Standard Pairing Model Research Articles

Impact of the pairing interaction on fission of U isotopes

The role of pairing interactions on the scission configurations, the total kinetic energy, and the mass distributions in U isotopes is investigated using the deformed mean-field plus standard pairing model. The total kinetic energy and the mass distributions of $^{232--238}\mathrm{U}$ isotopes obtained using the shape-dependent probability distribution expressed by the Wigner function reproduces the experimental data remarkably well. The model calculations show that the scission region is sensitive to the variation of the pairing interaction strength, particularly for the asymmetric and symmetric scission points. The apparent changes in the peak-to-valley ratio in mass distribution by varying the pairing interaction strength confirm that the pairing interaction plays an important role in achieving the scission process for $^{236}\mathrm{U}$ under the present model. These results also suggest that the pairing interaction strength in the current work should increase with the elongation of the nucleus to yield better fission products. Through numerical analysis, we provide possible microscopic pictures of spontaneous fission around the scission configurations in the exactly solvable pairing model.

Extended Bethe-Richardson-Gaudin ansatzes for the solution of a mean-field plus separable pairing model with three non-degenerate j-orbits

Based on the Bethe-Richardson-Gaudin ansatz for the standard pairing model, extended Bethe- Richardson-Gaudin ansatzes for eigenvectors of a spherical mean-field plus separable pairing model with three non-degenerate j-orbits are proffered. It is shown that the number of variables appearing in the general extended ansatz eigenvectors for given number of pairs N is N(N+1)/2. More importantly, when one of the j orbits is 1/2, there are only 2N variables involved in the alternative extended ansatz eigenvectors, which, like the standard pairing model, can be solved efficiently. Numerical results for an application of the model in the ds-shell up to its half-filling are presented, which serves to validate the procedure and illustrates the completeness of the solutions it renders.

Impact of pairing interactions on fission in the deformed mean-field plus standard pairing model

The role of pairing interactions on fission barriers and static fission paths in Th, U and Pu isotopes is investigated by using the deformed mean-field plus standard pairing model. It is found that pairing interaction plays different roles in different stages of the fission process. The first and second saddle point of the fission process are sensitive to the pairing interaction strength, while the ground-state and prescission points are mainly driven by nuclear deformation. It is also demonstrated that pairing interactions play a crucial role in the region of the inner and outer barriers. The barrier heights obtained by using the present model are found to reproduce the experimental data remarkably well and yield better results than those obtained by using the BCS approximation, thus indicating the importance of exact solutions of pairing interactions. Through numerical analysis, we provide possible microscopic pictures of spontaneous fission in the current pairing model.

Exact solutions of mean-field plus various pairing interactions and shape phase transitions in nuclei

The exact solutions of either spherical or deformed mean-field plus various types of pairing models are briefly reviewed. It is shown that, besides the standard pairing model, there are several special types of pairing interaction that can be solved exactly. In comparison to the standard pairing, the results of several pairing-driven quantities, such as pairing excitation energies, even–odd mass differences, the moment of inertia, etc., show that these types of pairing interaction can indeed be used to describe pairing correlations in nuclei either more accurately or efficiently. Moreover, the shape phase transitional behaviors of nuclei described by the consistent-Q formalism of the interacting boson model are summarized.

Exact solution of spherical mean-field plus multi-pair interaction model with two non-degenerate j-orbits

The exact solution of spherical mean-field plus multi-pair interaction model with two non-degenerate j-orbits, which is an extension of the widely used standard (two-body) pairing model, is derived based on the Bethe–Richardson–Gaudin approach. The Bethe–Richardson–Gaudin equations in determining eigenstates and the corresponding eigen-energies of the model are provided and exemplified with up to three-pair interactions. With a suitable parameterization of the overall multi-pair interaction strengths, the model with one adjustable parameter and valence nucleons confined in the $$1d_{5/2}$$ and $$0g_{7/2}$$ orbits is applied to fit binding energies of $$^{102{\text {--}}112}$$Sn. It is shown that the ground-state occupation probabilities of nucleon pairs calculated from this model and those from the standard pairing model are almost the same with perfect ground-state overlap of the two models. A noticeable feature of the multi-pair interactions is that the even–odd staggering of pairing interaction strength appearing in the standard pairing model due to the Pauli-blocking is suppressed. As the result, the pairing interaction strength of the model only depends on the number of valence nucleon pairs in the system.

An exact solution of spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits

An exact solution of nuclear spherical mean-field plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits is presented. The extended one-variable Heine–Stieltjes polynomials associated to the Bethe ansatz equations of the solution are determined, of which the sets of the zeros give the solution of the model, and can be determined relatively easily. A comparison of the solution to that of the standard pairing interaction with constant interaction strength among pairs in any orbit is made. It is shown that the overlaps of eigenstates of the model with those of the standard pairing model are always large, especially for the ground and the first excited state. However, the quantum phase crossover in the non-separable pairing model cannot be accounted for by the standard pairing interaction.

An exact solution of spherical mean-field plus a special separable pairing model

An exact solution of nuclear spherical mean-field plus a special orbit-dependent separable pairing model is studied, of which the separable pairing interaction parameters are obtained by a linear fitting in terms of the single-particle energies considered. The advantage of the model is that, similar to the standard pairing case, it can be solved easily by using the extended Heine–Stieltjes polynomial approach. With the analysis of the model in the ds- and pf-shell subspace, it is shown that this special separable pairing model indeed provides similar pair structures of the model with the original separable pairing interaction, and is obviously better than the standard pairing model in many aspects.

