Constraining the quadrupole deformation of atomic nuclei with relativistic nuclear collisions
Giuliano Giacalone
备注：
follows up on arXiv:1910.04673. v1: 18 pages; 14 figures; v2: 18 pages; 15 figures; Overall minor improvements. Removed preliminary STAR data points at the request of the experimental collaboration; v3: published version
Preliminary data by the STAR collaboration at the BNL Relativistic Heavy Ion Collider shows that the elliptic flow, $v_2$, and the average transverse momentum, $\langle p_t \rangle$, of final-state hadrons produced in high-multiplicity $^{238}$U+$^{238}$U collisions are negatively correlated. This observation brings experimental evidence of a significant prolate deformation, $β\approx 0.3$, in the colliding $^{238}$U nuclei. I show that a quantitative description of this new phenomenon can be achieved within the hydrodynamic framework of heavy-ion collisions, and that thus such kind of data in the context of high-energy nuclear experiments can help constrain the quadrupole deformation of the colliding species.
The self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. with the set DF3-a of parameters fixed previously is used to calculate three kinds of quadrupole moments. At first, we examined systematically quadrupole moments of odd neighbors of semi-magic lead and tin isotopes and $N=50,N=82$ isotones. Second, we found quadrupole moments of the first $2^+$ states in the same two chains of isotopes. Finally, we evaluated quadrupole moments of odd-odd nuclei neighboring to double magic ones. Reasonable agreement with available experimental data has bee…
Ground state quadrupole moments of odd-odd near double magic nuclei are calculated in the approximation of no interaction between odd particles. Under such a simple approximation, the problem is reduced to the calculations of quadrupole moments of corresponding odd-even nuclei. These calculations are performed within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. with the known DF3-a parameters. A reasonable agreement with the available experimental data has been obtained for odd-odd nuclei and odd near-magic nuclei investigated. The …
A very good linear correlation is found between the four-point charge radius relation $δR_{2p-2n}(Z,N)$ with that of quadrupole deformation data in even-even nuclei. This results in a further improved charge radius relation that holds in a precision of about 5$\times 10^{-3}$ fm. Such correlations are also seen in global nuclear models, their precisions, however, are not enough to be consistent with the experimental data. The new relation between charge radii and deformation of even-even nuclei can be generalized to the reduced electric quadrupole transition probability $B(E2)$ between the fir…
The quadrupole moments of odd neighbors of semi-magic lead and tin isotopes and $N=50,N=82$ isotones are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. Two sets of parameters, DF3 and DF3-a, fixed previously are used. They differ by the spin-orbit and effective tensor force parameters, the latter being significantly bigger in the DF3-a functional. Results for the two functionals turned out to be rather different. The functional DF3-a leads to quadrupole moments in reasonable agreement with the experimental ones for m…
We perform shell model calculations in odd-odd nuclei using a quadrupole-quadrupole interaction and single-particle splittings chosen so as to obtain the SU(3) results. Elliott had shown that such an interaction gives rotational bands for which the energies go as I(I+1). This is certainly true for eve-even and for odd-even or even-odd nuclei with $K\neq 1/2$. We have looked at odd-odd nuclei e.g $^{22}Na$ and found somewhat different behaviour. In $^{22}Na$ the $I=1^+_1 T=0$ and $I=0^+_1 T=1$ states are degenerate, and a rotational band built on the $I=0^+_1 T=1$ state behaves in a normal fash…
Excitation energies and transition probabilities of the first $2^+$ excitations in even tin and lead isotopes as well as the quadrupole moments of odd neighbors of these isotopes are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. The effect of the density dependence of the effective pairing interaction is analyzed in detail by comparing results obtained with volume and surface pairing. The effect is found to be noticeable. For example, the $2^+$-energies are systematically higher at 200-300 keV for the volume paring …
The influence of quadrupole and hexadecapole residual interactions on rotational bands is investigated in a single-j shell model. An exact shell-model diagonalization of quadrupole-plus-hexadecapole interaction can sometimes produce a staggering of energy levels in the yrast bands.
