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arXiv:nucl-th/0206016

The Pairing interaction in nuclei: comparison between exact and approximate treatments

J. Dukelsky, G.G. Dussel, Jorge G. Hirsch, Peter Schuck

Published Jun 6, 2002• 0 citations
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Abstract

As a model for a deformed nucleus the many level pairing model (picket fence model with ~100 levels) is considered in four approximations and compared to the exact solution given by Richardson long time ago. It is found that, as usual, the number projected BCS method improves over standard BCS but that it is much less accurate than the more sophisticated many-body-approaches which are Coupled Cluster Theory (CCT) in its SUB2 version or Self-Consistent Random Phase Approximation (SCRPA).

DOI
10.48550/arxiv.nucl-th/0206016

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As a model for a deformed nucleus the many level pairing model (picket fence model with ~100 levels) is considered in four approximations and compared to the exact solution given by Richardson long time ago. It is found that, as usual, the number projected BCS method improves over standard BCS but that it is much less accurate than the more sophisticated many-body-approaches which are Coupled Cluster Theory (CCT) in its SUB2 version or Self-Consistent Random Phase Approximation (SCRPA).
A comparison of different seniority zero solutions to the picket-fence model for the nuclear pairing hamiltonian problem is performed. These solutions are calculated, in the normal regime, within the self-consistent Random Phase Approximation (SCRPA) and various simplifications of this formalism, and also with the Tamm-Dancoff approach in the particle-particle-hole-hole channel (pphh-TDA). The latter formalism represents a first approximation to the earlier developped so-called P-Symmetric Many-Body method (PSY-MB). In the superfluid regime, the solutions are compared with the BCS results. By comparing the results with the exact ones, obtained by the Richardson method, it is shown that the PSY-MB method provides a powerful tool in solving the problem with good accuracy both in the normal and the superfluid regime, for single-particle space sizes adapted to typical nuclear structure calculations.
We study the structure of the number-projected BCS (PBCS) wave function in the particle-hole basis, displaying its similarities with coupled clusters theory (CCT). The analysis of PBCS together with several modifications suggested by the CCT wave function is carried out for the exactly solvable Richardson model involving a pure pairing Hamiltonian acting in a space of equally spaced, doubly degenerate levels. We point out the limitations of PBCS to describe the nonsuperconducting regime and suggest possible avenues for improvement.
Particle-number projection within the Lipkin-Nogami (LN) method is applied to the self-consistent quasiparticle random-phase approximation (SCQRPA), which is tested in an exactly solvable multilevel pairing model. The SCQRPA equations are numerically solved to find the energies of the ground and excited states at various numbers \ensuremath{\Omega} of doubly degenerate equidistant levels. The use of the LN method allows one to avoid the collapse of the BCS (QRPA) to obtain the energies of the ground and excited states as smooth functions of the interaction parameter GGG. The comparison between results given by different approximations such as the SCRPA, QRPA, LNQRPA, SCQRPA, and LNSCQRPA is carried out. Although the use of the LN method significantly improves the agreement with the exact results in the intermediate coupling region, we found that in the strong coupling region the SCQRPA results are closest to the exact ones.
The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.
Different steps leading to the new functional for pairing based on natural orbitals and occupancies proposed earlier [D. Lacroix and G. Hupin, Phys. Rev. B 82, 144509 (2010)] are carefully analyzed. Properties of quasiparticle states projected onto good particle numbers are first reviewed. These properties are used to (i) prove the existence of such a functional, (ii) provide an explicit functional through a 1/N1/N1/N expansion starting from the BCS approach, and (iii) give a compact form of the functional summing up all orders in the expansion. The functional is benchmarked in the case of the picket-fence pairing Hamiltonian where even and odd systems are studied, using the blocking technique, at various particle numbers and coupling strengths, with uniform and random single-particle level spacing. In all cases, very good agreement is found, with a deviation of $<$1% compared to the exact energy.