Approximate Particle Number Projection for Rotating Nuclei

In this paper a review is given of a class of sub-models of both approaches, characterized by the fact that they can be solved exactly, highlighting in the process a number of generic results related to both the nature of pair-correlated systems as well as collective modes of motion in the atomic nucleus.

While the no-core shell model is a state-of-the-art microscopic approach to low-energy nuclear structure, its intense computational requirements lead us to consider time-honored approximations such as the Hartree-Fock (HF) approximation and the random phase approximation (RPA). We review RPA and point out some common misunderstandings, then apply HF+RPA to the no-core shell model. Here the main issue is appropriate treatment of contamination by spurious center-of-mass motion.

We develop and test efficient approximations to estimate ground state correlations associated with low- and zero-energy modes. The scheme is an extension of the generator-coordinate-method (GCM) within Gaussian overlap approximation (GOA). We show that GOA fails in non-Cartesian topologies and present a topologically correct generalization of GOA (topGOA). An RPA-like correction is derived as the small amplitude limit of topGOA, called topRPA. Using exactly solvable models, t...

Beyond mean-field methods based on restoration of symmetries and configuration mixing by the generator coordinate method (GCM) enable to calculate on the same footing correlations in the ground state and the properties of excited states. Excitation energies are often largely overestimated, especially in nuclei close to magicity, even when transition probabilities are well-described. We analyse here the origin of this failure. The first part of the paper compares realistic pro...

A parameter free investigation of the moments of inertia of ground state rotational bands in well deformed rare-earth nuclei is carried out using Cranked Relativistic Hartree-Bogoliubov (CRHB) and non-relativistic Cranked Hartree-Fock-Bogoliubov (CHFB) theories. In CRHB theory, the relativistic fields are determined by the non-linear Lagrangian with the NL1 force and the pairing interaction by the central part of finite range Gogny D1S force. In CHFB theory, the properties in...

The evolution of the pairing correlations from closed shell to middle shell nuclei is analyzed with a Finite Range Density Dependent interaction in the Sn isotopes. As theoretical approaches we use the Hartree-Fock-Bogoliubov, the Lipkin-Nogami, their particle number projected counterparts and the full variation after particle number projection method. We find that whereas all approaches succeed rather well in the description of the total energy they differ significantly in t...

Pairing plays an essential role in describing nuclear spectra and attempts to describe it has a long history in nuclear physics. Many theoretical tools were developed to treat the pairing problem either exactly or at various levels of approximation. In this contribution to the proceedings of a conference honoring the career of Taka Otsuka, recent developments in the exact solutions of the pairing problem are described.

Cranked Relativistic Hartree-Bogoliubov (CRHB) theory is presented as an extension of Relativistic Mean Field theory with pairing correlations to the rotating frame. Pairing correlations are taken into account by a finite range two-body force of Gogny type and approximate particle number projection is performed by Lipkin-Nogami method. This theory is applied to the description of yrast superdeformed rotational bands observed in even-even nuclei of the $A\sim 190$ mass region....

A new stochastic number projection method is proposed. The component of the BCS wave function corresponding to the right number of particles is obtained by means of a Metropolis algorithm in which the weight functions are constructed from the single-particle occupation probability. Either standard BCS or Lipkin-Nogami probability distributions can be used, thus the method is applicable for any pairing strength. The accuracy of the method is tested in the computation of pairin...

The properties of pairing correlations in symmetric nuclear matter are studied in the relativistic mean field (RMF) theory with the effective interaction PK1. Considering well-known problem that the pairing gap at Fermi surface calculated with RMF effective interactions are three times larger than that with Gogny force, an effective factor in the particle-particle channel is introduced. For the RMF calculation with PK1, an effective factor 0.76 give a maximum pairing gap 3.2 ...