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The Conference “Chirality and Wobbling in Atomic Nuclei” (CWAN’23) was held in Huizhou, China, July 10-14, 2023. The scientific sessions will be held in the conference hall of the Kande International Hotel in Huizhou.
The conference was organized in a hybrid mode, with 89 registered participants. 30% of the talks have been performed remotely using the Zoom interface. The conference was devoted to the “Dynamics and statics of the nuclear triaxiality”, and celebrated 25 years of nuclear chirality. Contributions on other topics of nuclear structure were also presented.
Scientific program
The program consisted of plenary presentations by invited speakers and contributions selected among submitted abstracts, as well as of a poster session. The main topics were:
• Theoretical approaches for chirality and wobbling
• Experimental evidence for chirality and wobbling
• New modes for chiral nuclei
• Nature of triaxiality: Static shape or large fluctuations?
The breaking of symmetries in quantum systems is one of the key issues in nuclear physics. In particular, the spontaneous symmetry breaking in rotating nuclei leads to exotic collective modes, like the chiral and wobbling motions, which have been intensively studied in the last 25 years. Chiral bands have been identified in several mass regions of the Segrè chart of nuclei, and their properties were measured, including transition probabilities and magnetic moments. A plethora of bands based on configurations with two, thee, four and even six nucleons have been identified, and in some cases also in presence of other broken symmetries. Recent experimental results and hints on their interpretation will be presented.
The wobbling motion was also intensively studied in recent years, mainly after the introduction of the transverse wobbling concept. Latest experimental results on transverse or longitudinal wobbling at low-spin in odd-even nuclei and at medium spin in even-even nuclei will be discussed.
Experimental study of the nuclear chiral symmetry resembles the Schroedinger's cat problem where the direct observation of an object, here its handedness, is prohibited within the todays experimental methods. The nuclear chirality measurements are thus limited to indirect observables related to states superimposed over unobserved values. Such superpositions are built of the same amount of opposite handedness configurations that mutually cancel in the expectation values of handedness sensitive operators. However, there is still a possibility of a direct nuclear chirality measurement since the existence of the handedness only matters, not the value of the handedness itself. Thus the direct chirality measurement means the determination of the nucleus chiral structure regardless of the handedness present in the superimposed states. Such a direct measurement has been performed by observation of the magnetic dipole moment in $^{128}$Cs nucleus and the first results were published in 2018. The magnetic dipole moment can therefore serve as an ultimate test for chirality in nuclear physics. More details about the concept of the measurement, experimental techniques and obtained results will be present.
The concept of wobbling excitations of the rigid triaxial rotor is introduced by means of the topology of its classical orbits. The quantum-mechanical restrictions on the moments of inertia are discussed. The consequences of γ softness for the spectroscopic signatures are addressed and the available experimental examples for wobbling excitations based on the ground bands of even-even nuclei are presented.
The concept of transverse and longitudinal wobbling is introduced by means of the topology of the classical orbits of the total angular momentum of the triaxial rotor coupled to one, two, … high-j quasiparticles. Distillation of the topology from the quantal states by means of spin coherent states is demonstrated. The success and the limitations of the quasiparticle+triaxial rotor model will be illustrated by means of selected examples.
The Triaxial Projected Shell Model represents the most microscopic description of wobbling excitations, which avoids ambiguities in the parameter choice and errors due to core-particle exchange terms of the particle rotor model. Examples of the successful application are presented, and the open questions of interpreting the quantal results in terms of concepts of wobbling and quasiparticle excitations in a rotating potential are addressed.
The concept of chirality in atomic nuclei was first proposed in 1997 and has been extensively investigated for the past decades. In this talk, the theoretical efforts on nuclear chirality are reviewed. Recent progress including the early appearance of the chiral rotation induced by superfluidity, the microscopic dynamics of chiral excitations, Chirality-Parity (ChP) violation, collective Hamiltonian for chiral and wobbling modes, K plot and the azimuthal plot, searching for new observable characterizing chirality, will be emphasized.
Most investigations of collective excitations in medium-heavy and heavy nuclei are based on nuclear density functional theory. These methods allow the description of the interplay between collective and single-particle motion, which is important for chirality and wobbling motion. Very successful relativistic and non-relativistic functionals are available nowadays. However, most of them are phenomenological functionals. Therefore it is crucial to study the connection of such functionals to ab-initio nucleon-nucleon forces. Non-relativistic Brueckner-Hartree-Fock theory was a starting point of ab-initio investigations in nuclear structure in the fifties and sixties. However, it failed because three-body forces were not included at that time. Later it was found that the relativistic Brueckner-Hartree-Fock (RBHF) theory can reproduce the saturation properties of nuclear matter. In this contribution, we discuss recent developments in RBHF theory for infinite nuclear matter and for finite nuclei. In particular, we study the effective masses for protons and neutrons. This quantity and its comparison with experimental data provide a reliable measure for nuclear ab-initio theories.
