Ground-state Rotational Band Research Articles

Lambda motion and cluster states of [formula omitted] predicted via neural networks guided microscopic calculation

We investigate the Λ particle motion and cluster states of BeΛ9−11 using a novel multiple cooling approach guided by the Control Neural Networks method (Ctrl.NN). The numerical results are well reproduced, and the Ctrl.NN method shows rapid convergence concerning the set of the basis states in comparison to traditional generator coordinate method (GCM) calculations. Taking advantage of this merit, we calculate ground state energies and rotational bands of BeΛ9−11 which agree well with experimental observations. By analyzing the density distributions of the main basis, a slight short-range repulsive correlation between Λ particle and α clusters in directions perpendicular to the axis of the Be8 core is revealed. The Λ particle presents a significant broad distribution as a consequence of this correlation, which we call an “analog π-orbit” structure. Furthermore, we prove that this correlation originates from the competition between kinetic energy and Λ-N interaction, as indicated by the energy curves for different configurations of BeΛ9. In the ground states of BeΛ10 and BeΛ11, the Λ particle is confined in a cage-like structure of core nuclei formed by two α clusters and valence neutrons. These structures lead to the development of more compact Λ particle configurations, and the shrinkage of α-α relative distance is also well reproduced.

Moments of inertia in light deformed nuclei: Pairing and mean-field impacts

The dependence of the moment of inertia [Formula: see text] on the pairing and axial quadrupole deformation [Formula: see text] in [Formula: see text]Mg and [Formula: see text]Ne was investigated. The study is based on quadrupole-constrained calculations with three cranking approaches for [Formula: see text] (Inglis–Belyaev, Thouless–Valatin, adiabatic time-dependent Hartree–Fock) and a representative set of Skyrme forces (SVbas, SkM*, SLy6). At variance with macroscopic collective models, the calculations predict the specific regime [Formula: see text] at [Formula: see text] ([Formula: see text]Mg) and [Formula: see text] ([Formula: see text]Ne), where the pairing breaks down. This regime is explained by two effects: full break up of the pairing and specific evolution of a single dominant particle–hole (1ph) configuration with [Formula: see text]. The analysis of experimental data for the ground-state rotational bands in [Formula: see text]Mg and [Formula: see text]Ne shows that such regime is possible at low spins.

Fusion-fission analysis of C12+Cm248 and O16+Pu244 nuclear reactions across the Coulomb barrier

By using the symmetric-asymmetric Gaussian barrier distribution (SAGBD) model, Wong formula, and coupled channel approach, heavy ion fusion dynamics is investigated for $^{12}\mathrm{C}+^{248}\mathrm{Cm}$ and $^{16}\mathrm{O}+^{244}\mathrm{Pu}$ reactions at energies lying below and near the Coulomb barrier. Coupling of various channels linked with the structure of participants to the relative motion of the collision partners is done by considering a Gaussian type of weight function in the SAGBD model and cross sections are found to be enhanced relative to the calculations obtained by the simple barrier penetration model. In the SAGBD model, the channel coupling effects are calculated in terms of channel coupling parameter ($\ensuremath{\lambda}$) and percentage reduction in the height of apparent fusion barrier with respect to the Coulomb barrier (${V}_{\mathrm{CBRED}}$). The channel coupling parameter estimates the cumulative influence of dominant intrinsic channels, which are responsible for the sub-barrier fusion enhancement. The SAGBD calculations appropriately explain the dynamics of $^{12}\mathrm{C}+^{248}\mathrm{Cm}$ and $^{16}\mathrm{O}+^{244}\mathrm{Pu}$ reactions at energies lying around the Coulomb barrier. The coupled channel analysis of the present reactions is done by using the code ccfull and the coupled channel calculations unambiguously identify the dominant influences of the rotational states up to ${10}^{+}$ spin states of the ground state rotational band of target isotopes in both reactions. In addition, the couplings to higher order deformation, such as ${\ensuremath{\beta}}_{4}$ for target and low lying quantum states of the projectile, are necessarily required to reproduce the experimental data of $^{12}\mathrm{C}+^{248}\mathrm{Cm}$ and $^{16}\mathrm{O}+^{244}\mathrm{Pu}$ reactions. Apart from the fusion analysis, the dynamical cluster-decay model (DCM) is applied to understand the fission dynamics of the $^{260}\mathrm{No}^{*}$ nucleus formed via the above-mentioned reactions. The decay study is carried out at the center-of-mass energies spread, ${E}_{\mathrm{c}.\mathrm{m}.}$ ($\ensuremath{\approx}79$ to 109 MeV) by including the quadrupole deformations (${\ensuremath{\beta}}_{2}$) and optimum orientations (${\ensuremath{\theta}}_{i}^{\mathrm{opt}}$) of the decaying fragments. According to the experimental observation, the noncompound nucleus (nCN) fission component competes with the compound nucleus (CN) fission processes. Consequently, the possibility of nCN contribution is also explored in the decay of the $^{260}\mathrm{No}^{*}$ compound nucleus. With an aim to have a comprehensive analysis of CN and nCN fission mechanisms, the role of the center-of-mass energy (${E}_{\mathrm{c}.\mathrm{m}.}$) and angular momentum ($\ensuremath{\ell}$) is explored in terms of various parameters of DCM such as fragmentation potential, preformation probability, barrier modification, etc.

