We investigate the neutron pair transfer modes associated with the low-lying quadrupole states in neutron-rich Sn isotopes by means of the quasiparticle random phase approximation based on the Skyrme-Hartree-Fock-Bogoliubov mean field model. The transition strength of the quadrupole pair-addition mode feeding the $2_1^+$ state is enhanced in the Sn isotopes with $A \geq 132$. The transition density of the pair-addition mode has a large spatial extension in the exterior of nucleus, reaching far to $r\sim 12-13$ fm. The quadrupole pair-addition mode reflects sensitively a possible increase of the effective pairing interaction strength in the surface and exterior regions of neutron-rich nuclei.
The strength functions of quadrupole modes in the unstable oxygen isotopes 22O and 24O are calculated using an extended version of the time-dependent Hartree-Fock theory known as the time-dependent density-matrix theory (TDDM). It is found that TDDM gives the lowest quadrupole states which are energetically shifted upward and become significantly collective due to the coupling to two-body configurations. It is pointed out that these features of the lowest quadrupole states are similar to those obtained in the quasi-particle random phase approximation.
Quadrupole excitations of even-even Ru and Pd isotopes are described within microscopic approach based on the general collective Bohr model which includes the effect of coupling with the pairing vibrations. The excitation energies and E2 transition probabilities observed in 104-114Ru and 106-110Pd are reproduced in the frame of the calculation containing no free parameters.
The longstanding problem of characterization of the $0^+_2$ states in Gd isotopes is revisited by adopting the Nilsson$+$BCS mean field and the random-phase approximation. The interband electric quadrupole transition strengths varying almost two orders of magnitude are nicely reproduced at the same time as other observables. These results indicate that the $0^+_2$ states, in particular, those in lighter isotopes are well described as $β$ vibrations excited on top of deformed ground states without recourse to the shape-coexistence picture.
The interplay between the collective dynamics of the quadrupole and octupole deformation degree of freedom is discussed in a series of Sm and Gd isotopes both at the mean field level and beyond, including parity symmetry restoration and configuration mixing. Physical properties like negative parity excitation energies, E1 and E3 transition probabilities are discussed and compared to experimental data. Other relevant intrinsic quantities like dipole moments, ground state quadrupole moments or correlation energies associated to symmetry restoration and configuration mixing are discussed. For the…
The electric quadrupole response in $^{112,114}$Sn isotopes is investigated by energy-density functional (EDF) and three-phonon quasiparticle-phonon model (QPM) theory with special emphasis on 2$^+$ excitations located above the first collective quadrupole state and below 5 MeV. Additional quadrupole strength clustering as a sequence of states similar to the recently observed pygmy quadrupole resonance in $^{124}$Sn is found. The spectral distributions and transition densities of these 2$^+$ states show special features being compatible with oscillations of a neutron skin against the isospin-s…
Collective low lying levels of light and medium Xenon isotopes are deduced from the Generalized Bohr Hamiltonian (GBH). The microscopic seven functions entering into the GBH are built from a deformed mean field of the Woods-Saxon type. Theoretical spectra are found to be close to the ones of the experimental data taking into account that the calculations are completely microscopic, that is to say, without any fitting of parameters.
The angular momentum projected Generator Coordinate Method, with the quadrupole moment as collective coordinate and the Gogny force (D1S) as the effective interaction, is used to describe the properties of the ground state and low-lying excited states of the even-even Neon isotopes $^{20-34}$Ne, that is, from the stability valley up to the drip-line. It is found that the ground state of the N=20 nucleus $^{30}$Ne is deformed but to a lesser extent than the N=20 isotope of the Magnesium. In the calculations, the isotope $^{32}$Ne is at the drip-line in good agreement with other theoretical pred…
We study the couplings between collective vibrations such as the isovector giant dipole and isoscalar giant quadrupole resonances in tin isotopes in the framework of the time-dependent Hartree-Fock theory with a Skyrme energy density functional. These couplings are a source of anharmonicity in the multiphonon spectrum. In particular, the residual interaction is known to couple the isovector giant dipole resonance with the isoscalar giant quadrupole resonance built on top of it, inducing a nonlinear evolution of the quadrupole moment after a dipole boost. This coupling also affects the dipole m…
A key question concerning the spherical-vibrator attributes of states in Cadmium isotopes is addressed by means of a boson Hamiltonian encompassing U(5) partial dynamical symmetry. The U(5) symmetry is preserved in a segment of the spectrum and is broken in particular non-yrast states, and the resulting mixing with the intruder states is small. The vibrational character is thus maintained in the majority of low-lying normal states, as observed in $^{110}$Cd.