The contribution of quantum shape fluctuations to inertial properties of rotating nuclei has been analyzed for QQ-nuclear interaction using the random phase approximation (RPA). The different recipes to treat the cranking mean field plus RPA problem are considered. The effects of the dN=2 quadrupole matrix elements and the role of the volume conservation condition are discussed.
Properties of the ground and several collective excited states of the light nuclei ^{30,32,34}Mg are described in the framework of the angular momentum projected Generator Coordinate Method using the quadrupole moment as collective coordinate and the Gogny force as the effective interaction. The calculated excitation energies and B(E2) transition probabilities agree reasonably well with experiment. The results clearly indicate that both the restoration of the rotational symmetry and the quadrupole dynamics are key ingredients for the description of the properties of the above mentioned nuclei.
Transition probabilities of the ground-state bands in $γ$-soft nuclei are studied for the first time using the triaxial projected shell model approach. It is observed that the angular-momentum dependence of the transition quadrupole moment $Q_t$ is related to the triaxial deformation of the nuclear mean-field potential. The introduction of the $γ$-degree of freedom in the shell model basis is shown to have a little influence on the {\it constant} behavior of the low-spin $Q_t$ in a well-deformed nucleus. However, the {\it increasing} collectivity with spin for the low-spin states in a $γ$-soft…
The model of coherent quadrupole and octupole motion (CQOM) is applied to describe non-yrast split parity-doublet spectra in odd-mass nuclei. The yrast levels are described as low-energy rotation-vibration modes coupled to the ground single-particle (s.p.) state, while the non-yrast parity-doublet structures are obtained as higher-energy rotation-vibration modes coupled to excited s.p. states. It is shown that the extended model scheme describes both the yrast and non-yrast quasi parity-doublet spectra and the related B(E1) and B(E2) transition rates in different regions of heavy odd-A nuclei.…
Electric quadrupole (E2) matrix elements provide a measure of nuclear deformation and related collective structure. Ground-state quadrupole moments in particular are known to high precision in many p-shell nuclei. While the experimental electric quadrupole moment only measures the proton distribution, both proton and neutron quadrupole moments are needed to probe proton-neutron asymmetry in the nuclear deformation. We seek insight into the relation between these moments through the ab initio no-core configuration interaction (NCCI), or no-core shell model (NCSM), approach. Converged ab initio …
Elastic electron scattering from oriented odd-Aaxially deformed nuclei is studied in the plane-wave Born approximation. The nuclear structure is described within a microscopic selfconsistent Skyrme deformed Hartree-Fock approximation with pairing correlations. The interference form factors between monopole and quadrupole Coulomb terms that characterize the nuclear response with aligned nuclear targets are shown to increase or decrease the unpolarized cross-section, depending on the sign of the quadrupole deformation. This feature provides valuable information on the nuclear deformation that ca…
A model is presented that simultaneously describes shape coexistence and quadrupole and octupole collective excitations within a theoretical framework based on the nuclear density functional theory and the interacting boson model. An optimal interacting-boson Hamiltonian that incorporates the configuration mixing between normal and intruder states, as well as the octupole degrees of freedom, is identified by means of self-consistent mean-field calculations using a universal energy density functional and a pairing interaction, with constraints on the triaxial quadrupole and the axially-symmetri…
Magnetic dipole and electric quadrupole moments of the mirror nuclei 8Li and 8B are analysed in the framework of the multiparticle shell model by using two approaches : i) the one-particle spectroscopic factors and ii) the one-particle fractional parentage coefficients. These two approaches are compared both each to other and with a microscopic multicluster model. The one-particle nucleon states are calculated taking into account the continuum by the method of the expansion of the Sturm - Liouville functions. The experimental magnetic and quadrupole moments of 8Li and 8Bare reproduced well by …
The contribution of quantum shape fluctuations to inertial properties of rotating nuclei has been analysed within the self-consistent one-dimensional cranking oscillator model. It is shown that in even-even nuclei the dynamical moment of inertia calculated in the mean field approximation is equivalent to the Thouless-Valatin moment of inertia calculated in the random phase approximation if and only if the self-consistent conditions for the mean field are fulfilled.
