A particle rotor model with a quasi-neutron coupled with a triaxially deformed rotor is applied to study signature splitting for bands with intruder orbital $\nu7/2^{+} [633]$ and non-intruder orbital $\nu5/2^{-}[512]$ in $^{173}$W. Excellent agreement with the observed energy spectra has been achieved for both bands. Signature splitting for band $\nu7/2^{+} [633]$, and band $\nu5/2^{-}[512]$ before the onset of signature inversion, is satisfactorily reproduced by introducing the $γ$ degree of freedom. The phase and amplitude of signature splitting in band $\nu5/2^{-}[512]$ is attributed to strong competition between $2f_{7/2}$ and $1h_{9/2}$ components. However, the explanation of signature inversion in band $\nu5/2^{-}[512]$ self-consistently is beyond the present one quasi-neutron coupled with a triaxially deformed rotor.
The collective Bohr hamiltonian is solved for the soft triaxial rotor around $γ_0=π/6$ with a displaced harmonic oscillator potential in $γ$ and a Kratzer-like potential in $β$. The properties of the spectrum are outlined and a generalization for the more general triaxial case with $0<γ<π/6$ is proposed.
The spectroscopy properties and angular momentum geometry for the wobbling motion of a simple triaxial rotor are investigated within the triaxial rotor model up to spin $I=40\hbar$. The obtained exact solutions of energy spectra and reduced quadrupole transition probabilities are compared to the approximate analytic solutions by harmonic approximation formula and Holstein-Primakoff formula. It is found that the low lying wobbling bands can be well described by the analytic formulas. The evolution of the angular momentum geometry as well as the $K$-distribution with respect to the rotation and …
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 (M$χ$D) bands candidates with octupole correlations in $^{78}$Br. 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. The influence of the triaxial deformation $γ$ on the calculated $B(E1)$ is found to be significant. By changing $γ…
In this work we examine the possibility to describe yrast positive- and negative-parity excitations of deformed even-even nuclei through a collective rotation model in which the nuclear surface is characterized by triaxial quadrupole and octupole deformations. The nuclear moments of inertia are expressed as sums of quadrupole and octupole parts. By assuming an adiabatic separation of rotation and vibration degrees of freedom we suppose that the structure of the positive- and negative- parity bands may be determined by the triaxial-rigid-rotor motion of the nucleus. By diagonalizing the Hamilto…
The positive parity doublet bands based on the $πh_{11/2}\otimesνh_{11/2}$ configuration in $^{126}$Cs have been investigated in the two quasi-particles coupled with a triaxial rotor model. The energy spectra $E(I)$, energy staggering parameter $S(I)=[E(I)-E(I-1)]/2I$, $B(M1)$ and $B(E2)$ values, intraband $B(M1)/B(E2)$ ratios, $B(M1)_{\textrm{in}}/B(M1)_{\textrm{out}}$ ratios, and orientation of the angular momentum for the rotor as well as the valence proton and neutron are calculated. After including the pairing correlation, good agreement has been obtained between the calculated results an…
The triaxial dynamics of the quadrupole-deformed rotor model of both the rigid and the irrotational type have been investigated in detail. The results indicate that level patterns and E2 transitional characters of the two types of the model can be matched with each other to the leading order of the deformation parameter $β$. Especially, it is found that the dynamical structure of the irrotational type with most triaxial deformation ($γ=30^\circ$) is equivalent to that of the rigid type with oblate deformation ($γ=60^\circ$), and the associated spectrum can be classified into the standard rotat…
By taking the particle triaxial-rotor model with variable moment of inertia, we investigate the energy spectra, the deformations and the single particle configurations of the nuclei $^{183,185,187}$Tl systemically. The calculated energy spectra agree with experimental data quite well. The obtained results indicate that the aligned bands observed in $^{183,185,187}$Tl originate from the $[530]{1/2}^{-}$, $[532]{3/2}^{-}$, $[660]{1/2}^{+}$ proton configuration coupled to a prolate deformed core, respectively. Whereas, the negative parity bands built upon the ${9/2}^{-}$ isomeric states in $^{183…
The behavior of the collective rotor in the chiral motion of triaxially deformed nuclei is investigated using the particle rotor model by transforming the wave functions from the $K$-representation to the $R$-representation. After examining the energy spectra of the doublet bands and their energy differences as functions of the triaxial deformation, the angular momentum components of the rotor, proton, neutron, and the total system are investigated. Moreover, the probability distributions of the rotor angular momentum ($R$-plots) and their projections onto the three principal axes ($K_R$-plots…
