Isoscalar Pairing Correlations Research Articles

Proton-neutron pairing, α-like quarteting and ground/excited states of N=Z nuclei

Various studies have shown that the proton-neutron (pn) pairing correlations can be accurately described not by a condensate of Cooper pairs, as considered in the majority of mean-field calculations, but by a condensate of α-like quartets. After a short review of the quartet condensate model (QCM), it is discussed the effect of the pn pairing on the ground states of N = Z nuclei, analysed recently in the framework of Skyrme-HF+QCM calculations. An interesting aspect pointed out by these calculations is the strong interdependence between all types of pairing correlations. In particular, when the isoscalar pn pairing channel is switched on, the pairing correlations are redistributed among all the pairing channels without changing significantly the total pairing energy. Due to this reason, for the majority of N = Z nuclei, the binding energy is not affected much when the isoscalar pairing channel is switched on. Yet, in all calculations which include both the isovector and the isoscalar pairing forces, the isoscalar pairing correlations contribute significantly to the binding energies and coexist always with the isovector pn pairing. Finally it is presented a recent extension of the QCM approach to the excited stated of pn pairing Hamiltonians. It will be shown that the low-lying excited states of these Hamiltonians can be described by breaking a quartet from the ground state condensate and replacing it with an “excited” quartet, an approach which is analogous to the one-broken-pair approximation which is commonly used for like-particle systems.

Isoscalar pairing correlations by the tensor force in the ground states of C12, O16, Ne20 , and S32 nuclei

We investigate roles of the tensor force (TF) on the pairing correlations in the ground-state structure of $^{12}\mathrm{C}, ^{16}\mathrm{O}, ^{20}\mathrm{Ne}$, and $^{32}\mathrm{S}$ nuclei within a deformed BCS model. For pairing matrix elements, we exploit the Brueckner $G$ matrix derived from the charge-dependent Bonn potential. In particular, the isoscalar (IS) neutron-proton ($np$) pairing is studied in detail in relation to spin-triplet even TF in the nucleon-nucleon interaction. Detailed analyses of the $np$ pairing and possible enhancement of the IS channel are performed by focusing on roles of attractive spin-triplet even and repulsive spin-triplet odd channels in the TF for the ground states of the nuclei with some deformation. It is also shown that the TF may play a crucial role of properly interpreting the experimental data of a two-nucleon knockout reaction from $^{12}\mathrm{C}$.

Phase transition between the isovector to isoscalar pairing correlations in deformed N = Z nuclei

We investigate the pairing correlations on N = Z nuclei. In particular, the isoscalar (IS) pairing mode in neutron-proton (np) pairing is discussed in detail because the IS np pairing is closely related to the tensor force in the nucleon-nucleon interaction. Recent experiments demonstrated that the IS pairing might be enhanced from the M1 spin strength distribution data and deuteron, which is a consequence of the IS pairing, can be detected in 16O(p,p’d)14N reaction. The present study illustrates that the IS np pairing can be condensed not only in S=0, L = odd states but also in S=1, L=even states. The former is usually treated as T=0 np pairing. The latter is associated with deuteron and has not been fully treated in the BCS theory for the nuclear structure. In particular, the latter np pairing can be ejected as deuteron from the energetic probes in the experiments. Detailed analyses of the np pairing are performed mainly on 16O and extended to other N = Z nuclei.

Isospin Properties of Nuclear Pair Correlations from the Level Structure of the Self-Conjugate Nucleus Ru88

