Triaxial Rotor Research Articles

Possible chiral doublets in 60Ni

The open problem on whether or not the chirality exists in doublet bands M1 and M4 in light-mass even-even nucleus 60Ni is studied by adopting the recently developed fully quantal four-j shells triaxial particle rotor model. The corresponding experimental energy spectra, energy differences between doublet bands, and the available B(M1)/B(E2) values are successfully reproduced. The analyses on the basis of the angular momentum components, the azimuthal plots, and the K-plots suggest that the chiral modes exist at I≥12ħ in doublet bands M1 and M4.

Multiple chiral doublet bands with octupole correlations in reflection-asymmetric triaxial particle rotor model

A reflection-asymmetric triaxial particle rotor model (RAT-PRM) with a quasi-proton and a quasi-neutron coupled with a reflection-asymmetric triaxial rotor is developed and applied to investigate the multiple chiral doublet (MχD) bands candidates with octupole correlations in 78Br. The calculated excited energies, energy staggering parameters, and B(M1)/B(E2) ratios are in a reasonable agreement with the data of the chiral doublet bands with positive- and negative-parity. It is found that both the triaxial deformation γ and octupole deformation β3 influence the calculated B(E1) values. The chiral geometry based on the angular momenta for the rotor, the valence proton and valence neutron is discussed in details.

Two-phonon wobbling in 135Pr

The second-phonon (nω=2) wobbling band has been established in the nucleus 135Pr. Conclusive evidence for its wobbling nature comes from the ΔI=1, E2 character of the transitions between the new band and the previously identified transverse wobbler band (nω=1) in this nucleus. Theoretical calculations in the framework of the quasiparticle triaxial rotor and triaxial projected shell models are found to be in good agreement with the experimental results.

Two-dimensional collective Hamiltonian for chiral and wobbling modes. II. Electromagnetic transitions

The intraband electromagnetic transitions in the framework of collective Hamiltonian for chiral and wobbling modes are calculated. By going beyond the mean field approximation on the orientations of rotational axis, the collective Hamiltonian provides the descriptions on both yrast band and collective excitation bands. For a system with one $h_{11/2}$ proton particle and one $h_{11/2}$ neutron hole coupled to a triaxial rotor ($\gamma=-30^\circ$), the intraband electromagnetic transitions given by the one-dimensional and two-dimensional collective Hamiltonian are compared to the results by the tilted axis cranking approach and particle rotor model. Compared with the tilted axis cranking approach, the electromagnetic transitions given by the collective Hamiltonian have a better agreement with those by the particle rotor model, due to the consideration of the quantum fluctuations.

Alternating Parity Band in Octupole-Soft 140Xe with Axial Vibrational-Rotational Model and Triaxial Rigid Rotor Model**Supported by the National Natural Science Foundation of China under Grant Nos 11675094 and 11622540, and the Young Scholars Program of Shandong University under Grant No 2015WHWLJH01.

Energies of the yrast positive- and negative-parity excited states in 140Xe are reproduced by two different models considering quadrupole-octupole deformations, namely the axial vibrational-rotational model and the triaxial rigid rotor model, and compared with the stable octupole-deformed 222Th. The origin of the energy difference between the opposite parity sequences is considered from two different mechanisms, the vibration in axial deformed energy minima and the rotation considering the effective triaxial deformation. The success of reproducing the data in both the models implies that these two mechanisms are equivalent on some level for the octupole-soft nuclei. By investigating the probability distributions for projection of total angular momentum in the triaxial rigid rotor model, it is found that such an energy difference is associated with the difference of orientation of the rotational axis.

Multiple chiral doublets in four-j shells particle rotor model: Five possible chiral doublets in [formula omitted]Nd76

A particle rotor model, which couples nucleons in four single-j shells to a triaxial rotor core, is developed to investigate the five pairs of nearly degenerate doublet bands recently reported in the even-even nucleus 136Nd. The experimental energy spectra and available B(M1)/B(E2) values are successfully reproduced. The angular momentum geometries of the valence nucleons and the core support the chiral rotation interpretations not only for the previously reported chiral doublet, but also for the other four candidates. Hence, 136Nd is the first even-even candidate nucleus in which the multiple chiral doublets exist. Five pairs of chiral doublet bands in a single nucleus is also a new record in the study of nuclear chirality.

