Proton-neutron Interacting Boson Model Research Articles

Multiple Multi-Orbit Pairing Algebras in Nuclei

The algebraic group theory approach to pairing in nuclei is an old subject and yet it continues to be important in nuclear structure, giving new results. It is well known that for identical nucleons in the shell model approach with j − j coupling, pairing algebra is SU(2) with a complementary number-conserving Sp(N) algebra and for nucleons with good isospin, it is SO(5) with a complementary number-conserving Sp(2Ω) algebra. Similarly, with L − S coupling and isospin, the pairing algebra is SO(8). On the other hand, in the interacting boson models of nuclei, with identical bosons (IBM-1) the pairing algebra is SU(1, 1) with a complementary number-conserving SO(N) algebra and for the proton–neutron interacting boson model (IBM-2) with good F-spin, it is SO(3, 2) with a complementary number-conserving SO(ΩB) algebra. Furthermore, in IBM-3 and IBM-4 models several pairing algebras are possible. With more than one j or ℓ orbit in shell model, i.e., in the multi-orbit situation, the pairing algebras are not unique and we have the new paradigm of multiple pairing [SU(2), SO(5) and SO(8)] algebras in shell models and similarly there are multiple pairing algebras [SU(1, 1), SO(3, 2) etc.] in interacting boson models. A review of the results for multiple multi-orbit pairing algebras in shell models and interacting boson models is presented in this article with details given for multiple SU(2), SO(5), SU(1, 1) and SO(3, 2) pairing algebras. Some applications of these multiple pairing algebras are discussed. Finally, multiple SO(8) pairing algebras in shell model and pairing algebras in IBM-3 model are briefly discussed.

Quadrupole and octupole states in 152Sm using the proton–neutron interacting boson model

A scheme of solving the proton–neutron interacting boson model (IBM-2) in terms of the SU(3) basis is introduced, by which the IBM-2 coupled with an octupole boson is applied to describe the low-energy structure of the critical point nucleus, 152Sm. The results indicate that the spectral properties of both the positive-parity bands and negative-parity bands in this nucleus can be well captured by the IBM-2 calculations through a simple Hamiltonian, thus providing an example of the IBM-2 in a unified description of quadrupole and octupole states in a transitional system. In addition, a statistical analysis of the low-spin states in the model is also provided.

Algebraic solutions for [formula omitted] quantum phase transitions in the proton-neutron interacting boson model

A simple systematic procedure to construct the proton-neutron unitary, usdπν(12), orthogonal, osdπν(12), and quasi-spin susdπν(1,1) algebras of the sd bosonic system is presented. New algebraic substructures of these algebras are discussed and the explicit formulae for their generators and Casimir operators are given in the spherical tensor form. The complementarity relationship of the Casimir operators of the susdπν(1,1) and osdπν(12) is derived. The exact algebraic solutions of the quantum phase transition Hamiltonian between the osdπν(12) and usπν(2)⊗udπν(10) limits have been considered, for the first time, in the framework of affine susdπν(1,1) Lie algebra. The low lying energy spectra of the 70Ge, 76−78Se, 96−98Mo, and 100−102Ru isotopes are calculated using the osdπν(12)↔usπν(2)⊗udπν(10) transition Hamiltonian. The good agreement of our computation with empirical result in these isotopes emphasizes the importance of usπν(2)⊗udπν(10) limit. With this addition, symmetry can be extended to many nuclei.

IBM-2 configuration mixing in Mo nuclei

The even-even Mo isotopes are studied using an extension of the neutron-proton interacting boson model that accounts for the breaking of shell or sub-shell closures via two-particle-two-hole excitations. The energy levels and electromagnetic transitions are explored. The results back up the idea that these isotopes are interacting between two configurations.

Shape coexistence in 76Se within the neutron–proton interacting boson model

We have investigated the low-lying energy spectrum and electromagnetic transition strengths in even–even 76Se using the proton–neutron interacting boson model (IBM-2). The theoretical calculation for the energy levels and E2 and M1 transition strengths is in good agreement with the experimental data. Specifically, the excitation energy and E2 transition of state, which is intimately associated with shape coexistence, can be accurately reproduced. The analysis on low-lying states and the key structure indicators R 1, R 2, R 3 and R 4 and M1 transitions indicates that there is a coexistence between spherical shape and γ-soft shape in 76Se.

