Electric Quadrupole Transition Strengths Research Articles

Suppressed electric quadrupole collectivity in 49Ti

Single-step Coulomb excitation of 46,48,49,50Ti is presented. A complete set of E2 matrix elements for the quintuplet of states in 49Ti, centred on the 2+ core excitation, was measured for the first time. A total of nine E2 matrix elements are reported, four of which were previously unknown. 2249Ti27 shows a 20% quenching in electric quadrupole transition strength as compared to its semi-magic 2250Ti28 neighbour. This 20% quenching, while empirically unprecedented, can be explained with a remarkably simple two-state mixing model, which is also consistent with other ground-state properties such as the magnetic dipole moment and electric quadrupole moment. A connection to nucleon transfer data and the quenching of single-particle strength is also demonstrated. The simplicity of the 49Ti-50Ti pair (i.e., approximate single-j0f7/2 valence space and isolation of yrast states from non-yrast states) provides a unique opportunity to disentangle otherwise competing effects in the ground-state properties of atomic nuclei, the emergence of collectivity, and the role of proton-neutron interactions.

Electromagnetic properties of Λ hypernuclei with a beyond relativistic mean-field approach

We present a microscopic study of electromagnetic properties of the low-lying states of single-Λ hypernucleus Λ9Be using a quantum-number projected generator coordinate method (HyperGCM) based on covariant density functional theory. All the configurations are restricted to have axial symmetry, with the Λ hyperon in the lowest-energy state. The results are compared to the predictions of a simple particle-rotor model (PRM), for which analytic expressions are available. We show that the electromagnetic properties of Λ9Be from the HyperGCM calculation are close to those by the PRM. In particular, it is found that the magnetic moments and M1 transition strengths are less sensitive to the coupling strengths of the ΛN interaction than the electric quadrupole transition strengths. The latter may provide additional constraints on the ΛN interaction.

Study of S, Cl and Ar isotopes with $$N \ge Z$$ using microscopic effective sd-shell interactions

In the present work, newly developed microscopic effective $sd$-valence shell interactions such as chiral next-to-next-to-next-to-leading order (N3LO), $J$-matrix inverse scattering potential (JISP16), Daejeon16 (DJ16), and monopole-modified DJ16 (DJ16A) are employed to study the nuclear structural properties of sulphur, chlorine, and argon isotopes with $N \geq Z$. These interactions are derived using the \textit{ab initio} no-core shell-model and the OLS unitary transformation method. We calculate energy spectra and electromagnetic properties to test the predictive strength of the effective interactions for these heavier $sd$-shell nuclei. For a complete systematic study, we compare the microscopic results with the phenomenological USDB results and experimental data. By looking at the excitation energies of these nuclei, the DJ16A interaction is found to be {the} most suitable for these $sd$-shell nuclei among all microscopic interactions. The electric quadrupole transition strength and excitation energy of the first $2^+$ state data of even-even sulphur isotopes indicate the presence of the $N=20$ shell closure. Quadrupole moment predictions are also made using these interactions where experimental data are unknown. Magnetic moments are in excellent agreement with the experimental values. The root-mean-square deviations are also calculated to provide an idea of how accurate the interactions are.

Study of structure and radii for 20−31Na isotopes using microscopic interactions

In this work, Na isotopes with mass varying from A=20 to 31 have been studied using DJ16A, JISP16 and N3LO microscopic effective interactions in the sd-shell. These effective interactions are derived for sd-shell using no-core shell model wavefunctions and a unitary transformation method. We have also performed calculation with IMSRG effective interactions targeted for a particular nucleus. The studies include a detailed analysis of ground state binding energy, low-lying spectra, and electromagnetic properties such as reduced electric quadrupole transition strengths, quadrupole and magnetic dipole moments of the Na chain. The results obtained from these microscopic effective interactions were compared with the experimental data as well as with the results of phenomenological interaction USDB. The charge radii of Na isotopes are evaluated using shell model harmonic oscillator wave functions. In addition to charge radii, matter radii and neutron skin thickness in the Na chain are discussed with a focus on neutron-deficient Na isotopes.