Ground state phase transition in the Nilsson mean-field plus standard pairing model

The ground state phase transition in Nd, Sm, and Gd isotopes is investigated by using the Nilsson mean-field plus standard pairing model based on the exact solutions obtained from the extended Heine-Stieltjes correspondence. The results of the model calculations successfully reproduce the critical phenomena observed experimentally in the odd-even mass differences, odd-even differences of two-neutron separation energy, and the $\ensuremath{\alpha}$-decay and double ${\ensuremath{\beta}}^{\ensuremath{-}}$-decay energies of these isotopes. Since the odd-even effects are the most important signatures of pairing interactions in nuclei, the model calculations yield microscopic insight into the nature of the ground state phase transition manifested by the standard pairing interaction.

Quantum phase transition in the spherical mean-field plus quadrupole-quadrupole and pairing model in a single-jshell

The quantum-phase-transitional behavior of the spherical shell-model mean field plus the geometric quadrupole-quadrupole and standard pairing model within a single-$j$ shell is analyzed in detail. Various quantities, such as low-lying energy levels, some typical energy ratios, the overlaps of the excited states with those of the corresponding limiting cases, $B(E2)$ values and electric quadrupole moments of some low-lying states and their ratios, as functions of the control parameter of the model in a $j=15/2$ shell are calculated as an example, in which only a crossover occurs due to the Pauli exclusion. The results show that there are noticeable changes within the crossover region of the rotation-like to the pair-excitation (superconducting) phase transition, especially in the half-filling case. As an application to realistic nuclear systems, a chain of isotones $^{212}\mathrm{Rn}\text{\ensuremath{-}}^{213}\mathrm{Fr}\text{\ensuremath{-}}^{214}\mathrm{Ra}\text{\ensuremath{-}}^{215}\mathrm{Ac}$ is chosen to be described by the model with valence protons in the $1{h}_{9/2}$ shell. As far as the low-lying energy levels, the experimentally observed $B(E2)$ values, and the electric quadrupole moment within the yrast band are concerned, these nuclei seem fitted reasonably well. The results indicate that these nuclei are all within the rotation-like to the pair-excitation phase transition near the crossover point.

An exactly solvable spherical mean-field plus extended monopole pairing model

An extended pairing Hamiltonian that describes pairing interactions among monopole nucleon pairs up to an infinite order in a spherical mean field, such as the spherical shell model, is proposed based on the local E˜2 algebraic structure, which includes the extended pairing interaction within a deformed mean-field theory (Pan et al., 2004) [19] as a special case. The advantage of the model lies in the fact that numerical solutions of the model can be obtained more easily and with less computational time than the solutions to the standard pairing model. Thus, open-shell large-scale calculations within the model become feasible. As an example of the application, pairing contribution to the binding energy of 12–28O is estimated in the present model with neutron pairs allowed to occupy a no-core shell model space of 11 j-orbits up to the fifth major harmonic oscillator shell including excitations up to 14ħω for 12O and up to 40ħω for 28O. The results for 12O are also compared and found to be in agreement with those of ab initio calculations. It is shown that the pairing energy per particle in 12–28O ranges from 0.4 to 1.8 MeV/A with the strongest one observed for a small number of pairs.

Level statistical properties of the spherical mean-field plus standard pairing model

The effect of pairing correlations on the spectral statistical behavior of the spherical mean-field plus standard pairing model is investigated based on the exact solutions obtained from the extended Heine-Stieltjes correspondence. It is found that varying the pairing interaction strength $G$ in the model alters the statistical properties of the spectrum of the model. The spectrum is regular when $G$ is nonzero and either well below or well beyond the critical region of normal (localized) to superconducting (pair condensate) phase transition of the model, through which it is likely to display features of chaotic behavior. It is verified that the level statistics governed by the interplay between the single-particle energies and the pairing interaction for ${}^{48--53}$Ca calculated from the model with single-particle energies and a pairing strength extracted from experimental data indeed exhibits, for two of the Ca isotopes, spectral behaviors that deviate from the Poisson statistics within the phase transitional region, while most isotopes display regular spectra.

The Heine–Stieltjes correspondence and the polynomial approach to the Gaudin–Richardson models

The Heine–Stieltjes correspondence is extended and applied to solve Bethe ansatz equations of the Gaudin–Richardson models, from which the extended Heine–Stieltjes polynomial approach to these models is proposed. As examples for the application of this approach, exact solutions of the standard two-site Bose–Hubbard model and the standard pairing model for nuclei are formulated from the corresponding polynomials.

Nilsson mean-field plus the extended pairing model description of rare earth nuclei

The Nilsson mean-field plus the extended pairing model for well-deformed nuclei is applied to some representative rare earth examples. The binding energies, some low-lying pair-excited states and even-odd mass differences of Er, Yb and Hf isotopes are calculated systematically within the proton frozen-pair excitation limit. A comparison with experimental data for these nuclei shows that the results of the extended pairing model are better than those for the standard pairing model with the BCS approximation and the nearest-orbit pairing model.

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

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.