Large-amplitude dynamics of axial and triaxial quadrupole deformation in 24,26Mg, 24Ne, and 28Si is investigated on the basis of the quadrupole collective Hamiltonian constructed with use of the constrained Hartree-Fock-Bogoliubov plus the local quasiparticle random phase approximation method. The calculation reproduces well properties of the ground rotational bands, and beta and gamma vibrations in 24Mg and 28Si. The gamma-softness in the collective states of 26Mg and 24Ne are discussed. Contributions of the neutrons and protons to the transition properties are also analyzed in connection wit…
The effect of quadrupole-type surface vibrations on the quadrupole response function of heavy nuclei is studied by using a model based on the solution of the linearized Vlasov equation with moving-surface boundary conditions. By using a separable approximation for the residual interaction, an analytical expression is obtained for the moving-surface response function. Comparison of the fixed- and moving-surface strength functions shows that surface vibrations are essential in order to achieve a unified description of the two characteristic features of the quadrupole response: the giant resonanc…
The systematic study of isoscalar (IS) and isovector (IV) giant quadrupole responses (GQR) in normal and superfluid nuclei presented in [G. Scamps and D. Lacroix, Phys. Rev. 88, 044310 (2013)] is extended to the case of axially deformed and triaxial nuclei. The static and dynamical energy density functional based on Skyrme effective interaction are used to study static properties and dynamical response functions over the whole nuclear chart. Among the 749 nuclei that are considered, 301 and 65 are respectively found to be prolate and oblate while 54 do not present any symmetry axis. For these …
We analyze the one-photon mechanism of $e^+e^-$ pair production in the collision of nonrelativistic nuclei. The contribution of electric quadrupole radiation of virtual photon to the corresponding cross section is found. The effect of the finite nuclear size is considered in detail. A comparison is made with the contribution of electric dipole radiation of virtual photon and with the contribution of the two-photon pair production mechanism. It is shown that the contribution of electric quadrupole radiation is dominant in a wide range of relative velocities. The cross section for the production…
The electric quadrupole response is investigated theoretically by HFB and QPM calculations along the Sn isotopic chain with special emphasis on excitations above the first collective state and below the particle threshold. Depending on the asymmetry, additional strength clustering as a group of states similar to the known PDR mode is found. The spectral distributions and electric response functions are discussed. The transition densities of these Pygmy Quadrupole Resonances (PQR) states are closely related to the neutron excess and showing special features being compatible with an oscillation …
We propose a collective model formalism which describes the strong parity shift observed in low-lying spectra of nuclei with octupole deformations together with the fine rotational band structure developed at higher angular momenta. The parity effect is obtained by the Schroedinger equation for oscillations of the reflection asymmetric (octupole) shape between two opposite orientations in an angular momentum dependent double-well potential. The rotational structure is obtained by a collective quadrupole-octupole rotation Hamiltonian. The model scheme reproduces the complicated beat staggering …
We perform Shell Model Monte Carlo calculations of selected N=Z pf-shell nuclei with a schematic hamiltonian containing isovector pairing and quadrupole-quadrupole interactions. Compared to realistic interactions, this hamiltonian does not give rise to the SMMC ``sign problem'', while at the same time resembles essential features of the realistic interactions. We study pairing correlations in the ground states of N=Z nuclei and investigate the thermal dependence of selected observables for the odd-odd nucleus Co54 and the even-even nuclei Zn60 and Ni60. Comparison of the present results to tho…
In this work we examine the possibility to describe yrast positive- and negative-parity excitations of deformed even-even nuclei through a collective rotation model in which the nuclear surface is characterized by triaxial quadrupole and octupole deformations. The nuclear moments of inertia are expressed as sums of quadrupole and octupole parts. By assuming an adiabatic separation of rotation and vibration degrees of freedom we suppose that the structure of the positive- and negative- parity bands may be determined by the triaxial-rigid-rotor motion of the nucleus. By diagonalizing the Hamilto…
Calculations of B(E2)'s and quadruplole moments in the g_{9/2 region below ^{100} Sn are hampered by the fact that the inclusion of the g_{7/2}configuration leads to model spaces that are too large to handle. We therefore examine lighter nuclei if the fp region where one can easily include all the orbitals, f,{}_{7/2}, p_{3/2}, p_{1/2}and f_{5/2} . We perform such calculations but then take a step back and exclude the f_{5/2} orbital. By comparing the 2 calculations we can hope to get insight into the importance of the missing spin-orbit partner in other regions.
