Isovector Pairing Correlations Research Articles

Isovector pair correlations in analytically solvable models

The eigensolutions of the collective Hamiltonian with different potentials suggested for description of the isovector pair correlations are obtained, analyzed and compared with the experimental energies. It is shown that the isovector pair correlations in nuclei around [Formula: see text]Ni can be described as anharmonic pairing vibrations. The results obtained indicate the presence of the [Formula: see text]-particle type correlations in these nuclei and the existence of the interaction different from isovector pairing which also influences on the isospin dependence of the energies.

Isovector Pair Correlations of Nucleons in Nuclei

A finite boson representation of fermion operators is obtained. The collective Hamiltonian for a description of the pair excitation is constructed and the collective potential energy is analyzed. The collective modes related to the addition and removal of pair of particles are partitioned off the variety of nuclear degrees of freedom.

Collective treatment of the isovector pair correlations: Boson representation

The theoretical approach to consideration of the Hamiltonian with pairing forces using a technique of the finite boson representation is developed. It is shown that a simultaneous description of the pairing vibrational state in $^{56}$Ni and the pairing rotational states with $T$=0 in the neighboring $N=Z$ nuclei is possible if the pairing Hamiltonian takes into account only isovector monopole pairing. However, the calculated energies of the pairing rotational states of $N=Z$ nuclei removed from $^{56}$Ni by 12 and more nucleons exceed significantly the experimental values. The possible reason of this discrepancy is discussed.

Unified description of pairing and quarteting correlations within the particle-hole-boson approach

We study the description of single-species and isovector pairing correlations in the framework of the projected-BCS (PBCS) and the Quartet Condensation Model (QCM) from a particle-hole perspective and we introduce the representation of the QCM quartet condensate state in terms of particle-hole excitations with respect to the Hartree-Fock state. We also present a new bosonic approximation for both PBCS and QCM. In each case, the starting point is the reformulation of the pair/quartet condensate state in terms of particle-hole excitations with respect to the Hartree-Fock state. The main simplification of our approach is the assumption that the pair operators corresponding to both particle and hole states obey bosonic commutation relations. This simplifies tremendously the computations and allows for an analytic derivation of the averaged Hamiltonian on the condenstate state as a function of the mixing amplitudes. We study both the pure bosonic approach and the renormalized version, and compare the particle-hole bosonic version to the naive prescription of applying the boson approximation directly to the original condensate state. We compare the fermionic and the renormalized particle-hole bosonic approach in the case of a picket fence model of doubly degenerate states and in a realistic shell model space with an effective interaction for the $N=Z$ nuclei above $^{100}$Sn.

Analytical approach for the quartet condensation model

Within the Quartet Condensation Model (QCM), the isovector pairing correlations for $N = Z$ nuclei are described with a very high accuracy by a condensate of $\alpha$-like quartets. The usual approach involves cumbersome recurrence relations in order to compute numerically the relevant quantities of the model: the norm of the quartet states and the mean value of the isovector pairing Hamiltonian as functions of the pair mixing amplitudes. We present the final analytical expressions for the above mentioned quantities, for all cases up to four quartets in the valence shell. The analytical QCM expressions were obtained by a straightforward implementation of the SO(5) algebra in the symbolic computer algebra system Cadabra2. The norm of the quartet states and the mean value of the Hamiltonian are polynomial functions of the mixing amplitudes. The numerical implementation of the QCM model is thus made trivial as matter of copying and pasting the presented formulas. We introduce in this work the method of computer aided analytical calculus for a many body setting. In particular, we provide precise and easy to use tools for the description of isovector pairing correlations.