Occurrence of Staggering and Identical Energies in Ground State Rotational Bands in Some Actinide Isotopes within Two-Parameter Rotational Model

Occurrence of Staggering and Identical Energies in Ground State Rotational Bands in Some Actinide Isotopes within Two-Parameter Rotational Model

Angular momentum projection in the deformed relativistic Hartree-Bogoliubov theory in continuum

The angular momentum projection (AMP) method is implemented in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the point-coupling density functional. The wave functions of angular momentum projected states are expanded in terms of the Dirac Woods-Saxon (WS) basis, providing a proper description of the asymptotic behavior of the wave functions. The contribution of continuum induced by the pairing is considered by treating the pairing correlation with the Bogoliubov transformation. We present the formulas and numerical checks for the DRHBc$+$AMP approach in detail and use it to study low-lying excited states of axially deformed nuclei. Our calculations show that neutron-rich magnesium isotopes $^{36,38,40}\mathrm{Mg}$ are all well-deformed nuclei. The low-lying spectra of these three nuclei are obtained. The ground-state rotational bands of $^{36,38,40}\mathrm{Mg}$ are reproduced reasonably well by using this new DRHBc$+$AMP approach with the density functional PC-F1.

Possible cluster states in heavy and superheavy nuclei

The $\ensuremath{\alpha}$-core structure has been constructed to successfully interpret the band spectra of a series of nuclei above the double shell closure, plus the extension to the heavier cluster structure in the actinide region. By refining the binary cluster model, we show that such kind of picture can be applicable to more heavy nuclei, implying that there can exist cluster states in any heavy nuclei. As for superheavy nuclei, the involved cluster, analogous to the heavy particle radioactivity, is allowed to be above $Z>28$ plus the core $^{208}\mathrm{Pb}$. In this sense, besides some predictions on energy spectra of cluster states in superheavy nuclei, the very recently reported ${2}^{+}$ state of $^{282}\mathrm{Cn}$ is diagnosed to be not in the ground state rotational band via the present method. During the whole procedure, the uncertainty of the cluster model is determined for the first time, by the nonparametric resampling strategy, to make the whole theoretical results reliable and complete.

Nuclear Dynamics and Reactions in the Ab Initio Symmetry-Adapted Framework

We review the ab initio symmetry-adapted (SA) framework for determining the structure of stable and unstable nuclei, along with related electroweak, decay, and reaction processes. This framework utilizes the dominant symmetry of nuclear dynamics, the shape-related symplectic [Formula: see text] symmetry, which has been shown to emerge from first principles and to expose dominant degrees of freedom that are collective in nature, even in the lightest species or seemingly spherical states. This feature is illustrated for a broad scope of nuclei ranging from helium to titanium isotopes, enabled by recent developments of the ab initio SA no-core shell model expanded to the continuum through the use of the SA basis and that of the resonating group method. The review focuses on energies, electromagnetic transitions, quadrupole and magnetic moments, radii, form factors, and response function moments for ground-state rotational bands and giant resonances. The method also determines the structure of reaction fragments that is used to calculate decay widths and α-capture reactions for simulated X-ray burst abundance patterns, as well as nucleon–nucleus interactions for cross sections and other reaction observables.

