A triaxial core rotating around the middle axis, i.e. 2-axis, is cranked around the 1-axis, due to the coupling of an odd proton from a high j orbital. Using the Bargmann representation of a new and complex boson expansion of the angular momentum components, the eigenvalue equation of the model Hamiltonian acquires a Schrödinger form with a fully separated kinetic energy. From a critical angular momentum, the potential energy term exhibits three minima, two of them being degenerate. Spectra of the deepest wells reflects a chiral-like structure. Energies corresponding to the deepest and local minima respectively, are analytically expressed within a harmonic approximation. Based on a classical analysis, a phase diagram is constructed. It is also shown that the transverse wobbling mode is unstable. The wobbling frequencies corresponding to the deepest minimum are used to quantitatively describe the wobbling properties in $^{135}$Pr. Both energies and e.m. transition probabilities are described.
The influence of triaxial deformation $γ$ on the purely collective form of wobbling motion in even-even nuclei are discussed based on the triaxial rotor model. It is found that the harmonic approximation is realized well when $γ=30^{\circ}$ for the properties of energy spectra and electric quadrupole transition probabilities, while this approximation gets bad when $γ$ deviates from $30^{\circ}$. A recent data from Coulomb excitation experiment, namely $3_1^+$ and $2_2^+$ for the $^{110}$Ru are studied and might be suggested as the bandhead of the wobbling bands. In addition, two types of angul…
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 have 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. Due to 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 ob…
The reported transverse wobbling band in odd-mass $^{105}$Pd has been reinvestigated by the triaxial particle rotor model. Employing different parameter sets of moment of inertia (MOI), several calculated results could be in good agreement with the experimental data, which show distinct modes of rotational excitation, respectively. These modes are sensitive to the ratio between the MOI at intermediate and short axis. With the increase of this ratio, a wobble about the short axis of the total angular momentum is gradually changed to a wobble about the intermediate axis. In addition, it is exhib…
A new interpretation for the wobbling bands in the even-odd Lu isotopes is given within a particle-triaxial rotor semi-classical formalism. While in the previous papers the bands TSD1, TSD2, TSD3 and TSD4 are viewed as the ground, one, two and three phonon wobbling bands, here the corresponding experimental results are described as the ground band with spin equal to I=R+j, for R=0,2,4,...(TSD1), the ground band with I=R+j and R=1,3,5,...(TSD2), the one phonon excitations of TSD2 (TSD3), with the odd proton moving in the orbit $j=i_{13/2}$, and the ground band of I=R+j, with R=1,3,5,... and $j=…
A particle-triaxial rigid core Hamiltonian is semi-classically treated. The coupling term corresponds to a particle rigidly coupled to the triaxial core, along a direction that does not belong to any principal plane of the inertia ellipsoid.The equations of motion for the angular momentum components provide a sixth-order algebraic equation for one component and subsequently equations for the other two. Linearizing the equations of motion, a dispersion equation for the wobbling frequency is obtained. The equations of motion are also considered in the reduced space of generalized phase space coo…
A unitary description for wobbling motion in even-even and even-odd nuclei is presented. In both cases compact formulas for wobbling frequencies are derived. The accuracy of the harmonic approximation is studied for the yrast as well as for the excited bands in the even-even case. Important results for the structure of the wave function and its behavior inside the two wells of the potential energy function corresponding to the Bargmann representation are pointed out. Applications to $^{158}$Er and $^{163}$Lu reveal a very good agreement with available data. Indeed, the yrast energy levels in t…