Absolute values of EM transition probabilities in partner bands provide a stringent test of nuclear chirality. I discuss eight nuclei, $^{128}$Cs, $^{126}$Cs, $^{124}$Cs, $^{130}$Cs, $^{135}$Nd, $^{76}$Br, $^{80}$Br and $^{194}$Tl, for which such data were obtained through lifetime measurements using the Doppler Shift Attenuation Method. Analysis of the DSAM data was performed with the software developed by Alexandr A. Pasternak: COMPA, GAMMA, SHAPE (4 cases) and/or with the LINESHAPE program (5 cases). Important part of the DSAM analysis is information on a stopping power of studied nuclei. In the case of $^{128}$Cs, $^{126}$Cs, $^{124}$Cs the stopping power was specially measured by the semi-thick target method. For other was used the SRIM code of Ziegler or the the tables of Northcliffe and Schilling, corrected for shell effects.
One of important signatures of nuclear chirality, a special pattern of B(M1) staggering is expected when the so-called S symmetry is preserved, which requires, among others, that particle (proton) and hole (neutron) occupy the same level. Such situation occurs in $^{124,126,128,130}$Cs (j$_{11/2}$ level) and in $^{76,80}$Br (g$_{9/2}$ level). For $^{135}$Nd and $^{194}$Tl the partner bands are interpreted through many quasiparticle configurations (3qp and 4qp, respectively) and the S symmetry is not preserved. That is why there is no B(M1) staggering in these nuclei.
Nuclei with triaxial shape can rotate around their three axes, which produces a number of excited rotational bands for each single-particle configuration. This three-dimensional rotation can be visualised in the angular moment space, it represents a precession of the total angular momentum along an axis, and looks like the precession of a rotating top. In even-even nuclei the precession gives rise to the ground-state and the γ bands. In odd-mass nuclei the rotation is coupled to the angular momentum of the valence nucleon producing a precession of the total angular momentum, and giving rise to an yrast and a number of excited rotational bands with the same nucleon configuration.
The nature of the predicted rotational bands in odd-mass triaxial nuclei with πh$_{11/2}$ configurations using the quasiparticle-plus-triaxial rotor model will be discussed. Limits with free and frozen single-particle alignment will be considered to evaluate the impact of the single-particle degree of freedom. The rotational nature of the excited bands produced within frozen single-particle alignment will be highlighted. The difference in the features of such excited rotational bands and those produced by a vibrational excitation (e.g. a wobbling phonon) will be evaluated and linked to the features of excited rotational bands in even-even triaxial nuclei. The discussion will be extended to odd-odd triaxial nuclei with πh$_{11/2}$$\otimes$h$_{11/2}$ chiral configuration and the striking similarity in the excitation energy patterns of partner bands with πh11/2$\otimes$h11/2 and with πh$_{11/2}$ nature will be discussed.
Shapes and the related spectroscopy of odd-A and odd-odd nuclei are studied using the framework of nuclear density functional theory and particle-boson coupling scheme. The Hamiltonian of the interacting boson-fermion model describing the low-lying structure of odd nuclei is constructed based on the microscopic self-consistent mean-field calculations with a choice of universal energy density functional. This presentation will focus on recent IBFM studies on $\gamma$-soft nuclei, concerning the low-spin bands of odd-A systems that have been interpreted as wobbling bands, the chiral doublet bands in the odd-odd Cs region, and the beta-decay properties.
Chirality is a well-known phenomenon in many fields, such as chemistry, biology, molecular physics, and particle physics. The study of nuclear chirality has attracted a lot of attention since it was originally suggested for triaxial nuclei in 1997. The experimental evidence of nuclear chirality is the so-called chiral doublet bands, which have been observed in many nuclei. Theoretically, there have been many approaches employed to study the chiral nuclei including the phenomenological approaches and the microscopic ones. In a series of our recent works we have developed the three-dimensional cranking relativistic density functional theory to study the chirality in triaxial nuclei. In particular, by overcoming the variational collapse and the fermion doubling problem, relativistic density functional theory has been solved in three-dimensional lattice space, and the corresponding time-dependent relativistic density functional theory has been established. It allows a unified description of the static and dynamic properties of nuclei without assuming any spatial symmetry restrictions. In this talk, I will review recent progress in the development of time-dependent relativistic density functional theory in space lattice and its application for the chiral structure and dynamics in triaxial nuclei. In particular, the chiral dynamics in triaxially soft nuclei will be discussed.
Chirality and wobbling are two unique fingerprints for a triaxial nuclear shape. In both of case, the triaxial deformation parameters and the configuration information are of importance. The triaxial deformation determines whether the chiral and wobbling mode exists or not, while the configuration determines what kind of the modes. Therefore, a reliable theoretical approach is needed to get the information of the configuration and deformation for a specific nuclei. Over the past years, we combined the constrained triaxial covariant density functional theory combined with quantum particle rotor model calculations to investigate the chirality and wobbling and achieve some progresses. In this talk, some recent examples will be shown.
A system composed of a triaxial core with non-axial rigid quasiparticle alignments is investigated in a semiclassical approach. The classical features of the chiral geometry and wobbling motion emerging in atomic nuclei and the associated dynamics are investigated for various core deformations and single-particle alignments. Distinct dynamical characteristics are identified in specific angular momentum ranges, triaxiality and alignment conditions. Quantum observables are recovered from a Schroedinger equation, constructed from the classical picture, which has the total angular momentum projection as a continuous variable. The separation of a potential energy as a function of the chosen variable allows the phenomenological interpretation of the experimental spectra in terms of wobbling or chiral oscillations and tilted axis rotations.
