], [2], [3]. While the Bardeen–Cooper–Schrieffer (BCS) and the more refined Hartree–Fock–Bogolyubov (HFB) approximations provide a simple and clear demonstration of the role of pairing correlations in nuclei [2], [4], [5], tremendous efforts have been made in finding accurate solutions to the problem [6], [7], [8], [9], [10], [11] to overcome serious drawbacks in the BCS and the HFB, such as spurious states, nonorthogonal solutions, etc., resulting from particle number-nonconservation effects in these approximations [9], [11], [12], [13]. It is known that either spherical or deformed mean-field plus the standard pairing interaction can be solved exactly by using the Gaudin–Richardson method [14], [15], [16]. It has been shown that the set of Gaudin–Richardson equations can be solved more easily by using the extended Heine–Stieltjes polynomial approach [17], [18]. However, numerical work in finding roots of the Gaudin–Richardson equations increases with increasing the number of orbits and the number of valence nucleon pairs, which limits the application of the theory within a shell model subspace. On the other hand, the deformed mean-field plus the extended pairing model can be solved more easily than the standard pairing model, especially when both the number of valence nucleon pairs and the number of single-particle orbits are large [19]. It has been proven [20] that the extended pairing model is equivalent to the standard pairing model at a first-step approximation, in which only the lowest energy eigenstate of the standard pure pairing interaction part is considered for solutions of the standard pairing model. The low-lying states obtained in the extended pairing model thus display pair structures similar to the ones of the standard pairing model.
However, the extended pairing interaction proposed in [19] augments a deformed mean-field theory, which cannot be used directly to describe pairing interactions among valence nucleon monopole pairs in spherical mean-field theories, in which the total angular momentum of the system is always conserved. The purpose of this work is to establish an extended pairing Hamiltonian to describe pairing interactions among valence nucleon monopole pairs in a spherical mean field similar to the deformed case [19], with which large-scale calculations within the model become feasible.
Section snippets
The extended monopole pairing model
Similarly to the extended pairing model for deformed nuclei [19], the Hamiltonian of a spherical mean-field plus the extended monopole pairing model may be written as where is a set of single-particle energies generated from any spherical mean-field theory, such as those of the spherical harmonic oscillator (HO) shell model, , in which () is the creation (annihilation) operator for a nucleon with angular momentum quantum number j and that of its
Comparison of the extended pairing with the standard pairing
In the standard pairing Hamiltonian [15], [16], the pairing correlations and their effects are mainly driven by the pairing interactions among different orbits accounted by the non-diagonal part of the Hamiltonian, similar to the role performed by shown in (1). Hence, while the non-diagonal part of the standard-pairing effects is not present in the first two terms of (1), the effectively accounts for it. Specifically, the ground state of the pure standard pairing Hamiltonian
Application to the ground state of 12–28O isotopes
As a nontrivial application, binding energies and even–odd mass differences of 12–28O are calculated by the spherical shell model plus the extended monopole pairing (1). In order to demonstrate the basic features of the model, we consider neutron orbits up to the fifth HO major shell in the spherical no-core shell model with 11 j-orbits: , , , , , , , , , , and , though the results for the ground state are similar when only a few less or
Conclusions
The extended pairing Hamiltonian to describe pairing interactions among nucleon monopole pairs up to infinite order in a spherical mean field, such as the spherical shell model, is proposed based on the local algebraic structure, which includes the extended pairing interaction model within a deformed mean-field theory proposed in [19] as a special case. The advantage of the model lies in the fact that numerical solutions of the model can be obtained much more easily with less CPU time than
Acknowledgements
Support from the U.S. National Science Foundation (OCI-0904874), U.S. Department of Energy (DE-SC0005248), the Southeastern Universities Research Association, the China–U.S. Theory Institute for Physics with Exotic Nuclei (CUSTIPEN) (DE-FG02-13ER42025), the National Natural Science Foundation of China (11175078, and 11375080), and the LSU–LNNU joint research program (9961) is acknowledged.
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