A particle rotor model (PRM) with a quasi-proton and a quasi-neutron coupled with a triaxial rotor is developed and applied to study chiral doublet bands with configurations of a $h_{11/2}$ proton and a $h_{11/2}$ quasi-neutron. With pairing treated by the BCS approximation, the present quasi-particle PRM is aimed at simulating one proton and many neutron holes coupled with a triaxial rotor. After a detailed analysis of the angular momentum orientations, energy separation between the partner bands, and behavior of electromagnetic transitions, for the first time we find aplanar rotation or equi…
The wobbling motion of a triaxial rotor coupled to a high-j quasiparticle is treated semiclassi- cally. Longitudinal and transverse coupling regimes can be distinguished depending on, respectively, whether the quasiparticle angular momentum is oriented parallel or perpendicular to the rotor axis with the largest moment of inertia. Simple analytical expressions for the wobbling frequency and the electromagnetic E2 and M1 transition probabilities are derived assuming rigid alignment of the quasiparticle with one of the rotor axes and harmonic oscillations (HFA). Transverse wobbling is characteri…
Adopting the fully quantal triaxial particle rotor model, the candidate chiral doublet bands in odd-$A$ nuclei $^{103}$Rh and $^{105}$Rh with $πg_{9/2}^{-1}\otimesνh^{2}_{11/2}$ configuration are studied. For the doublet bands in both nuclei, agreement is excellent for the observed energies over entire spin range and $B(M1)/B(E2)$ at higher spin range. The evolution of the chiral geometry with angular momentum is discussed in detail by the angular momentum components and their probability distributions. Chirality is found to change from chiral vibration to nearly static chirality at spin $I=37…
We discuss in depth the application of the classical concepts for interpreting the quantal results from the triaxial rotor core without and with odd-particle. The corresponding limitations caused by the discreteness and finiteness of the angular momentum Hilbert space and the extraction of the relevant features from the complex wave function and distributions of various angular momentum components are discussed in detail. New methods based on spin coherent states and spin squeezed states are introduced. It is demonstrated that the spin coherent state map is a powerful tool to visualize the ang…
A solution of the Bohr collective hamiltonian for the $β-$soft, $γ-$soft triaxial rotor with $γ\sim π/6$ is presented making use of a harmonic potential in $γ$ and Coulomb-like and Kratzer-like potentials in $β$. It is shown that, while the $γ-$angular part in the present case gives rise to a straightforward extension of the rigid triaxial rotor energy in which an additive harmonic term appears, the inclusion of the $β$ part results instead in a non-trivial expression for the spectrum. The negative anharmonicities of the energy levels with respect to a simple rigid model are in qualitative agr…
A microscopic quantum ideal rotor-model intrinsic Hamiltonian for triaxial rotation is derived from the nuclear Schrodinger equation by applying a rotation operator to a deformed nuclear ground state. This Hamiltonian is obtained only when a rigid-flow prescription is used for the three rotation angles in the rotation operator. Using Hartree-Fock variational and second quantization methods, the rotor Hamiltonian is transformed into that of a self-consistent triaxial cranking model (MSCRM-3) with a self-consistent angular-velocity vector, plus residual terms associated with the square of the an…
The Kerman-Klein formulation of the equations of motion for a nuclear shell model and its associated variational principle are reviewed briefly. It is then applied to the derivation of the self-consistent particle-rotor model and of the self-consistent cranking model, for both axially symmetric and triaxial nuclei. Two derivations of the particle-rotor model are given. One of these is of a form that lends itself to an expansion of the result in powers of the ratio of single-particle angular momentum to collective angular momentum, that is essentual to reach the cranking limit. The derivation a…
The wobbling spectrum of $^{163}$Lu is described through a novel approach, starting from a triaxial rotor model within a semi-classical picture, and obtaining a new set of equations for all four rotational bands that have wobbling character. Redefining the band structure in the present model is done by adopting the concepts of Signature Partner Bands and Parity Partner Bands. Indeed, describing a wobbling spectrum in an even-odd nucleus through signature and parity quantum numbers is an inedited interpretation of the triaxial super-deformed bands.