We present a study on the isoscalar quadrupole strength in tin nuclei, focusing mainly on the low-energy region. The calculations are performed using the Skyrme type energy density functionals within the fully self-consistent quasiparticle random phase approximation, allowing for a good description of the experimental data for the first 2$^+$ state and the isoscalar giant quadrupole resonance. It is found that the first $2^+$ state and the low-energy quadrupole states between 3 and 6 MeV display an opposite behavior with increasing neutron number. While the strength of the first $2^+$ state de…
The nuclear quadruple moment is a fundamental character associated to the nuclei, this moment is related to the not purely spherical distribution in the nuclei, indeed its measure allows us to survey the geometric deformation of the nuclei of its spherical shape. The measurement methods of the quadruple moment is to study the electrical energy hyperfine interaction between the quadruple moment and the electric field gradient due to atomics electrons, one of the methods is the nuclear quadruple resonance NQR which is observed at the transitions between energy levels splits by the effect of the …
Recent studies at Jefferson Lab show that there are a certain proportion of nucleons in nuclei have momenta greater than the so-called nuclear Fermi momentum $p_{F}$. Based on the transport model of nucleus-nucleus collisions at intermediate energies, nucleon high momentum caused by the neutron-proton short-range correlations in nuclei is constrained by comparing with $π$ and photon experimental data and considering some uncertainties. The high momentum cutoff value $p_{max}$ $\leq$ 2$p_{F}$ is obtained.
The generalized Bohr Hamiltonian was used to describe the low-lying collective excitations in even-even isotopes of Ru, Pd, Te, Ba and Nd. The Strutinsky collective potential and cranking inertial functions were obtained using the Nilsson potential. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not contain any free parameter.
The existence of permanent electric dipole moments (EDMs) and magnetic quadrupole moments (MQMs) violate both time reversal invariance ($T$) and parity ($P$). Following the $CPT$ theorem they also violate combined $CP$ symmetry. Nuclear EDM are completely screened in atoms and molecules while interaction between electrons and MQM creates atomic and molecular EDM which can be measured and used to test CP-violation theories. Nuclear MQM are produced by the nucleon-nucleon $T,P$-odd interaction and by nucleon EDM. In this work we study the effect of enhancement of the nuclear MQM due to the nucle…
Nuclear forces violating parity and time reversal invariance (${\cal P},{\cal T}$-odd) produce ${\cal P},{\cal T}$-odd nuclear moments, for example, the nuclear Schiff moment. In turn, this moment can induce the electric dipole moment in the atom. The nuclear Schiff moment is predicted to be enhanced in nuclei with static quadrupole and octupole deformation. The analogous suggestion of the enhanced contribution to the Schiff moment from the soft collective quadrupole and octupole vibrations in spherical nuclei is tested in this article in the framework of the quasiparticle random phase approxi…
A detailed theoretical investigation of polarization effects in superdeformed nuclei is performed. In the pure harmonic oscillator potential it is shown that when one particle (or hole) with the mass single-particle quadrupole moment q_{nu} is added to a superdeformed core, the change of the electric quadrupole moment can be parameterized as q_{eff}=e(bq_{nu}+a), and analytical expressions are derived for the two parameters, $a$ and $b$. Simple numerical expressions for q_{eff}(q_ν}) are obtained in the more realistic modified oscillator model. It is also shown that quadrupole moments of nucle…
The phenomenological classification of collective quadrupole excitations by means of the Bohr Hamiltonian is reviewed with focus on signatures for triaxility. The variants of the microscopic Bohr Hamiltonian derived by means of the Adiabatic Time Dependent Mean Field theory from the Pairing plus Quadrupole-Quadrupole interaction, the Shell Correction Method, the Skyrme Energy Density Functional, the Relativistic Mean Field Theory, and the Gogny interaction are discussed and applications to concrete nuclides reviewed. The Generator Coordinate Method for the five dimensional quadrupole deformati…
The degrees of freedom associated with shape fluctuations and space orientation of atomic nuclei are analyzed with effective forces and large configuration spaces. A pedagogical theoretical introduction to the topic of symmetries restoration broken in the mean field approach as well as to the concept of generator coordinate is presented. We discuss global properties, like binding energies, two neutron separation energies and quadrupole moments, as well as spectroscopic ones like excitation energies or transition probabilities. The analysis covers selected examples of light and heavy nuclei, as…
