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Physical Review C

Thermal and quantal fluctuations in the pairing plus quadrupole model

G. Puddu

Published Mar 1, 1993• 25 citations
Publisher

Abstract

Thermal and quantal fluctuations are included in the partition function of the pairing and quadrupole model. The formalism is applied to the calculation of the thermal response functions. The effects of both thermal and quantal fluctuations to the level densities are studied. The limits and the physical content of the scheme are analyzed.

DOI
10.1103/physrevc.47.1067

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Thermal and quantal fluctuations are included in the partition function of the pairing and quadrupole model. The formalism is applied to the calculation of the thermal response functions. The effects of both thermal and quantal fluctuations to the level densities are studied. The limits and the physical content of the scheme are analyzed.
The nuclear level density for solvable model Hamiltonians is calculated as a function of the excitation energy, taking into account both large-amplitude thermal and small-amplitude quantal fluctuations. The formalism, which is applied to simple systems displaying a phase transition (deformed to spherical, superfluid to normal) as a function of the temperature, accurately reproduces the exact solutions. The contribution of thermal fluctuations changes the level density by an order of magnitude. Quantal fluctuations are essential to reproduce the absolute energies of the system and thus to relate the excitation energy to temperature.
The effects of quantal and thermal fluctuations beyond the BCS quasiparticle mean field on the chemical potential are studied within a model, which consists of NNN particles distributed amongst $\ensuremath{\Omega}$ doubly folded equidistant levels interacting via a pairing force with parameter GGG. The results obtained at zero and finite temperatures TTT within several approaches, which include the fluctuations beyond the BCS theory, are compared with the exact results. The chemical potential, defined as the Lagrangian multiplier to preserve the average number of particles, is compared with the corresponding quantity, which includes the effect from fluctuations of particle and quasiparticle numbers beyond the BCS quasiparticle mean field. The analysis of the results shows that the latter differs significantly from the former as functions of GGG and TTT. The chemical potential loses its physical meaning in the system with a fixed number of particles or after eliminating quantal fluctuations of particle (quasiparticle) numbers by means of particle number projection. The validity of the criterion for the signature of the transition to Bose-Einstein condensation, which occurs in infinite systems when the chemical potential hits the bottom of the energy spectrum, is reexamined for the finite multilevel model.
Abstract. The mean-field approximation predicts pairing and shape phase transitions in nuclei as a function of temperature or excitation energy. However, in the finite nucleus the singularities of these phase transitions are smoothed out by quantal and thermal fluctuations. An interesting question is whether signatures of these transitions survive despite the large fluctuations. The shell model Monte Carlo (SMMC) approach enables us to calculate the statistical properties of nuclei beyond the mean-field approximation in model spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods. We have extended the SMMC method to heavy nuclei and used it to study the transition from vibrational (spherical) to rotational (deformed) nuclei in families of rare-earth isotopes. We have calculated collective enhancement factors of level densities as a function of excitation energy and found that the decay of the vibrational and rotational enhancements is well correlated with the pairing and shape phase transitions, respectively.
The mean-field approximation predicts pairing and shape phase transitions in nuclei as a function of temperature or excitation energy. However, in the finite nucleus the singularities of these phase transitions are smoothed out by quantal and thermal fluctuations. An interesting question is whether signatures of these transitions survive despite the large fluctuations. The shell model Monte Carlo (SMMC) approach enables us to calculate the statistical properties of nuclei beyond the mean-field approximation in model spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods. We have extended the SMMC method to heavy nuclei and used it to study the transition from vibrational (spherical) to rotational (deformed) nuclei in families of rare-earth isotopes. We have calculated collective enhancement factors of level densities as a function of excitation energy and found that the decay of the vibrational and rotational enhancements is well correlated with the pairing and shape phase transitions, respectively.
Variational calculations within the pairing plus quadrupole model are carried out for a set of particles moving in a single j -shell and for values of the coupling constants giving rise to the main features of strongly deformed nuclei. Two calculations are carried out and the corresponding results compared. In the first, particles are allowed to align in the mean field (particle aligned model). In the second, pairs of particles coupled to angular momentum λ = 0 and λ = 2 align themselves in the average field (pair aligned model). Although many features of the results are qualitatively similar, even for well deformed systems, there are important quantitative differences. In particular, the single-particle occupancies predicted by the pair aligned model display a diffusivity around the Fermi surface (effective pairing gap) which is ∼ 1.7 times larger than that predicted by the particle aligned model. The quadrupole moment predicted by the pair aligned model at maximum deformation is ∼ 1.4 smaller than that predicted by the particle aligned model. The pair aligned model and the particle aligned model thus describe two many-body systems which are rather different. In the first pairing correlations are as important as quadrupole correlations while in the second the quadrupole deformation plays the dominant role. The effect of allowing pairs of particles to couple to angular momenta λ > 2 is found to be essential to achieve quantitative agreement between the two models.