The low-lying energy spectrum of the extremely neutron-deficient self-conjugate (N=Z) nuclide _{44}^{88}Ru_{44} has been measured using the combination of the Advanced Gamma Tracking Array (AGATA) spectrometer, the NEDA and Neutron Wall neutron detector arrays, and the DIAMANT charged particle detector array. Excited states in ^{88}Ru were populated via the ^{54}Fe(^{36}Ar,2nγ)^{88}Ru^{*} fusion-evaporation reaction at the Grand Accélérateur National d'Ions Lourds (GANIL) accelerator complex. The observed γ-ray cascade is assigned to ^{88}Ru using clean prompt γ-γ-2-neutron coincidences in anticoincidence with the detection of charged particles, confirming and extending the previously assigned sequence of low-lying excited states. It is consistent with a moderately deformed rotating system exhibiting a band crossing at a rotational frequency that is significantly higher than standard theoretical predictions with isovector pairing, as well as observations in neighboring N>Z nuclides. The direct observation of such a "delayed" rotational alignment in a deformed N=Z nucleus is in agreement with theoretical predictions related to the presence of strong isoscalar neutron-proton pair 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.

Neutrinoless ββ decay mediated by the exchange of light and heavy neutrinos: the role of nuclear structure correlations

Neutrinoless decay nuclear matrix elements calculated with the shell model and energy-density functional theory typically disagree by more than a factor of two in the standard scenario of light-neutrino exchange. In contrast, for a decay mediated by sterile heavy neutrinos the deviations are reduced to about 50%, an uncertainty similar to the one due to short-range effects. We compare matrix elements in the light- and heavy-neutrino-exchange channels, exploring the radial, momentum transfer and angular momentum-parity matrix element distributions, and considering transitions that involve correlated and uncorrelated nuclear states. We argue that the shorter-range heavy-neutrino exchange is less sensitive to collective nuclear correlations, and that discrepancies in matrix elements are mostly due to the treatment of long-range correlations in many-body calculations. Our analysis supports previous studies suggesting that isoscalar pairing correlations, which affect mostly the longer-range part of the neutrinoless decay operator, are partially responsible for the differences between nuclear matrix elements in the standard light-neutrino-exchange mechanism.

Isovector and isoscalar pairing in odd–odd N = Z nuclei within a quartet approach

The quartet condensation model (QCM) is extended for the treatment of isovector and isoscalar pairing in odd-odd N=Z nuclei. In the extended QCM approach the lowest states of isospin T=1 and T=0 in odd-odd nuclei are described variationally by trial functions composed by a proton-neutron pair appended to a condensate of 4-body operators. The latter are taken as a linear superposition of an isovector quartet, built by two isovector pairs coupled to the total isospin T=0, and two collective isoscalar pairs. In all pairs the nucleons are distributed in time-reversed single-particle states of axial symmetry. The accuracy of the trial functions is tested for realistic pairing Hamiltonians and odd-odd N=Z nuclei with the valence nucleons moving above the cores $^{16}$O, $^{40}$Ca and $^{100}$Sn. It is shown that the extended QCM approach is able to predict with high accuracy the energies of the lowest T=0 and T=1 states. The present calculations indicate that in these states the isovector and the isoscalar pairing correlations coexist together, with the former playing a dominant role.

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.

Low-energy collective Gamow-Teller states and isoscalar pairing interaction

The Gamow-Teller (GT) strength distributions and isobaric analog resonance (IAR) states of several $N=Z+2$ nuclei with mass number $A=42--58$ are studied by using a self-consistent Skyrme Hartree-Fock-Bogoliubov method plus quasiparticle random phase approximation (HFB+QRPA) formalism. The isoscalar spin-triplet pairing interaction is included in QRPA on top of the isovector spin-singlet one in the HFB method. It is found that the isoscalar pairing correlations mix largely the $(\ensuremath{\nu}{j}_{>}\ensuremath{\rightarrow}\ensuremath{\pi}{j}_{<})$ configurations into the low-energy states, and this mixing plays an important role in the formation and in the collectivity of these low-energy states. Furthermore, the observed excitation energy of the low-energy GT state with respect to the IAR can be well reproduced when the strength of isoscalar pairing is about $1.0--1.05$ times that of the isovector pairing, irrespective of the adopted Skyrme interactions. In $N=Z+2$ nuclei in the middle of the $pf$-shell, a mutual cooperative effect of isoscalar pairing and tensor interaction is found; namely, the tensor force reduces the spin-orbit splittings and enhances the effect of the isoscalar pairing.