Evolution of low-lying M1 modes in germanium isotopes

Magnetic dipole strength functions are determined for the series of germanium isotopes from $N=Z=32$ to $N$ = 48 on the basis of a large number of transition strengths calculated within the shell model. The evolution of the strength with increasing neutron number in the $1{g}_{9/2}$ orbital is analyzed. A bimodal structure comprising an enhancement toward low transition energy and a resonance in the region of the scissors mode is identified. The low-energy enhancement is strongest near closed shells, in particular at the almost completely filled $1{g}_{9/2}$ orbital, while the scissorslike resonance is most pronounced in the middle of the open shell, which correlates with the magnitude of the also deduced electric quadrupole transition strengths. The results are consistent with previous findings for the shorter series of iron isotopes and prove the occurrence and correlation of the two low-lying magnetic dipole modes as a global structural feature.

Coulomb excitation of Kr80,82 and a change in structure approaching N=Z=40

Background: Nuclei approaching N=Z=40 are known to exhibit strongly deformed structures and are thought to be candidates for shape coexistence. In the krypton isotopes, Kr80,82 are poorly characterized, preventing an understanding of evolving deformation approaching N=40. Purpose: The present work aims to determine electric quadrupole transition strengths and quadrupole moments of Kr80,82 in order to better characterize their deformation. Methods: Sub-barrier Coulomb excitation was employed, impinging the isotopes of krypton on Pt196 and Pb208 targets. Utilizing a semiclassical description of the safe Coulomb-excitation process E2 matrix elements could then be determined. Results: Eleven new or improved matrix elements are determined in Kr80 and seven in Kr82. The new B(E2;01+→21+) value in Kr82 disagrees with the evaluated value by 3σ, which can be explained in terms of deficiencies in a previous Coulomb-excitation analysis. Conclusions: Comparison of measured Qs(21+) and B(E2;01+→21+) values indicates that neutron-deficient (N≤42) isotopes of krypton are closer to axial deformation than other isotopic chains in the mass region. A continuation of this trend to higher Z may result in Sr and Zr isotopes exhibiting near-axial prolate deformation.1 MoreReceived 19 July 2021Revised 1 September 2021Accepted 28 September 2021DOI:https://doi.org/10.1103/PhysRevC.104.044313©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsElectromagnetic transitionsSpectroscopic factors & electromagnetic momentsProperties59 ≤ A ≤ 89Nuclear Physics

Calculation of nuclear charge radii with a trained feed-forward neural network

A feed-forward neural network model is trained to calculate the nuclear charge radii. The model trained with input data set of proton and neutron number $Z,N$, the electric quadrupole transition strength $B(E2)$ from the first excited 2$^+$ state to the ground state, together with the symmetry energy. The model reproduces well not only the isotope dependence of charge radii, but also the kinks of charge radii at the neutron magic numbers $N=82$ for Sn and Sm isotopes, and also $N=126$ for Pb isotopes. The important role of $B(E2)$ value is pointed out to reproduce the kink of the isotope dependence of charge radii in these nuclei. Moreover, with the inclusion of the symmetry energy term in the inputs, the charge radii of Ca isotopes are well reproduced. This result suggests a new correlation between the symmetry energy and charge radii of Ca isotopes. The Skyrme HFB calculation is performed to confirm the existence of this correlation in a microscopic model.

Microscopic calculations of the electric Quadrupole transition strengths of Be isotopes (9, 10, 12, 14)

Electric Quadrupole transitions are calculated for beryllium isotopes (9, 10, 12 and 14). Calculations with configuration mixing shell model usually under estimate the measured E2 transition strength. Although the consideration of a large basis no core shell model with 2ℏtruncations for 9,10,12 and14 where all major shells s, p, sd are used, fail to describe the measured reduced transition strength without normalizing the matrix elements with effective charges to compensate for the discarded space. Instead of using constant effective charges, excitations out of major shell space are taken into account through a microscopic theory which allows particle–hole excitations from the core and model space orbits to all higher orbits with 2ℏw excitations which are called core-polarization effects. The two body Michigan sum of three ranges Yukawa potential (M3Y) is used for the core-polarization matrix element. The simple harmonic oscillator potential is used to generate the single particle matrix elements of all isotopes considered in this work. The b value of each isotope is adjusted to reproduce the experimental matter radius, These size parameters of the harmonic oscillator almost reproduce all the root mean square (rms) matter radii for 9,10,12,14Be isotopes within the experimental errors. Almost same effective charges are obtained for the neutron- rich Be isotopes which are smaller than the standard values. The major contribution to the transition strength comes from the core polarization effects. The present calculations of the neutron-rich 12,14Beisotopes show a deviation from the general trends in accordance with experimental and other theoretical studies. The configurations arises from the shell model calculations with core-polarization effects reproduce the experimental B(E2) values.