Octupole and quadrupole modes in radon isotopes using the proton-neutron interacting boson model

We have performed a systematic study of the spectroscopic properties of the isotopic chain $^{214\ensuremath{-}226}\mathrm{Rn}$ using the $spdf$-IBM-2 interacting boson model. This model includes octupole degrees of freedom, what allows the description of negative- and positive-parity states with the same Hamiltonian. We discuss how this model is able to describe the transition from vibrational to rotational spectra in these nuclei with a rather small set of parameters, which were obtained to reproduce the properties of $^{214\ensuremath{-}222}\mathrm{Rn}$. Then the calculations regarding $^{224,226}\mathrm{Rn}$ can be considered a prediction, which compares very well with recent experimental data.

E2 decay characteristics of the M1 scissors mode of Sm152

The $E2/M1$ multipole mixing ratio of the ${1}_{\text{sc}}^{+}\ensuremath{\rightarrow}{2}_{1}^{+}\phantom{\rule{4pt}{0ex}}\ensuremath{\gamma}$-ray transition and therefore the $F$-vector $E2$ decay of the scissors mode of the shape-phase transitional nucleus $^{152}\mathrm{Sm}$ have been measured with the nuclear resonance fluorescence method at the High-Intensity $\ensuremath{\gamma}$-ray Source ($\mathrm{HI}\ensuremath{\gamma}\mathrm{S}$). Furthermore, parity quantum numbers of several dipole-excited states have been remeasured, making use of the polarized $\ensuremath{\gamma}$-ray beam at $\mathrm{HI}\ensuremath{\gamma}\mathrm{S}$, and partially reassigned. The new data enable an unambiguous determination of the proton and neutron effective boson quadrupole charges within the proton-neutron interacting boson model. With the well-constrained parameter set, a new mixed-symmetry state is predicted.

A unified description of shape coexistence and mixed-symmetry states in 98Mo using IBM-2**Supported by the National Natural Science Foundation of China under Grant Nos. 11947410, 11147148 and U1832139 and the Natural Science Foundation of Zhejiang Province under grant No. LY19A050002.

Level structure and electromagnetic transitions in 98Mo have been investigated on the basis of the proton-neutron interacting boson model (IBM-2) by considering the energy difference between neutron boson εν and proton boson επ. The results are compared with the recent experimental data and it is observed that they are in good agreement. In particular, the strongest M1 transition from state to can be well reproduced, from which one can determine the as an mixed-symmetry (MS) state. We have calculated the electric monopole strength , and our result agrees with the experimental one. The calculation indicates that shape coexistence and MS states are simultaneously well described using IBM-2.

Electromagnetic Properties in Samarium Isotopes Within the Framework of the IBM-2

Electromagnetic transitions between low-lying states in deformed even-even 148–154Sm nuclei were investigated within the framework of the SU(3) limit of the neutron-proton interacting boson model. To explain the M1 transitions between low-lying states and the E2 transition between interband states, we applied first-order perturbation theory by adding the proton-neutron quadrupole interaction Qπ· Qv that breaks F-spin symmetry in the dynamical symmetric SU(3) Hamiltonian as a perturbed term. We calculated the transition probabilities of the M1 and the E2 transition operators between low-lying interband states by using the 1/N expansion methods. The matrix elements of the proton-neutron quadrupole interaction Qπ· Qv were directly calculated, as shown in a previous study, by applying the F-spin operator Qπ· Qa instead of Qπ· Qv. The results of the present study for electromagnetic transitions were applied to some deformed even-even Samarium nuclei, and these theoretical results could effectively describe the experimental data.

Description of the critical point symmetry in 124Te by IBM-2 * * Supported by the National Natural Science Foundation of China (11475062, 11147148, 11747312)

Based on the neutron and proton degrees of freedom, low-lying energy levels, , , and transition strengths of nucleus 124Te have been calculated by the neutron-proton interacting boson model. The calculated results are reasonably consistent with the experimental data. By comparing the key observables of the states at the critical point of - transition with the experimental data and calculated results, we show that the 124Te is a possible nucleus at the critical point of the second-order phase transition from vibration to unstable rotation, and such a critical point exhibits slight triaxial rotation. The 0 state of 124Te can be interpreted as the lowest state of the first-excited family of the intrinsic levels in the critical point symmetry.

