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 $E_{\textrm{wob}}$ and spin-rotational frequency relations $ω(I)$ are better described than with the leading order Hamiltonian.
The decay rate of a triaxially-deformed proton emitter is calculated in a particle-rotor model, which is based on a deformed Woods-Saxon potential and includes a deformed spin-orbit interaction. The wave function of the $I=7/2^{-}$ ground state of the deformed proton emitter $^{141}$Ho is obtained in the adiabatic limit, and a Green's function technique is used to calculate the decay rate and branching ratio to the first excited 2$^{+}$ state of the daughter nucleus. Only for values of the triaxial angle $γ$ $<5^{\circ}$ is good agreement obtained for both the total decay rate and the 2$^{+}$ …
To study shape fluctuations of nuclei in transitional regions, the collective Hamiltonian method has often been employed. We intend to construct the quadrupole collective Hamiltonian with the collective inertial functions given by the local quasiparticle random-phase approximation (QRPA) based on the Skyrme energy density functional. For this purpose, we first construct a practical framework of Skyrme QRPA for triaxial nuclear shapes with the finite amplitude method (FAM). We show quadrupole strength functions for a triaxial superfluid nucleus $^{188}$Os and the Thouless-Valatin rotational mom…
The impurity effect of hyperon on atomic nuclei has received a renewed interest in nuclear physics since the first experimental observation of appreciable reduction of $E2$ transition strength in low-lying states of hypernucleus $^{7}_Λ$Li. Many more data on low-lying states of $Λ$ hypernuclei will be measured soon for $sd$-shell nuclei, providing good opportunities to study the $Λ$ impurity effect on nuclear low-energy excitations. We carry out a quantitative analysis of $Λ$ hyperon impurity effect on the low-lying states of $sd$-shell nuclei at the beyond-mean-field level based on a relativi…
The effective field theory (EFT) for triaxially deformed even-even nuclei is generalized to include the vibrational degrees of freedom. The pertinent Hamiltonian is constructed up to next-to-leading order. The leading order part describes the vibrational motion, and the NLO part couples rotations to vibrations. The applicability of the EFT Hamiltonian is examined through the description of the energy spectra of the ground state bands, $γ$-bands, and $K=4$ bands in the $^{108, 110, 112}$Ru isotopes. It is found that by taking into account the vibrational degrees of freedom, the deviations for h…
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…
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…
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 implemented a variation after projection (VAP) algorithm based on a triaxially deformed Hartree-Fock-Bogoliubov vacuum state. This is the first projected mean field study that includes all the quantum numbers (except parity), i.e., spin ($J$), isospin ($T$) and mass number ($A$). Systematic VAP calculations with $JTA$-projection have been performed for the even-even $sd$-shell nuclei with the USDB Hamiltonian. All the VAP ground state energies are within 500 keV above the exact shell model values. Our VAP calculations show that the spin projection has two important effects: (1) the spin pro…
The five-dimensional quadrupole collective model based on energy density functionals (EDF) has often been employed to treat long-range correlations associated with shape fluctuations in nuclei. Our goal is to derive the collective inertial functions in the collective Hamiltonian by the local quasiparticle random phase approximation (QRPA) that correctly takes into account time-odd mean-field effects. Currently, practical framework to perform the QRPA calculation with the modern EDFs on the $(β,γ)$ deformation space is not available. Toward this goal, we develop an efficient numerical method to…
The survey of different configurations near Fermi surface of 138Nd results in 12 lowest configurations, at both positive- and negative-deformations. These are calculated to be the energetically lowest configurations. The results show that, for both EDFs, the rotational states based on positive-minimum, which is at gamma~35, are lower than the respective configurations with negative-deformation. The general trends of the spin-versus-omega curve, and the energy-versus-spin curve reproduce well those of the experimental data. Further, for the observed bands `T1-T8', the calculated results using S…
A fully symmetry unrestricted Time-Dependent Density Functional Theory extended to include pairing correlations is used to calculate properties of the isovector giant dipole resonances of the deformed open-shell nuclei 172Yb (axially deformed), 188Os (triaxially deformed), and 238U (axially deformed), and to demonstrate good agreement with experimental data on nuclear photo-absorption cross-sections for two different Skyrme force parametrizations of the energy density functional: SkP and SLy4.
