Spin-isospin SU Research Articles

Effects of the Coulomb and the spin-orbit interaction in a deformed mean field on the pairing correlations in N=Z nuclei

We perform systematic calculations regarding the effects by Coulomb and/or spin-orbit (SO) interaction on like- and unlike-pairing correlations in $sd$-, $pf$-, and $sdgh$-shell $N=Z$ nuclei. The former two interactions are comprised in a deformed mean-field potential and the latter pairing correlations are treated by residual interactions in the mean field. We make use of two different pairing matrix elements (PMEs) for the residual interactions : constant and state-dependent Brueckner $G$ matrix. The constant PME may give rise to meaningful information on the pairing correlations under the Wigner spin-isospin SU(4) symmetry in the absence of the Coulomb and the SO interaction. The state-dependent Brueckner PME takes into account the nuclear medium effect based on a realistic nucleon-nucleon interaction. In this work, through the analyses of the Coulomb and the SO interaction effects on the pairing correlations, we discuss in detail the isoscalar pair condensation and the coexistence of the isoscalar and the isovector pairs in the unlike pairing of the $N=Z$ nuclei. Our results show that the Coulomb and the SO interaction as well as nuclear deformation affect the single-particle state evolution in a deformed mean field and, as a result, give significant impacts on the pairing correlations and the smearing of occupation probabilities near Fermi energy. In particular, the pairing gaps in the heavy nuclei are largely disturbed by the Coulomb interaction as well as the SO interaction.

Precise Mass Calculations for Atomic Nuclei of Isospin Tz = 51/2, 26, 53/2, 55/2, 28, and 57/2 with Restored Spin-Isospin SU(4) Symmetry

Via simulating the contribution of spin-orbit interaction to nuclear masses by orthogonal Chebyshev polynomials, the mass excesses are calculated for a group of atomic nuclei characterized by the isospins of Tz = 51/2, 26, 53/2, 55/2, 28, and 57/2 and by the presence of restored Wigner’s spin-isospin SU(4) symmetry. The root-mean-square deviation of the results of these calculations from their experimental counterparts is σ = 140 keV. The potential of the proposed method is discussed.

Partial restoration of spin-isospin SU(4) symmetry and the one-quasiparticle random-phase approximation method in double- β decay

The one-QRPA method is used to describe simultaneously both double decay beta modes, giving special attention to the partial restoration of spin-isospin SU(4) symmetry. To implement this restoration and to fix the model parameters, we resort to the energetics of Gamow-Teller resonances and to the minima of the single $\beta^+$-decay strengths. This makes the theory predictive regarding the $\beta\beta_{2\nu}$-decay, producing the $2\nu$ moments in $^{48}$Ca, $^{76}$Ge, $^{82}$Se, $^{96}$Zr, $^{100}$Mo, $^{128,130}$Te, and $^{150}$Nd, that are of the same order of magnitude as the experimental ones; however, the agreement with $\beta\beta_{2\nu}$ data is only modest. To include contributions coming from induced nuclear weak currents, we extend the $\beta\beta_{0\nu}$-decay formalism employed previously in C. Barbero et. al, Nuc. Phys. A628, 170 (1998). The numerical results for the $\beta\beta_{0\nu}$ moments in the above mentioned nuclei are similar to those obtained in other theoretical studies although smaller on averag by $\sim 40\%$. We attribute this difference basically to the one-QRPA-method, employed here for the first time, instead of the currently used two-QRPA-method. The difference is partially due to the way of carrying out the restoration of the spin-isospin symmetry. It is hard to say which is the best way to make the restoration, since the $\beta\beta_{0\nu}$ moments are not experimentally measurable. The numerical uncertainties in the $\beta\beta$ moments, related with i) their strong dependence on the residual interaction in the p-p channel when evaluated within the QRPA, and ii) lack of proper knowledge of single-particle energies, have been quantified. It is concluded that the partial restoration of the SU (4) symmetry is crucial in the description of the $\beta\beta$-decays, regardless of the nuclear model used.

The restoration of Wigner’s SU(4)-symmetry in the superheavy nucleus and its correlation with the problem “island of stability”, or does the “island of stability” exist?

This paper reviews the history and stages of experimental verification of the hypothesis of Wigner’s spin–isospin SU(4)-symmetry restoration in the field of heavy atomic nuclei and its implications on hypothesis of the “island of stability”. Energies of [Formula: see text]-decay of a number of [Formula: see text]-chains of new superheavy nuclei were calculated based on Wigner’s mass formula without contribution of spin–orbit interaction that correspond to the restoration of Wigner’s spin–isospin symmetry. Calculated energies of the [Formula: see text]-decay fit the experimental data better than other theoretical approaches. It is concluded that there is a need to continue theoretical research of the “island of stability” taking into account mechanisms of restoration of Wigner’s spin–isospin SU(4)-symmetry.

