Nuclear Pairing Correlations Research Articles

Energy competition and pairing effect for the fission path with a microscopic model

We studied the fission barrier of 236U with a microscopic mean-field model employing Skyrme-type effective interaction. It has been known that the microscopic mean-field calculation had a trend of overestimating the fission barriers derived from the fission cross section, and our results were found to be in accord with it. To reveal a major factor of the discrepancy, we investigated various components of the Skyrme energy-density functional building of the fission barrier height by a static mean-field model, including nuclear pairing correlation. We found that the spin-orbit and pairing terms affected the fine structure of the fission barrier as a function of elongation of the nucleus. Therefore, we investigated the sensitivity of the fission barrier height on the pairing strength, considering the change of level density along the calculated fission path.

Consistency of two different approaches to determine the strength of a pairing residual interaction in the rare-earth region

Two fits of the pairing residual interaction in the rare-earth region are independently performed. One is made on the odd-even staggering of masses by comparing measured and explicitly calculated three-point binding-energy differences centered on odd-even nuclei. Another deals with the moments of inertia of the first 2+ states of well deformed even-even nuclei upon comparing experimental data with the results of Inglis-Belyaev moments (supplemented by a crude estimate of the so-called Thouless-Valatin corrections). The sample includes 24 even-even and 31 odd-mass nuclei selected according to two criteria: they should have good rotor properties and should not correspond to low pairing-correlation regimes in their ground states. Calculations are performed in the self-consistent Hartree-Fock plus BCS framework (implementing a self-consistent blocking in the case of odd-mass nuclei). The Skyrme SIII parametrization is used in the particle-hole channel and the fitted quantities are the strengths of |Tz|=1 proton and neutron seniority residual interactions. As a result, the two fits yield sets of strengths in excellent agreement: about 0.1% for the neutron parameters and 0.2% for protons. In contrast when one performs such a fit on odd-even staggering from quantities deduced from BCS gaps or minimal quasiparticle energies in even-even nuclei, as is traditional, one obtains results significantly different from those obtained in the same nuclei by a fit of moments of inertia. As a conclusion, beyond providing a phenomenological tool for microscopic calculations in this region, we have illustrated the proposition made in the seminal paper of Bohr, Mottelson and Pines that moments of inertia and odd-even staggering in selected nuclei were excellent measuring sticks of nuclear pairing correlations.

Dependence of two-proton radioactivity on nuclear pairing models

Sensitivity of two-proton emitting decay to nuclear pairing correlation is discussed within a time-dependent three-body model. We focus on the $^6$Be nucleus assuming $\alpha + p + p$ configuration, and its decay process is described as a time-evolution of the three-body resonance state. For a proton-proton subsystem, a schematic density-dependent contact (SDDC) pairing model is employed. From the time-dependent calculation, we observed the exponential decay rule of a two-proton emission. It is shown that the density dependence does not play a major role in determining the decay width, which can be controlled only by the asymptotic strength of the pairing interaction. This asymptotic pairing sensitivity can be understood in terms of the dynamics of the wave function driven by the three-body Hamiltonian, by monitoring the time-dependent density distribution. With this simple SDDC pairing model, there remains an impossible trinity problem: it cannot simultaneously reproduce the empirical $Q$ value, decay width, and the nucleon-nucleon scattering length. This problem suggests that a further sophistication of the theoretical pairing model is necessary, utilizing the two-proton radioactivity data as the reference quantities.

Particle entropy and depairing in hot nuclei

The nuclear level densities and single particle entropies are predicted for nuclei in the mass region [Formula: see text] within a framework of statistical theory of hot nuclei method. In this method, particle-number and energy conservation as well as nuclear pairing correlations are included in the partition function of grand canonical ensemble. The suppression of pairing correlations is distinctly noticed in temperature dependence of entropies between the critical temperatures [Formula: see text] MeV and [Formula: see text] MeV for [Formula: see text], [Formula: see text] and [Formula: see text] isotopes of the elements. These structural thermodynamic entropies are interpreted as a remarkable signature of the superfluid to normal phase transition connected to the vanishing of pairing gap. The calculated level densities are compared with recent experimental values. In addition, the single particle entropy of intermediate-mass nuclei is depicted as half of the entropy of mid-shell nuclei in the rare-earth region. As a consequence, the [Formula: see text] shell closure of [Formula: see text]V carries low entropy at low excitation energy presents an interesting analogy to the [Formula: see text] shell closure of [Formula: see text]Ni. Merely, in the case of odd–even [Formula: see text] has higher entropy than the even–even [Formula: see text] nucleus.