We investigate the spectral statistics of the asymmetric rotor model (triaxial rigid rotator). The asymmetric top is classically integrable and, according to the Berry-Tabor theory, its spectral statistics should be Poissonian. Surprisingly, our numerical results show that the nearest neighbor spacing distribution $P(s)$ and the spectral rigidity $Δ_3(L)$ do not follow Poisson statistics.
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 $γ$ 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 $γ$ band in the low spin region are well describ…
The lowest six rotational bands have been studied in the particle-rotor model with the particle-hole configuration $πh^1_{11/2}\otimesνh^{-1}_{11/2}$ and different triaxiality parameter $γ$. Both constant and spin-dependent variable moments of inertial (CMI and VMI) are introduced. The energy spectra, electromagnetic transition probabilities, angular momentum components and $K$-distribution have been examined. It is shown that, besides the band 1 and band 2, the predicted band 3 and band 4 in the calculations of both CMI and VMI for atomic nuclei with $γ=30^\circ$ could be interpreted as chira…
Magnetic and chiral bands have been a hot subject for more than twenty years. Therefore, quite large volumes of experimental data as well as theoretical descriptions have been accumulated. Although some of the formalisms are not so easy to handle, the results agree impressively well with the data. The objective of this paper is to review the actual status of both experimental and theoretical investigations. Aiming at making this material accessible to a large variety of readers, including young students and researchers, I gave some details on the schematic models which are able to unveil the m…
The triaxiality of odd-mass nuclei is investigated by coupling a quasiparticle to an even-even core through the core-quasiparticle coupling model. Both soft and rigid triaxial cores are considered. The "soft core" is described by the collective model with rotation-vibrational motion, while the "rigid core" is described by the triaxial rotor model, which is a limiting case of the collective model with only rotational motion. We show that the presence of the odd quasiparticle modifies the collective quadrupole dynamics of the core to appear more "rigid".
The signature splitting of the $γ$-vibrational band of several Ru, Pd, Xe, Ba, Os and Pt isotopes is analyzed in the framework of the interacting boson model (IBM). The nuclei studied are close to the $γ$-unstable SO(6) limit of the IBM and have well-known $γ$ bands. It is shown that in most nuclei the signature splitting is better reproduced by the inclusion of a three-body interaction between the $d$ bosons. In none of the nuclei evidence for a stable, triaxial ground-state shape is found.
By the very definition the triaxial deformation parameter $γ$ is related to the expectation values of the K=0 and K=2 components of the intrinsic quadrupole tensor operator. On the other hand, using the same symbol "$γ$", various different parametrizations of triaxial deformation have been employed, which are suitable for various types of the mean-field potentials. It is pointed out that the values of various "$γ$" are quite different for the same actual triaxial deformation, especially for the large deformation; for example, the difference can be almost a factor two for the case of the triaxi…
A reflection-asymmetric triaxial particle rotor model with three quasiparticles and a reflection-asymmetric triaxial rotor is developed, and applied to investigate the observed multiple chiral doublets (M\c{hi}D) candidates with octupole correlations in 131Ba, i.e., two pairs of positive-parity bands D3-D4 and D5-D6, as well as one pair of negative-parity bands D7-D8. The energy spectra, the energy staggering parameters, the B(M1)/B(E2) ratios, and the B(E1)/B(E2) ratios are reproduced well. The chiral geometries for these M\c{hi}D candidates are examined by the azimuthal plots, and the evolut…
Within the low-lying spectrum of 10Be, multiple rotational bands are found, with strikingly different moments of inertia. A proposed interpretation has been that these bands variously represent triaxial rotation and prolate axially-deformed rotation. The bands are well-reproduced in ab initio no-core configuration interaction (NCCI) calculations. We use the calculated wave functions to elucidate the nuclear shapes underlying these bands, by examining the Elliott SU(3) symmetry content of these wave functions. The ab initio results support an interpretation in which the ground-state band, along…
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 $^{136}$Nd. 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, $^{136}$Nd is the first even-even candidate nucleus in whic…