Structure of the two-neutrino double-βdecay matrix elements within perturbation theory

The two-neutrino double-$\ensuremath{\beta}$ Gamow-Teller and Fermi transitions are studied within an exactly solvable model, which allows a violation of both spin-isospin SU(4) and isospin SU(2) symmetries, and is expressed with generators of the SO(8) group. It is found that this model reproduces the main features of realistic calculation within the quasiparticle random-phase approximation with isospin symmetry restoration concerning the dependence of the two-neutrino double-$\ensuremath{\beta}$ decay matrix elements on isovector and isoscalar particle-particle interactions. By using perturbation theory an explicit dependence of the two-neutrino double-$\ensuremath{\beta}$ decay matrix elements on the like-nucleon pairing, particle-particle $T=0$ and $T=1$, and particle-hole proton-neutron interactions is obtained. It is found that double-$\ensuremath{\beta}$ decay matrix elements do not depend on the mean field part of Hamiltonian and that they are governed by a weak violation of both SU(2) and SU(4) symmetries by the particle-particle interaction of Hamiltonian. It is pointed out that there is a dominance of two-neutrino double-$\ensuremath{\beta}$ decay transition through a single state of intermediate nucleus. The energy position of this state relative to energies of initial and final ground states is given by a combination of strengths of residual interactions. Further, energy-weighted Fermi and Gamow-Teller sum rules connecting $\mathrm{\ensuremath{\Delta}}Z=2$ nuclei are discussed. It is proposed that these sum rules can be used to study the residual interactions of the nuclear Hamiltonian, which are relevant for charge-changing nuclear transitions.

Embedded Random Matrix Ensembles with Lie Symmetries: Results from U(Ω) Wigner-Racah algebra

Random matrix ensembles for a system of m number of fermions or bosons in Ω number of single particle levels each r-fold degenerate and interacting with two-body forces are considered. The spectrum generating algebra for these systems is U(rΩ) and a subalgebra of interest is U(rΩ) ⊃ U(Ω) ⊗ SU(r) algebra. Now, for random two-body interactions preserving SU(r) symmetry, one can introduce embedded Gaussian unitary ensemble of random matrices with U(Ω)⊗SU(r) embedding and this class of ensembles are denoted by EGUE(2)-SU(r). Ensembles with r = 1,2 and 4 for fermions correspond to spinless fermions, fermions with spin and fermions with Wigner's spin-isospin SU(4) symmetry respectively. Similarly, for bosons r = 1, 2 and 3 correspond to spinless bosons, two species boson systems and bosons with spin one respectively. The distinction between fermions and bosons is in the U(Ω) irreducible representations. General formulation based on Wigner-Racah algebra for lower order moments of the one- and two-point functions in eigenvalues generated by EGUE(2)- SU(r) is briefly reviewed. The final formulas for the moments involve only SU(Ω) Racah coefficients. For the fourth moment of the one-point function for r > 1 and for the higher order (> 4) bivariate moments of the two-point function for r ≥ 1, formulas are not available for the SU(Ω) Racah coefficients that are needed. It is necessary to derive analytical formulas for these or develop methods that give asymptotic results (an example for this is given in the paper) or develop methods that allow for their numerical evaluation. This important open problem is discussed in some detail.

Embedded Gaussian unitary ensembles with U(Ω)⊗SU(r) embedding generated by random two-body interactions with SU(r) symmetry

Following the earlier studies on embedded unitary ensembles generated by random two-body interactions [EGUE(2)] with spin SU(2) and spin-isospin SU(4) symmetries, developed is a general formulation, for deriving lower order moments of the one- and two-point correlation functions in eigenvalues, that is valid for any EGUE(2) and BEGUE(2) (“B” stands for bosons) with U(Ω)⊗SU(r) embedding and with two-body interactions preserving SU(r) symmetry. Using this formulation with r = 1, we recover the results derived by Asaga et al. [Ann. Phys. (N.Y.) 297, 344 (2002)]10.1006/aphy.2002.6248 for spinless boson systems. Going further, new results are obtained for r = 2 (this corresponds to two species boson systems) and r = 3 (this corresponds to spin 1 boson systems).

Analysis of experimental data on nuclear masses within Wigner spin-isospin SU(4) symmetry

The problem of the realization of Wigner spin-isospin SU(4) symmetry in nuclei is analyzed on the basis of available experimental data on nuclide masses in the mass-number range 1 ≤ A ≤ 257. Empirical expressions are obtained for the universal functions in the Wigner mass formula. The experimental values of the energy of spin-orbit interaction are determined for the aforementioned nuclides. An alternative mechanism of the origin of the odd-even effect in nuclei having an even mass number associated with a specific property of the Casimir operator is proposed. The results obtained in this study suggest that SU(4) symmetry is broken predominantly by spin-orbit interaction.