Octupole deformations in high-Kisomeric states of heavy and superheavy nuclei

We study the effects of quadrupole-octupole deformations on the energy and magnetic properties of high-K isomeric states in even-even heavy and superheavy nuclei. The neutron two-quasiparticle (2qp) isomeric energies and magnetic dipole moments are calculated within a deformed shell model with the Bardeen-Cooper- Schrieffer (BCS) pairing interaction over a wide range of quadrupole and octupole deformations. We found that in most cases the magnetic moments exhibit a pronounced sensitivity to the octupole deformation, while the 2qp energies indicate regions of nuclei in which the presence of high-K isomeric states may be associated with the presence of octupole softness or even with octupole deformation. In the present work we also examine the influence of the BCS pairing strength on the energy of the blocked isomer configuration. We show that the formation of 2qp energy minima in the space of quadrupole-octupole and eventually higher multipolarity deformations is a subtle effect depending on nuclear pairing correlations.

Pairing phase transition: A finite-temperature relativistic Hartree-Fock-Bogoliubov study

Background: The relativistic Hartree-Fock-Bogoliubov (RHFB) theory has recently been developed and it provides a unified and highly predictive description of both nuclear mean field and pairing correlations. Ground state properties of finite nuclei can accurately be reproduced without neglecting exchange (Fock) contributions. Purpose: Finite-temperature RHFB (FT-RHFB) theory has not yet been developed, leaving yet unknown its predictions for phase transitions and thermal excitations in both stable and weakly bound nuclei. Method: FT-RHFB equations are solved in a Dirac Woods-Saxon (DWS) basis considering two kinds of pairing interactions: finite or zero range. Such a model is appropriate for describing stable as well as loosely bound nuclei since the basis states have correct asymptotic behaviour for large spatial distributions. Results: Systematic FT-RH(F)B calculations are performed for several semi-magic isotopic/isotonic chains comparing the predictions of a large number of Lagrangians, among which are PKA1, PKO1 and DD-ME2. It is found that the critical temperature for a pairing transition generally follows the rule $T_c = 0.60\Delta(0)$ for a finite-range pairing force and $T_c = 0.57\Delta(0)$ for a contact pairing force, where $\Delta(0)$ is the pairing gap at zero temperature. Two types of pairing persistence are analysed: type I pairing persistence occurs in closed subshell nuclei while type II pairing persistence can occur in loosely bound nuclei strongly coupled to the continuum states. Conclusions: This first FT-RHFB calculation shows very interesting features of the pairing correlations at finite temperature and in finite systems such as pairing re-entrance and pairing persistence.

Nuclear pairing within a configuration-space Monte Carlo approach

Pairing correlations in nuclei play a decisive role in determining nuclear drip lines, binding energies, and many collective properties. In this work a new configuration-space Monte Carlo (CSMC) method for treating nuclear pairing correlations is developed, implemented, and demonstrated. In CSMC the Hamiltonian matrix is stochastically generated in Krylov subspace, resulting in the Monte Carlo version of Lanczos-like diagonalization. The advantages of this approach over other techniques are discussed; the absence of the fermionic sign problem, probabilistic interpretation of quantum-mechanical amplitudes, and ability to handle truly large-scale problems with defined precision and error control are noteworthy merits of CSMC. The features of our CSMC approach are shown using models and realistic examples. Special attention is given to difficult limits: situations with nonconstant pairing strengths, cases with nearly degenerate excited states, limits when pairing correlations in finite systems are weak, and problems when the relevant configuration space is large.