A possible scheme of realizing shell model calculations for heavy nuclei is based on a deformed basis and the projection technique. Here we present a new development for odd-odd nuclei, in which one starts with triaxially-deformed multi-quasi-particle configurations, builds the shell-model space through exact three-dimensional angular-momentum-projection, and diagonalizes a two-body Hamiltonian in this space. The model enables us to study the old problem of signature inversion from a different view. With an excellent reproduction of the experimental data in the mass-130 region, the results ten…
The University of Tokyo（东京大学）；University of Tsukuba（筑波大学）；RIKEN Nishina Center（理化学研究所西崎中心）；Grand Accélérateur National d’Ions Lourds（大离子加速器国家设施）；Technische Universität Darmstadt（达姆施塔特工业大学）；CEA（法国原子能委员会）；CNRS（法国国家科学研究中心）；IN2P3（粒子物理与核物理研究所）；Institut für Kernphysik, Technische Universität Darmstadt（达姆施塔特工业大学核物理研究所）；IRFU, CEA, Université Paris-Saclay（巴黎-萨克雷大学CEA IRFU）；Japan Atomic Energy Agency（日本原子力研究开发机构）；Keio University（庆应义塾大学）
We show that the lowest-energy solution of the Hartree-Fock-Bogoliubov (HFB) equation has the even particle-number parity as long as the time-reversal symmetry is conserved in the HFB Hamiltonian without null eigenvalues. Based on this finding, we give a rigorous foundation of a method for solving the HFB equation to describe the ground state of odd-mass nuclei by employing a time-reversal anti-symmetric constraint operator to the Hamiltonian, where one obtains directly the ground state as a self-consistent solution of the cranked-HFB-type equation. Numerical analysis is done for the neutron-r…
We develop an effective field theory (EFT) for deformed odd-mass nuclei. These are described as an axially symmetric core to which a nucleon is coupled. In the coordinate system fixed to the core the nucleon is subject to an axially symmetric potential. Power counting is based on the separation of scales between low-lying rotations and higher-lying states of the core. In leading order, core and nucleon are coupled by universal derivative terms. These comprise a covariant derivative and gauge potentials which account for Coriolis forces and relate to Berry-phase phenomena. At leading order, the…
An article by Covello, Gargano and Itaco [Phys.Rev. C 56, 3092 (1997)] tries to find evidence for the important role of the residual tensor force between the valence proton and neutron in doubly odd deformed nuclei. It is shown that observable effects discussed by these authors do not fully justify their rather strong conclusions.
A new theoretical framework named as deformed base antisymmetrized molecular dynamics that uses the localized triaxially deformed Gaussian as the single particle wave packet is presented. The model space enables us to describe sufficiently well the deformed mean-field structure as well as the cluster structure and their mixed structure within the same framework. The improvement over the original version of the antisymmetrized molecular dynamics which uses the spherical Gaussian is verified by the application to $^{20}{\rm Ne}$ nucleus. The almost pure $α+ ^{16}{\rm O_{g.s}}$ cluster structure …
The boson-pair coherent state developed in Part (I) is generalized to the case in which the state is a mixture of even-boson and odd-boson number states. A general framework is shown and its three concrete examples are discussed. Main idea is in the application of the deformed boson scheme presented by the present authors.
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.
In order to describe the exotic nuclear structure in unstable odd-$A$ or odd-odd nuclei, the deformed relativistic Hartree Bogoliubov theory in continuum has been extended to incorporate the blocking effect due to the odd nucleon. For a microscopic and self-consistent description of pairing correlations, continuum, deformation, blocking effects, and the extended spatial density distribution in exotic nuclei, the deformed relativistic Hartree Bogoliubov equations are solved in a Woods-Saxon basis in which the radial wave functions have a proper asymptotic behavior at large $r$. The formalism an…
We have investigated the effects of the neutron-proton interaction in several doubly odd deformed nuclei within the framework of the particle-rotor model. In this paper, we show some selected results of our study which evidence the importance of the tensor-force effects.
An idea of the deformed boson scheme developed by the present authors is applied to the case of boson-pair coherent state. In this state, even and odd boson number difference is stressed and various concrete examples of the boson-pair coherent states are shown. Concerning some of them, mutual relations are investigated in connection to the su(1,1)-algebra. A formulation in terms of the MYT boson mapping is also performed.