Nuclear spin features relevant to ab initio nucleon-nucleus elastic scattering

Background: Effective interactions for elastic nucleon-nucleus scattering from first principles require the use of the same nucleon-nucleon interaction in the structure and reaction calculations, as well as a consistent treatment of the relevant operators at each order. Purpose: Previous work using these interactions has shown good agreement with available data. Here, we study the physical relevance of one of these operators, which involves the spin of the struck nucleon, and examine the interpretation of this quantity in a nuclear structure context. Methods: Using the framework of the spectator expansion and the underlying framework of the no-core shell model, we calculate and examine spin-projected, one-body momentum distributions required for effective nucleon-nucleus interactions in $J=0$ nuclear states. Results: The calculated spin-projected, one-body momentum distributions for $^4$He, $^6$He, and $^8$He display characteristic behavior based on the occupation of protons and neutrons in single particle levels, with more nucleons of one type yielding momentum distributions with larger values. Additionally, we find this quantity is strongly correlated to the magnetic moment of the $2^+$ excited state in the ground state rotational band for each nucleus considered. Conclusions: We find that spin-projected, one-body momentum distributions can probe the spin content of a $J=0$ wave function. This feature may allow future \textit{ab initio} nucleon-nucleus scattering studies to inform spin properties of the underlying nucleon-nucleon interactions. The observed correlation to the magnetic moment of excited states illustrates a previously unknown connection between reaction observables such as the analyzing power and structure observables like the magnetic moment.

New insights into sub-barrier fusion of 28Si + 100Mo

Fusion cross sections of the 28Si + 100Mo system have been measured near and below the Coulomb barrier by detecting the evaporation residues at forward angles. The excitation function has an overall smoother trend than what obtained in a previous experiment, and a large discrepancy is found for the lowest-energy region, where we observe a tendency of the S factor to develop a maximum, which would be a clear indication of hindrance. The results have been compared with the theoretical prediction of coupled-channels calculations using a Woods–Saxon nuclear potential, and including the low-energy excitation modes of both nuclei. Good agreement with data is found by including, in the coupling scheme, the three lowest members of the ground state rotational band of the oblate deformed 28Si, and two-phonons of the strong quadrupole vibration of 100Mo. The additional coupling, in a schematic way, of the two-neutron pick-up between ground states (Q-value = +4.86 MeV) has a minor effect on calculated cross sections, and does not essentially improve the data fit. The excitation function of 28Si + 100Mo has been compared with that of (1) the heavier system 60Ni + 100Mo having analogous features, and (2) several near-by 28Si, 32S + Zr, Mo systems with various nuclear structures and transfer Q-values. The role of quadrupole and octupole excitation modes, as well as of transfer channels, in affecting the fusion dynamics, are clarified to some extent. Systematic measurements of fusion barrier distributions and CC calculations properly including transfer couplings, are necessary, in order to shed full light on the influence of the various coupled channels on the fusion cross sections.

Nucleon-pair coupling scheme in Elliott's SU(3) model

Elliott's SU(3) model is at the basis of the shell-model description of rotational motion in atomic nuclei. We demonstrate that SU(3) symmetry can be realized in a truncated shell-model space if constructed in terms of a sufficient number of collective $S$, $D$, $G$, $\dots$ pairs (i.e., with angular momentum zero, two, four, $\dots$) and if the structure of the pairs is optimally determined either by a conjugate-gradient minimization method or from a Hartree-Fock intrinsic state. We illustrate the procedure for 6 protons and 6 neutrons in the $pf$ ($sdg$) shell and exactly reproduce the level energies and electric quadrupole properties of the ground-state rotational band with $SDG$ ($SDGI$) pairs. The $SD$-pair approximation without significant renormalization, on the other hand, cannot describe the full SU(3) collectivity. A mapping from Elliott's fermionic SU(3) model to systems with $s$, $d$, $g$, $\dots$ bosons provides insight into the existence of a decoupled collective subspace in terms of $S$, $D$, $G$, $\dots$ pairs.

