Solutions of nuclear pairing

Abstract

We give the exact solution of orbit dependent nuclear pairing problem between two nondegenerate energy levels using the Bethe ansatz technique. Our solution reduces to previously solved cases in the appropriate limits including Richardson's treatment of reduced pairing in terms of rational Gaudin algebra operators.Pair correlations are manifest in a remarkable range of quantum many-body systems. Originally the idea of pairing and the methods were developed in the context of superconductivity in macroscopic systems by Bardeen, Cooper, and Schrieffer [1] and by Bogoliubov [2]. In nuclear physics, it has been long known that the independent particle picture must be improved with the addition of a nondiagonal two-particle force as evidenced by the absence of single particle excitations at low energies [3] and soon the idea of the pairing was carried over [4,5]. But methods borrowed from superconductivity in infinite systems were inconvenient for the finite nucleus because the former is based on wave functions with indefinite number of particles. Although it is known that the pairing effects play a major role in determining nearly all nuclear properties including the excitation spectra, the transition probabilities and the equilibrium shape, an exact number-conserving solution to nuclear pairing problem is still lacking except in three limits. The limit in which single particle energy levels are degenerate and all pairing strengths are the same was solved by Kerman using the quasispin algebra [6]. Later Richardson solved the limit in which the single particle energy levels are nondegenerate but the pairing strengths are the same [7]. Finally, the limit in which the single particle levels have different pairing strengths but are all degenerate is solved by Pan et al. [8] and by Balantekin et al. [9,10]. On the other hand, in many cases it is the interplay between the one-body and the two-body effects which determine the equilibrium properties of the nucleus and a full solution of the nuclear pairing problem is highly desirable as a realistic model. The purpose of this Rapid Communication is to present an exact solution of the pairing problem away from the above mentioned limits, i.e., with orbit dependent pairing strengths and nondegenerate single particle energies, in the presence of two nuclear levels.The problem is described by the Hamiltonian H P

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