The nucleus is a strongly correlated quantum many-body system. The large number of nucleons and the complex interaction between nucleons make the nucleus exhibit rich deformation characteristics, such as spherical, quadrupole deformation, octupole deformation, etc., and even shape coexistence occurs in specific regions. Nuclear shapes and related collective motions are reflected in the nuclear low-lying spectra, which can be measured directly in the experiments. Theoretically, we have developed microscopic collective Hamiltonian based on the covariant density functional theory (CDFT), and applied to the description of nuclear low-lying spectra and shape coexistence.
1) Using the five-dimensional collective Hamiltonian model based on CDFT (5DCH-CDFT), the mass, low-lying spectrum and quadrupole shape invariants of even-even nuclei are systematically studied. The known shape coexistence nuclear regions are reproduced, and new coexistence nuclear regions are predicted.
2) The quadrupole-octupole collective Hamiltonian (QOCH) and the core-quasiparticle coupling (CQC) model including octupole coupling based on CDFT are used to systematically study the octupole shape evolution and the parity-doublet structures of odd-A nuclei of actinide nuclei.
3) Very recently, we have established a seven-dimensional collective Hamiltonian including both octupole and triaxial degrees of freedom based on a multi-dimensional constraint CDFT. An illustrative calculation is done for the nucleus $^{144}$Ba.
In the last decade, chiral symmetry in atomic nuclei has attracted significant attention and become one of the hot topics in current nuclear physics frontiers. This presentation provides a review of experimental studies on nuclear chirality in Shandong University. These studies found a new chiral mass region (A~80), tested the chiral geometry of $^{80}$Br and $^{76}$Br by lifetime measurements. In addition, simultaneous breaking of chirality and other symmetries have been studied in $^{74}$As, $^{76}$Br, $^{78}$Br, $^{80}$Br, and $^{81}$Kr.
Chirality and wobbling are two direct evidences of triaxial deformed nuclei. Following with the experimental explorations of chirality and wobbling in 80 mass region in Shandong University, we studied the corresponding issues on the theoretical side. The nuclear deformation in 80 mass region has been investigated using covariant density functional theory, and the candidates of chiral nuclei in bromine and rubidium isotopes have been suggested. The influence of the moments of inertia on different wobbling modes has been investigated using the particle rotor model, and our understanding for the wobbling motion are exhibited. The group theory for the two phenomena are also mentioned briefly.
A comprehensive investigation of γ-deformation observed in atomic nuclei is performed for more than 30 nuclei in different regions of the periodic chart using the microscopic triaxial projected shell model approach. It is demonstrated that apart from signature splitting of the observed g-bands, the quadrupole transitions of these bands are also sensitive to the nature of the γ-deformation. Further, it is shown that mixing of the quasiparticle excitations into the collective vacuum state can transform the system from γ-rigid to that of γ-soft.
Different kinds of shape deformations may simultaneously manifest in certain regions of the nuclear chart entailing corresponding collective modes and specific structure of excitation spectra. In particular, in the mass region of Barium, Cerium, Neodymium and Samarium isotopes one may observe different degrees of manifestation of pear-shape quadrupole-octupole (reflection-asymmetric) modes, either stiff, e.g. in $^{144,146}$Ba, or soft in other nuclei and, at the same time, manifestation of quadrupole triaxiality, again, soft or stiff enough to form chiral structures in the spectrum, such as, e.g., in $^{136}$Nd. In this talk we address the manifestation of reflection-asymmetric degrees of freedom in coexistence with triaxial quadrupole modes. We discuss the possibility for the presence of non-axial octupole deformation and the way to identify it in the structure of alternating-parity spectra by using a collective quadrupole-octupole rotation model. We show that the possible appearance of triaxial modes may stronger affect the higher angular momenta by counteracting the parity-shift effect with a simultaneous stabilization of the overall quadrupole-octupole shape.
Nuclear shape transitions from spherical to deformed, which accompany variation of nucleon numbers around the doubly-magic shell closures, are often manifested by octupole shapes, $\alpha_{3,\mu} \neq 0$, at vanishing quadrupole deformations and not -- as often thought -- exclusively by the prolate/oblate quadrupole shapes. Corresponding exotic nuclear geometry is described by the point-group symmetries known in molecular physics: $C_{2v}$ ($\alpha_{31} \neq 0$), $T_d$ ($\alpha_{32} \neq 0$), $D_{3h}$ ($\alpha_{33} \neq 0$) and $D_{2d}$ ($\alpha_{32} \neq 0, \alpha_{20} \neq 0$) and, needles to say, are different from the tri-planar quadrupole ellipsoids.
Large scale mean field theory calculations focussing on heavy and super-heavy nuclei (analogous mechanisms are predicted also for the lighter mass zones) predict the presence of strong shell gaps simultaneously for $\alpha_{30}\neq 0$, $\alpha_{31}\neq 0$, $\alpha_{32}\neq 0$ and $\alpha_{33}\neq 0$, wherefrom the term {\em four-fold octupole magic numbers}. Moreover, they apply for several symmetries at the same time -- in contrast to the traditional double shell gaps at spherical geometry -- wherefrom the term {\em universal magic numbers} used by the cited authors.