Empirical model with independent variable moments of inertia for triaxial nuclei applied to Ge76 and Os192

An empirical model with independent variable moments of inertia for triaxial nuclei is devised and applied to $^{76}\mathrm{Ge}$ and $^{192}\mathrm{Os}$. Three intrinsic moments of inertia, ${J}_{1}$, ${J}_{2}$, and ${J}_{3}$, are varied independently as a particular function of spin $I$ within a revised version of the triaxial rotor model so as to reproduce the energy levels of the ground-state, $\ensuremath{\gamma}$, and (in the case of $^{192}\mathrm{Os}$) ${K}^{\ensuremath{\pi}}={4}^{+}$ bands. The staggering in the $\ensuremath{\gamma}$ band is well reproduced in both phase and amplitude. Effective $\ensuremath{\gamma}$ values are extracted as a function of spin $I$ from the ratios of the three moments of inertia. The eigenfunctions and the effective $\ensuremath{\gamma}$ values are subsequently used to calculate the ratios of $B(E2)$ values associated with these bands. Good agreement between the model calculation and the experimental data is obtained for both $^{76}\mathrm{Ge}$ and $^{192}\mathrm{Os}$.

The magnification of structural anomalies with Grodzins systematic in the framework of Asymmetric Rotor Model

The dependence of Grodzins systematic as shape fluctuation energy product ESF⁎B(E2)↑ and rotational energy product EROT⁎B(E2)↑ on the Asymmetry parameter γ0 is carried out in the Z = 50–82, N = 82–126 major shell space. The Asymmetry parameter γ0, varying from 0° to 60°, reflects the change in nuclear structure from prolate to oblate. Strong anomalies are highlighted in the shape transitional isotopes. The product ESF⁎B(E2)↑ evolves from low negative values for vibrator nuclei, passing close to zero and then substantially increasing towards triaxial rotor limit with γ0∼30°. However, the product EROT⁎B(E2)↑ decreases as a function of γ0 for all the nuclei approaching towards triaxiality from Z = 50–82, N = 82–126. Anomalies are also noticed for the N>104 region where the product EROT⁎B(E2)↑ decreases in zigzag phase for 188–196Pt isotopes corresponding to γ0∼25–30° and this reflects the breakdown of coherence between rotational energy EROT and excitation strength B(E2)↑. The product EROT⁎B(E2)↑ indicates the shape phase transition for Pt isotopic chain from spherical to γ−soft to slightly triaxial. We have studied for the first time the role of Grodzins systematic ESF and EROT in the framework of Asymmetric Rotor Model.

Spectroscopic study of the possibly triaxial transitional nucleus 75Ge

The collective structures of Ge75 have been studied for the first time via the Ge74(α,2p1n)Ge75 fusion-evaporation reaction. Two negative-parity bands and one tentative positive-parity band built on the νp1/2,νf5/2, and νg9/2 states, respectively, are established and compared with the structures in the neighboring N=43 isotones. According to the configuration-constrained potential-energy surface calculations, a shape transition from oblate to prolate along the isotopic chain in odd-A Ge isotopes is suggested to occur at Ge75. The properties of the bands in Ge75 are analyzed in comparison with the triaxial particle rotor model calculations.Received 23 February 2017Revised 20 November 2017DOI:https://doi.org/10.1103/PhysRevC.97.034322©2018 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNuclear reactionsNuclear structure & decaysNucleon distributionProperties59 ≤ A ≤ 89Nuclear Physics

Coherent states for the quantum complete rigid rotor

Abstract Motivated by the possibility to describe orientations of quantum triaxial rigid rotors, such as molecules, with respect to both internal (body-fixed) and external (laboratory) frames, we go through the theory of coherent states and design the appropriate family of coherent states on T ∗ SO ( 3 ) , the classical phase space of the freely rotating rigid body (the Euler top). We pay particular attention to the resolution of identity property in order to establish the explicit relation between the parameters of the coherent states and classical phase-space variables, actions and angles.