Isoscalar neutron-proton pairing and SU(4)-symmetry breaking in Gamow-Teller transitions

The isoscalar neutron-proton pairing is thought to be important for nuclei with equal number of protons and neutrons but its manifestation in structure properties remains to be understood. We investigate the Gamow-Teller (GT) transitions for the f7/2-shell nuclei in large-scale shell-model calculations with the realistic Hamiltonian. We show that the isoscalar T=0, J=1+ neutron-proton pairing interaction plays a decisive role for the concentration of GT strengths at the first-excited 1+ state in 42Sc, and that the suppression of these strengths in 46V, 50Mn, and 54Co is mainly caused by the spin-orbit force supplemented by the quadrupole-quadrupole interaction. Based on the good reproduction of the charge-exchange reaction data, we further analyze the interplay between the isoscalar and isovector pairing correlations. We conclude that even for the most promising A=42 nuclei where the SU(4) isoscalar-isovector-pairing symmetry is less broken, the probability of forming an isoscalar neutron-proton pairing condensation is less than 60% as compared to the expectation at the SU(4)-symmetry limit.

Isoscalar-isovector proton-neutron pairing and quartet condensation inN=Znuclei

We show that the correlations generated in the ground state of $N=Z$ nuclei by the isovector and isoscalar pairing forces can be treated with high precision as a condensate of alpha-like quartets. To treat these correlations, the quartet condensation model (QCM) is extended to the treatment of spherically symmetric isovector $(T=1,J=0)$ and isoscalar $(T=0,J=1)$ pairing forces . Within QCM we discuss the competition between $T=1$ and $T=0$ pairing correlations in the case of a two-level model and for $N=Z$ nuclei with nucleons moving in the open shells above $^{16}$O, $^{40}$Ca and $^{100}$Sn. We show that in $N=Z$ systems isovector and isoscalar proton-neutron pairing correlations always coexist.

Proton–neutron pairing in N = Z nuclei: Quartetting versus pair condensation

The isoscalar proton–neutron pairing and isovector pairing, including both isovector proton–neutron pairing and like-particle pairing, are treated in a formalism which conserves exactly the particle number and the isospin. The formalism is designed for self-conjugate (N=Z) systems of nucleons moving in axially deformed mean fields and interacting through the most general isovector and isoscalar pairing interactions. The ground state of these systems is described by a superposition of two types of condensates, i.e., condensates of isovector quartets, built by two isovector pairs coupled to the total isospin T=0, and condensates of isoscalar proton–neutron pairs. The comparison with the exact solutions of realistic isovector–isoscalar pairing Hamiltonians shows that this ansatz for the ground state is able to describe with high precision the pairing correlation energies. It is also shown that, at variance with the majority of Hartree–Fock–Bogoliubov calculations, in the present formalism the isovector and isoscalar pairing correlations coexist for any pairing interactions. The competition between the isovector and isoscalar proton–neutron pairing correlations is studied for N=Z nuclei with the valence nucleons moving in the sd and pf shells and in the major shell above 100Sn. We find that in these nuclei the isovector pairing prevail over the isoscalar pairing, especially for heavier nuclei.

Effects of deformation on the coexistence between neutron-proton and particle-like pairing inN=Zmedium-mass nuclei

A model combining self-consistent mean-field and shell-model techniques is used to study the competition between particle like and proton-neutron pairing correlations in fp-shell even-even self-conjugate nuclei. Results obtained using constant two-body pairing interactions as well as more sophisticated interactions are presented and discussed. The standard BCS calculations are systematically compared with more refined approaches including correlation effects beyond the independent quasi-particle approach. The competition between proton-neutron correlations in the isoscalar and isovector channels is also analyzed, as well as their dependence on the deformation properties. Besides the expected role of the spin-orbit interaction and particle number conservation, it is shown that deformation leads to a reduction of the pairing correlations. This reduction originates from the change of the single-particle spectrum and from a quenching of the residual pairing matrix elements. The competition between isoscalar and isovector pairing in the deuteron transfer is finally addressed. Although a strong dependence the isovector pairing correlations with respect to nuclear deformation is observed, they always dominate over the isoscalar ones.

Isoscalar and isovector pairing in a formalism of quartets

Isoscalar (T=0, J=1) and isovector (T=1, J=0) pairing correlations in the ground state of self-conjugate nuclei are treated in terms of alpha-like quartets built by two protons and two neutrons coupled to total isospin T=0 and total angular momentum J=0. Quartets are constructed dynamically via an iterative variational procedure and the ground state is represented as a product of such quartets. It is shown that the quartet formalism describes accurately the ground state energies of realistic isovector plus isoscalar pairing Hamiltonians in nuclei with valence particles outside the 16O, 40Ca and 100Sn cores. Within the quartet formalism we analyze the competition between isovector and isoscalar pairing correlations and find that for nuclei with the valence nucleons above the cores 40Ca and 100Sn the isovector correlations account for the largest fraction of the total pairing correlations. This is not the case for sd-shell nuclei for which isoscalar correlations prevail. Contrary to many mean-field studies, isovector and isoscalar pairing correlations mix significantly in the quartet approach.