[Background] The BE2 rates of the Sn isotopes for $N\le 64$ exhibit enhancements hitherto unexplained. The same is true for the Cd isotopes. [Purpose] Describe the electromagnetic properties of the Sn and Cd isotopes [Method] Shell model calculations with a minimally renormalized realistic interaction, supplemented by Quasi and Pseudo-SU3 symmetries and Nilsson-SU3 selfconsistent calculations. [Results for $N\le 64$] Shell model calculations with the neutron effective charge as single free parameter describe well the BE2(2>0) and BE2(4>2) rates for $N\le 64$ in the Cd and Sn isotopes. The form…
A realistic shell-model study is performed for neutron-deficient tin isotopes up to mass A=108. All shell-model ingredients, namely two-body matrix elements, single-particle energies, and effective charges for electric quadrupole transition operators, have been calculated by way of the many-body perturbation theory, starting from a low-momentum interaction derived from the high-precision CD-Bonn free nucleon-nucleon potential. The focus has been put on the enhanced quadrupole collectivity of these nuclei, which is testified by the observed large B(E2;0+ -> 2+)s. Our results evidence the crucia…
In this work the excited energy levels, reduced transition probabilities, $B(E2)\uparrow$, intrinsic quadrupole moments and deformation parameters have been calculated for $^{62-68}$Zn isotopes with neutrons (N=32,34,36 and 38 ). Nushellx code has been applied for all energy states of (fp-shell) nuclei. Shell-model calculations for the zinc isotopes have been carried out with active particles distributed in the lp$_{3/2}$, 0f$_{5/2}$ and lp$_{1/2}$ orbits outside doubly-magic closed "56Ni"core nucleus. By using (f5p) model space and (f5pvh) interaction, the theoretical results have been obtain…
The angular momentum projected Generator Coordinate Method using the quadrupole moment as collective coordinate and the Gogny force as the effective interaction is used to describe the properties of the ground state and low-lying excited states of the neutron rich light nuclei 32,34,36Si. It is found that the ground state of the nucleus 34Si is spherical. However, this is not only due to the N=20 shell closure as the ground state of 34Si contains a significant amount of the intruder f7/2 neutron orbital. On the other hand, rather good agreement with experimental data for many observables is ob…
The spherical-to-prolate shape transition in neutron-rich Cr isotopes from N = 34 to 42 is studied by solving the collective Schr\"odinger equation for the five-dimensional quadrupole collective Hamiltonian. The collective potential and inertial functions are microscopically derived with use of the constrained Hartree-Fock-Bogoliubov plus local quasiparticle random-phase approximation method. Nature of the quadrupole collectivity of low-lying states is discussed by evaluating excitation spectra and electric quadrupole moments and transition strengths. The result of calculation indicates that …
Shell model calculations with the neutron effective charge as single free parameter describe well the \bet and \bef rates for $N\le 64$ in the Cd and Sn isotopes. The former exhibit weak permanent deformation corroborating the prediction of a pseudo SU3 symmetry, which remains of heuristic value in the latter, though the pairing force erodes the quadrupole dominance. Calculations in $10^7$ and $10^{10}$-dimensional spaces exhibit almost identical patterns: A vindication of the shell model. For $N\ge 64$ quadrupole dominance is accentuated in the Cd isotopes and gives way to seniority dominance…
