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Journal of Physics Conference Series

Pairing and specific heat in161,162Dy and171,172Yb isotopes

D. Gambacurta, Denis Lacroix A, Nicolae Sandulescu

Published Sep 10, 2014• 1 citation
Publisher

Abstract

The finite-temperature variation after projection BCS approach (FT-VAP) is employed to study the pairing phase-transition for the 161,162 Dy and 171,172 Yb isotopes. Due to the restoration of particle number conservation, the pairing gap and the specific heat calculated in the FT-VAP approach vary smoothly with the temperature, indicating a gradual transition from the superfluid to the normal phase, as expected in finite systems. FT-VAP results are compared with those obtained in the exact canonical approach as well as with experimental data.

DOI
10.1088/1742-6596/533/1/012012

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The finite-temperature variation after projection BCS approach (FT-VAP) is employed to study the pairing phase-transition for the 161,162 Dy and 171,172 Yb isotopes. Due to the restoration of particle number conservation, the pairing gap and the specific heat calculated in the FT-VAP approach vary smoothly with the temperature, indicating a gradual transition from the superfluid to the normal phase, as expected in finite systems. FT-VAP results are compared with those obtained in the exact canonical approach as well as with experimental data.
A systematic comparison is conducted for pairing properties of finite systems at nonzero temperature as predicted by the exact solutions of the pairing problem embedded in three principal statistical ensembles, as well as the unprojected (FTBCS1+++SCQRPA) and Lipkin+++Nogami projected (FTLN1+++SCQRPA) theories that include the quasiparticle number fluctuation and coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation. The numerical calculations are performed for the pairing gap, total energy, heat capacity, entropy, and microcanonical temperature within the doubly folded equidistant multilevel pairing model. The FTLN1+++SCQRPA predictions agree best with the exact grand-canonical results. In general, all approaches clearly show that the superfluid-normal phase transition is smoothed out in finite systems. A novel formula is suggested for extracting the empirical pairing gap in reasonable agreement with the exact canonical results.
To investigate the pairing phase transition in finite systems with the particle-number conservation, we formulate a new BCS-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. This theory enables us to distinguish the symmetry-restoring fluctuation (SRF) from additional quantum fluctuations. By numerical calculations, it is found that the phase transition is compatible with the conservation law in this theory, and that the SRF shifts up the critical temperature. Having correct zero-temperature limit, this theory also gives new interpretation on the Pauli blocking effect in systems with odd particle number.
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.
We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (Tcr{T}^{\mathrm{cr}}Tcr). This shift of Tcr{T}^{\mathrm{cr}}Tcr occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.
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.
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.