Relativistic Dirac analyses of proton scattering from $$^{92}{\text {Zr}}$$

Relativistic coupled channel analyses based on the Dirac equation are presented for polarized 800-MeV proton inelastic scatterings from an axially symmetric deformed nucleus $$^{92}{\text {Zr}}$$ by using an optical potential model and the first-order collective model. The sequential iteration method using a computer program is employed to solve the complicated Dirac coupled channel equations. The optical potential parameters of the Woods–Saxon shape and the deformation parameters of several low-lying excited states are determined phenomenologically and analyzed. The coupled channel effect between the excited states that belong to the ground-state rotational band is investigated.

Rotational excitations in rare-earth nuclei: A comparative study within three cranking models with different mean fields and treatments of pairing correlations

High-spin rotational bands in rare-earth Er ($Z=68$), Tm ($Z=69$) and Yb ($Z=70$) isotopes are investigated by three different nuclear models. These are (i) the cranked relativistic Hartree-Bogoliubov (CRHB) approach with approximate particle number projection by means of the Lipkin-Nogami (LN) method, (ii) the cranking covariant density functional theory (CDFT) with pairing correlations treated by a shell-model-like approach (SLAP) or the so called particle-number conserving (PNC) method, and (iii) cranked shell model (CSM) based on the Nilsson potential with pairing correlations treated by the PNC method. A detailed comparison between these three models in the description of the ground state rotational bands of even-even Er and Yb isotopes is performed. The similarities and differences between these models in the description of the moments of inertia, the features of band crossings, equilibrium deformations and pairing energies of even-even nuclei under study are discussed. These quantities are considered as a function of rotational frequency and proton and neutron numbers. The changes in the properties of the first band crossings with increasing neutron number in this mass region are investigated. On average, a comparable accuracy of the description of available experimental data is achieved in these models. However, the differences between model predictions become larger above the first band crossings. Because of time-consuming nature of numerical calculations in the CDFT-based models, a systematic study of the rotational properties of both ground state and excited state bands in odd-mass Tm nuclei is carried out only by the PNC-SCM. With few exceptions, the rotational properties of experimental 1-quasiparticle and 3-quasiparticle bands in $^{165,167,169,171}$Tm are reproduced reasonably well.