The B(E2:0_1^+ -> 2_1^+) values of the Ba isotopes (Z=56) exhibit a sharp increase in deformation as the neutron numbers approach the mid-shell value of N=66. This behavior is anomalous because the 2_1^+ level energies are very similar to those of the neighboring isotopes. By means of the axially-symmetric deformed Woods-Saxon (WS) hamiltonian plus the BCS method, we investigated the systematics of B(E2) of the Ba isotopes. We showed that 15% of the B(E2) values at N=66 was due to the level crossing, occurring at the deformation with beta being nearly 0.3, between the proton orbits originating…
Microscopic description of quadrupole-hexadecapole coupling in radium, thorium, uranium and plutonium isotopes with the Gogny energy density functional
Recently, charge radii and ground-state electromagnetic moments of Li and Be isotopes were measured precisely. We have performed large-scale ab initio no-core shell model calculations for these isotopes using high-precision nucleon-nucleon potentials. The isotopic trends of our computed charge radii and quadrupole and magnetic-dipole moments are in good agreement with experimental results with the exception of the 11Li charge radius. The magnetic moments are in particular well described, whereas the absolute magnitudes of the quadrupole moments are about 10% too small. The small magnitude of t…
In the random-interaction ensembles, electromagnetic moments of Cd $I^π=11/2^-$ isomers predominately present linear systematics as changing the neutron number, which has been reported in realistic nuclear system. Quadrupole-like and $δ$-like $pn$ interaction are responsible for such linear systematics of quadrupole and magnetic moments, respectively.
Low-frequency quadrupole vibrational modes in deformed $^{36,38,40}$Mg close to the neutron drip line are studied by means of the quasiparticle-random-phase approximation based on the coordinate-space Hartree-Fock-Bogoliubov formalism. Strongly collective $K^π=0^{+}$ and $2^{+}$ excitation modes carrying 10-20 Weisskopf units in the intrinsic isoscalar quadrupole transition strengths are obtained at about 3 MeV. There are two reasons for the enhancement of the transition strengths. First, the quasiparticle wave functions generating these modes possess spatially very extended structure. The asy…
The lowest positive- and negative-parity bands of $^{20}$Ne and neutron-rich even-even Ne isotopes are investigated using a theoretical framework based on energy density functionals. Starting from a self-consistent relativistic Hartree-Bogoliubov calculation of axially-symmetric and reflection-asymmetric deformation energy surfaces, the collective symmetry-conserving states are built using projection techniques and the generator coordinate method. Overall a good agreement with the experimental excitation energies and transition rates is obtained. In particular, the model provides an accurate d…
Quadrupole moments and charge radii in cadmium isotopes are studied from a microscopic perspective. The results obtained from self-consistent deformed Hartree-Fock+BCS calculations with Skyrme forces are compared with isomer and isotope shifts measured from high resolution laser spectroscopy experiments. The microscopic calculations reproduce fairly well the main features observed in those isotopes that include the linear increase with the neutron number of the quadrupole moments of the 11/2- isomers, as well as the parabolic behavior of the mean-square charge radii difference between isomers …
A method of calculating static moments of excited states and transitions between excited states is formulated for non-magic nuclei within the Green function formalism. For these characteristics, it leads to a noticeable difference from the standard QRPA approach. Quadrupole moments of the first 2+ states in Sn and Pb isotopes are calculated using the self-consistent TFFS based on the Energy Density Functional by Fayans et al. with the set of parameters DF3-a fixed previously. A reasonable agreement with available experimental data is obtained.