A comprehensive decay scheme of $^{93}$Nb below 2 MeV has been constructed from information obtained with the $^{93}$Nb(n,n$^\prime$$γ$) and $^{94}$Zr(p,2n$γ$$γ$)$^{93}$Nb reactions. Branching ratios, lifetimes, transition multipolarities and spin assignments have been determined. From $M1$ and $E2$ strengths, fermionic-bosonic excitations of isoscalar and isovector character have been identified from the weak coupling $\pi1g_{9/2}$$\otimes$$^{92}_{40}$Zr and $\pi2p_{1/2}^{-1}$$\otimes$$^{94}_{42}$Mo configurations. A microscopic interpretation of such excitations is attained from shell-model …
Partial dynamical symmetry (PDS) is shown to be relevant for describing the odd-even staggering in the $γ$-band of $^{156}$Gd while retaining solvability and good SU(3) symmetry for the ground and $β$ bands. Several classes of interacting boson model Hamiltonians with SU(3) PDS are surveyed.
A basic idea is proposed for extending the formalism of Part (II) of the series of our paper to the case of the parameter-dependent deformation. It is stressed that, through this extension, the variety of the application increases. Further, an answer for the problem mentioned in Part (II) is presented.
We have investigated the Gamow-Teller transitions from ${}^{22}\textrm{Ne}$ to ${}^{22}\textrm{Na}$ by applying the isospin projected antisymmetrized molecular dynamics combined with generator coordinate method. We have found that the GT strength from $^AZ(J^πT)={}^{22}\textrm{Ne}(0_1^+1)$ is fragmented into two final states ${}^{22}\textrm{Na}(1_{1,2}^+0)$, which belong to $K=0$ and $K=1$ bands constructed by a prolately deformed $^{20}$Ne core with a $S=1$ proton-neutron ($pn$) pair. Coupling of the intrinsic-spin of the $pn$ pair with the core deformation play an important role in the GT fr…
In two previous papers, the Kerman-Klein-Donau-Frauendorf (KKDF) model was used to study rotational bands of odd deformed nuclei. Here we describe backbending for odd nuclei using the same model. The backbending in the neighboring even nuclei is described by a phenomenological two band model, and this core is then coupled to a large single-particle space, as in our previous work. The results obtained for energies and M1 transition rates are compared with experimental data for 165Lu and for energies alone to the experimental data for 179W. For the case of 165Lu comparison is also made with prev…
We study the local violation of P and CP invariance in heavy ion collisions as observed at RHIC and LHC using a simple "deformed QCD" model. This model is a weakly coupled gauge theory, which however has all the relevant crucial elements allowing us to study difficult and nontrivial questions which are known to be present in real strongly coupled QCD. Essentially, we want to understand the physics of long range order in form of coherent low dimensional vacuum configurations observed in Monte Carlo lattice simulations. Apparently precisely such kind of configurations are responsible for suffici…
The odd-even differences in stability peninsula for $106 \leqslant Z \leqslant 112$ region with the deformed relativistic Hartree-Bogoliubov theory in continuum
Examples of the change of neutron shell-structure in both weakly-bound and resonant neutron one-particle levels in nuclei towards the neutron drip line are exhibited. It is shown that the shell-structure change due to the weak binding may lead to the deformation of those nuclei with the neutron numbers $N \approx$ 8, 20, 28 and 40, which are known to be magic numbers in stable nuclei. Nuclei in the "island of inversion" are most easily and in a simple manner understood in terms of deformation. As an example of spectroscopic properties other than single-particle energies, magnetic moments of so…
Inter-band B(E2) transition strengths between different normal parity bands in 163Dy and 165Er are described using the pseudo-SU(3) model. The Hamiltonian includes Nilsson single-particle energies, quadrupole-quadrupole and pairing interactions with fixed, parametrized strengths, and three extra rotor terms used to fine tune the energy spectra. In addition to inter-band transitions, the energy spectra and the ground state intra-band B(E2) strengths are reported. The results show the pseudo-SU(3) shell model to be a powerful microscopic theory for a description of the normal parity sector in he…
While the angular-momentum projection is a common tool for theoretical nuclear structure studies, a large amount of computations are required particularly for triaxially deformed states. In the present work, we clarify the conditions of the exactness of quadratures in the projection method. For efficient computation, the Lebedev quadrature and spherical $t$-design are introduced to the angular-momentum projection. The accuracy of the quadratures is discussed in comparison with the conventional Gauss-Legendre and trapezoidal quadratures. We found that the Lebedev quadrature is the most efficien…
Beta-decay properties of neutron-rich Zr and Mo isotopes are investigated within a microscopic theoretical approach based on the proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual separable particle-hole and particle-particle forces are also included in the formalism. The structural evolution in these isotopic chains including both even and odd isotopes is analyzed in terms of the equilibrium deformed shapes. Gamow-Teller strength distributions, beta…