Isovector proton-neutron pairing and Wigner energy in Hartree-Fock mean field calculations

We propose a new approach for the treatment of isovector pairing in self-consistent mean field calculations which conserves exactly the isospin and the particle number in the pairing channel. The mean field is generated by a Skyrme-HF functional while the isovector pairing correlations are described in terms of quartets formed by two neutrons and two protons coupled to the total isospin T=0. In this framework we analyse the contribution of isovector pairing to the symmetry and Wigner energies. It is shown that the isovector pairing provides a good description of the Wigner energy, which is not the case for the mean field calculations in which the isovector pairing is treated by BCS-like models.

Three-body model calculations for N = Z odd-odd nuclei with T = 0 and T = 1 pairing correlations

We study the interplay between the isoscalar (T=0) and isovector (T=1) pairing correlations in N=Z odd-odd nuclei from 14N to 58Cu by using three-body model calculations. The strong spin-triplet T=0 pairing correlation dominates in the ground state of 14N, 18F, 30P, and 58Cu with the spin-parity J^{\pi}=1+, which can be well reproduced by the present calculations. The magnetic dipole and Gamow-Teller transitions are found to be strong in 18F and 42Sc as a manifestation of SU(4) symmetry in the spin-isospin space. We also discuss the spin-quadrupole transitions in these nuclei.

Number projected isovector neutron–proton pairing effect in odd-mass nuclei

A formalism which enables one to strictly conserve the number of particles when taking into account the isovector pairing correlations is presented in the case of odd-mass nuclei. With this aim, we had to first establish the expression of the projector for such systems. Expressions of the ground state and its energy have been exhibited. The model has been numerically tested in the framework of a schematic model.

Number-projected isovector neutron-proton pairing effect in odd mass nuclei

Abstract A formalism which enables one to strictly conserve the number of particles when taking into account the isovector pairing correlations is presented in the case of odd mass nuclei. With this aim, we had to first establish the expression of the projector for such systems. Expressions of the ground state and its energy have been exhibited. The model has been numerically tested in the framework of a schematic model. PACS 21.60.-n; 21.30.Fe

Quartet condensation and isovector pairing correlations inN=Znuclei

We propose a simple quartet condensation model (QCM) which describes with very high accuracy the isovector pairing correlations in self-conjugate nuclei. The quartets have an $\ensuremath{\alpha}$-like structure and are formed by collective isovector pairs. The accuracy of the QCM is tested for $N=Z$ nuclei for which exact shell model diagonalizations can be performed. The calculations are done with two isovector pairing forces, one extracted from standard shell model interactions and the other of seniority type, acting, respectively, upon spherical and axially deformed single-particle states. It is shown that for all calculated nuclei the QCM gives very accurate values for the pairing correlations energies, with errors which do not exceed 1$%$. These results show clearly that the correlations induced by the isovector pairing in self-conjugate nuclei are of quartet type and also indicate that QCM is the proper tool to calculate the isovector proton-neutron correlations in mean field pairing models.

Pairing and alpha-like quartet condensation in N = Z nuclei

We discuss the treatment of isovector pairing by an alpha-like quartet condensate which conserves exactly the particle number, the spin and the isospin. The results show that the quartet condensate describes accurately the isovector pairing correlations in the ground state of systems with an equal number of protons and neutrons.