Nuclear chiral doublet bands data tables

Since the prediction of nuclear chirality in 1997, tremendous progresses both theoretically and experimentally have been achieved. Experimentally, 59 chiral doublet bands in 47 chiral nuclei (including 8 nuclei with multiple chiral doublets) have been reported in A∼80,100,130, and 190 mass regions. The spins, parities, energies, ratios of the magnetic dipole transition strengths to the electric quadrupole transition strengths, and related references for these nuclei are compiled and listed. For these nuclei with the magnetic dipole transition strengths and the electric quadrupole transition strengths measured, the corresponding results are given. A brief discussion is provided after the presentation of energy E, energy difference ΔE, energy staggering parameter S(I), rotational frequency ω, kinematic moment of inertia J(1), dynamic moment of inertia J(2), and ratio of the magnetic dipole transition strength to the electric quadrupole transition strength B(M1)∕B(E2) versus spin I in each mass region.

Quadrupole collectivity beyond N=50 in neutron-rich Se and Kr isotopes

We report $B(E2;{0}_{1}^{+}\ensuremath{\rightarrow}{2}_{n}^{+})$ strengths for the neutron-rich $^{88,90}\mathrm{Kr}$ and $^{86}\mathrm{Se}$ isotopes from intermediate-energy Coulomb excitation. The electric quadrupole transition strengths to the first ${2}^{+}$ states complete, with considerably improved uncertainties, the evolution of quadrupole collectivity in the Kr and Se isotopes approaching $N=60$, for which $^{90}\mathrm{Kr}$ and $^{86}\mathrm{Se}$ had previously been the most uncertain. We also report significant excitation strength to several higher-lying ${2}^{+}$ states in the krypton isotopes. This fragmentation of $B(E2)$ strength in $^{88,90}\mathrm{Kr}$ confirms shell-model calculations in the $\ensuremath{\pi}(fpg)\text{\ensuremath{-}}\ensuremath{\nu}(sdgh)$ shell with an only minimally tuned shell-model setup that is based on a nucleon-nucleon interaction derived from effective field theory and effective charges adjusted to $^{86}\mathrm{Kr}$.

Effective field theory for vibrations in odd-mass nuclei

Heavy even-even nuclei exhibit low-energy collective excitations that are separated in scale from the microscopic (fermion) degrees of freedom. This separation of scale allows us to approach nuclear vibrations within an effective field theory (EFT). In odd-mass nuclei collective and single-particle properties compete at low energies, and this makes their description more challenging. In this article we describe odd-mass nuclei with ground-state spin $I=\sfrac{1}{2}$ by means of an EFT that couples a fermion to the collective degrees of freedom of an even-even core. The EFT relates observables such as energy levels, electric quadrupole ($E2$) transition strengths, and magnetic dipole ($M1$) moments of the odd-mass nucleus to those of its even-even neighbor, and allows us to quantify theoretical uncertainties. For isotopes of rhodium and silver the theoretical description is consistent with data within experimental and theoretical uncertainties. Several testable predictions are made.

Sensitivities and correlations of nuclear structure observables emerging from chiral interactions

Starting from a set of different two- and three-nucleon interactions from chiral effective field theory, we use the importance-truncated no-core shell model for ab initio calculations of excitation energies as well as electric quadrupole (E2) and magnetic dipole (M1) moments and transition strengths for selected p-shell nuclei. We explore the sensitivity of the excitation energies to the chiral interactions as a first step towards and systematic uncertainty propagation from chiral inputs to nuclear structure observables. The uncertainty band spanned by the different chiral interactions is typically in agreement with experimental excitation energies, but we also identify observables with notable discrepancies beyond the theoretical uncertainty that reveal insufficiencies in the chiral interactions. For electromagnetic observables we identify correlations among pairs of E2 or M1 observables based on the ab initio calculations for the different interactions. We find extremely robust correlations for E2 observables and illustrate how these correlations can be used to predict one observable based on an experimental datum for the second observable. In this way we circumvent convergence issues and arrive at far more accurate results than any direct ab initio calculation. A prime example for this approach is the quadrupole moment of the first 2^+ state in C-12, which is predicted with an drastically improved accuracy.