Bohr Hamiltonian for $\gamma = 30^{\circ}$ γ = 30 ∘ with Davidson potential

A $\gamma$ -rigid solution of the Bohr Hamiltonian for $\gamma = 30^{\circ}$ is constructed with the Davidson potential in the $\beta$ part. This solution is going to be called $Z(4)$ -D. The energy eigenvalues and wave functions are obtained by using the analytic method developed by Nikiforov and Uvarov. The calculated intraband and interband $B(E2)$ transitions rates are presented and compared with the $Z(4)$ model predictions. The staggering behavior in $\gamma$ -bands is considered to search Z(4) -D candidate nuclei. A variational procedure is applied to demonstrate that the $Z(4)$ model is a solution of the critical point at the shape phase transition from spherical to rigid triaxial rotor.

Tilted-axis wobbling in odd-mass nuclei

A triaxial rotor Hamiltonian with a rigidly aligned high-$j$ quasiparticle is treated by a time-dependent variational principle, using angular momentum coherent states. The resulting classical energy function has three unique critical points in a space of generalized conjugate coordinates, which can minimize the energy for specific ordering of the inertial parameters and a fixed angular momentum state. Because of the symmetry of the problem, there are only two unique solutions, corresponding to wobbling motion around a principal axis and, respectively, a tilted axis. The wobbling frequencies are obtained after a quantization procedure and then used to calculate $E2$ and $M1$ transition probabilities. The analytical results are employed in the study of the wobbling excitations of $^{135}\mathrm{Pr}$ nucleus, which is found to undergo a transition from low angular momentum transverse wobbling around a principal axis toward a tilted-axis wobbling at higher angular momentum.

Effective field theory for triaxially deformed nuclei

Effective field theory (EFT) is generalized to investigate the rotational motion of triaxially deformed even-even nuclei. A Hamiltonian, called the triaxial rotor model (TRM), is obtained up to next-to-leading order (NLO) within the EFT formalism. Its applicability is examined by comparing with a five-dimensional collective Hamiltonian (5DCH) for the description of the energy spectra of the ground state and $\gamma$ band in Ru isotopes. It is found that by taking into account the NLO corrections, the ground state band in the whole spin region and the $\gamma$ band in the low spin region are well described. The results presented here indicate that it should be possible to further generalize the EFT to triaxial nuclei with odd mass number.

Lifetimes and electromagnetic transition strengths in Dy157

Lifetimes of excited states of $^{157}\mathrm{Dy}$ were measured by recoil distance Doppler-shift (RDDS) and Doppler-shift attenuation (DSAM) methods. The $\ensuremath{\gamma}$-ray coincidence data were analyzed by the differential decay curve method. The level scheme and the newly deduced transition strengths were simultaneously described in the framework of the particle plus triaxial rotor model (PTRM) and the rigid-rotor model.

Global Correlations for Low-Lying Collective 2+ States

By using the triaxial rotor model and the anharmonic vibrator model with phonon mixing, we derive a global correlation between the quadrupole moments of the two lowest 2+ states in collective nuclei that had previously been observed in experimental data across the periodic table. We then derive other electromagnetic properties for these two models of nuclear structure and compare them globally with experimental data. We find that both models are able to robustly describe the experimental data across the region of nuclei for which the models are applicable, including a large number that they have in common. We then show that there seems to exists a robust orthogonal transformation between these two models for realistic nuclear systems, suggesting that these two seemingly diverse descriptions of quadrupole collective phenomena seem to act in a similar model space and may therefore have a common origin.