Beyond mean-field methods are very successful tools for the description of large-amplitude collective motion for even-even atomic nuclei. The state-of-the-art framework of these methods consists in a Generator Coordinate Method based on angular-momentum and particle-number projected triaxially deformed Hatree-Fock-Bogoliubov (HFB) states. The extension of this scheme to odd-mass nuclei is a long-standing challenge. We present for the first time such an extension, where the Generator Coordinate space is built from self-consistently blocked one-quasiparticle HFB states. One of the key points for…
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…
Systematic study on α-decay fine structure is presented for the first time in the case of odd-even nuclei in the range 83 \leq Z \leq 101. The model used for the study is the recently proposed Coulomb and proximity potential model for deformed nuclei (CPPMDN), which employs deformed Coulomb potential, deformed two term proximity potential and centrifugal potential. The computed partial half lives, total half lives and branching ratios are compared with experimental data and are in good agreement. The standard deviation of partial half-life is 1.08 and that for branching ratio is 1.21. Our form…
The low-energy spectra and B(E2) electromagnetic transition strengths of 159Eu, 159Tb and 159Dy are described using the pseudo SU(3) model. Normal parity bands are built as linear combinations of SU(3) states, which are the direct product of SU(3) proton and neutron states with pseudo spin zero (for even number of nucleons) and pseudo spin 1/2 (for odd number of nucleons). Each of the many-particle states have a well-defined particle number and total angular momentum. The Hamiltonian includes spherical Nilsson single-particle energies, the quadrupole-quadrupole and pairing interactions, as wel…
Pseudo-SU(3) shell-model results are reported for M1 excitation strengths in 157-Gd, 163-Dy and 169-Tm in the energy range between 2 and 4 MeV. Non-zero pseudo-spin couplings between the configurations play a very important role in determining the M1 strength distribution, especially its rapidly changing fragmentation pattern which differs significantly from what has been found in neighboring even-even systems. The results suggest one should examine contributions from intruder levels.
In this paper, we present some results of a particle-rotor model study of 176Lu. In particular, we consider the two lowest K=1+ balds, which present a rather large odd-even staggering. This effect, which may be traced to direct Coriolis coupling with Newby-shifted K=0+ bands, is of great interest since it gives information on the proton-neutron interaction. We use both zero-range and finite-range interactions with particular attention focused on the role of the tensor force. Comparison of the calculated results with experimental data evidences the importance of the tensor-force effects in the …
Spectroscopy of nuclei in the mass range A=76 to A=130, participating in double beta decay processes are studied in the framework of the self-consistent deformed Hartree-Fock (HF) and angular momentum (J) projection model. Spectra of ground bands have been studied and compared with available experimental results for even-even parent and daughter as well as for intermediate odd-odd nuclei. To test the reliability of the wave functions we have also calculated the reduced E2 transition matrix elements, electric quadrupole moments and magnetic dipole moments for these nuclei. The calculated result…
The M1 transitions between low-lying T=1 and T=0 states in deformed odd-odd N=Z nuclei are analyzed in the frames of the rotor-plus-particle model. Using the representation of an explicit coupling of angular momenta we show that strong coupling of the quasideuteron configurations to the axially deformed core results in a distribution of the total 0+ --> 1+ strength among a few low-lying 1+ states. Simple analytical formulae for B(M1) values are derived. The realization of the M1 sum rule for the low-lying 1+,T=0 states is indicated. The calculated B(M1) values are found to be in good agreement…
Possible deformation of odd-N nuclei with N $\approx$ 28 towards the neutron drip line is investigated using the Nilsson diagram based on deformed Woods-Saxon potentials. Both weakly-bound and resonant one-particle levels are properly obtained by directly solving the Schrödinger equation in mesh of space coordinate with the correct boundary condition. If we use the same diffuseness of the potential as that of $β$-stable nuclei, the energy difference between the neutron 2p$_{3/2}$ and 1f$_{7/2}$ levels becomes very small or the N=28 energy gap almost disappears, as the binding energies of those…
The microscopic analysis of experimental data in $^{160,162}$Tm is presented within the two-quasiparticle-phonon model. The model includes the interaction between odd quasiparticles and their coupling with core vibrations. The coupling explains naturally the attenuation of the Coriolis interaction in rotating odd-odd nuclei. It is shown that the competition between the Coriolis and neutron-proton interactions is responsible for the signature inversion phenomenon.