Novel triaxial structure in low-lying states of neutron-rich nuclei aroundA≈100

Background: In recent years, the study of triaxiality in the low-lying states of atomic nuclei with transition character or shape coexistence has been of great interest. Previous studies indicate that the neutron-rich nuclei in the $A\ensuremath{\sim}100$ mass region with $Z\ensuremath{\sim}40,N\ensuremath{\sim}60$ serve as good grounds for examining the role of triaxiality in nuclear low-lying states.Purpose: The aim of this work is to provide a microscopic study of low-lying states for nuclei in the $A\ensuremath{\sim}100$ mass regions and to examine in detail the role of triaxiality in the shape-coexistence phenomena and the variation of shape with the isospin and spin values at the beyond mean-field level.Method: The starting point of our method is a set of relativistic mean-field plus BCS wave functions generated with a constraint on triaxial deformations $(\ensuremath{\beta},\ensuremath{\gamma})$. The excitation energies and electric multipole transition strengths of low-lying states are calculated by solving a five-dimensional collective Hamiltonian (5DCH) with parameters determined by the mean-field wave functions.Results: The low-lying states of Mo isotopes and of $N=60$ isotones in the $A\ensuremath{\sim}100$ mass region are calculated. The results indicate that triaxiality is essential to reproduce the data of excitation energies and electric quadrupole transition strengths in low-lying states and plays an important role in the shape evolution as a function of nucleon number. However, the decrease of nuclear collectivity with the increase of angular momentum in neutron-rich Mo isotopes has not been reproduced.Conclusions: The evolution of nuclear collectivity in the low-lying states of neutron-rich nuclei in the $A\ensuremath{\sim}100$ mass region as a function of nucleon number is governed by the novel triaxial structure. However, the mechanism that governs the variation of nuclear shape with spin in Mo isotopes remains unclear and deserves further investigation by taking into account the effects other than the collective motions.

Quadrupole and monopole transition properties of 0+2in Gd isotopes

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 $\beta$ vibrations excited on top of deformed ground states without recourse to the shape-coexistence picture.

Shape coexistence and evolution inSr98

Shape coexistence between the strongly deformed ground state and the weakly deformed ${0}_{2}^{+}$ state in $^{98}\mathrm{Sr}$ has been a major topic of interest due to the energy difference of 215 keV, which is the smallest in all even-even nuclei. The electric monopole transition strength ${\ensuremath{\rho}}^{2}(E0)$ is an important quantity that can relate the deformation difference and the shape mixing between the two ${0}^{+}$ states, which are admixtures of the vibrational (S) and the rotational (D) states in a simple mixing model. In a $\ensuremath{\beta}$-decay spectroscopy experiment, the experimental ${\ensuremath{\rho}}^{2}(E0)$ was measured. A value of 0.053(5) is consistent with the previous measurement and was combined with known electric quadrupole transition strengths $B(E2)$ in calculations of a two-state mixing model. Based on a systematic study on neighboring Kr, Zr, and Mo isotopes, the mixing of the ${0}^{+}$ and ${2}^{+}$ states in $^{98}\mathrm{Sr}$ was determined to be 8.6% and 1.3%, respectively, corresponding to deformation parameters ${\ensuremath{\beta}}_{\text{D}}=0.38(1)$ and ${\ensuremath{\beta}}_{\text{S}}=\ensuremath{-}0.23(2)$. These parameters reproduce experimental transition strengths well except for the ${4}_{1}^{+}\ensuremath{\rightarrow}{2}_{1}^{+}$ transition, which suggests a smaller D-band deformation for $J\ensuremath{\ge}4$.

Electric Monopole Transition Strengths in62Ni

Excited states in 62 Ni were populated with a (p, p') reaction using the 14UD Pelletron accelerator at the Australian National University. Electric monopole transition strengths, ρ 2 (E0), were measured through simultaneous detection of the internal conversion electrons and γ rays emitted from the de-excitation of popu- lated states, using the Super-e spectrometer coupled with a germanium detector. The strength of the 0 + to 0 + transition has been measured to be 77 +23 −34 × 10 −3 and agrees with previously reported values. Upper limits have been placed on the 0 + to 0 + and 0 + to 0 + transitions. The measured ρ 2 (E0) value of the 2 + to 2 + transition in 62 Ni has been measured for the first time and found to be one of the largest ρ 2 (E0) values measured to date in nuclei heavier than Ca. The low-lying states of 62 Ni have previously been classified as one- and two-phonon vibrational states based on level energies. The measured electric quadrupole transition strengths are consistent with this interpretation. However as electric monopole transitions are forbidden between states which differ by one phonon number, the simple harmonic quadrupole vibrational picture is not sufficient to explain the large ρ 2 (E0) value for the 2 + to 2 + transition.