The interplay between quadrupole and octupole degrees of freedom is discussed in a series of neutron-rich actinides and superheavy nuclei with $92 \le$ Z $\le 110$ and $186 \le$ N $\le 202$. In addition to the static Hartree-Fock-Bogoliubov approach, dynamical beyond-mean-field correlations are taken into account via both parity restoration and symmetry-conserving Generator Coordinate Method calculations based on the Gogny-D1M energy density functional. Physical properties such as correlation energies, negative-parity excitation energies as well as reduced transition probabilities $B(E1)$ and …
We introduce a local-scaling point transformation to allow for modifying the asymptotic properties of the deformed three-dimensional Cartesian harmonic oscillator wave functions. The resulting single-particle bases are very well suited for solving the Hartree-Fock-Bogoliubov equations for deformed drip-line nuclei. We then present results of self-consistent calculations performed for the Mg isotopes and for light nuclei located near the two-neutron drip line. The results suggest that for all even-even elements with $Z$=10--18 the most weakly-bound nucleus has an oblate ground-state shape.
Recent results of the description of quadrupole moments of odd semi-magic nuclei are briefly reviewed. They are based on the self-consistent theory of finite Fermi systems with account for the phonon-particle coupling (PC) effects. The self-consistent model for describing the PC effects was developed previously for magnetic moments. Account for the non-pole diagrams is an important ingredient of this model. In addition to previously reported results for the odd In and Sb isotopes, which are the proton-odd neighbors of even tin nuclei, we present new results for odd Bi isotopes, the odd neighbo…
The interrelation between the octupole phonon and the low-lying proton-neutron mixed-symmetry quadrupole phonon in near-spherical nuclei is investigated. The one-phonon states decay by collective E3 and E2 transitions to the ground state and by relatively strong E1 and M1 transitions to the isoscalar 2+ state. We apply the proton-neutron version of the Interacting Boson Model including quadrupole and octupole bosons (sdf-IBM-2). Two F-spin symmetric dynamical symmetry limits of the model, namely the vibrational and the γ-unstable ones, are considered. We derive analytical formulae for excitati…
The model, introduced in a previous paper, for the description of the octupole and quadrupole degrees of freedom in conditions close to the axial symmetry, is used to describe the negative-parity band based on the first octupole vibrational state in nuclei close to the critical point of the U(5) to SU(3) phase transition. The situation of 150Nd and 152Sm is discussed in detail. The positive parity levels of these nuclei, and also the in-band E2 transitions, are reasonably accounted for by the X(5) model. With simple assumptions on the nature of the octupole vibrations, it is possible to descri…
Quantum effects in the quadrupole rotor picture of ultra-relativistic ion-ion collisions
CEA, DES, IRESNE, DER, SPRC, LEPh（法国原子能及替代能源委员会）；School of Physics and Astronomy, Sun Yat-sen University（中山大学物理学院）；Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Sun Yat-Sen University（中山大学广东省量子计量与传感重点实验室）；Technische Universität Darmstadt（达姆施塔特工业大学）；Université Paris-Saclay – CEA – IRFU（巴黎萨克雷大学-法国原子能及替代能源委员会-核物理研究所）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
The additivity principle of the extreme shell model stipulates that an average value of a one-body operator be equal to the sum of the core contribution and effective contributions of valence (particle or hole) nucleons. For quadrupole moment and angular momentum operators, we test this principle for highly and superdeformed rotational bands in the A~130 nuclei. Calculations are done in the self-consistent cranked non-relativistic Hartree-Fock and relativistic Hartree mean-field approaches. Results indicate that the additivity principle is a valid concept that justifies the use of an extreme s…
The self-consistent model, developed previously to describe phonon coupling (PC) effects in magnetic moments of odd magic and semi-magic nuclei, is extended to quadrupole moments. It is based on the theory of finite Fermi systems with the use of the perturbation theory in $g_L^2$, where $g_L$ is the vertex creating the $L$-phonon. Accounting for the phonon tadpole diagrams is an important ingredient of this model. The calculation scheme is based on the Fayans energy density functional DF3-a and does not contain any adjusted parameters. The odd In and Sb isotopes are considered, which are the p…
The properties of the low-lying 2^+ states in the even-even nuclei around 132Sn are studied within the quasiparticle random phase approximation. Starting from a Skyrme interaction in the particle-hole channel and a density-dependent zero-range interaction in the particle-particle channel, we use the finite rank separable approach in our investigation. It is found that the fourth 2^+ state in 132Te could be a good candidate for a mixed-symmetry state.
