Competition of isoscalar and isovector proton-neutron pairing in nuclei

We use shell model techniques in the complete pf shell to study pair correlations in nuclei. Particular attention is paid to the competition of isoscalar and isovector proton-neutron pairing modes which is investigated in the odd-odd N = Z nucleus ⁴⁶V and in the chain of even Fe-isotopes. We confirm the dominance of isovector pairing in the ground states. An inspection of the level density and pair correlation strength in ⁴⁶V, however, shows the increasing relative importance of isoscalar correlations with increasing excitation energy. In the Fe-isotopes we find the expected strong dependence of the proton-neutron isovector pairing strength on the neutron excess, while the dominant J = 1 isoscalar pair correlations scale much more gently with neutron number. We demonstrate that the isoscalar pair correlations depend strongly on the spin-orbit splitting.     

Calculations of the β-decay half-lives of neutron-deficient nuclei

In this work,β~+/EC decays of some medium-mass nuclei are investigated within the extended quasiparticle random-phase approximation(QRPA),where neutron-neutron,proton-proton and neutron-proton(np) pairing correlations are taken into consideration in the specialized Hartree-Fock-Bogoliubov(HFB) transformation.In addition to the pairing interaction,the Br¨uckner G-matrix obtained with the charge-dependent Bonn nucleon-nucleon force is used for the residual particle-particle and particle-hole interactions.Calculations are performed for even-even proton-rich isotopes ranging from Z =24 to Z =34.It is found that the np pairing interaction plays a significant role inβ-decay for some nuclei far from stability.Compared with other theoretical calculations,our calculations show good agreement with the available experimental data.Predictions of β-decay half-lives for some very neutron-deficient nuclei are made for reference.     

Band crossing phenomena inN=Z nuclei a probe toT=0 pairing correlations?

The structure of theN=Z nuclei⁶⁰Zn,⁶²Ga,⁶⁴Ge, and⁷²Kr has been investigated at GASP. Total Routhian surface (TRS) calculations have been performed for theN=36, 38 and 40 Kr isotones. In the case of⁷²Kr the four quasi-particleg _9/2 alignment is observed to be significantly delayed in rotational frequency with respect to the heavier Kr isotopes. Such a delay contradicts the predictions of mean field calculations and may be viewed as the first sign of additional correlations in theT=0 pairing channel.     

An investigation of the influence of the pairing correlations on the properties of the isobar analog resonances inA = 208 nuclei

Within the quasi-particle random phase approximation (QRPA), the method of the self-consistent determination of the isovector effective interaction which restores a broken isotopic symmetry for the nuclear part of the Hamiltonian is given. The effect of the pairing correlations between nucleons on the following quantities were investigated for theA = 208 nuclei: energies of the isobar analog 0⁺ states, the isospin admixtures in the ground state of the even-even nuclei, and the differential cross-section for the²⁰⁸Pb(³He,t)²⁰⁸Bi reaction atE(³He)=450 MeV. Both couplings of the excitation branches withT ^z = T⁰ ± 1, and the analog state with isovector monopole resonance (IVMR) in the quasi-particle representation were taken into account in our calculations. As a result of these calculations, it was seen that the pairing correlations between nucleons have no considerable effect on theT = 23 isospin admixture in the ground state of the²⁰⁸Pb nucleus, and they cause partially an increase in the isospin impurity of the isobar analog resonance (IAR). It was also established that these correlations have changed the isospin structure of the IAR states, and shifted the energies of the IVMR states to the higher values.     

Non-empirical pairing functional

The present contribution reports the first systematic finite-nucleus calculations performed using the Energy Density Functional method and a non-empirical pairing functional derived from low-momentum interactions. As a first step, the effects of Coulomb and the three-body force are omitted while only the bare two-nucleon interaction at lowest order is considered. To cope with the finite-range and non-locality of the bare nuclear interaction, the ¹S₀ channel of V_{low k} is mapped onto a convenient operator form. Neutron-neutron and proton-proton pairing correlations generated in finite nuclei by the direct term of the two-nucleon interaction are characterized in a systematic manner. Eventually, such predictions are compared to those obtained from empirical local functionals derived from density-dependent zero range interactions. The characteristics of the latter are analyzed in view of that comparison and a specific modification of their isovector density dependence is suggested to accommodate Coulomb effects and the isovector trend of neutron gaps at the same time.     

The effect of neutron-proton pairing correlations on the transfer of a neutron-proton pair

A theory of neutron-proton pairing interaction is developed considering bothJ=0T=1 andJ≠0T=0 correlations. The model of a singlej-shell is investigated explicitly forN=Z nuclei. Instead of solving the full HFB (Hartree Fock Bogoliubov) problem a variational method is used for determining the ground state energy and wavefunction. Our model shows that the best solutions contain either onlyT=0 or onlyT=1 correlations. A solution mixingT=0 andT=1 is energetically worse. It is estimated in PWBA (Plane Wave Born Approximation) that for ground state transitions at light nuclei in the transfer of a neutron-proton pair the cross section is enhanced up to a factor 3 by pairing correlations compared with shell model calculations.     