Nuclear pairing correlations within and beyond HFB-BCS models

Pairing correlations in nuclear systems play a crucial role in several aspects, i.e. binding energies and odd-even effects, superfluid phenomena and pair transfer mechanisms, just to quote few of them. On the theoretical side, the standard description of these features is done by using BCS or HFB models which allow to describe in a simple way pairing effects. However, due to the explicit breaking of the particle number, these theories present some limitations which can be cured by using particle number projection techniques. In this paper, we will show the merits of these techniques and their accuracy in treating pairing correlations. First, a beyond BCS-HFB method is introduced where the effect of four quasi-particle states is included perturbatively and subsequently the particle number is restored. We will then show the need of restoring the good particle number also for excited states that are essential in the pair transfer process between superfluid systems. Applications to the Richardson model are shown and discussed.

Low-lying2+states generated bypn-quadrupole correlation andN=28shell quenching

The quadrupole vibrational modes of neutron-rich $N$=28 isotones ($^{48}$Ca, $^{46}$Ar, $^{44}$S and $^{42}$Si) are investigated using the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory with several choice of energy density functionals, including nuclear pairing correlation. It is found that the quenching of $N$=28 shell gap and the proton holes in the $sd$-shell trigger quadrupole correlation and increase the collectivity of the low-lying 2$^+$ state in $^{46}$Ar. It is also found that the pairing correlation plays an important role to increase the collectivity. We also demonstrate that the same mechanism to enhance the low-lying collectivity applies to other $N$=28 isotones $^{44}$S and $^{42}$Si, and it generates a couple of low-lying 2$^+$ states which can be associated with the observed $2^+$ states.

Examining nuclear pairing correlations in the continuum via a Monte Carlo algorithm

Examining nuclear pairing correlations in the continuum via a Monte Carlo algorithm

Building Monte Carlo Models of Glasses Using Neutron and/or X-ray Diffraction Data

Neutron and X-ray diffraction are key techniques that are used to investigate the atomic and nanometric mesoscale structure of glasses and amorphous materials. These experimental methods probe the nuclear (neutron) or atomic (X-ray) pair correlation func- tions between atoms. For a glass containing N atom types, the information content of the data is low, considering that the data are a weighted sum of N(N+1)/2 partial pair correlation terms. This complexity can often make direct interpretation of results difficult or impossible. Modern computational methods can now rapidly refine atomistic models of disordered materials that satisfy the constraints imposed by diffraction data. These models can then be used to investigate how the partial pair correlation functions contribute to the total scattering data, given a chosen set of underlying physico-chemical constraints, and allow us to extract many structural functions of interest such as bond angle distributions and coordination number histograms. To illustrate these capabilities the technique of Empirical Potential Structure Refinement (EPSR), has been applied to a range of results from a selection of oxide-glass systems and the results provide a set of reference parameters that can be used in future studies on similar glass systems where EPSR is the goal.

Simulation of heavy ion collision using a time-dependent density functional theory including nuclear superfluidity

We carried out a simulation of heavy ion collision using a time-dependent density functional theory. We call it the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB) which can describe nuclear dynamics in three-dimensional coordinate space, treating nuclear pairing correlation. We simulate 20O+20O collision using the Cb-TDHFB with a contact-type pairing functional, and show the behavior of gap energy which is decreasing and vibrating while colliding.

Two-Particle Transfer and Pairing Correlations: Interplay of Reaction Mechanism and Structure Properties

Two-particle transfer processes induced by light and heavy ions are known to be an ideal dynamical tool for extracting information on nuclear pairing correlations. The procedure is however not unique and different reaction mechanism models (ranging from microscopic correlated successive one-particle transfer to collective macroscopic models) can be introduced to establish a link with the structural aspects of initial and final states. We briefly review the subject with special attention to the novel features arising in systems close to the drip lines from the weak binding situation and the consequent role of continuum states.

Heavy-ion two-particle transfer reactions as a probe of pairing correlations

Two-particle transfer processes induced by light and heavy ions are known to be an ideal dynamical tool for extracting information on nuclear pairing correlations. The procedure is however not unique and different reaction mechanism models (ranging from microscopic correlated successive one-particle transfer to collective macroscopic models) can be introduced to establish a link with the structural aspects of initial and final states. In the case of reactions involving weakly-bound nuclei the low-density features characterizing the vicinity of the drip line and the suggested consequent enhancement of the pairing correlations should show up as clear signature in the two-particle transfer cross sections. From a theoretical point of view in the weakly-bound case the focus is centered on the specific role of continuum states, both at the nuclear structure level, to give properly correlated wave functions, and at the reaction level, for example to treat the unbound states in the intermediate A±1 system.