Nuclei with an octupole deformation have a non-zero electric octupole moment, electric dipole moment (EDM), Schiff moment and magnetic quadrupole moment (MQM) in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by parity (P) and time reversal invariance (T) conservation, meaning their expectation values vanish due to nuclear rotation. However, nuclei with an octupole deformation have doublets of close opposite parity rotational states with the same spin, which are mixed by T,P-odd nuclear forces. …
Excitation energies and transition probabilities of the first 2+ excitations in even lead, tin and nickel isotopes are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. A reasonable agreement with available experimental data is obtained. The effect of the density dependence of the effective pairing interaction is analyzed in detail by comparing results obtained with volume and surface pairing. The effect is found to be noticeable, especially for the 2+ energies which are systematically higher at 200-300 keV for the volu…
Deformed nuclei possess enhanced moments violating time reversal invariance ($T$) and parity ($P$). Collective magnetic quadrupole moments (MQM) appear in nuclei with a quadrupole deformation (which have ordinary $T$,$P$-conserving collective electric quadrupole moments). Nuclei with an octupole deformation have a collective electric octupole moment, electric dipole moment (EDM), Schiff moment and MQM in the intrinsic frame which rotates with the nucleus. In a state with definite angular momentum in the laboratory frame, these moments are forbidden by $T$ and $P$ conservation, meaning their ex…
The isoscalar (IS) and isovector (IV) quadrupole responses of nuclei are systematically investigated using the time-dependent Skyrme Energy Density Functional including pairing in the BCS approximation. Using two different Skyrme functionals, Sly4 and SkM*, respectively 263 and 304 nuclei have been found to be spherical along the nuclear charts. The time-dependent evolution of these nuclei has been systematically performed giving access to their quadrupole responses. It is shown that the mean-energy of the collective high energy state globally reproduces the experimental IS and IV collective e…
The present communication is a continuation of the review, the first part of which was presented in nucl-th/0109033. The second part dealswith the manifistation of Chaotic dynamics at quantum level. The variations of statistical properties of energy spectrum in the process of R-C-R transition have been studied in detail. We proved that the type of the classical motion is correlated with the structure of the eigenfunctions of highly excited states in the R-C-R transition. Shell structure destruction induced by the increase of nonintegrable perturbation was analyzed.
We show that the potential in the radial equation in the model of coherent quadrupole-octupole motion (CQOM) in nuclei generates a sequence of superpotentials and subsequent series of effective potentials which satisfy the shape-invariance condition and correspond to a SUSY-QM hierarchy of Hamiltonians. On this basis we suggest that the original CQOM level scheme possesses a generic supersymmetric structure of the spectrum inherent for the coherent quadrupole-octupole mode. We outline the mechanism in which the real quadrupole-octupole spectra in even-even and odd-even nuclei deviate from the …
We have perfomed Time Dependant Hartree-Fock (TDHF) calculations on the non linear response of nuclei. We have shown that quadrupole (and dipole) motion produces monopole (and quadrupole) oscillations in all atomic nuclei. We have shown that these findings can be interpreted as a large coupling between one and two phonon states leading to large anharmonicities.