A close look at the competition of isovector and isoscalar pairing in A=18 and 20 even-even N≈Z nuclei

competition of isovector and isoscalar pairing in A=18 and 20 even-even N≈Z nuclei is analyzed in the framework of the mean-field plus the dynamic quadurpole-quadurpole, pairing and particle-hole interactions, whose Hamiltonian is diagonalized in the basis U(24) ?(U(6) ? S U(3) ? S O(3))■(U(4) ? S US(2)■ S UT(2)) in the L = 0 configuration subspace. Besides the pairing interaction, it is observed that the quadurpole-quadurpole and particlehole interactions also play a significant role in determining the relative positions of low-lying excited 0~+ and 1~+ levels and their energy gaps, which can result in the ground state first-order quantum phase transition from J = 0 to J = 1.The strengths of the isovector and isoscalar pairing interactions in these even-even nuclei are estimated with respect to the energy gap and the total contribution to the binding energy. Most importantly, it is shown that although the mechanism of the particle-hole contribution to the binding energy is different, it is indirectly related to the Wigner term in the binding energy.     

The Calculation of 2νββ Matrix Elements Within the Collective Description

The problem of particles coupled by an isovector pairing interaction is solved by means of a collective treatment. The Hamiltonian and the beta-decay operators acting within the np-subspace are constructed. The procedure reproduces exact results for a schematic model. The realistic calculation of the Fermi amplitude for the double-beta decay of ⁷⁶Ge confirms the current expectation about its smallness relative to the Gamow-Teller decay.     

Pairing and quarteting correlations in nuclei

The Gorkov approach to the pairing problem has been generalized in order to take into account pairing and quarteting correlations with a model Hamiltonian including a charge independent pairing interaction and a n-p isopairing term. The single quasi-particle and quasi-hole energies calculated for all the doubly even nuclei of the $\text{7}\text{2}$^? shell show the importance of quarteting correlations in nuclei.     

Conventional BCS,unconventional BCS,and non-BCS hidden dineutron phases in neutron matter

The nature of pairing correlations in neutron matter is re-examined. Working within the conventional approximation in which the nn pairing interaction is provided by a realistic bare nn potential fitted to scattering data, it is demonstrated that the standard BCS theory fails in regions of neutron number density, where the pairing constant λ, depending crucially on density, has a non-BCS negative sign. We are led to propose a non-BCS scenario for pairing phenomena in neutron matter that involves the formation of a hidden dineutron state. In low-density neutron matter, where the pairing constant has the standard BCS sign, two phases organized by pairing correlations are possible and compete energetically: a conventional BCS phase and a dineutron phase. In dense neutron matter, where λ changes sign, only the dineutron phase survives and exists until the critical density for termination of pairing correlations is reached at approximately twice the neutron density in heavy atomic nuclei.     

Properties of isovector 1+ states in A=28 nuclei and nuclear muon capture

A method is proposed for taking into account, in a calculation of partial rates of muon capture by nuclei, experimental information about strength functions for Gamow-Teller and isovector M1 transitions. The method, which amounts to choosing an orthogonal transformation that acts in the subspace of wave functions for excited states, requires neither modifying transition operators nor introducing effective charges. The matrix of the above transformation is constructed as a product of the matrices of reflection in a plane. All calculations are performed on the basis of the multiparticle shell model. Numerical results are obtained for isovector states in A=28 nuclei. Strength functions for Gamow-Teller and isovector M1 transitions in ²⁸Si are considered, and the lifetimes of 1⁺ states in ²⁸Al and the branching fractions for gamma decays of this state are calculated. Owing to taking into account experimental information about the properties of isovector states, the branching fractions for the γ decays of the 1⁺ state at 2.201 MeV in ²⁸Al are successfully described for the first time. The above transformation of the wave functions changes substantially the distribution of partial rates of allowed muon capture by a ²⁸Si nucleus among the 1⁺ states of the final nucleus ²⁸Al in relation to the results of the calculations with the eigenfunctions of the Hamiltonian of the multiparticle shell model. The muon-capture rates calculated with the transformed functions agree well with experimental data.     

On the origin of the Wigner energy

Surfaces of experimental masses of even-even and odd-odd nuclei exhibit a sharp slope discontinuity at N = Z. This cusp (Wigner energy), reflecting an additional binding in nuclei with neutrons and protons occupying the same shell model orbitals, is usually attributed to neutron-proton pairing correlations. A method is developed to extract the Wigner term from experimental data. Both empirical arguments and shell-model calculations suggest that the Wigner term can be traced back to the isospin T = 0 part of nuclear interaction. The structure of the Wigner energy is analyzed in terms of neutron-proton pairs of a given angular momentum and isospin. In particular, we find that the Wigner term cannot be solely explained in terms of correlations between the neutron-proton J = 1, T = 0 (deuteron-like) pairs.     