Nuclear surface dynamics and pairing correlations

We solve the Nambu-Gor'kov equations in 120Sn, demonstrating that the coupling of quasiparticles with collective surface vibrations renormalizes the low-lying part of the nuclear spectrum and influences pairing correlations in a substantial way.

Pairing Reentrance Phenomenon in Heated Rotating Nuclei in the Shell-Model Monte Carlo Approach

Rotational motion of heated 72Ge is studied within the microscopic shell-model Monte Carlo approach. We investigate the angular momentum alignment and nuclear pairing correlations associated with J^{π} Cooper pairs as a function of the rotational frequency and temperature. The reentrance of pairing correlations with temperature is predicted at high rotational frequencies. It manifests itself through the anomalous behavior of specific heat and level density.

The effects of temperature on finite nuclei

We study the effects of temperature on finite nuclei [1] and look at nuclear deformation and pairing correlations in a self–consistent way. We include temperature in the the Dirac–Hartree–Bogoliubov (DHB) theory using the Matsubara formalism [2]. In order to simplify the calculation we exclude the Fock term and use a zero–range approximation to the relativistic pairing tensor to obtain pure proton–proton and neutron–neutron pairing. We use the Bonche et. al [3–5] vapor subtraction procedure to account for unbound states and remove long range Coulomb repulsion between the hot nucleus and gas, i.e, remove the contributions of the external gas.

Hydrogen molecule ion: Path-integral Monte Carlo approach

The path-integral Monte Carlo approach is used to study the coupled quantum dynamics of the electron and nuclei in hydrogen molecule ion. The coupling effects are demonstrated by comparing differences in adiabatic Born-Oppenheimer and nonadiabatic simulations, and inspecting projections of the full three-body dynamics onto the adiabatic Born-Oppenheimer approximation. Coupling of the electron and nuclear quantum dynamics is clearly seen. The nuclear pair correlation function is found to broaden by 0.040a{sub 0}, and the average bond length is larger by 0.056a{sub 0}. Also, a nonadiabatic correction to the binding energy is found. The electronic distribution is affected less than the nuclear one upon inclusion of nonadiabatic effects.

Variational theory of hot nucleon matter

We develop a variational theory of hot nuclear matter in neutron stars and supernovae. It can also be used to study charged, hot nuclear matter which may be produced in heavy-ion collisions. This theory is a generalization of the variational theory of cold nuclear and neutron star matter based on realistic models of nuclear forces and pair correlation operators. The present approach uses microcanonical ensembles and the variational principle obeyed by the free energy. In this paper we show that the correlated states of the microcanonical ensemble at a given temperature $T$ and density \ensuremath{\rho} can be orthonormalized preserving their diagonal matrix elements of the Hamiltonian. This allows for the minimization of the free energy without corrections from the nonorthogonality of the correlated basis states, similar to that of the ground state energy. Samples of the microcanonical ensemble can be used to study the response, and the neutrino luminosities and opacities of hot matter. We present methods to orthonormalize the correlated states that contribute to the response of hot matter.

Theory of electron spin decoherence by interacting nuclear spins in a quantum dot

We present a quantum solution to the electron spin decoherence by a nuclear pair-correlation method for the electron-nuclear spin dynamics under a strong magnetic field and a temperature high for the nuclear spins but low for the electron. The theory incorporates the hyperfine interaction, the intrinsic (both direct and indirect) nuclear interactions, and the extrinsic nuclear coupling mediated by the hyperfine interaction with the single electron in question. The last is shown to be important in free-induction decay (FID) of the single electron spin coherence. The spin echo eliminates the hyperfine-mediated decoherence but only reduces the decoherence by the intrinsic nuclear interactions. Thus, the decoherence times for single spin FID and ensemble spin echo are significantly different. The decoherence is explained in terms of quantum entanglement, which involves more than the spectral diffusion.