CONSTRAINTS ON THE HYPERON-SIGMA MESON COUPLING BY GW OBSERVATIONS

The next generation of gravitational wave observatories are promissing candidates to make the first detections. Once the detection occurs the GW characteristics permit to extract some information about the gravitational wave source. In the present work we focus on waves produced by neutron stars which can give stringent constraints on the nuclear matter equation of state. The microscopic description is based on a nonlinear field-theoretical model in order to construct such an equation of state. The model has free parameters, which from actual knowledge may not be pinned down by direct nuclear matter experiments. An important example is the hyperon-sigma meson coupling constant, currently determined by the spin-isospin SU(6) scheme. The coupling constant is of significant relevance for the structure of the equation of state, controlling its rigidity and, consequently, the properties of neutron stars and gravitational wave signals. We show, in this work, how one can constrain the hyperon-sigma meson coupling constant assuming the detection of a gravitational wave.

Intrinsic state for an extended version of the interacting boson model

An intrinsic-state formalism for IBM-4 is presented. A basis of deformed bosons is introduced which allows the construction of a general trial wave function which has Wigner's spin-isospin SU(4) symmetry as a particular limit. Intrinsic-state calculations are compared with exact ones showing good agreement.

O(36) Symmetry Limit of IBM-4 with Good s, d, and sd Boson Spin–Isospin Wigner's SU(4)∼O(6) Symmetries for N≈Z Odd–Odd Nuclei

Led by the current interest in nuclear structure experiments at the N=Z line and the problem of competing isospin T=0 and T=1 pairing modes in such nuclei, it has become important to investigate the properties of the spin–isospin invariant interacting boson model (IBM-4) of Elliott and White. Following this, in the present work the IBM-4 OsdST(36)⊃OSsTs(6)⊕OdST(30)⊃OL(3)⊕OST(6) symmetry limit with not only good sd boson spin–isospin O(6) symmetry but also good s and d boson space O(6) symmetries (i.e., Wigner's spin–isospin SU(4)'s) is identified and analyzed. This symmetry limit is expected to carry signatures typical of some heavy (proton-rich) N≈Z odd–odd nuclei. For this symmetry limit group chain, irrep labels, group generators, quadratic Casimir operators, and energy formula are given. Both in the symmetry limit and allowing for mixing, the number of T=0 pairs in the ground states as a function of boson number and isospin is studied for N=Z nuclei and the results are consistent with known shell model predictions. For N=Z odd–odd nuclei, in the symmetry limit analytical formulas are derived for B(E2)'s in low-lying levels (with O(36) quantum number ω=N, where N is boson number) and also for the yrast band with (ST)=(01) (say as in 74Rb). It is seen that the pattern of B(E2)'s in low-lying levels depend sensitively on the structure of the E2 operator with respect to OST(6) and the symmetry limit generates a isospin dictated ΔJ=4 staggering in (ST)=(01) yrast B(E2)'s. Finally, it is shown that two-nucleon transfer strengths from even–even to odd–odd N=Z nuclei give information about high-lying states with O(36) quantum number ω=N−2.

Pseudo-SU(4) Symmetry inpf-Shell Nuclei

As an alternative to Wigner's spin-isospin SU(4) symmetry, known to be broken in heavier nuclei, a pseudo-SU(4) symmetry is proposed which follows from the combined invariance in pseudospin and isospin. The pseudo-SU(4) scheme is tested in $N,Zg28$ nuclei, and its consequences with regard to Gamow-Teller $\ensuremath{\beta}$ decay are pointed out.

Ikeda sum rule, self-consistency and double-beta decay in the renormalized quasiparticle random phase approximation

The effect of the inclusion of ground-state correlations into the QRPA equation of motion for the two-neutrino double-beta ( ββ 2 ν ) decay is carefully analyzed. The resulting model, called renormalized QRPA (RQRPA), does not collapse near the physical value of the nuclear force strength in the particle-particle channel, as happens with the ordinary QRPA. Still, the ββ 2 ν transition amplitude is only slightly less sensitive on this parameter in the RQRPA than that in the plain QRPA. It is argued that this fact reveals once more than the characteristic behavior of the ββ 2 ν transition amplitude within the QRPA is not an artifact of the model, but a consequence of the partial restoration of the spin-isospin SU(4) symmetry. It is shown that the price paid for bypassing the collapse in the RQRPA is the violation of the Ikeda sum rule.