Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nucl-th/0503066 · Physical Review C

Effects of particle-number conservation on heat capacity of nuclei

K. Esashika, Hitoshi Nakada, K. Tanabe

Published Oct 12, 2005• 29 citations
arXivHTMLPDF

Abstract

By applying the particle-number projection to the finite-temperature Bardeen-Cooper-Schrieffer (BCS) theory, the S-shaped heat capacity, which has recently been claimed to be a fingerprint of the superfluid-to-normal phase transition in nuclei, is reexamined. It is found that the particle-number (or number-parity) projection gives S shapes in the heat capacity of nuclei that look qualitatively similar to the observed ones. These S shapes are accounted for as effects of the particle-number conservation on the quasiparticle excitations and occur even when we keep the superfluidity at all temperatures by assuming a constant gap in the BCS theory. The present study illustrates significance of the conservation laws in studying phase transitions of finite systems.

DOI
10.1103/physrevc.72.044303

Related papers

New Bardeen-Cooper-Schrieffer-type theory at finite temperature with particle-number conservation

Hitoshi Nakada, K. Tanabe

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 (${T}^{\mathrm{cr}}$). This shift of ${T}^{\mathrm{cr}}$ 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.

arXiv:quant-ph/0603113•Physical Review C•Dec 6, 2006•19 citations

Particle-number-projected thermal pairing

Nguyen Dinh Dang

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 $^{120}\mathrm{Sn}$ 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.

Physical Review C•Dec 28, 2007•22 citations

BCS-TYPE THEORY IN CANONICAL ENSEMBLES

Hitoshi Nakada, K. Tanabe

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.

International Journal of Modern Physics E•Nov 1, 2006•3 citations

Particle-number projection in the finite-temperature mean-field approximation

P. Fanto, Y. Alhassid, G. F. Bertsch

Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei $^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}$, and $^{150}\mathrm{Sm}$. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.

arXiv:1610.08954•Physical Review C•Jul 7, 2017•18 citations

Superfluidity of heated Fermi systems in the static fluctuation approximation

Airat A. Khamzin, А. С. Никитин, Аirat Sitdikov

Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.

Physics of Atomic Nuclei•Oct 1, 2015•5 citations

Study of nuclear structure by using many-body theory

Nguyễn Quang Hưng

Atomic nucleus consists of nucleons (protons and neutrons) interacting each other with the so-called nucleon-nucleon (NN) forces. The structure of atomic nucleus can be studied via many-body theories. Some of them are the well-known random-phase approximation (RPA) and/or the Bardeen-Cooper-Schrieffer (BCS) theory. These theories produce very well the properties of infinite systems such as superconductors, where the quantal and thermal fluctuations are zero or very small. However, our recent studies have shown that these fluctuations are significant in finite systems such as atomic nuclei. As the result, the physical properties of nuclear systems are changed due to the effects of quantal and thermal fluctuations. Therefore, the conventional many body theories need to be modified to include the effects of these fluctuations when they are applied to atomic nucleus. Recently, we have proposed a self-consistent quasiparticle RPA (SCQRPA) taking into account the effects of quantal and thermal fluctuations. The SCQRPA has been tested and the results obtained are in reasonable agreements with the exact solutions of the model cases as well as those obtained within the quantum Monte-Carlo method for realistic nuclei.

•Jan 1, 2011•0 citations
By applying the particle-number projection to the finite-temperature Bardeen-Cooper-Schrieffer (BCS) theory, the S-shaped heat capacity, which has recently been claimed to be a fingerprint of the superfluid-to-normal phase transition in nuclei, is reexamined. It is found that the particle-number (or number-parity) projection gives S shapes in the heat capacity of nuclei that look qualitatively similar to the observed ones. These S shapes are accounted for as effects of the particle-number conservation on the quasiparticle excitations and occur even when we keep the superfluidity at all temperatures by assuming a constant gap in the BCS theory. The present study illustrates significance of the conservation laws in studying phase transitions of finite systems.
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.
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.
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.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.
Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.
Atomic nucleus consists of nucleons (protons and neutrons) interacting each other with the so-called nucleon-nucleon (NN) forces. The structure of atomic nucleus can be studied via many-body theories. Some of them are the well-known random-phase approximation (RPA) and/or the Bardeen-Cooper-Schrieffer (BCS) theory. These theories produce very well the properties of infinite systems such as superconductors, where the quantal and thermal fluctuations are zero or very small. However, our recent studies have shown that these fluctuations are significant in finite systems such as atomic nuclei. As the result, the physical properties of nuclear systems are changed due to the effects of quantal and thermal fluctuations. Therefore, the conventional many body theories need to be modified to include the effects of these fluctuations when they are applied to atomic nucleus. Recently, we have proposed a self-consistent quasiparticle RPA (SCQRPA) taking into account the effects of quantal and thermal fluctuations. The SCQRPA has been tested and the results obtained are in reasonable agreements with the exact solutions of the model cases as well as those obtained within the quantum Monte-Carlo method for realistic nuclei.