GCM analysis of the collective properties of lead isotopes with exact projection on particle numbers

We present a microscopic analysis of the collective behaviour of the lead isotopes in the vicinity of ²⁰⁸Pb. In this study, we rely on a coherent approach based on the Generator Coordinate Method (GCM) including exact projection on N and Z numbers within a collective space generated by means of the constrained Hartree-Fock BCS method. With the same Hamiltonian used in HF + BCS calculations, we have performed a comprehensive study including monopole, quadrupole and octupole excitations as well as pairing vibrations. We find that, for the considered nuclei, the collective modes which modify the most the conclusions drawn from the mean-field theory are the octupole and pairing vibrations. Received: 31 May 2001 / Accepted: 23 August 2001     

Application of a microscopic interacting boson model to single-closed-shell nuclei

The microscopic treatment of an extended IBM including the pairing vibrational modes is applied to single-closed-shell nuclei with N = 50 or Z = 50. The low-lying spectra in the N = 50 isotones are well described in terms of the quadrupole phonon and the pairing vibration. The Sn isotopes roughly resemble the N = 50 isotones. The behavior of the 0₂⁺, 0₃⁺ and 2₁⁺ levels can be explained fairly well by the present model. Introduction of the pairing-vibrational modes into the IBM is essential for describing the single-closed-shell nuclei.     

Alpha clusters in nuclei with 41 ≦ A ≦ 45

The effect of pairing and surface α-clustering structure was studied by means of a Hamiltonian with pairing and four-body interaction terms in a simple two-level model in odd nuclei with 41 ≦ A ≦ 45. It was shown that the dynamical α-structures can be induced by the pairing correlation alone, but with an unusually large value of the strength G of the pairing interaction. The addition of a small four-body interaction term allowed us to diminish the G-value to a physically reasonable one and, at the same time, to get meaningful α-clustering. The four-body term added to the Hamiltonian is a simplified way to improve, in this respect, the two-body pairing interaction (itself a simplified model interaction). For example, in the ground state of ⁴⁵Ti the “α-clusters” appeared with larger probability than that for separate nucleons, while in ⁴⁵Ca both probabilities are approximately equal. The calculated binding energy was also applied to reproduce the experimental difference of the masses of neighbouring nuclei. With the eigenstates of the Hamiltonian, schematic spectroscopic factors for two-nucleon transfer, α-transfer, and elastic scattering of α-particles were also calculated. Spectroscopic factors for the transitions from ground state to ground state and from ground state to excited state were then used to distinguish between superfluid or normal behaviour of these nuclei. In addition to the superfluidity induced by pairing forces, a superfluidity due to four-body correlations was also predicted.     

The isospin admixture of the ground state and the properties of the isobar analog resonances in medium and heavy mass nuclei

Within the framework of quasiparticle random phase approximation (QRPA), Pyatov-Salamov method [23] for the self-consistent determination of the isovector effective interaction strength parameter, restoring a broken isotopic symmetry for the nuclear part of the Hamiltonian, is used. The isospin admixtures in the ground state of the parent nucleus, and the isospin structure of the isobar analog resonance (IAR) state were investigated with the inclusion of the pairing correlations between nucleons for the medium and heavy mass regions: 80 <A < 90, 102 <A < 124, and 204 <A < 214. It was determined that the influence of the pairing interaction between nucleons on the isospin admixtures in the ground state and the isospin structure of the IAR state is more pronounced for the light isotopes (N ≈ Z) of the investigated nuclei     

Shell structure of <Superscript>107,109</Superscript>Ag nuclei and spin–parities nuclei in which the 1<Subscript><Emphasis Type="Italic">g</Emphasis>9/2</Subscript> orbit is being filled

Occupation numbers and energies of proton and neutron orbits in the isotopes ¹⁰⁷Ag and ¹⁰⁹Ag were obtained on the basis of data on nucleon-stripping and nucleon-pickup reactions. A special data-analysis method reducing systematic and statistical uncertainties was applied in relevant calculations. It is shown that, in this region, the spin–parity of nuclei is determined by shells, pairing, and polarization. The schemes of proton and neutron (sub)orbits were constructed. These schemes explain the ground-state spin–parities of N, Z = 39–49 nuclei.     

Effect of Nucleon Pairing in the Spectra of <Emphasis Type="Italic">N</Emphasis> = 50 Isotones

The emergence of the pairing effect of identical nucleons in the j = 9/2 state in low-lying excited states of nuclei near ⁹⁰Zr (N = 50, Z = 40) is discussed. Multiplets of states with seniority s ≥ 2, the splitting of which is determined by the proton pairing energy, are clearly visible in the nuclear spectra for a chain of N = 50 isotones. A comparison of the spectra of ground state multiplets, calculated in the δ-interaction approximation, with experimental data and results from other theoretical calculations shows this approach can be used to describe the structure of spectra and level positions with high J values.