Fusion of C12+Mg24 far below the barrier: Evidence for the hindrance effect

Background: The phenomenon of fusion hindrance may have important consequences on the nuclear processes occurring in astrophysical scenarios, if it is a general behavior of heavy-ion fusion at extreme subbarrier energies, including reactions involving lighter systems, e.g., reactions in the carbon and oxygen burning stages of heavy stars. The hindrance is generally identified by the observation of a maximum of the $S$ factor vs energy. Whether there is an $S$-factor maximum at very low energies for systems with a positive fusion $Q$ value is an experimentally challenging question.Purpose: Our aim has been to search for evidence of fusion hindrance in $^{12}\mathrm{C}+^{24}\mathrm{Mg}$ which is a medium-light system with positive $Q$ value for fusion, besides the heavier cases where hindrance is recognized to be a general phenomenon. $^{12}\mathrm{C}+^{24}\mathrm{Mg}$ is very close to the $^{16}\mathrm{O}+^{16}\mathrm{O}$ and $^{12}\mathrm{C}+^{12}\mathrm{C}$ systems that are important for the late evolution of heavy stars.Methods: The experiment has been performed in inverse kinematics using the $^{24}\mathrm{Mg}$ beam from the XTU Tandem accelerator of LNL in the energy range 26--52 MeV with an intensity of 4--8 pnA. The targets were $^{12}\mathrm{C}$ evaporations $50\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{g}/{\mathrm{cm}}^{2}$ thick, isotopically enriched to $99.9%$. The fusion-evaporation residues were detected at small angles by a $E\ensuremath{-}\mathrm{\ensuremath{\Delta}}E$-ToF detector telescope following an electrostatic beam deflector.Results: Previous measurements of fusion cross section for $^{12}\mathrm{C}+^{24}\mathrm{Mg}$ were limited to above-barrier energies. In the present experiment the excitation function has been extended down to $\ensuremath{\simeq}15\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{b}$ and it appears that the $S$ factor develops a clear maximum vs energy, indicating the presence of hindrance. This is the first convincing evidence of an $S$ factor maximum in a medium-light system with a positive fusion $Q$ value. These results have been fitted following a recently suggested method and a detailed analysis within the coupled-channels model that has been performed using a Woods-Saxon potential and including the ground state rotational band of $^{24}\mathrm{Mg}$. The coupled-channels calculations give a good account of the data near and above the barrier but overpredict the cross sections at very low energies.Conclusions: The hindrance phenomenon is clearly observed in $^{12}\mathrm{C}+^{24}\mathrm{Mg}$, and its energy threshold is in reasonable agreement with the systematics observed for several medium-light systems. The fusion cross sections at the hindrance threshold show that the highest value (${\ensuremath{\sigma}}_{s}=1.6\phantom{\rule{0.28em}{0ex}}\mathrm{mb}$) is indeed found for this system. Therefore it may even be possible to extend the measurements further down in energy to better establish the position of the $S$-factor maximum.

Investigation of a large change in deformation for the γ -soft nucleus Sm136

Lifetimes of excited states of the ground-state rotational band in the $^{136}\mathrm{Sm}$ nucleus have been measured up to ${I}^{\ensuremath{\pi}}={20}^{+}$ using the Doppler-shift attenuation method. The states were populated in the reaction $^{107}\mathrm{Ag}$($^{32}\mathrm{S}$, $1p2n$) at 145-MeV beam energy and the $\ensuremath{\gamma}$ rays emitted from the excited states were detected using the Indian National Gamma Array at the Tata Institute of Fundamental Research, Mumbai. The extracted transitional quadrupole moments indicate a reduction of collectivity with increasing spin after the band crossing. The results have been compared with the predictions of the cranked shell model as well as the triaxial projected shell-model calculations and indicate that the nucleus evolves from prolate $\ensuremath{\gamma}$-soft to a stable triaxially deformed shape after the first and the second crossing involving $\ensuremath{\pi}{h}_{11/2}^{2}$ and $\ensuremath{\pi}{h}_{11/2}^{2}\ensuremath{\bigotimes}\ensuremath{\nu}{h}_{11/2}^{2}$ configurations, respectively.

Testing the core-cluster model calculations for some heavy deformed nuclei

In this work, the spectra of some even–even isotopes are studied by selecting core-cluster decomposition of the parent nucleus. The considered nuclei lie in the rare-earth and the transition metal regions. The Schrödinger equation can be solved using Bohr–Sommerfeld relation and the modified Woods–Saxon beside Coulomb potentials to reproduce the spectra of these isotopes with mass number [Formula: see text]. The theoretical calculations of the excitation energies of the ground state rotational band are compared to the experimental data. The cluster model calculations show a good agreement with the experimental data for the transitional and rotational nuclei more than the vibrational nuclei. Some negative parity bands of the chosen nuclei are studied. The core-cluster charge products are correlated with the transition probability [Formula: see text].

High-K isomer and the rotational properties in the odd-Z neutron-rich nucleus M163Eu * * Supported by National Natural Science Foundation of China (11775112) and the Priority Academic Program Development of Jiangsu Higher Education Institutions

The newly observed isomer and ground-state band in the odd-Z neutron-rich rare-earth nucleus 163Eu are investigated by using the cranked shell model (CSM), with pairing treated by the particle-number conserving (PNC) method. This is the first time detailed theoretical investigations are performed of the observed 964(1) keV isomer and ground-state rotational band in 163Eu. The experimental data are reproduced very well by the theoretical results. The configuration of the 964(1) keV isomer is assigned as the three-particle state . More low-lying multi-particle states are predicted in 163Eu. Due to its significant effect on the nuclear mean field, the high-order deformation plays an important role in the energy and configuration assignment of the multi-particle states. Compared to its neighboring even-even nuclei 162Sm and 164Gd, there is a 10%~15% increase of of the one-particle ground-state band in 163Eu. This is explained by the pairing reduction due to the blocking of the nucleon on the proton [413] orbital in 163Eu.

