Quadrupole Pairing Interaction Research Articles

Magnetic anisotropy in uranium monosulfide probed by magnetic torque measurements

We have studied the magnetic torque in uranium monosulfide (US) single crystals to explore the magnetic anisotropy in this material. Uranium monosulfide crystallizes in cubic, NaCl-type of crystal structure and exhibits the largest magneto-crystalline anisotropy observed in cubic systems. By performing detailed torque measurements we observe a strongly anisotropic behavior in the paramagnetic phase due to crystal defects and quadrupolar pair interactions. Our studies also confirm the presence of a large anisotropy in the ferromagnetic state of the US system with the <100>, <111>, and <110> directions being hard, easy, and intermediate axis, respectively. Furthermore, the anisotropy in the paramagnetic phase shows similar characteristics to the anisotropy observed in the ferromagnetic phase, as characterized by second and fourth rank susceptibility terms. The similarity of the anisotropic behaviors in paramagnetic and ferromagnetic phases is the consequence of strong magneto-elastic properties in this system, which possibly lead to the rhombohedral structural distortion, not only in the ferromagnetic phase but also in the paramagnetic phase (induced by applied magnetic fiield).

Quantum-number-projected generator-coordinate-method analysis of low-lying states in nuclei around mass 80

The quantum-number-projected generator coordinate method (QNPGCM) is applied to the neutron-rich Ge and Se isotopes where the monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an effective interaction. The energy spectra calculated with both axial and triaxial deformations are compared to the shell-model results and the experimental data. In both cases, the QNPGCM reproduces well the energy levels of the even-spin yrast states. However, the QNPGCM results by only assuming axial deformations are not satisfactory enough to reproduce the energy levels of the quasi-$\ensuremath{\gamma}$ bands. Taking account of triaxial deformations is essentially important to describe the yrast and quasi-$\ensuremath{\gamma}$ bands simultaneously.

Electromagnetic properties of neutron-rich Ge isotopes * * Supported by National Natural Science Foundation of China (11875188), Open Project Fund of Shanghai Key Laboratory of Particle Physics and Cosmology (18DZ2271500-2), Sichuan Science and Technology Program (2019JDRC0017), and the Doctoral Program of Southwest University of Science and Technology (18zx7147)

The electric quadrupole moment and the magnetic moment (or the factor) of low-lying states in even-even nuclei 72-80Ge and odd-mass nuclei 75-79Ge are studied in the framework of the nucleon pair approximation (NPA) of the shell model, assuming the monopole and quadrupole pairing plus quadrupole-quadrupole interaction. Our calculations reproduce well the experimental values of and for 72,74,76Ge, as well as the yrast energy levels of these isotopes. The structure of the states and the contributions of the proton and neutron components in and are discussed in the -pair truncated shell-model subspace. The overall trend of and as a function of the mass number , as well as their signs, are found to originate essentially from the proton contribution. The negative value of in 72,74Ge is suggested to be due to the enhanced quadrupole-quadrupole correlation and configuration mixing.

Understanding nuclear dynamics in the SD-pair shell model: From pre-vibration to collective rotation

Influences of some important residual interactions considered in the shell model on nuclear dynamics and the shape phase evolution have been investigated in the SD-pair shell model for an identical nucleon system. The results indicate that the collective modes including the spherical vibration, prolate rotation, and X(5)-like critical dynamics are mainly driven by the competition between the monopole-pairing interaction and the quadrupole-quadrupole interaction, and the main features of the U(5)-SU(3) phase transition can be well reproduced within the SD-pair model. In addition, it is also shown that the quadrupole-pairing interaction cannot be ignored in reproducing an ideal rotational spectrum with a realistic single-particle configuration.