We present a new method by which wavefunctions with simple structure are renormalized so as to contain more complicated structure. This method, called $H^n$-cooling method, is applied to the study of the quadrupole collective motion of $^{56}$Fe, $^{54}$Cr, $^{58}$Fe and $^{56}$Cr. The shell-model wavefunctions of lowest-lying states of these nuclei are well treated by this method. By using the wavefunctions obtained via the $H^n$-cooling method, IBM-2 parameters are derived from a realistic shell-model hamiltonian and transition operators. The Majorana interaction becomes sizably repulsive, p…
Collective effects in deformed atomic nuclei present possible avenues of study on the non-spherical distribution of neutrons and the violation of the local Lorentz invariance. We introduce the weak quadrupole moment of nuclei, related to the quadrupole distribution of the weak charge in the nucleus. The weak quadrupole moment produces tensor weak interaction between the nucleus and electrons and can be observed in atomic and molecular experiments measuring parity nonconservation. The dominating contribution to the weak quadrupole is given by the quadrupole moment of the neutron distribution, t…
The generalized Bohr Hamiltonian is applied to a description of low-lying collective excitations in even-even isotopes of Te, Xe, Ba, Ce, Nd and Sm. The collective potential and inertial functions are determined by means of the Strutinsky method and the cranking model, respectively. A shell-dependent parametrization of the Nilsson potential is used. An approximate particle-number projection is performed in treatment of pairing correlations. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not co…
Nuclei with spin $I \ge 1$ have a weak quadrupole moment which leads to tensor contribution to the parity non-conserving interaction between nuclei and electrons. We calculate this contribution for Yb$^+$, Fr and Ra$^+$ and found it to be small. In contrast, in many lanthanides (e.g., Nd, Gd, Dy, Ho, Er, Pr, Sm) and Ra close levels of opposite parity lead to strong enhancement of the effect making it sufficiently large to be measured. Another possibility is to measure the PNC transitions between the hyperfine components of the ground state of Bi. Since nuclear weak charge is dominated by neutr…
We present a comprehensive study on the low-lying states of neutron-rich Er, Yb, Hf, and W isotopes across the $N=126$ shell with a multi-reference covariant density functional theory. Beyond mean-field effects from shape mixing and symmetry restoration on the observables that are relevant for understanding quadrupole collectivity and underlying shell structure are investigated. The general features of low-lying states in closed-shell nuclei are retained in these four isotopes around $N=126$, even though the shell gap is overall quenched by about 30\% with the beyond mean-field effects. These …
We present recent investigations on dipole and quadrupole excitations in spherical skin nuclei, particular exploring their connection to the thickness of the neutron skin. Our theoretical method relies on density functional theory, which provides us with a proper link between nuclear many-body theory of the nuclear ground state and its phenomenological description. For the calculation of the nuclear excited states we apply QPM theory. A new quadrupole mode related to pygmy quadrupole resonance (PQR) in tin isotopes is suggested.
The interacting boson model is extended to describe the spectroscopy of superdeformed bands. Microscopic structure of the model in the second minimum is discussed and superdeformed bosons are introduced as the new building blocks. Solutions of a quadrupole Hamiltonian are implemented through the $1/N$ expansion method. Effects of the quadrupole parameters on dynamic moment of inertia and electric quadrupole transition rates are discussed and the results are used in a description of superdeformed bands in the Hg-Pb and Gd-Dy regions.
Evolution of quadrupole deformations in $sd$ and $pf$ shell nuclei with mass A= 18$\sim$56 is studied by using deformed Skyrme Hartree-Fock (HF) model with pairing correlations. We point out that the quadrupole deformations of the nuclei with the isospin T=0 and T=1 show strong mass number dependence as a clear manifestation of dynamical evolution of deformation in nuclear many-body systems. The competition between the deformation driving particle-vibration coupling and the closed shell structure is shown in a systematic study of the ratios between the proton and neutron deformations in nuclei…