Rotation of triaxially deformed nucleus has been an interesting subject in the study of nuclear structure. In the present series of work, we investigate wobbling motion and chiral rotation by employing the microscopic framework of angular-momentum projection from cranked triaxially deformed mean-field states. In this first part the wobbling motion is studied in detail. The consequences of the three dimensional cranking are investigated. It is demonstrated that the multiple wobbling rotational bands naturally appear as a result of fully microscopic calculation. They have the characteristic prop…
An alternative interpretation of the recently reported low-lying excited bands in $γ$-soft odd-mass nuclei as wobbling bands is presented in terms of the interacting boson-fermion model. The model Hamiltonian is determined based on the mean-field calculations with the nuclear energy density functionals. The predicted mixing ratios of the $Δ{I}=1$ electric quadrupole to magnetic dipole transition rates between yrast bands and those yrare bands previously interpreted as wobbling bands in $^{135}$Pr, $^{133}$La, $^{127}$Xe, and $^{105}$Pd are consistently smaller in magnitude than the experimenta…
Precession is a unique motion in which the orientation of the rotational axis of a rotating body is not fixed but moving, and it generally exists in the Universe from giant stars through tiny atomic nuclei. In principle, the precession of an atomic nuclide can be approximately described as wobbling motion, arising from the coupling of a rotation and a harmonic vibration. Recently, a number of wobbling bands were reported at low spin, which violate the wobbling approximation that can be valid only at high spin. Here we explore the nature of the reported low-spin wobbling bands. Via a new experi…
The existence of one-phonon and possible two-phonon transverse wobbling bands is proposed for the first time in two even-even nuclei, $^{134}$Ce and $^{136}$Nd. The predominant $E2$ character of the $ΔI = 1$ transitions connecting the one-phonon wobbling band in $^{134}$Ce to the two-quasiparticle yrast band supports the wobbling interpretation. The extracted wobbling frequencies decrease with increasing spin, indicating the transverse character of the wobbling motion, with the angular momenta of the two quasiparticles aligned perpendicular to the axis of collective rotation. A candidate for t…
Two newly observed bands built on a two-quasiparticle configuration in 130Ba have been investigated for the first time with the microscopic projected shell model. The experimental energy spectra and the available electromagnetic transition probabilities are well reproduced. The wobbling character of the higher band is revealed by the angular momentum projected wavefunctions via the K plot and the azimuthal plot. This provides the first strong microscopic evidence for wobbling motion based on a two-quasiparticle configuration in even-even nuclei.
A new interpretation for the wobbling bands in $^{163}$Lu is given within a particle-triaxial rotor semi-classical formalism. While in the previous papers the bands TSD1, TSD2, TSD3 and TSD4 are viewed as the ground, one, two and three phonon wobbling bands, here the corresponding experimental results are described as the ground band with spin equal to I=R+j, for R=0,2,4,...(TSD1), the ground band with I=R+j and R=1,3,5,...(TSD2), the one phonon excitations of TSD2 (TSD3), with the odd proton moving in the orbit $j=i_{13/2}$, and the ground band of I=R+j, with R=1,3,5,... and $j=h_{9/2}$ (TSD4…
We found that in 156Dy and 162Yb the lowest odd spin gamma-vibrational states transform to the wobbling excitations after the backbending, associated with the transition from axially-symmetric to nonaxial shapes. The analysis of quadrupole electric transitions determines uniquely the sign of the gamma-deformation in both nuclei after the transition point.