Odd-even staggering of binding energy for nuclei in thesdshell

In this paper we study odd-even staggering phenomena of binding energy in the framework of the nuclear shell model for nuclei in the $sd$ shell. We decompose the USDB effective interaction into the monopole interaction and multipole (residual) interactions. We extract the empirical proton-neutron interaction, the Wigner energy, and the one-neutron separation energy using calculated binding energies. The monopole interaction, which represents the spherical mean field, provides contributions to the empirical proton-neutron interaction, the symmetry energy, and the Wigner energy. It does not induce odd-even staggering of the empirical proton-neutron interaction or the one-neutron separation energy. Isovector monopole and quadrupole pairing interactions and isoscalar spin-1 pairing interactions play a key role in reproducing an additional binding energy in both even-even and odd-odd nuclei. The Wigner energy coefficients are sensitive to residual two-body interactions. The nuclear shell structure has a strong influence on the evolution of the one-neutron separation energy, but not on empirical proton-neutron interactions. The so-called three-point formula is a good probe of the shell structure.

A new renormalization procedure of the quasiparticle random phase approximation

The ground state of a many body Hamiltonian considered in the quasiparticle representation is redefined by accounting for the quasiparticle quadrupole pairing interaction. The residual interaction of the newly defined quasiparticles is treated by the quasiparticle random phase approximation (QRPA). Solutions of the resulting equations exhibit specific features. In particular, there is no interaction strength where the first root is vanishing. A comparison with other renormalization methods is presented. Application to a single [Formula: see text]-shell allows for the results interpretation by comparing them with those obtained by exact calculations.

Quasi-particle structure of proton–hole cobalt isotopes

Projected Shell Model calculations have been employed for the description of the ground band in odd mass 57–67Co isotopes. In the present work, quadrupole and monopole pairing interactions as well as quadrupole–quadrupole interactions are effectively included in the Hamiltonian for obtaining various nuclear structure properties using the angular momentum projection technique. The yrast spectra of these isotopes are described as interplay between the angular momentum projected states around the Fermi level. The quasi-particle structure of these nuclei is found to be comprised of different intrinsic K-quantum numbers. Rotational alignments in terms of kinetic moment of inertia (ℑ(1)) have also been discussed. The electromagnetic transition probabilities [B(E2) and B(M1)] are also obtained in the present work and are found to be in good agreement with the available experimental as well as the other theoretical data, which tests the consistency of the applied projected shell model. The present PSM calculations also report the existence of low lying deformed structure along with the spherical structure at N=40.

Application of the generator coordinate method to neutron-rich Se and Ge isotopes

The quantum-number projected generator coordinate method (GCM) is ap- plied to the neutron-rich Se and Ge isotopes, where the monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an effective interaction. The en- ergy spectra obtained by the GCM are compared to both the shell model results and the experimental data. The GCM reproduces well the energy levels of high-spin states as well as the low-lying states. The structure of the low-lying collective states is analyzed through the GCM wave functions.

Band structure of neutron rich Se and Ge isotopes

The band structure of the neutron-rich Se and Ge isotopes has been studied in terms of the full-fledged shell model. The monopole and quadrupole pairing plus quadrupole-quadrupole interaction is employed as an effective interaction. The model reproduces well the energy levels of high-spin states as well as the low-lying states. In order to investigate the structure of the high-spin states and low-lying collective states, the energy spectra in the shell model are compared with those in the quantum-number-projected generator coordinate method. It is shown that the triaxial components play essential roles in describing the γ bands.

B(E2;01+→21+)values of even-even Sn isotopes

The ${2}_{1}^{+}$ energies and $B(E2;{0}_{1}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})$ values of even-even Sn nuclei with mass numbers between 102 and 130 have been calculated within the framework of the $SD$-pair approximation of the nuclear shell model. We employ monopole and quadrupole pairing plus quadrupole-quadrupole interaction with optimized parameters. Good agreement is obtained between the calculated results and experimental data. The robustness of our calculated ${2}_{1}^{+}$ level and $E2$ transition rates with respect to different sets of effective charge and different forms of the Hamiltonian is studied.