Nonadiabatic quasiparticle approach for rotation-particle coupling in triaxial odd-Anuclei

We discuss the formulation of a nonadiabatic approach to study the rotational states in triaxially deformed odd-$A$ nuclei. The rotation-particle coupling is treated microscopically by coupling the triaxial rotor states of the even-even core with the states of the valence particle in order to obtain the matrix elements of the odd-$A$ system. We arrive at a nonadiabatic quasiparticle approach where the rotational states can have contributions from various quasiparticle states near the Fermi level. We bring out the advantages of this approach over the conventional particle rotor model with a fixed or variable moment of inertia. One clear evidence favoring our approach is the rotation alignment phenomenon which is demonstrated in the case of $^{137}\mathrm{Pm}$. We discuss our results for $^{136}\mathrm{Nd}$ and $^{137}\mathrm{Pm}$, and justify that this approach is suitable also for studying nuclei away from stability.

Triaxiality near the 110Ru ground state from Coulomb excitation

A multi-step Coulomb excitation measurement with the GRETINA and CHICO2 detector arrays was carried out with a 430-MeV beam of the neutron-rich 110Ru (t1/2=12s) isotope produced at the CARIBU facility. This represents the first successful measurement following the post-acceleration of an unstable isotope of a refractory element. The reduced transition probabilities obtained for levels near the ground state provide strong evidence for a triaxial shape; a conclusion confirmed by comparisons with the results of beyond-mean-field and triaxial rotor model calculations.

Electromagnetic transitions in multiple chiral doublet bands**Supported by National Natural Science Foundation of China (11675094, 11622540, 11545011, 11405096, 11461141001, U1432119), Shandong Natural Science Foundation (ZR2014AQ012), and Young Scholars Program of Shandong University, Weihai (2015WHWLJH01)

Multiple chiral doublet bands (MχD) in the 80, 130 and 190 mass regions are studied by the model of γ = 90° triaxial rotor coupled with identical symmetric proton-neutron configurations. By selecting a suitable basis, the calculated wave functions are explicitly exhibited to be symmetric under the operator Â, which is defined as rotation by 90° about the 3-axis with the exchange of valance proton and neutron. We found that both M1 and E2 transitions are allowed between levels with different values of A, while they are forbidden between levels with same values of A. Such a selection rule holds true for MχD in different mass regions.

Two-dimensional collective Hamiltonian for chiral and wobbling modes

A two-dimensional collective Hamiltonian (2DCH) on both azimuth and polar motions in triaxial nuclei is proposed to investigate the chiral and wobbling modes. In the 2DCH, the collective potential and the mass parameters are determined from three-dimensional tilted axis cranking (TAC) calculations. The broken chiral and signature symmetries in the TAC solutions are restored by the 2DCH. The validity of the 2DCH is illustrated with a triaxial rotor ($\gamma=-30^\circ$) coupling to one $h_{11/2}$ proton particle and one $h_{11/2}$ neutron hole. By diagonalizing the 2DCH, the angular momenta and energy spectra are obtained. These results agree with the exact solutions of the particle rotor model (PRM) at high rotational frequencies. However, at low frequencies, the energies given by the 2DCH are larger than those by the PRM due to the underestimation of the mass parameters. In addition, with increasing angular momentum, the transitions from the chiral vibration to chiral rotation and further to longitudinal wobbling motion have been presented in the 2DCH.

Global correlations between electromagnetic and spectroscopic properties of collective21+and22+states

By using the general triaxial rotor model (TRM) and the phonon-configuration mixing scheme within an anharmonic-vibrator(AHV) framework, a series of global correlations between electromagnetic properties of nuclear $2^+_1$ and $2^+_2$ states are analytically established. The correlations from both models can roughly describe the experimental data involving quadrupole collectivity with few exceptions. Furthermore, there seems to be a robust orthogonal transformation between the AHV and TRM bases for realistic nuclear systems, suggesting that the two models may in fact be describing the collective features of nuclear low-lying states in similar model spaces.