In this project we present a series of new theory predictions addressing exotic shape competition and isomerism, as a follow-up of our recent articles. The employed nuclear structure approach uses theory tools which are among the most powerful in the domain, such as nuclear mean field theory, inverse problem theory, group and group-representation theories and graph theory. The newly obtained parameterisations of the phenomenological mean field Hamiltonians are free from parametric correlations known to destabilise the nuclear shape predictions and systematically concord with the corresponding experimental data.
Using group representation theory, powerful spectroscopic criteria addressing the issue of identification and the structure of rotational bands built on the corresponding exotic symmetry equilibrium shapes are constructed and illustrated. Characteristically, the bands in question involve states of both parities and spin-parity combinations as well as multiple degeneracies of levels. All these features are discussed in detail establishing the strategic lines of the new spectroscopy principles. In particular, the analysis in terms of the static shapes as opposed to the large deformation fluctuations are considered and illustrated.
All these features provide enriched quantum environment for the coupling between the collective rotation angular momentum and angular momenta related to the odd-proton and neutron orbitals underlying and manifesting chirality, since they involve more exotic quantum symmetry structures as compared to those used often in the past.
Collective shape degrees of freedom have been a major direction in the study of the nuclear finite many-body problem for over 50 years. There is widespread evidence for quadrupole deformations, primarily of large prolate spheroidal deformation with axially symmetric rotor degrees of freedom. This naturally leads to the question of whether axially asymmetric rotor degrees of freedom are exhibited by any nuclei, with the implication of triaxial shapes. An overview of ground-state empirical moments of inertia for the 1-, 2-, and 3-axes will be given, extracted from experimental 2$^+_{g,\gamma}$ energies and electric quadrupole matrix elements determined from multi-step Coulomb excitation data. The results are extracted from the latest data and compared to expectations based on rigid and irrotational inertial flow; this is an update on the initial view provided in Ref. [PLB 767, 226 (2017)].
*This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics.
In the past twenty years, great efforts have been devoted to chiral symmetry, from both theoretical and experimental sides. As the frontier of such researches, multiple chiral doublet bands (M$\chi$D) have been observed in several nuclei.
From an experiment performed in Legnaro, Italy, we found a level structure exhibiting multiple chiral doublet bands in $^{131}$Ba, which include 2 negative-parity bands and 4 positive-parity bands. Many linking transitions have been found among the four positive-parity bands, which may be attributed to both chiral symmetry and pseudo-spin symmetry. A series of E1 linking transitions have been found between the positive and negative bands, which should be a result of octupole correlations. Comparing this structure with those in $^{133}$Ce and $^{135}$Nd, it seems that the effects of pseudo-spin symmetry and octupole correlations in $^{131}$Ba are enhanced.
Nuclei with mass A <120 are perfectly placed to study the shape-driving properties of different quasiparticle configurations. For these nuclei, the Fermi surface for the protons lies close to the low-Ω h11/2 orbitals which drives the nucleus towards prolate shape, while the neutron fermi surface lies near the mid-Ω h$_{11/2}$ orbitals which induces an oblate deformation. Due to the conflicting deformation driving properties, the nuclei observe prolate, oblate or triaxial shape in this mass region. Also, the presence of octupole driving h$_{11/2}$ and d$_{5/2}$ orbitals near the Fermi surface make them suitable to exhibit octupole correlation. In this mass region, octupole correlations have been reported earlier in several isotopes of Xe-Cs-Ba having N < 70. In previous high spin gamma ray spectroscopy measurements in 118Xe nuclei though the octupole correlations have been reported in Refs. but in almost all the cases a precise data on parity assignments, was missing. For example, though the issue of octupole collectivity has been discussed in relation to the observed inter-band transitions, 1022 keV (7$^−$ → 6$^+$), 846 keV (9$^−$ → 8$^+$), 726 keV (11$^−$ → 10$^+$) and 924 keV (8$^−$ → 8$^+$), however, the quoted B(E1) values have errors in the range from 4% to 28%. In the present work, the excited states in 118Xe nucleus were reinvestigated with the aim: 1) to update the level scheme with inclusion of more γ transition in the non-yrast bands (if any), 2) to fix the missing parities with polarisation measurements and 3) to provide a more precise data on the octupole collectivity (transition). We have also performed the triaxial projected shell model (TPSM) calculations to investigate the observed band structures further.
High spin states in $^{118}$Xe have been populated via $^{93}$Nb ($^{28}$Si, xpyn) $^{118}$Xe fusion-evaporation reaction at a beam energy of 115 MeV provided by the 15 UD pelletron accelerator facility at the Inter University Accelerator Center, New Delhi. In the experiment, several new γ-transitions have been found and are placed appropriately in the level scheme. Theoretical study using the triaxial projected shell model (TPSM) approach suggests that the first band-crossing is due to the alignment of two neutrons, and a parallel band tracking the yrast configuration is the γ-band built on the two-quasiparticle state. Enhanced E1 transition rates have been obtained between opposite parity bands, involving νh$_{11/2}$ and νd$_{5/2}$ orbitals having ∆j = ∆l = 3, indicates the presence of octupole correlation in this nucleus. More details of the analysis and the physics outcomes will be discussed during the presentation.