The simplicity in the nuclear quadrupole moments reported recently in $^{107-129}$Cd, i.e., a linear increase of the ${11/2}^-$ quadrupole moments, is investigated microscopically with the covariant density functional theory. Using the newly developed functional PC-PK1, the quadrupole moments as well as their linear increase tendency with the neutron number are excellently reproduced without any {\it ad hoc} parameters. The core polarization is found to be very important and contributes almost half of the quadrupole moments. The simplicity of the linear increase is revealed to be due to the pa…
This work studies the systematic reduced transition probabilities B(E2), intrinsic quadrupole moments and deformation parameters of Pd isotopes with even neutrons from N= 62 to 66. The downward reduced transition probabilities B(E2) from gamma transition 8+ to 6+, 6+ to 4+, 4+ to 2+ and 2+ to 0+ states of even-even 108-112Pd isotopes were calculated by the Interacting Boson Model (IBM-1) and compared with the available previous experimental results. The ratio of the excitation energies of the first 4+ and the first 2+ excited states, R4/2, is also studied for the classification of symmetry of …
A numerical method to integrate the time-dependent Hartree-Fock Bogoliubov (TDHFB) equations with Gogny interaction is proposed. The feasibility of the TDHFB code is illustrated by the conservation of the energy, particle numbers, and center-of-mass in the small amplitude vibrations of oxygen 20. The TDHFB code is applied to the isoscalar quadrupole and/or isovector dipole vibrations in the linear (small amplitude) region in oxygen isotopes (masses A = 18,20,22 and 24), titanium isotopes (A = 44,50,52 and 54), neon isotope (A = 26), and magnesium isotopes (A = 24 and 34). The isoscalar quadrup…
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 core polarization effect for the electric quadrupole moment of the neutron-rich $^{31}$Al, $^{33}$Al and $^{35}$Al isotopes in the vicinity of the island of inversion are investigated by means of the microscopic particle-vibration coupling model in which the Skyrme Hartee-Fock-Bogoliubov and quasiparticle-random-phase approximation are used to calculate the single-quasiparticle wave functions and the excitation modes. It is found that the polarization charge for the proton $1d_{5/2}$ hole state in $^{33}$Al is quite sensitive to coupling to the neutrons in the $pf$-shell associated with th…
We analyse deformation mechanism in neutron-rich Cr, Fe and Ti isotopes with N=32-44 by means of a Skyrme-Hartree-Fock-Bogoliubov mean-field code employing a two-dimensional mesh representation in the cylindrical coordinate system. Evaluating systematically the quadrupole deformation energy, we show that the Skyrme parameter set SkM* gives a quadrupole instability around the neutron number N$\sim$38-42 in Cr isotopes, where the deformation energy curve suggests a transitional behavior with a shallow minimum extending to a large prolate deformation. Roles of a deformed N=38 gap and the position…
We present systematic calculations on the spectroscopy and transition properties of even-even Te isotopes by using the large-scale configuration interaction shell model approach with a realistic interaction. These nuclei are of particular interest since their yrast spectra show a vibrational-like equally-spaced pattern but the few known E2 transitions show anomalous rotational-like behavior, which cannot be reproduced by collective models. Our calculations reproduce well the equally-spaced spectra of those isotopes as well as the constant behavior of the $B(E2)$ values in $^{114}$Te. The calcu…