Q-Deformation of symplectic dynamical symmetries in algebraic models of nuclear structure

With a view toward further nuclear structure applications of approaches based on quantum-deformed (or q-deformed) algebras, introduced to the authors by Yu.F. Smirnov, we construct a q analog of a boson realization of the symplectic noncompact sp(4, R) algebra together with a q analog of a fermion realization of the symplectic compact sp(4) algebra. The first study, on the q-deformed Sp(4,R) symmetry, is applied to the development of a q analog of the two-dimensional Interacting Boson Model with q-deformed SU(3) the underpinning dynamical symmetry group. An explicit realization in terms of q-tensor operators with respect to the standard suq(2) algebra is given. The group-subgroup structure of this framework yields the physical interpretation of the generators of the groups under consideration. The second symplectic algebra, the q-deformed sp(4), is applied to studying isovector pairing correlations in atomic nuclei. A specific q deformation of the sp(4) algebra is realized in terms of q deformed fermion creation and annihilation operators of the shell model. The generators of the algebra close on four distinct realizations of the uq(2) subalgebra. These reductions, which correspond to different types of pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q deformation is based on a comparison of the results for energies of the lowest isovector-paired 0+ states in the deformed and nondeformed cases.

Collectivity in the light xenon isotopes: A shell model study

The lightest Xenon isotopes are studied in the framework of the Interacting Shell Model (ISM). The valence space comprises all the orbits lying between the magic closures N=Z=50 and N=Z=82. The calculations produce collective deformed structures of triaxial nature that encompass nicely the known experimental data. Predictions are made for the (still unknown) N=Z nucleus 108-Xe. The results are interpreted in terms of the competition between the quadrupole correlations enhanced by the pseudo-SU(3) structure of the positive parity orbits and the pairing correlations brought in by the 0h11/2 orbit. We have studied as well the effect of the excitations from the 100-Sn core on our predictions. We show that the backbending in this region is due to the alignment of two particles in the 0h11/2 orbit. In the N=Z case, one neutron and one proton align to J=11 and T=0. In 110-Xe and 112-Xe the alignment begins in the J=10 T=1 channel and it is dominantly of neutron neutron type. Approaching the band termination the alignment of a neutron and a proton to J=11 and T=0 takes over. In a more academic mood, we have explored the role of the isovector and isoscalar pairing correlations on the structure on the yrast bands of 108-Xe and 110-Xe and examined the role of the isovector and isoscalar pairing condensates in these N~Z nuclei.

NUMBER-PROJECTED BETA TRANSITION PROBABILITIES WITHIN A NEUTRON–PROTON PAIRING FRAMEWORK

Particle-number fluctuations effects on the beta transition probabilities are studied in the neutron–proton pairing framework. The Hamiltonian of the system has been considered in its most general form and has been diagonalized by means of the linearization method. However, since the generalized Bogoliubov–Valatin transformation obtained in this way leads to a quasi-particle Hamiltonian which is still nondiagonal, a rediagonalization has been performed. The corresponding wave functions have been projected on both the good neutron and proton numbers using a recently proposed method. Expressions of the beta transition probabilities which strictly conserve the particle-number have then been established. As a first step, the model has been numerically tested within the framework of the schematic one-level model. As a second step, nuclei such as N = Z has been studied using the single-particle energies and eigenstates of the Woods–Saxon deformed mean field. It has thus been shown the necessity of: (i) including the isovector pairing correlations, (ii) performing a rediagonalization of the Hamiltonian, (iii) performing a particle-number projection, (iv) carefully choice the pairing-strength values, when calculating the transition probabilities.

Spin-isospin nuclear response using the existing microscopic Skyrme functionals

Our paper aims at providing an answer to the question of whether one can reliably describe the properties of the most important spin-isospin nuclear excitations by using the available nonrelativistic Skyrme energy functionals. Our method, which has been introduced in a previous publication devoted to the isobaric analog states, is the self-consistent quasiparticle random-phase approximation (QRPA). The inclusion of pairing is instrumental for describing a number of experimentally measured spherical systems which are characterized by open shells. We discuss the effect of isoscalar and isovector pairing correlations. Based on the results for the Gamow-Teller resonance in $^{90}\mathrm{Zr}$, $^{208}\mathrm{Pb}$, and a few Sn isotopes, we draw definite conclusions on the performance of different Skyrme parametrizations, and we suggest improvements for future fits. We also use the spin-dipole resonance as a benchmark of our statements.