Recently the DESPEC nuclear spectroscopy campaign started at the SIS18/FRS facility at GSI as part of the FAIR/NUSTAR Phase-0 program. It aims at the investigation of exotic heavy nuclei produced in fragmentation reactions employing detectors and instrumentation developed for the FAIR facility. With the unique availability of new lanthanide primary beams, i.e. $^{170}$Er, secondary ion beams of very neutron-rich nuclei can be produced at the GSI/FAIR fragmentation facilities with unprecedented intensities. These nuclei mid-way between the proton closed shells at Z=50 and 82 and neutron closed shells at N=82 and 126 are expected to exhibit some of the largest collectivities known. Thus, a plethora of interesting collective phenomena is being expected.
The production cross sections are sufficient for decay studies at the current SIS18/FRS facility. For that purpose the DESPEC collaboration has developed the DEGAS HPGe array, enabling high-resolution gamma-spectroscopy with highest efficiency. Complementary lifetime measurements are possible with the FATIMA LaBr3 fast-timing array. First experiments are currently in preparation for data taking from 2024 on. With the advent of the SIS18/SIS100/Super-FRS facility at FAIR still higher secondary beam intensities are envisaged. They will enable in addition in-beam studies based on relativistic Coulomb excitation and few nucleon knock-out reactions. For such studies, the AGATA HPGe tracking array will be employed.
The physics plans and experimental details will be reported.
The decay of the excited states of the atomic nucleus is almost always companied by gamma radiation. Large gamma detection platform, which could give the most important observables of the nucleus, is proved to be one of the essential instruments of the laboratories around the world. A new detector array consisting of 32 detection elements of the early phase, including 16 coaxial HPGe detectors, 8 Clover HPGe detectors and 8 LaBr detectors, was built at Institute of Modern Physics of CAS. It is one of the most efficient gamma detector arrays in Asia, with capability of measuring hyperfine spectroscopies containing lifetimes, and sustaining the most frontier nuclear studies. In addition, reliable auxiliary equipments such as high speed/capacity electronic, DAQ, and automatic cooling system have been developed for the detector array. The array is currently serving at the low energy beam line of the HIRFL facility for the in-beam gamma spectroscopy. Additionally, it is traveling around China and world for other collaborative projects. The details of the array will be presented.
The Research Center for Nuclear Physics, Osaka University, is an accelerator facility with an AVF cyclotron and a ring cyclotron, which can be used for various nuclear physics research. To advance nuclear structure research by gamma-ray spectroscopy, we have initiated the CAGRA project, based on the international collaboration among Japan, the U.S., and China. CAGRA array consists of up to 16 Ge Clover detectors with BGO Compton shields and a digital data acquisition system employing GRETINA digitizers is used, which enables high-rate data taking. At RCNP cyclotron facility, there are various experimental capabilities such as high-resolution spectrometer, Grand Raiden, low-energy RI beam facility, EN beam line, and the DC muon beam facility, MuSIC. By combining the CAGRA with these devices, many physics opportunities will be provided. So far, CAGRA campaign experiments at EN beam line and Grand Raiden were successfully performed. The CAGRA experiment at JAEA to study heavy actinoid have also been performed. Experiments at RIBF will provide for further experimental studies of unstable nuclei. In this talk, an overview of gamma-ray spectroscopic studies at RCNP, including recent experimental results, and future prospect will be presented.
In the core-particle coupling scheme, a collective model is proposed to describe high-spin phenomena in the triaxial odd-A and odd-odd systems with their even-even cores assumed to follow a critical point symmetry. It is shown that the model Hamiltonian can be approximately solved with the solutions being expressed in terms of the Bessel functions of irrational orders, which in turn generate a dynamical structure involving collective multiple chiral doublets bands or wobbling bands. An example for $^{132,133}$La will be discussed.
The relativistic configuration-interaction density functional theory, which combines the advantages of configuration-interaction shell model and relativistic density functional theory and allows to treat nuclear spectroscopic properties in the laboratory frame, is applied to investigate the chiral rotation in $^{130}$Cs. Both energy spectra and transition probabilities of the chiral doublets are well reproduced. The chiral geometries of the chiral doublets in $^{130}$Cs are clearly illustrated by the K plot and azimuthal plot. The present work provides the first microscopic and quantized description for the nuclear chiral rotation.
Chirality is a subject of general interest in natural science. Nuclear chirality was first predicted in 1997, and up to now 62 candidate chiral doublet bands in 49 nuclei have been reported. Based on the covariant density functional theory, a phenomenon named multiple chiral doublets (MχD), i.e., more than one pair of chiral doublet bands in one single nucleus, was predicted in 2006, which has attracted extensive attentions. In 2016, the MχD with octupole correlations were reported in $^{78}$Br. This observation indicates that nuclear chirality can be robust against the octupole correlations, which encourages the exploration of the simultaneous chiral and reflection symmetry breaking in a reflection-asymmetric triaxial nucleus.
In this talk, I will briefly introduce the developed reflection-asymmetric triaxial particle rotor model (RAT-PRM). This model is applied to investigate the nuclear Chirality-Parity (ChP) violation, a simultaneous breaking of chiral and reflection symmetries in the intrinsic frame. A new symmetry for an ideal ChP violation system is found and the corresponding selection rules of the electromagnetic transitions are derived. The fingerprints for the ChP violation including the nearly degenerate quartet bands and the selection rules of the electromagnetic transitions are provided. These fingerprints are examined for ChP quartet bands by taking a two-j shell $h_{11/2}$ and $d_{5/2}$ with typical energy spacing for A = 130 nuclei.