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）
The results obtained for $^{165,167}$Lu with a semi-classical formalism are presented. Properties like excitation energies for the super-deformed bands TSD1, TSD2, TSD3, in $^{165}$Lu, and TSD1 and TSD2 for $^{167}$Lu, inter- and intra-band B(E2) and B(M1), the mixing ratios, transition quadrupole moments are compared either with the corresponding experimental data or with those obtained for $^{163}$Lu. Also alignments, dynamic moments of inertia, relative energy to a reference energy of a rigid symmetric rotor with an effective moment of inertia and the angle between the angular momenta of th…
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 ($γ=-30^\circ$) coupling to one $h_{11/2}$ proton particle and one $h_{11/2}$ neutron hole. By diagonalizing the 2DCH, the angular momenta and energ…
A systematic investigation of the nuclear observables related to the triaxial degree of freedom is presented using the multi-quasiparticle triaxial projected shell model (TPSM) approach. These properties correspond to the observation of $γ$-bands, chiral doublet bands and the wobbling mode. In the TPSM approach, $γ$-bands are built on each quasiparticle configuration and it is demonstrated that some observations in high-spin spectroscopy that have remained unresolved for quite some time could be explained by considering $γ$-bands based on two-quasiparticle configurations. It is shown in some C…
The effective field theory for collective rotations of triaxially deformed nuclei is generalized to odd-mass nuclei by including the angular momentum of the valence nucleon as an additional degree of freedom. The Hamiltonian is constructed up to next-to-leading order within the effective field theory formalism. The applicability of this Hamiltonian is examined by describing the wobbling bands observed in the lutetium isotopes $^{161,163,165,167}$Lu. It is found that by taking into account the next-to-leading order corrections, quartic in the rotor angular momentum, the wobbling energies $E_{\t…
Some developments of nuclear-structure physics uniquely related to Copenhagen School are sketched based on theoretical considerations versus experimental findings and one-particle versus collective aspects. Based on my personal overview I pick up the following topics; (1) Study of vibration in terms of particle-vibration coupling; (2) One-particle motion in deformed and rotating potentials, and yrast spectroscopy in high-spin physics; (3) Triaxial shape in nuclei: wobbling motion and chiral bands; (4) Nuclear structure of drip line nuclei: in particular, shell-structure (or magic numbers) chan…
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）
After identifying the nuclei in the regions near A =80 and A = 160 for which beta-decay rates have the greatest effect on weak and main r-process abundance patterns, we apply the finite-amplitude method (FAM) with Skyrme energy-density functionals (EDFs) to calculate beta-decay half-lives of those nuclei in the quasiparticle random-phase approximation (QRPA). We use the equal filling approximation to extend our implementation of the charge-changing FAM, which incorporates pairing correlations and allows axially symmetric deformation, to odd-A and odd-odd nuclei. Within this framework we find d…
In the sequel of the present study, we have investigated the rotational motion of triaxially deformed nucleus by using the microscopic framework of angular-momentum projection. The Woods-Saxon potential and the schematic separable-type interaction are employed as a microscopic Hamiltonian. As the first example nuclear wobbling motion was studied in detail in the part~I of the series. This second part reports on another interesting rotational mode, chiral doublet bands: two prototype examples, $^{128}$Cs and $^{104}$Rh, are investigated. It is demonstrated that the doublet bands naturally appea…
We use the finite amplitude method (FAM), an efficient implementation of the quasiparticle random phase approximation, to compute beta-decay rates with Skyrme energy-density functionals for 3983 nuclei, essentially all the medium-mass and heavy isotopes on the neutron rich side of stability. We employ an extension of the FAM that treats odd-mass and odd-odd nuclear ground states in the equal filling approximation. Our rates are in reasonable agreement both with experimental data where available and with rates from other global calculations.
The nuclear shell model assumes an effective mean-field plus interaction Hamiltonian in a specific configuration space. We want to understand how various interaction matrix elements affect the observables, the collectivity in nuclei and the nuclear level density for odd-A and odd-odd nuclei. Using the sd and pf shells, we vary specific groups of matrix elements and study the evolution of energy levels, transition rates and the level density. In all cases studied, a transition between a "normal" and a collective phase is induced, accompanied by an enhancement of the level density in the collect…
Two types of average neutron-proton interaction formulas are compared: In the first type, neutron-proton interactions for even-even and odd-$A$ nuclei extracted from experimental binding energies show a smooth behavior as a function of mass number $A$ and are dominated by the contribution from the symmetry energy. Whereas in the second type large systematic staggering is seen between even-$A$ and odd-$A$ nuclei. This deviation is understood in term of the additional neutron-proton interaction in odd-odd nuclei relative to the neighboring even-even and odd-$A$ systems. We explore three possible…