Masses of superheavy (SH) nuclei with $Z=98-128$, including odd and odd-odd nuclei, are systematically calculated within the microscopic-macroscopic model based on the deformed Woods-Saxon potential. Ground states are found by minimizing energy over deformations and configurations. Pairing in odd particle-number systems is treated either by blocking or by adding the BCS energy of the odd quasiparticle. Three new parameters are introduced which may be interpreted as the constant mean pairing energies for even-odd, odd-even and odd-odd nuclei. They are adjusted by a fit to masses of heavy nuclei…
Shell structure and shape transition in odd-$Z$ superheavy nuclei with proton numbers $Z=117, 119$: insights from deformed relativistic Hartree-Bogoliubov in continuum
It is proposed that the odd-even staggering (OES) in the $γ$- bands of heavy deformed nuclei can be reasonably characterized by a discrete approximation of the fourth derivative of the odd-even energy difference as a function of angular momentum $L$. This quantity exhibits a well developed staggering pattern (zigzagging behavior with alternating signs) in rare earth nuclei and actinides with long $γ$- bands ($L\geq 10$). It is shown that the OES can be interpreted reasonably as the result of the interaction of the $γ$ band with the ground band in the framework of a Vector Boson Model with SU(3…
In the present work we have reported comprehensive analysis of recently available experimental data [H.M. David et al., Phys. Lett. B {\bf 726}, 665 (2013)] for high-spin states up to $17^+$ with $T=0$ in the odd-odd $N=Z$ nucleus $^{62}$Ga using shell model calculations within the full $f_{5/2}pg_{9/2}$ model space and deformed shell model based on Hartee-Fock intrinsic states in the same space. The calculations have been performed using jj44b effective interaction developed recently by B.A. Brown and A.F. Lisetskiy for this model space. The results obtained with the two models are similar an…
Ground-state properties of deformed proton-rich odd-Z nuclei in the region $59 \leq Z \leq 69$ are described in the framework of Relativistic Hartree Bogoliubov (RHB) theory. One-proton separation energies and ground-state quadrupole deformations that result from fully self-consistent microscopic calculations are compared with available experimental data. The model predicts the location of the proton drip-line, the properties of proton emitters beyond the drip-line, and provides information about the deformed single-particle orbitals occupied by the odd valence proton.
The coordinate-space Hartree-Fock-Bogoliubov (HFB) approach with quasiparticle blocking has been applied to study the odd-A weakly bound nuclei $^{17,19}$B and $^{37}$Mg, in which halo structures have been reported in experiments. The Skyrme nuclear forces SLy4 and UNEDF1 have been adopted in our calculations. The results with and without blocking have been compared to demonstrate the emergence of deformed halo structures due to blocking effects. In our calculations, $^{19}$B and $^{37}$Mg have remarkable features of deformed halos.
We consider deformed nuclei within an effective theory that exploits the small ratio between rotational and vibrational excitations. For even-even nuclei, the effective theory predicts small changes in the rotational constants of bands built on multi-phonon excitations that are linear in the number of excited phonons. In 166Er and 168Er, this explains the main variations of the rotational constants of the two-phonon gamma vibrational bands. In 232Th, the effective theory correctly explains the trend that the rotational constants decrease with increasing spin of the band head. We also study the…