Using the Hartree-Fock-Bogoliubov (HFB) mean field theory in coordinate space, we investigate ground state properties of the sulfur isotopes from the line of stability up to the two-neutron dripline ($^{34-52}S$). In particular, we calculate two-neutron separation energies, quadrupole moments, and rms-radii for protons and neutrons. Evidence for shape coexistence is found in the very neutron-rich sulfur isotopes. We compare our calculations with results from relativistic mean field theory and with available experimental data. We also study the properties of neutron-rich zirconium ($^{102,104}Z…
The evolution of quadrupole and octupole collectivity in neutron-deficient Xe, Ba, Ce, and Nd nuclei near the "octupole magic" neutron number $N=56$ is investigated within the mapped $sdf$-IBM framework. Microscopic input is obtained via quadrupole and octupole constrained Hartree-Fock-Bogoliubov calculations, based on the parametrization D1M of the Gogny energy density functional. Octupole-deformed mean-field ground states are predicted for Ba and Ce isotopes near $N=56$. Excitation energies of positive- and negative-parity states as well as electric transition rates are computed with wave fu…
The ground-state deformations of the Ge isotopes are investigated in the framework of Gogny-Hartree-Fock-Bogoliubov (HFB) and Skyrme Hartree-Fock plus pairing in the BCS approximation. Five different Skyrme parametrizations are used to explore the influence of different effective masses and spin-orbit models. There is generally good agreement for binding energies and deformations (total quadrupole moment, triaxiality) with experimental data where available (i.e., in the valley of stability). All calculations agree in predicting a strong tendency for triaxial shapes in the Ge isotopes with only…
We study the predictive power of Skyrme forces with respect to low lying quadrupole spectra along the chains of Sn, Cd, and Te isotopes. Excitation energies and B(E2) values for the lowest quadrupole states are computed from a collective Schroedinger equation which as deduced through collective path generated by constraint Skyrme-Hartree-Fock (SHF) plus self-consistent cranking for the dynamical response. We compare the results from four different Skyrme forces, all treated with two different pairing forces (volume versus density-dependent pairing). The region around the neutron shell closure …
We solve the Hartree-Fock-Bogoliubov (HFB) equations for deformed, axially symmetric even-even nuclei in coordinate space on a 2-D lattice utilizing the Basis-Spline expansion method. Results are presented for the neutron-rich zirconium isotopes up to the two-neutron dripline. In particular, we calculate binding energies, two-neutron separation energies, normal densities and pairing densities, mean square radii, quadrupole moments, and pairing gaps. Very large prolate quadrupole deformations (beta2=0.42,0.43,0.47) are found for the (102,104,112)Zr isotopes, in agreement with recent experimenta…
The ground and low-lying states of neutron-rich exotic Te and Sn isotopes are studied in terms of the nuclear shell model by the same Hamiltonian used for the spherical-deformed shape phase transition of Ba isotopes, without any adjustment. An anomalously small value is obtained for $B(E2;0^+_1\to 2^+_1)$ in $^{136}$Te, consistently with a recent experiment. The levels of $^{136}$Te up to yrast $12^+$ are shown to be in agreement with observed ones. It is pointed out that $^{136}$Te can be an exceptionally suitable case for studying mixed-symmetry 1$^+$, 2$^+$ and 3$^+$ states, and predictions…
The systematic investigation of ground state shape evolution from gamma unstable O(6) to spherical U(5) for Te isotopes has been presented by using the quadrupole moment constrained Hartree Fock Bogoliubov (HFB) method with the Skyrme force SLy4. 124Te has been pointed out as to be the possible critical point nucleus with E(5) symmetry.