Following the reports of candidate chiral doublet bands observed in cesium isotopes, the possible chiral candidates and the evolution of three-dimensional rotation in $^{120-134}{\textrm{Cs}}$ are investigated within the microscopic three-dimensional tilted axis cranking covariant density functional theory (3DTAC-CDFT). By investigating the evolution of the polar angle $\theta$ and azimuth angle $\varphi$ as a function of rotational frequency $\hbar\omega$, the transition from the planar rotation to the chiral rotation has been found in $^{121-133}{\textrm{Cs}}$. The corresponding critical rotational frequency $\omega_{\textrm{crit}}$ of the appearance of chiral aplanar rotation decreases as neutron number increases, which can be attributed to the neutrons in $(gd)$ and $(sd)$ shells having smaller angular momentum components along both the short and long axes, and larger components along medium axis, respectively. In comparison, only planar rotation has been obtained in $^{120,134}{\textrm{Cs}}$. With these interpretations, the obtained $I\sim\hbar\omega$ and energy spectra as well as $B(M1)/B(E2)$ values show reasonable agreement with the available experimental data. In addition, the evolution of quadrupole deformation $\beta$ and triaxial deformation $\gamma$ are also discussed.
Based on covariant density functional theory, the pairing correlations are studied with the shell-model-like approach (SLAP), in which the particle numbers are conserved strictly, and the blocking effect is handled precisely.
The temperature-dependent S-shaped heat capacity curve was obtained for even-even nuclei, which predicts the occurrence of the pairing phase transition near the critical temperature. Our calculations yield a smooth temperature-dependent pairing gap, representing the gradual transition from the superfluid to the normal phase. We also calculate the temperature dependence of the different seniority components and investigate the contribution of the different seniority states to the thermodynamic quantities, such as entropy and energy density. We analyze the microscopic mechanism of the onset of the pairing correlation phase transition and find that one pair of particle-broken states and two pairs of particle-broken states play a crucial role in the appearance of the S-shaped heat capacity curve for even-even nuclei.
In the case of odd-A nuclei, we find that one pair of particle-broken states still lead to an S-shaped heat capacity curve. By studying the structure of the single-particle energy level of the odd-A nucleus and the precise blocking of the single-particle level near the Fermi surface, our work concludes that the S-shaped heat capacity of the odd-A nucleus may be suppressed by its blocking effect. However, due to the influence of the single-particle level structure, the S-shaped heat capacity curve of $^{171}$Yb appears, which should have been smoothed out by the blocking effect.
Also, since Ehrenfest's definition does not apply to small systems, Lee-Yang's theorem is applied to classify the pairing phase transition by analyzing the distribution of zeros of the partition function in the complex temperature plane. The distribution of zeros of the partition function converges with increasing particle numbers and illustrates the characteristics of the phase transition. In our calculations, we determine the first order of the phase transition near the critical temperature. Different seniority states show the pairing phase transition from a superfluid to a normal phase, ranging from fully paired states to completely unpaired states.
The microscopic understanding on the influence of the pairing correlations or the superfluidity on the nuclear chiral rotation has been a longstanding and challenging problem. Based on the three-dimensional cranking covariant density functional theory, a shell-model-like approach with exact particle number conservation is implemented to take into account the pairing correlations and applied for the chiral doublet bands built on the configuration $\pi h_{11/2}^2\otimes\nu h_{11/2}^{-1}$ in $^{135}$Nd. The data available, including the $I−\omega$ relation, as well as the electromagnetic transition probabilities $B(M1)$ and $B(E2)$, are well reproduced. It is found that the superfluidity can reduce the critical frequency and make the chiral rotation easier. The mechanism is that the particle/hole alignments along the short/long axis are reduced by the pairing correlations, resulting in the enhanced preference of the collective rotation along the intermediate axis, and inducing the early appearance of the chiral rotation.
Results from recent experiments studying nuclei in the $^{78}$Ni region suggest that the N = 50 shell closure persists, in agreement with state-of-the-art shell model calculations. However, how collectivity manifests and evolves in this region of the Segrè chart is still an open question, particularly concerning phenomena such as vibrational modes, triaxiality and shape coexistence. This is especially true in the Zn isotopic chain in the neutron-rich region, in which even definitive spin assignments are unavailable except for the very low-lying states.
In this talk, I will present the results of a recent experiment performed at the TRIUMF laboratory (Vancouver, Canada) using the GRIFFIN γ–ray spectrometer. The excited states of $^{74}$Zn were investigated via γ–ray spectroscopy following $^{74}$Cu β-decay. By exploiting γ-γ angular correlation analysis, the $2^+_2$, $3^+_1$, $0^+_2$ and $2^+_3$ states in $^{74}$Zn were firmly identified. The γ–ray branching and E2/M1 mixing ratios for transitions de-exciting the $2^+_2$, $3^+_1$ and $2^+_3$ states were measured, allowing for the extraction of relative B(E2) values. In particular, the $2^+_3 \to 0^+_2$ and $2^+_3 \to 4^+_1$ transitions were observed for the first time. The levels observed were organized into rotational-like bands and the results were compared with large-scale shell-model calculations from which the shapes of individual states were determined. Enhanced axial shape asymmetry (triaxiality) is suggested to characterize $^{74}$Zn in its ground state. Furthermore, an excited K = 0 band with different configurations is identified. A shore of the N = 40 island of inversion appears to manifest above Z = 28, previously thought as its northern limit in the nuclide chart.