Analysis of α + 9Be Scattering with a Semimicroscopic Potential

The analysis of the available data on the α + 9Be elastic scattering in the energy range from 28 to 104 MeV, including recent measurements at energies of 30, 40, and 90 MeV is carried out. The parameters of the semi-microscopic potential are obtained in the framework of the dispersion optical model, in which the exchange components of the average field potential were calculated using the previously proposed pseudo-oscillator approximation for the single-particle density matrix. The found potential is tested using the distorted wave method on the analysis of inelastic scattering in the considered energy region with excitation of the 5/2− (2.43 MeV) and 7/2− (6.38 MeV) levels of the ground-state rotational band. The potential parameters used for the output channel were estimated on the basis of the energy dependence. A satisfactory description of the angular distributions and the values of the deformation length is obtained.

Study of A = 100–150 superdeformed mass region by using two-parameterformula

Empirical formulae of rotational spectra consisting two parameters, such as single-term energy formula, E = aJb for spin J , and ab formula, were used to study the different features of superdeformed band in A = 100-150 mass region nuclei. The nuclear kinematic and dynamic moment of inertia for the ground-state rotational bands were calculated for this purpose and both showed gradual rise with rotational frequency. The study of ΔI = 2 staggering effects in the γ-ray energies, where the two sequences J = 4i; 4i + 1 and J = 4i + 2, (i = 0; 1; ...) are bifurcated, was also done. We also calculated the variation of the gamma ray energies from a smooth reference using the fourth derivative of the gamma ray energies at a given spin. The excellent agreement between the observed and calculated transition energies are in good support of the two-parameter formula.

Surface diffuseness parameter with quasi-elastic scattering for some heavy-ion systems

Itemised studies of the surface property of the inter-nucleus potential in heavy-ion reactions have been achieved using large-angle quasi-elastic scattering at sub-barrier energies close to the Coulomb barrier height for 63Li + 6430Zn and 73Li + 6430Zn systems. In this paper, the nuclear potential has been expressed using Wood Saxon (WS) formula. The effect of rotational deformation was included for the nucleus 6430Zn with ground state rotational band up to the 4+ states. The Single-Channel (SC) and Coupled-Channels (CC) calculations have been carried out to elicit the diffuseness parameter of the nuclear potential as well as the potential depth, these calculations have been performed by using (CQEL) program which is considered the latest version of computer code (CCFULL). The chi square method, χ², played an important role in determining the best-fitted value of the diffuseness parameter. Through CC calculations with inert projectile and rotational target for 63Li + 6430Zn and 73Li + 6430Zn systems, full compatibility of diffuseness parameter has been achieved with the standard value 0.63 fm with χ² = 0.130 and χ² = 0.163 respectively, while the SC calculations give 0.64 fm and 0.65 fm, respectively.

Dirac Phenomenological Analyses of 1.047-GeV Proton Inelastic Scatterings from 62Ni and 64Ni

Unpolarized 1.047 GeV proton inelastic scatterings from Ni isotopes, $^{62}$Ni and $^{64}$Ni are analyzed phenomenologically employing an optical potential model and the first order collective model in the relativistic Dirac coupled channel formalism. The Dirac equations are reduced to the Schr\"{o}dinger-like second-order differential equations and the effective central and spin-orbit optical potentials are analyzed by considering mass number dependence. The multistep excitation via $2^+$ state is found to be important for the $4^+$ state excitation in the ground state rotational band at the proton inelastic scatterings from the Ni isotopes. The calculated deformation parameters for the 2$^+$, the 4$^+$ states of the ground state rotational band and the first 3$^-$ state are found to agree pretty well with those obtained in the nonrelativistic calculations.