Experimentally observed heaviest $N \approx Z$ nuclei, Ru isotopes, are investigated by the shell model on a spherical basis with the extended $P+QQ$ Hamiltonian. The energy levels of all the Ru isotopes can be explained by the shell model with a single set of force parameters. The calculations indicate enhancement of quadrupole correlations in the $N=Z$ nucleus $^{88}$Ru as compared with the other Ru isotopes, but the observed moments of inertia seem to require much more enhancement of quadrupole correlations in $^{88}$Ru. It is discussed that the particle alignment takes place at $8^+$ in $^…
The recently measured experimental data of $^{79,81,83}$Se isotopes have been interpreted in terms of shell model calculations. The calculations have been performed in ${f_{5/2}pg_{9/2}}$ space with the recently derived interactions, namely with JUN45 and jj44b. To study the importance of the proton excitations across the $Z=28$ shell in this region. We have also performed calculation in ${fpg_{9/2}}$ valence space using an ${fpg}$ effective interaction with $^{48}$Ca core and imposing a truncation. Excitation energies, $B(2)$ values, quadrupole moments and magnetic moments are compared with e…
In this paper, we have studied the shapes coexistence in the 180-190Hg isotopes. The SO(6) representation of eigenstates and a transitional Hamiltonian in the Interacting Boson Model are used to consider the evolution from prolate to oblate shapes for systems with total boson number N = 9-12. Parameter free (up to overall scale factors) predictions for energy spectra and quadrupole transition rates are found to be in good agreement with experimental counterparts. The results for the control parameter of transitional Hamiltonian offer a combination of spherical and deformed shapes in these Hg i…
Effect of ground-state deformation on the Isoscalar Giant Monopole Resonance and the first observation of overtones of the Isoscalar Giant Quadrupole Resonance in rare-earth Nd isotopes
The transition rates for the 2_{1}^{+} states in 62,64,66Fe were studied using the Recoil Distance Doppler-Shift technique applied to projectile Coulomb excitation reactions. The deduced E2 strengths illustrate the enhanced collectivity of the neutron-rich Fe isotopes up to N=40. The results are interpreted by the generalized concept of valence proton symmetry which describes the evolution of nuclear structure around N=40 as governed by the number of valence protons with respect to Z~30. The deformation suggested by the experimental data is reproduced by state-of-the-art shell calculations wit…
The recent RCNP $(α, α')$ data on the Isoscalar Giant Monopole Resonance (ISGMR) and Isoscalar Giant Quadrupole Resonance (ISGQR) in $^{92,94,96,98,100}$Mo are analyzed within a fully self-consistent Quasiparticle Random Phase Approximation (QRPA) approach with Skyrme interactions, in which pairing correlations and possible axial deformations are taken into account. The Skyrme sets SkM*, SLy6, SVbas and SkP$^δ$, that explore a diversity of nuclear matter properties, are used. We discuss the connection between the line shape of the monopole strength ISGMR and the deformation-induced coupling be…
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…
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 …
Spectroscopic properties that characterize the shape phase transitions in krypton isotopes with the mass $A\approx80$ region are investigated within the framework of the nuclear density functional theory. Triaxial quadrupole constrained self-consistent mean-field calculations that employ relativistic energy density functionals and a pairing interaction are carried out for the even-even nuclei $^{76-86}$Kr. The spectroscopic properties are computed by solving the triaxial quadrupole collective Hamiltonian, with the ingredients, i.e., the deformation-dependent moments of inertia and mass paramet…
The soft dipole modes in neutron rich even-even Ni-isotopes are investigated in the quasiparticle relativistic random phase approximation. We study the evolution of strengths distribution, centroid energies of dipole excitation in low-lying and normal GDR regions with the increase of the neutron excess. It is found in the present study that the centroid energies of the soft dipole strengths strongly depend on the thickness of neutron skin along with the neutron rich even-even Ni-isotopes.
Low energy quadrupole excitations of the 84-110Mo nuclei are studied in the frame of the general Bohr collective model based on the ATDHFB method with the Skyrme effective interaction. Obtained energies of the 2_1, 4_1, 2_2 and 0_2 levels and B(E2) values for the 2_1 -> 0_g.s transitions are in good agreement with experimental data.
We study the quadrupole moments and the B(E2; $2^+ \rightarrow 0^+$) values for the ${11/2}^-$ states and the first $2^+$ states, respectively, by using a multi-j generalized seniority approach in the Cd (Z = 48), Sn (Z = 50) and Te (Z = 52) isotopic chains. The g-factor trends have also been discussed. Although, Cd and Te isotopes represent two-proton hole and two-proton particle systems, thus involving both kind of particles (protons and neutrons) in contrast to Sn (Z = 50) where only neutrons play a role, we find that a similar model based on neutron valence space alone is able to explain n…
Continuum coupling correction to binding energies in the neutron rich oxygen and fluorine isotopes is studied using the Shell Model Embedded in the Continuum. We discuss the importance of different effects, such as the position of one-neutron emission threshold, the effective interaction or the number of valence particles on the magnitude of this correction.