Inspired by the newly discovered experimental data, the nuclear structure of the light superheavy nuclei are studied. The single-particle structure, high-K isomers, rotational properties and α-decay energies of the light superheavy nuclei are investigated within the framework of the cranked shell model (CSM) with pairing correlation treated by a particle-number-conserving (PNC) method. High-order deformation ε$_6$ plays an important role both in the single-particle orbitals and in the multi-particle states of the light superheavy mass region. A reverse of the single-particle energy levels is resulted by including ε$_6$ deformation, based on which the microscopic mechanism of the identical bands between Lr isotopes is explained. The reflection asymmetric octupole deformation is used to explain the variation of the rotational bands versus the rotational frequency in U and Pu isotopes. Pairing reduction of the multi-particle bands in the light superheavynuclei is discussed in details.
High-spin rotational bands in rare-earth Er, Tm and Yb isotopes are investigated by (1) the cranked relativistic Hartree-Bogoliubov approach with Lipkin-Nogami method, (2) the cranking covariant density functional theory with pairing correlations treated by a shell-model-like approach or the so called particle-number conserving (PNC) method, and (3) cranked shell model (CSM) with pairing correlations treated by the PNC method. A detailed comparison between these three models in the description of the ground state bands of even-even Er and Yb isotopes is performed. The similarities and differences between these models in the description of the moments of inertia, the band crossings, equilibrium deformations and pairing energies of even-even nuclei under study are discussed. On average, a comparable accuracy of the description of available experimental data is achieved in these models. However, the differences between model predictions become larger above the first band crossings. Because of time-consuming nature of the two CDFT-based models, systematic study of the rotational properties of both ground state and excited state bands in odd-mass Tm nuclei is carried out only by the PNC-SCM. With few exceptions, the rotational properties of experimental 1-quasiparticle and 3-quasiparticle bands in $^{165,167,169,171}$Tm are reproduced reasonably well.
PSM calculation for A~130 region
Several new bands have been identified in $^{130}$Ba, among which there is one with band-head spin 8$^+$. Its properties are in agreement with the Fermi-aligned νh$^2_{11/2}$, 7/2$^−$[523] ⊗ 9/2$^−$[514] Nilsson configuration. This is the first observation of a two-quasiparticle t-band in the A=130 mass region in experiment. The observed bands are discussed using several theoretical models including projected shell model (PSM). With the new results, $^{130}$Ba presents one of the best and most complete sets of collective excitations that a γ-soft nucleus can manifest at medium and high spins, revealing a diversity of shapes and rotations for the nuclei in the A = 130 mass region.
PSM calculation for A~160 region
There have been compelling evidences indicating that deformed single-particle (SP) states in neutron-rich regions are different from those in the stable region. It was pointed out that for the light rare-earth ($_{60}$Nd, $_{62}$Sm, and $_{64}$Gd), neutron rich (N=98−102) nuclei, the Woods–Saxon potential, the Nilsson modified oscillator potential with ‘universal’ parameters, and the folded Yukawa potential all failed to describe the correct ordering of neutron SP states. The location and size of deformed shell gaps in this mass region are under current debate. We propose a modification for the ‘standard’ Nilsson parameters of Bengtsson and Ragnarsson introduced in 1985. The proposed N-dependent spin–orbit interaction causes directly changes in deformed shell gaps with neutron number and deformation. By applying the modified Nilsson parameters to generate deformed bases for the projected shell model, we demonstrate that the calculation can explain consistently the current experimental data, including the ground state configuration in odd-neutron nuclei, upbending of the yrast moment of inertia at higher spins and the energies of 2-quasineutron 6$^−$ and 4$^−$ isomers in even–even nuclei.
The exotic collective excitation modes called chirality and wobbling have been experimentally investigated in the $^{135,136,137}$Nd and $^{135}$Pr nuclei. Many new bands have been identified which were explored by the constrained tilted axis cranking covariant density functional theory, as well as by newly developed particle-rotor models. A good agreement between experimental data and the results of models calculations is achieved, supporting thus the existence of multiple chiral doublet bands phenomenon in the $^{135,136,137}$Nd nuclei. The wobbling motion in the $^{136}$Nd and $^{135}$Pr nuclei will also be discussed in the present talk.
The very neutron-deficient strongly-deformed $^{119}$Cs, $^{119}$Ba, and $^{118}$Cs nuclei have been studied with the JUROGAM 3 + MARA recoil-mass separator setup. Two most complete level schemes from low to high spin in the odd-even and odd-odd proton-rich cesium nuclei, $^{119}$Cs and $^{118}$Cs, were observed. One new rotational band and several low-lying states were newly identified in $^{119}$Ba. The configurations of the observed bands were assigned based on the analysis of the alignment properties of the bands, on systematics and on particle number conserving cranked shell model (PNC-CSM) calculations.