Description of Shape Coexistence and Mixed-Symmetry States in 96Mo Using IBM-2**Supported by the National Natural Science Foundation of China under Grant Nos 11475062, 11575059, 11747312 and 11147148.

We investigate the level structure and E2, M1 electromagnetic transition properties in an even–even 96Mo nucleus within the framework of the proton–neutron interacting boson model (IBM-2). The calculated results of the IBM-2 can reproduce the recent new experimental data on 96Mo both qualitatively and quantitatively. It is found that both shape coexistence and mixed-symmetry states in 96Mo can be simultaneously described very well with the IBM-2 by taking into account that the relative energy of d neutron bosons is different from that of proton bosons.

Nuclear structure of 76Ge from proton-neutron interacting boson model calculations

The properties of low-lying states in $^{76}$Ge, especially the characteristics of themixed-symmetry states, have been investigated within the neutron-proton interactingboson model (IBM-2). By considering the relative energy of $d$ proton boson to bedifferent from that of neutron boson, the low-lying positive parity levels and $M1$, $E2$transition strengths have been calculated. The IBM-2 calculated results are in goodagreement with the experimental data. Particularly, the mixed-symmetry states have beenreproduced quite well. The calculation and systematic analysis demonstrated that thecollective character of $^{76}$Ge lies closest to the $\textrm{SU}^{\ast}_{\pi\nu}(3)$, with some possible $\textrm{O}_{\pi\nu}(6)$ dynamic symmetry in IBM-2 viewpoint.

Description of spectrum and electromagnetic transitions in 94Mo through the proton-neutron interacting boson model

We investigated the properties of low-lying states in 94Mo within the framework of the proton-neutron interacting boson model (IBM-2), with special focus on the characteristics of mixed-symmetry states. We calculated level energies and M1 and E2 transition strengths. The IBM-2 results agree with the available quantitative and qualitative experimental data on 94Mo. The properties of mixed-symmetry states can be well described by IBM-2 given that the energy of the d proton boson is different from that of the neutron boson, especially for the transition of B(M1; 4 2 + → 4 1 + ).

Description of mixed symmetry states in 96Ru using IBM-2

We have investigated the properties of low-lying states in 96Ru within the framework of the neutron-proton interacting boson model (IBM-2), with special attention paid to the characteristics of the mixed symmetry states. By considering the relative energy of d proton boson to be different from that of neutron boson, the level energies and M1, E2 transition strengths have been calculated. The IBM-2 calculation is consistent with the experimental data of 96Ru both quantitatively and qualitatively. Particularly, the strong M1 transition between the 42+ and 41+ states has been reproduced nicely. The calculated results show that the M1 transition strength of B(M1; 42+ → 41+) in 96Ru can be described successfully by the IBM-2.

Relativistic Coulomb excitation of Kr88

To investigate the systematics of mixed-symmetry states in N=52 isotones, a relativistic Coulomb excitation experiment was performed during the PreSPEC campaign at the GSI Helmholtzzentrum für Schwerionenforschung to determine E2 transition strengths to 2+ states of the radioactive nucleus Kr88. Absolute transition rates could be measured towards the first and third 2+ states. For the latter a mixed-symmetry character is suggested on the basis of the indication for a strong M1 transition to the fully symmetric 21+ state, extending the knowledge of the N=52 isotones below Z=40. A comparison with the proton-neutron interacting boson model and shell-model predictions is made and supports the assignment.Received 10 February 2015Revised 15 February 2016DOI:https://doi.org/10.1103/PhysRevC.94.054323©2016 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsElectromagnetic transitionsLow & intermediate energy heavy-ion reactionsProperties59 ≤ A ≤ 89TechniquesModels based on symmetriesRadioactive beamsNuclear Physics

Coexistence of Prolate and Oblate Shapes in 98Sr Nuclei Using IBM2**Supported by the National Natural Science Foundation of China under Grant Nos 11475062, 11547312 and 11147148.

The properties of the low-lying states and the shape coexistence in 98Sr are investigated within the framework of the proton—neutron interacting boson model (IBM2). By considering the relative energy of the d proton boson to be different from that of the neutron boson, it is found that the calculated energy levels and the B(E2) transition strengths agree with the experimental data perfectly. Particularly, the second 0+ state, which is associated with the shape coexistence phenomenon and has the lowest energy E(02+) among all known even—even nuclei, is reproduced very well. The behavior of the calculated quadrupole shape invariants is consistent with the experimental results.