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.

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.

Relation between Wigner energy and proton-neutron pairing

The linear term proportional to $|N-Z|$ in the nuclear symmetry energy (Wigner energy)is obtained in a model that uses isovector pairing on single particle levels from a deformed potential combined with a $\vec T^2$ interaction. The pairing correlations are calculated by numerical diagonalization of the pairing Hamiltonian acting on the six or seven levels nearest the $N=Z$ Fermi surface. The experimental binding energies of nuclei with $N\approx Z$ are well reproduced. The Wigner energy emerges as a consequence of restoring isospin symmetry. We have found the Wigner energy to be insensitive to the presence of moderate isoscalar pair correlations.

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.

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.

Skyrme force-like extension of the nuclear pairing interaction

In this paper, isovector and isoscalar pairing correlations were investigated using generalized Bardeen–Cooper–Shrieffer theory and a Skyrme force-like form of the residual interaction. The calculations are done for even–even N∼Z germanium nuclei in an axially symmetric Skyrme–Hartree–Fock + BCS scheme with SIII force and time-reversal invariance. A coexistence of T=0 and T=1 superfluid phases is observed.

Reappearance of the pairing correlations at finite temperature

Rotational and deformation dependence of isovector and isoscalar pairing correlations at finite temperature are studied in an exactly solvable cranked deformed shell model Hamiltonian. It is shown that isovector pairing correlations, as expected, decrease with increasing deformation and the isoscalar pairing correlations remain constant at temperature, T=0. However, it is observed that at finite temperature both isovector and isoscalar pairing correlations are enhanced with increasing deformation, which contradict the mean-field predictions. It is also demonstrated that the pair correlations, which are quenched at T=0 and high rotational frequency re-appear at finite temperature. The changes in the individual multipole pairing fields as a function of rotation and deformation are analyzed in detail.

Competition between isoscalar and isovector pairing correlations inN=Znuclei

We study the isoscalar (T=0) and isovector (T=1) pairing correlations in N=Z nuclei. They are estimated from the double difference of binding energies for odd-odd N=Z nuclei and the odd-even mass difference for the neighboring odd-mass nuclei, respectively. The empirical and BCS calculations based on a T=0 and T=1 pairing model reproduce well the almost degeneracy of the lowest T=0 and T=1 states over a wide range of even-even and odd-odd N=Z nuclei. It is shown that this degeneracy is attributed to competition between the isoscalar and isovector pairing correlations in N=Z nuclei. The calculations give an interesting prediction that the odd-odd N=Z nucleus 82Nb has possibly the ground state with T=0.

Cranking in isospace

The response of isovector and isoscalar pair fields to generalized rotation in isospace is studied. Analytical expressions for constant gap solutions for different limiting cases of the model are derived. In particular, the connections between gauge angles among pairing gaps and the position of the iso-cranking axis are investigated in N = Z nuclei. The two domains of collective and noncollective rotation in space are generalized to isospace. The amplitudes for pair-transfer of T = 0 and T = 1 pairs are calculated. It is shown that the structure of the T = 0 state in odd-odd nuclei prevents any enhancement of pair transfer also in the presence of strong isoscalar pairing correlations. The energy differences of the T = 0 and T = 1 excitations in odd-odd nuclei are qualitatively reproduced by Total-Routhian-Surface calculations.

Proton-neutron pairing in the deformed BCS approach

We examine isovector and isoscalar proton-neutron pairing correlations for the ground state of even-even Ge isotopes with mass number A=64-76 within the deformed BCS approach. For N=Z 64Ge the BCS solution with only T=0 proton-neutron pairs is found. For other nuclear systems (N>Z) a coexistence of a T=0 and T=1 pairs in the BCS wave function is observed. A problem of fixing of strengths of isoscalar and isovector pairing interactions is addressed. A dependence of number of like and unlike pairs in the BCS ground state on the difference between number of neutrons and protons is discussed. We found that for nuclei with N much bigger than Z the effect of proton-neutron pairing is small but not negligible.