We observed the chiral bands built on a configuration with only protons in the transient backbending regime for the first time in $^{119}$Cs. One new band which is nearly degenerate to the strongly-coupled $\pi g_{9/2}[404]9/2$ band of $^{119}$Cs was identified. It exhibits a backbending similar to that of one known band, at nearly constant rotation frequency. Tilted axis cranking covariant density functional theory with pairing correlations and PNC-CSM calculations show that the backbending is induced by the rotational alignment of two $h_{11/2}$ protons, whose angular momenta reorient from the short to the intermediate axis, in a plane orthogonal to the angular momentum of the strongly-coupled $g_{9/2}$ proton which keeps aligned along the long axis. The total spin points in 3D, inducing the breaking of the chiral symmetry and giving rise to nearly degenerate doublet bands. We can conclude that the chirality in nuclei is a general phenomenon, being robust and present not only in a two-component quantum many-body system, but also in a one-component quantum many-body system as well.
It was recently proposed that odd-mass triaxial nuclei can exhibit wobbling motion at low spins. Exited bands identified as wobbling were reported in several nuclei, including in the two gold isotopes of $^{183,187}$Au. The Au isotopes are good candidates for studying such phenomena as the nuclei in this mass region are strongly affected by the triaxial degree of freedom. An excited band with $\pi$h$_{9/2}$ nature in $^{183}$Au was associated with transverse wobbling, where the odd proton is oriented along the short nuclear axis. In contrast in $^{187}$Au an excited band with $\pi$h$_{9/2}$ nature was associated with longitudinal wobbling where the proton is oriented along the intermediate axis. The proposed different alignment of the angular momentum of the valence proton in these two Au isotopes is quite interesting, as the proton Fermi level is expected to be similar. The most important experimental evidence supporting the proposed wobbling nature of the excited bands in these two Au isotopes was the large magnitude of the measured mixing ratios of transitions linking the excited and the yrast $\pi$h$_{9/2}$ bands. Recently, the mixing ratios for these linking transitions in $^{187}$Au were measured again. The new results suggested that the M1 component was dominant, thus ruling out the previously proposed wobbling nature. These contrasting results highlight the difficulty of such measurements and suggest that more investigations are needed, including in particular the Au isotopes.
In the present work excited low-spin states of $^{193}$Au were studied using the tape station set-up at iThemba LABS. These states were populated in the $\beta$-decay that followed the $^{197}$Au(p,5n)$^{193}$Hg reaction at E$_p$=50 MeV. The emitted γ rays were detected with three Compton-suppressed clover detectors and one Compton-suppressed segmented clover. In addition a Si(Li) detector was used for the emitted internal conversion elections. More than 100 new transitions were placed in $^{193}$Au at low spins. Mixing ratios were measured for several transitions in $^{193}$Au through internal conversion analysis. Experimental results on the low-spin states in $^{193}$Au will be presented and discussed.
The nuclei in the mass region A ∼ 125 are susceptible to shape changes due to presence of the unique parity h$_{11/2}$ orbital. Owing to the larger angular momentum of this orbital compared to the other subshells near the Fermi surface, it serves an essential role in the building up of angular momentum. It is also responsible for the observation of shape changes as protons occupying h$_{11/2}$ orbitals drive the nucleus to a prolate shape while the neutrons drive it to an oblate shape. The interplay of the opposing shape-driving effects leads to the observation of various shape-change phenomena in these nuclei.
The structure of $^{125}$Te is characterized by the coupling of a neutron hole to the even-even $^{126}$Te core. The low-spin region of the level scheme exhibits a vibrational structure. In the framework of the IBFM, this nucleus lies in the transitional region between the SU(5) (vibrational) and O(6) (γ-unstable) limits, and a transition to a triaxial shape can be expected at certain spin values. In our present work, we have populated the nuclear levels of $^{125}$Te via the reaction $^{124}$Sn(α, 3n), with the 35 MeV α beam provided by the K-130 cyclotron at VECC, India. The de-exciting γ rays were detected by the Indian National Gamma Array (INGA) spectrometer, consisting of 7 Compton-suppressed HPGe detectors- 4 detectors at 90◦, 2 at 125◦ and one detector at 40◦. The data were processed using a digital data acquisition system based on PIXIE-16(XIA LLC, USA) 12-bit 250 MHz digitizer modules. Subsequent analysis, including building the level scheme and assigning spin-parity to the levels, was done using the programs in the RADWARE package.
The previously obtained level scheme was expanded by placing 30 new γ transitions. Based on the positive parity 1/2 and 3/2 levels, two structures have been extended up to spin 19/2$^+$. These two structures have been identified as the favored and unfavored partners of the one-quasiparticle band based on the s$_{1/2}$ orbital. A comparison of the energy levels of the favored band with the ground-state band of the even-even core ($^{126}$Te) showed that this band was of a coupled nature, as also indicated by the moderate magnitude of signature splitting. Additionally, a signature inversion has been observed at spin I=15/2$\hbar$. Signature inversion is observed in nuclei that are soft w.r.t γ deformations and indicative of a rotational alignment of the quasiparticles along the total angular momentum and change to a non-axial shape. Thus, there are indications of the presence of a triaxial deformation in $^{125}$Te. Our work is currently in the preliminary stage, and further calculations are being carried out to further explore the nature of triaxiality in $^{125}$Te.