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Journal of Physics Conference Series

Particle-number conservation in thermodynamic systems within the Richardson model

Naziha Benhamouda, N. H. Allal, M. Fellah, M. R. Oudih

Published Aug 1, 2008• 0 citations
Publisher

Abstract

To describe thermal pairing systems, an approach that combines the modified BCS method and the Lipkin-Nogami particle-number projection method is proposed. The expressions of the thermal pairing gap and internal energy are derived. The latter are numerically studied as a function of the temperature within the Richardson model.
DOI
10.1088/1742-6596/128/1/012031

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We propose a description of pairing properties in finite systems within the canonical and microcanonical ensembles. The approach is derived by solving the BCS and self-consistent quasiparticle random-phase approximation with the Lipkin-Nogami particle-number projection at zero temperature. The obtained eigenvalues are embedded into the canonical and microcanonical ensembles. The results obtained are found in quite good agreement with the exact solutions of the doubly-folded equidistant multilevel pairing model as well as the experimental data for 56Fe^{56}\mathrm{Fe}56Fe nucleus. The merit of the present approach resides in its simplicity and its application to a wider range of particle number, where the exact solution is impracticable.
Particle-number projection is applied to the modified BCS (MBCS) theory. The resulting particle-number-projected MBCS theory, taking into account the effects due to fluctuations of particle and quasiparticle numbers at finite temperature, is tested within the exactly solvable multilevel model for pairing as well as the realistic 120Sn^{120}\mathrm{Sn}120Sn nucleus. The signature of the pseudogap in the crossover region above the critical temperature of superfluid-normal phase transition is discussed in terms of the pairing spectral function.
The pairing interaction is one of the most important contribution of the residual interaction and then, it is of major importance for the study of many-body systems. One can get solutions of the pairing Hamiltonian throught the Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximations but, the pairing Hamiltonian admit exact solution worked out by Richardson. Nuclei far away from the stability line have important correlations with the continuum part of the energy spectrum, due that the Fermi level is very close to the contiuum thershold. The correlations with the continuum can be included in the many-body description through the complex energy states, called Gamow states. In this work we compare the approximates and exact solutions of the pairing Hamiltonian in real and complex-energy representations. In the application of this formulation to the symmetric Lipkin model, we found that the LN solution is in a good agreement with the exact one; besides, the extension of the BCS solution to the complex energy plane gives solution even for strength below the critical one, which is purely imaginary.
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 pairing energies of model and real systems.
Abstract A method for the treatment of pairing correlations at finite temperature is proposed within the path integral formalism, based on the square root extraction of the pairing term in the Hamiltonian of the system. Gap equations and expressions for the pairing gap parameter Δ, energy E, and heat capacity C are established. The formalism is first tested using the Richardson model, which enables comparison with an exact solution. The results obtained using this formalism are also compared with the finite temperature BCS (FTBCS) results. An improvement over the FTBCS model is noted, especially at low temperatures. Indeed, the agreement between the Δ values of this study and the exact values is good at low temperatures. This leads to better agreement between the values of E and C of this model and the exact values than with the FTBCS values. However, a critical value of temperature remains. Subsequently, realistic cases are considered using single-particle energies of a deformed Woods-Saxon mean-field for the nuclei Dy and Yb. In the framework of the current approach, pairing effects persist beyond the FTBCS critical temperature. Moreover, at low temperatures, a good agreement between the model and semiexperimental values of the heat capacity is observed, and a clear improvement compared to the FTBCS method is noted. This is no more the case at higher temperatures.
A canonical description of the thermodynamical pairing properties of small systems is achieved by using the Variation After Projection approach at finite temperature. The minimization of the free energy is made by a direct evaluation of the energy and full diagonalization of the entropy. We use the Richardson – pairing model whose exact solution allows to study the reliability of different approaches. We show that the Projection After Variation approach, that is usually performed at zero temperature with rather good success, provides a quite poor description at finite temperature. On the contrary, the Variation After Projection applied at finite temperature provides a perfect reproduction of the exact canonical properties of odd or even systems from very low to high temperature.