E2transitions in Sn isotopes within the quasiparticle-phonon model

The evolution of the structure of the first ${{2}_{1}}^{+}$ state along the chain of Sn isotopes is investigated within the quasiparticle-phonon model through the analysis of the energies and the $B(E2;{{0}_{1}}^{+}\ensuremath{\rightarrow}{{2}_{1}}^{+})$ strengths. The calculation reproduces the trend of the energies and, partly, the observed deviations of the $E2$ strengths from the parabolic behavior obtained in some large-scale shell-model calculations. Such an asymmetric trend is shown to be the outcome of several factors: single-particle energies, polarization of the $N=Z=50$ core, interplay between pairing plus quadrupole, and quadrupole pairing interactions.

Validity of pair approximations for nuclei in open shells

We study the validity of pair approximations of the nuclear shell model (SM) for a few realistic nuclei (${}^{130\ensuremath{-}131}$Te and ${}^{132}$I here) in open shells (i.e., nuclei with both valence protons and valence neutrons). We take a phenomenological shell model Hamiltonian, which includes the single-particle energy term, monopole and quadrupole pairing interactions for like particles, and quadrupole-quadrupole interactions between all valence nucleons. We make comparisons of energy levels and $B(E2)$ values calculated by pair approximations with those in the SM calculations. The overlaps between wave functions of the nucleon pair approximation (NPA) and those of the SM are presented explicitly for the low-lying states. Our calculated results demonstrate that the NPA is a remarkable approximation of the SM for these nuclei. The important role played by pairs with negative parity in some low-lying states is discussed.

Low-lying structure of neutron-rich Zn and Ga isotopes

Low-lying states of even-even Zn and odd-mass Ga nuclei with neutron numbers between 42 and 50 have been calculated within the framework of the $SDG$-pair approximation of the nuclear shell model. We employ a monopole and quadrupole pairing plus quadrupole-quadrupole interaction with optimized parameters, which are assumed to be constants for nuclei with the same proton number or neutron number. We calculate low-lying level schemes, electric quadrupole and magnetic dipole moments, and $E2$ and $M1$ transition rates. Reasonable agreement is achieved between the calculated results and experimental data. Dominant configurations in the ground states of odd-mass Ga nuclei are discussed in terms of pair correlations. The weak-coupling picture for some states of odd-mass Ga nuclei is studied.

Pair-truncated shell-model analysis of nuclei around mass 130

Low-lying states for even-even, odd-mass, and doubly odd nuclei in the mass $A~130$ region are systematically investigated using a pair-truncated shell model. In this model the collective nucleon pairs with angular momenta zero and two are the basic ingredients for even-even nuclei. Additional unpaired nucleons are added to the even-even core for a description of odd-mass and doubly odd nuclei. The effective interactions consist of single-particle energies and monopole and quadrupole pairing plus quadrupole-quadrupole interactions, whose strengths are assumed to be linearly changed as functions of the number of nucleons so as to describe the level schemes of the even-even and odd-mass nuclei. Energy levels of the low-lying collective states for even-even Xe, Ba, Ce, and Nd isotopes are reproduced very well along with intraband and interband $B(E2)$ values, which simulate the typical features of the O(6) limit of the interacting boson model. For odd-mass and doubly odd nuclei, complicated level schemes and electromagnetic moments are in excellent agreement with the experimental data.

Microscopic description of large-amplitude shape-mixing dynamics with inertial functions derived in local quasiparticle random-phase approximation

On the basis of the adiabatic self-consistent collective coordinate method, we develop an efficient microscopic method of deriving the five-dimensional quadrupole collective Hamiltonian and illustrate its usefulness by applying it to the oblate-prolate shape coexistence/mixing phenomena in proton-rich $^{68,70,72}\mathrm{Se}$. In this method, the vibrational and rotational collective masses (inertial functions) are determined by local normal modes built on constrained Hartree-Fock-Bogoliubov states. Numerical calculations are carried out using the pairing-plus-quadrupole Hamiltonian including the quadrupole-pairing interaction within the two major-shell active model spaces both for neutrons and protons. It is shown that the time-odd components of the moving mean-field significantly increase the vibrational and rotational collective masses in comparison with the Inglis-Belyaev cranking masses. Solving the collective Schr\"odinger equation, we evaluate excitation spectra, quadrupole transitions, and moments. The results of the numerical calculation are in excellent agreement with recent experimental data and indicate that the low-lying states of these nuclei are characterized as an intermediate situation between the oblate-prolate shape coexistence and the so-called $\ensuremath{\gamma}$ unstable situation where large-amplitude triaxial-shape fluctuations play a dominant role.