Description of the shape coexistence in neutron-deficient 74,76Kr with IBM2

The shape deformation and shape coexistence in $^{74,76}$Kr isotopes are investigated within the framework of the proton-neutron interacting boson model (IBM2). By considering the relative energy of the $d$ proton boson to be different from that of the neutron boson, the low-lying energy spectrum is in good agreement with experimental results both qualitatively and quantitatively. In particular, the low-lying 0$^+_2$ states associated with the shape-coexistence phenomenon are reproduced quite well. The calculated key sensitive quantities of$B(E2)$ transition branch ratios are fairly consistent with the experimental data except for $R_4$. The predicated deformation parameter is very similar for the ground states in $^{74}$Kr and $^{76}$Kr, showing good agreement with the experimental result, and the calculated deformation parameter for the second 0$^+$ state in $^{74}$Kr is close to the experimental data. The calculated results of the triaxiality parameter indicated an almost purely prolate shape for the ground state of $^{76}$Kr and a mostly prolate shape with a little triaxiality for the ground state of$^{74}$Kr. The calculations also show an oblate triaxial shape for the second 0$^+$ state in $^{76}$Kr and maximum triaxiality for the second 0$^+$ state in $^{74}$Kr. These results confirm the importance of the triaxial deformation for the description of such shape coexistence.

Phase diagram of the two-fluid Lipkin model: A “butterfly” catastrophe

Background: In the last few decades quantum phase transitions have been of great interest in Nuclear Physics. In this context, two-fluid algebraic models are ideal systems to study how the concept of quantum phase transition evolves when moving into more complex systems, but the number of publications along this line has been scarce up to now. Purpose: We intend to determine the phase diagram of a two-fluid Lipkin model, that resembles the nuclear proton-neutron interacting boson model Hamiltonian, using both numerical results and analytic tools, i.e., catastrophe theory, and to compare the mean-field results with exact diagonalizations for large systems. Method: The mean-field energy surface of a consistent-Q-like two-fluid Lipkin Hamiltonian is studied and compared with exact results coming from a direct diagonalization. The mean-field results are analyzed using the framework of catastrophe theory. Results: The phase diagram of the model is obtained and the order of the different phase-transition lines and surfaces is determined using a catastrophe theory analysis. Conclusions: There are two first order surfaces in the phase diagram, one separating the spherical and the deformed shapes, while the other separates two different deformed phases. A second order line, where the later surfaces merge, is found. This line finishes in a transition point with a divergence in the second order derivative of the energy that corresponds to a tricritical point in the language of the Ginzburg-Landau theory for phase transitions.

Study of nuclear structure of 76-86Sr isotopes in the pn interacting boson model

The proton neutron interacting boson model (IBM-2) has been used to make a systematic study of Strontium isotopes in this mass region of A ∼ 80 with 38 N 48 and Z = 38. The three-term Talmi–Otsuka general Hamiltonian in the framework of the neutron proton version of the Interaction boson model was used to perform the calculations. The yrast levels energy are reproduced. The beta and gamma band energy levels also matched well. The reduced transition probabilities were also calculated and were found to be in agreement with the experimental values. In addition, g-factor for the state was evaluated. Possible candidates for mixed symmetry states were also predicted for several nuclei in this isotopic chain.

Description of the Shape Coexistence in 98Mo with IBM2**Supported by the National Natural Science Foundation of China under Grant No 11475062, and the Natural Science Foundation of Zhejiang Province under Grant No Y6100135.

We investigate the properties of the low-lying states and the relevant shape dynamics of 98Mo within the framework of the proton-neutron interacting boson model (IBM2). By considering the relative energy of the d proton boson to be different from that of the neutron bosons, the low-lying levels and the key observable B(E2) transition branching ratios are calculated. The characteristic feature of the energy spectrum and the most crucial available structure indicator indicate that the substantial mixing between the spherical-vibrational and γ-unstable shapes in 98Mo. The calculation results of the overall deformation in 98Mo are almost the same for both the ground and the first excited 0+ states, showing a weak deformation. While the triaxiality parameter indicates that the mostly triaxial shape with some oblate for the ground state, and the triaxial shape with some prolate for the excited 0+2 state, being equilibrium shapes of spherical-vibrational and γ-unstable in 98Mo.