The triaxial deformation in $^{10}$Be is investigated using a microscopic $α+α+n+n$ model. The states of two valence neutrons are classified based on the molecular-orbit (MO) model, and the $π$-orbit is introduced about the axis connecting the two $α$-clusters for the description of the rotational bands. There appear two rotational bands comprised mainly of $K^π= 0^+$ and $K^π= 2^+$, respectively, at low excitation energy, where the two valence neutrons occupy $K^π= 3/2^-$ or $K^π= 1/2^-$ orbits. The triaxiality and the $K$-mixing are discussed in connection to the molecular structure, particu…
A particle rotor model is developed which couples several valence protons and neutrons to a rigid triaxial rotor core. It is applied to investigating the chirality in odd-$A$ nucleus $^{135}$Nd with $πh_{11/2}^2\otimesνh^{-1}_{11/2}$ configuration for the first time in a fully quantal approach. For the two chiral sister bands, the observed energies and the $B(M1)$ and $B(E2)$ values for the in-band as well as interband transitions are reproduced excellently. Root mean square values of the angular momentum components and their probability distributions are used for discussing in detail the chir…
We extend the microscopic particle-rotor model for hypernuclear low-lying states by including the derivative and tensor coupling terms in the point-coupling nucleon-$Λ$ particle ($NΛ$) interaction. Taking $^{13}_{~Λ}$C as an example, we show that a good overall description for excitation spectra is achieved with four sets of effective $NΛ$ interaction. We find that the $Λ$ hyperon binding energy decreases monotonically with increasing the strengths of the high-order interaction terms. In particular, the tensor coupling term decreases the energy splitting between the first $1/2^-$ and $3/2^-$ s…
The criteria for chiral doublet bands based on one particle and one hole coupled to a triaxial rotor have been summarized. Two representative cases in $A\sim100$ odd-odd nuclei, nearly degenerate $ΔI =1$ doublet bands in $^{104}$Rh and $^{106}$Rh are checked against these chiral criteria. It is shown that $^{106}$Rh possesses better chiral geometry than $^{104}$Rh, although the energy near degeneracy is achieved in $^{104}$Rh in comparison with the constant energy separation of doublet bands in $^{106}$Rh.
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 ($γ=-30^\circ$), the intraband electromagnetic transitions given by the one-dimensional and two-dimensional collective Hamiltonian are compared to the results by the tilt…
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 collecti…
Multiple chiral doublet bands (M$χ$D) in the $80$, 130 and $190$ mass regions are studied by the model of $γ$=90$^{\circ}$ triaxial rotor coupled with identical symmetric proton-neutron configurations. By selecting the suitable basis, the calculated wave functions are explicitly exhibited to be symmetric under the operator $\hat{A}$, which is defined as rotation by $90^{\circ}$ about 3-axis with the exchange of valance proton and neutron. We found that both $M1$ and $E2$ transitions are allowed between the levels with different values of $A$, while are forbidden between the levels with same va…
Evolution of chirality from transverse wobbling in $^{135}$Pr
University of Notre Dame（圣母大学）；University of North Carolina Chapel Hill（北卡罗来纳大学教堂山分校）；Triangle Universities Nuclear Laboratory, Duke University（杜克大学三角大学核实验室）；East China Normal University（华东师范大学）；National Nuclear Data Center, Brookhaven National Laboratory（布鲁克海文国家实验室国家核数据中心）；Argonne National Laboratory（阿贡国家实验室）
Novel chiral-wobbling excitation mode in atomic nuclei
Islamic University of Science and Technology（伊斯兰科学技术大学）；University of Kashmir（克什米尔大学）；Institute of Physics (IOP)（物理研究所（IOP））；Homi Bhabha National Institute（霍米·巴巴国家研究所）；Government Degree College Shopian（肖皮安政府学位学院）；Government Degree College Kulgam（库尔加姆政府学位学院）；Shandong University（山东大学）
Shape evolution in neutron-rich odd-even $^{105-109}$Nb isotopes
Universidad Complutense de Madrid（马德里康普顿斯大学）；Universite Claude Bernard Lyon 1（里昂第一大学）；Shandong University（山东大学）；Vanderbilt University（范德堡大学）；GANIL, CEA/DRF-CNRS/IN2P3（大尺度加速器国家实验室）；Université de Strasbourg（斯特拉斯堡大学）
New transitions in neutron rich $^{100}$Y have been identified in a $^9$Be+$^{238}$U experiment with mass- and Z- gates to provide full fragment identification. These transitions and high spin levels of $^{100}$Y have been investigated by analyzing the high statistics $γ$-$γ$-$γ$ and $γ$-$γ$-$γ$-$γ$ coincidence data from the spontaneous fission of $^{252}$Cf at the Gammasphere detector array. Two new bands, 14 new levels and 23 new transitions have been identified. The $K^π=4^+$ new band decaying to an 1s isomeric state is assigned to be the high-$K$ Gallagher-Moszkowski (GM) partner of the kn…