Signature splitting of the πh11/2 band and high-spin structures in 155Tb

Seven experimentally observed bands of 155Tb are analyzed in detail, using the particle-number-conserving method for treating the cranked shell model with monopole and quadrupole pairing interactions. We satisfactorily reproduce the experimental alignments and especially focus on the microscopic mechanism of the second back-bending and the influence of pair interaction on ultrahigh spins. Our calculated results show that the πi13/2 orbitals are too high to give a contribution to the moment of inertia below ħω ≈ 0.7 MeV. Instead, the crossing between the π[541]1/2 and other proton orbitals is responsible for the second back-bending. We assign a possible configuration to the decoupled band found in 155Tb and predict eleven bands which are experimentally unobserved.

Low-lying states of heavy nuclei within the nucleon pair approximation

In this article we perform systematic calculations on low-lying states of 33 nuclei with A=202-212, using the nucleon pair approximation of the shell model. We use a phenomenological shell-model Hamiltonian that includes single-particle energies, monopole and quadrupole pairing interactions, and quadrupole-quadrupole interactions. The building blocks of our model space include one J=4 valence neutron pair, and one J=4,6,8 valence proton pair, in addition to the usual S and D pairs. We calculate binding energies, excitation energies, electric quadrupole and magnetic dipole moments of low-lying states, and E2 transition rates between low-lying states. Our calculated results are reasonably consistent with available experimental data. The calculated quadrupole moments and magnetic moments, many of which have not yet been measured for these nuclei, are useful for future experimental measurements.

Microscopic Derivation of Collective Hamiltonian by Means of the Adiabatic Self-Consistent Collective Coordinate Method: Shape Mixing in Low-Lying States of 68Se and 72Kr

The microscopic dynamics of oblate-prolate shape coexistence/mixing phenomena in 68 Se and 72 Kr are studied by means of the adiabatic self-consistent collective coordinate (ASCC) method in conjunction with the pairing-plus-quadrupole (P+Q) Hamiltonian, including the quadrupole pairing interaction. A quantum collective Hamiltonian is constructed, and excitation spectra, spectroscopic quadrupole moments and quadrupole transition properties are evaluated. The effect of the time-odd pair field on the collective mass (inertia function) of the large-amplitude vibration and the rotational moments of inertia about three principal axes is evaluated. It is found that the basic properties of the shape coexistence/mixing are qualitatively reproduced. The results of the calculation indicate that the oblate-prolate shape mixing decreases as the angular momentum increases.

Systematic calculations of low-lying states in odd-Anuclei within the nucleon pair approximation

Systematic calculations on low-lying states are carried out for eighty odd-$A$ nuclei with proton number ranging from 50 to 58 and neutron number from 74 to 90 by using the nucleon pair approximation. We use a separable phenomenological Hamiltonian that includes the monopole pairing, quadrupole pairing plus quadrupole-quadrupole interactions, with strength parameters exactly as those of the neighboring even-even nuclei. Our calculated energy spectra, electrical quadrupole moments, and magnetic $g$ factors agree well with experimental data. Electromagnetic properties for low-lying states of these nuclei are discussed in detail.

Neutron [formula omitted] alignments in the odd–odd nucleus 154Tb

The experimentally observed four bands of 154Tb are analyzed in detail using the particle number conserving method for treating cranked shell model with monopole and quadrupole pairing interactions. Besides analyzing their behaviors at higher frequencies beyond the observed frequency range, we further predict the sixteen bands based on the low excited states in 154Tb. Our calculated results show that below the rotational frequency ℏ ω ≈ 0.45 MeV , the ν i 13 / 2 and their interferences are responsible for all band crossings and above the frequency ℏ ω ≈ 0.45 MeV , besides the ν i 13 / 2 orbitals, the π [ 660 ] 1 / 2 and ν [ 505 ] 11 / 2 orbitals begin to influence rotational alignments.