Antimagnetic rotation and sudden change of electric quadrupole transition strength in 143Eu

Lifetimes of the states in the quadrupole structure in 143Eu have been measured using the Doppler shift attenuation method and the parity of the states in the sequence has been firmly identified from polarization measurements using the Indian National Gamma Array. The decreasing trends of the deduced quadrupole transition strength B(E2) with spin, along with increasing J(2)/B(E2) values before the band crossing, conclusively establish the origin of these states as arising from antimagnetic rotation. The abrupt increase in the B(E2) values after the band crossing in the quadrupole band, a novel feature observed in the present experiment, may possibly indicate the crossing of different shears configurations resulting in the re-opening of a shears structure. The results are reproduced well by numerical calculations within the framework of a semi-classical geometric model.

Microscopic study of low-lying spectra ofΛhypernuclei based on a beyond-mean-field approach with a covariant energy density functional

We present a detailed formalism of the microscopic particle-rotor model for hypernuclear low-lying states based on a covariant density functional theory. In this method, the hypernuclear states are constructed by coupling a hyperon to low-lying states of the core nucleus, which are described by the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this method to study in detail the low-lying spectrum of $_{\mathrm{\ensuremath{\Lambda}}}^{13}\mathrm{C}$ and $_{\mathrm{\ensuremath{\Lambda}}}^{21}\mathrm{Ne}$ hypernuclei. We also briefly discuss the structure of $_{\mathrm{\ensuremath{\Lambda}}}^{155}\mathrm{Sm}$ as an example of heavy deformed hypernuclei. It is shown that the low-lying excitation spectra with positive-parity states of the hypernuclei, which are dominated by $\mathrm{\ensuremath{\Lambda}}$ hyperon in the $s$ orbital coupled to the core states, are similar to that for the corresponding core states, while the electric quadrupole transition strength, $B(E2)$, from the ${2}_{1}^{+}$ state to the ground state is reduced according to the mass number of the hypernuclei. Our study indicates that the energy splitting between the first $1/{2}^{\ensuremath{-}}$ and $3/{2}^{\ensuremath{-}}$ hypernuclear states is generally small for all the hypernuclei which we study. However, their configurations depend much on the properties of a core nucleus, in particular on the sign of deformation parameter. That is, the first $1/{2}^{\ensuremath{-}}$ and $3/{2}^{\ensuremath{-}}$ states in ${}_{\mathrm{\ensuremath{\Lambda}}}^{13}\mathrm{C}$ are dominated by a single configuration with $\mathrm{\ensuremath{\Lambda}}$ particle in the $p$-wave orbits and thus provide good candidates for a study of the $\mathrm{\ensuremath{\Lambda}}$ spin-orbit splitting. On the other hand, those states in the other hypernuclei exhibit a large configuration mixing and thus their energy difference cannot be interpreted as the spin-orbit splitting for the $p$ orbits.

Shape transition and fluctuations in neutron-rich Cr isotopes aroundN=40

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 Cr isotopes around 64Cr are prolately deformed but still possess transitional character; large-amplitude shape fluctuations dominate in their low-lying states.

Triaxial superdeformation inAr40

Superdeformed (SD) states in $^{40}\mathrm{Ar}$ have been studied using the deformed-basis antisymmetrized molecular dynamics. Low-energy states were calculated by the parity and angular momentum projection and the generator coordinate method (GCM). Basis wave functions were obtained by the energy variation with a constraint on the quadrupole deformation parameter $\ensuremath{\beta}$, while other quantities such as triaxiality $\ensuremath{\gamma}$ were optimized by the energy variation. By the GCM calculation, an SD band was obtained just above the ground-state band. The SD band involves a ${K}^{\ensuremath{\pi}}={2}^{+}$ side band due to the triaxiality. The calculated electric quadrupole transition strengths of the SD band reproduce the experimental values appropriately. Triaxiality is significant for understanding low-lying states.