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Signatures of phase transitions in nuclei at finite excitation energies

Y. Alhassid, C. Özen, Hitoshi Nakada

Published Aug 17, 2016• 0 citations
OpenAlex

Abstract

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.

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MANY-BODY CORRELATIONS IN NUCLEI AND QUANTUM DOTS

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Confined quantum many-body systems of a given particle number exhibit a variety of intrinsic shape characteristics as a function of increasing external field and internal thermal excitation. The shell model is an important tool for the theoretical description of these various structures and transitions in nuclei. Another system in which correlations beyond the mean field may play an important role is semiconductor quantum dots. In this Proceedings, the author compares nuclei and quantum dots and their various deformation properties. The author reports on shell-model calculations in nuclei and some recent mean-field calculations of the thermal properties of quantum dots.

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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.
The auxiliary-field Monte Carlo (AFMC) method, also known as the shell model Monte Carlo (SMMC), enables us to calculate microscopically statistical and collective properties of nuclei in the presence of strong correlations. These calculations can be carried out in shell model spaces that are many orders of magnitude larger than spaces that can be treated with conventional diagonalization methods. A recent major development has been the extension of the AFMC approach to heavy nuclei. Such applications to heavy nuclei have been a major challenge. On the conceptual level, a crucial question is whether a truncated spherical shell model Hamiltonian can describe the proper collectivity observed in heavy nuclei and, in particular, the rotational character of strongly deformed nuclei. On the technical level, the low excitation energies make it necessary to perform calculations down to much lower temperatures. At such low temperatures the propagator becomes ill-conditioned and requires the introduction of stabilization methods.
The shell model Monte Carlo (SMMC) method enables calculations in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods, and is particularly suitable for the calculation of level densities in the presence of correlations. We review recent advances and applications of SMMC for the microscopic calculation of level densities. Recent developments include (i) a method to calculate accurately the ground-state energy of an odd-mass nucleus, circumventing a sign problem that originates in the projection on an odd number of particles, and (ii) a method to calculate directly level densities, which, unlike state densities, do not include the spin degeneracy of the levels. We calculated the level densities of a family of nickel isotopes 59−64Ni and of a heavy deformed rare-earth nucleus 162Dy and found them to be in close agreement with various experimental data sets.
Methods which include the effect of mean-field fluctuations on the nuclear partition function and level density of hot nuclei are tested by applying them to a many-body system for which exact solutions are available, the Lipkin model. It is found that for finite systems fluctuation corrections to the mean field partition function and level density can be substantial, and especially so near a shape phase transition where these fluctuations smooth out discontinuities in the mean-field quantities. The model calculations suggest that only time-independent fluctuations are important at high temperatures or at high excitation energies. Moreover, even when fluctuations are restricted to a small set of constraints, a significant improvement over the mean-field calculations can result. This is thus a practical approximation since it requires only knowledge of the deformation free energy surfaces at finite temperature.
The heat capacity of iron isotopes is calculated within the interacting shell model using the complete (pf+0g9/2)({pf+0g}_{9/2})(pf+0g9/2​) shell. We identify a signature of the pairing transition in the heat capacity that is correlated with the suppression of the number of spin-zero neutron pairs as the temperature increases. Our results are obtained by a novel method that significantly reduces the statistical errors in the heat capacity calculated by the shell model Monte Carlo approach. The Monte Carlo results are compared with finite-temperature Fermi gas and BCS calculations.
Confined quantum many-body systems of a given particle number exhibit a variety of intrinsic shape characteristics as a function of increasing external field and internal thermal excitation. The shell model is an important tool for the theoretical description of these various structures and transitions in nuclei. Another system in which correlations beyond the mean field may play an important role is semiconductor quantum dots. In this Proceedings, the author compares nuclei and quantum dots and their various deformation properties. The author reports on shell-model calculations in nuclei and some recent mean-field calculations of the thermal properties of quantum dots.