Ground Rotational Band Research Articles

Theoretical investigation of α -like quasimolecules in heavy nuclei

Quasimolecular $\ensuremath{\alpha}$-like ground rotational bands were evidenced a long time ago in light nuclei, but they cannot be detected in heavy nuclei due to large Coulomb barriers. In order to search for rotational bands built on excited states in these nuclei, we investigate the shape of an $\ensuremath{\alpha}$-nucleus quasimolecular potential matched to a realistic external $\ensuremath{\alpha}$-daughter interaction by using as input data $\ensuremath{\alpha}$-decay widths. It turns out that its Gaussian length parameter lies in a narrow interval, ${b}_{0}\ensuremath{\in}[0.6,0.8]$ fm, and the equilibrium radius is slightly larger than the predicted Mott transition point from nucleonic to the $\ensuremath{\alpha}$-cluster phase in finite nuclei, confirming that $\ensuremath{\alpha}$ clusters are born on the nuclear surface at low densities. We point out that the $\ensuremath{\alpha}$ emitters above magic nuclei have the largest spectroscopic factors ${S}_{\ensuremath{\alpha}}\ensuremath{\sim}10%$. In addition, we predict that for nuclei with ${b}_{0}g0.75$ fm, the first excited vibrational resonant state in the quasimolecular potential is close to the Coulomb barrier and therefore the rotational band built on it can be evidenced by the structure of the $\ensuremath{\alpha}$-scattering cross section versus energy. Moreover, its detection by a highly sensitive $\ensuremath{\gamma}$-ray beam produced by laser facilities would provide an additional proof for the existence of $\ensuremath{\alpha}$ molecules in heavy nuclei.

Influence of 1+ states on the gR-factors of ground band in isotopes of 160Dy and 170, 174Yb

In this work, non-adiabatic effects manifested in the magnetic properties of ground band in even–even deformed nuclei are studied. A simple phenomenological model, which takes into account the Coriolis mixing of states of the ground and bands, is proposed. Using perturbation theory, corrections to the wave functions of states are determined. In addition, an analytical expression to determine the gR-factor of the states of ground rotational band is obtained. The decrease in gR-factor with increasing angular momentum I is discussed. The accuracy of the approximation is verified. The determined values of wave functions of different methods are found to be close to each other.

Angular correlation between proton and neutron rotors

A brief review is given on the controversy and its solution about the fact that the angular momentum vector of protons and that of neutrons in well-deformed nuclei at low total angular momenta have a strong correlation that they are oriented in opposite directions. In a simple two-rotor model in 2-dimensional space, this fact is explained as originating from the quantum mechanical uncertainty relation between the angle and the angular momentum for the relative rotation of the two rotors. As the second topic, a more realistic model consisting of two triaxial rotors in 3-dimensional space coupled with a QQ interaction is employed to investigate a possible shears-band-like collective rotation predicted by T. Otsuka, in which the angle at which the angular momentum of protons and that of neutrons intersect changes continuously from 180° at spin zero toward 0° at high spins within the same rotational band. The probability distributions of the angle between the two angular momenta and the angle between the longest principal axes of two rotors are calculated to examine the participation of the scissors mode in the evolution of the ground rotational band versus spin.

Ground-state, β and K = 11/2− γ bands in 163, 165Er

The Bohr Hamiltonian prescription for collective behavior in heavy nuclei is used with the ground state, vibrational and rotational bands each allowed unique mass parameter values. Excited-state energies and E2 transition probabilities of the nuclei 163, 165Er are calculated, these nuclei being selected for their particular relevance to the model, and their topical nature in both theoretical and experimental studies.

Dependence of Quasi-Particle Alignment on Proton Number in N = 44 Isotones 82Sr, 83Y, 84Zr and 85Nb

The g-factors of Ground Rotational Band states of N = 44 isotones 82Sr, 83Y, 84Zr and 85Nb have been measured by the transient-magnetic-field ion implantation perturbed angular distribution (TMF-IMPAD) method. The measured g-factors of 82Sr increase with the increase of spin I, indicating a proton alignment only. Positive peaks appear in the variation of g-factors with spin for 83Y and 84Zr at spin 21/2+ and 10+ respectively, indicating a proton alignment followed by a neutron alignment. A negative peak occurs for 85Nb at the spin 25/2+, indicating a neutron alignment followed by a proton alignment.

Triaxial quadrupole deformation dynamics insd-shell nuclei aroundMg26

Large-amplitude dynamics of axial and triaxial quadrupole deformation in $^{24,26}\mathrm{Mg}$, $^{24}\mathrm{Ne}$, and $^{28}\mathrm{Si}$ is investigated on the basis of the quadrupole collective Hamiltonian constructed with the 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 $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ vibrations in $^{24}\mathrm{Mg}$ and $^{28}\mathrm{Si}$. The $\ensuremath{\gamma}$ softness in the collective states of $^{26}\mathrm{Mg}$ and $^{24}\mathrm{Ne}$ are discussed. Contributions of the neutrons and protons to the transition properties are also analyzed in connection with the large-amplitude quadrupole dynamics.

Evidence onΔI=4bifurcation in ground bands of even-even nuclei and the theoretical explanation with the interacting boson model

All the data available for the ground rotational bands of even-even actinide nuclei and rare-earth nuclei (with band crossing spins I{sub c}{ge}16 ) are studied in terms of the staggering parameter, introduced in the discussion of superdeformed bands. Evidence of {Delta}I=4 bifurcation ({Delta}I=2 staggering) is observed in many cases. We study theoretically this bifurcation phenomenon in terms of the interacting boson model (IBM) by using the variation before projection (VBP) method. In our theoretical framework, the staggering phenomenon is found to originate from the finite boson number effect. The IBM explains the opposite signs of the staggering patterns between two neighboring nuclei, which are found experimentally. We perform also the full diagonalization of the IBM Hamiltonian and obtain consistent results with the VBP method. By introducing the g bosons, we are able to reproduce even irregular or localized staggering patterns in the {Delta}I=4 bifurcation spectrum observed in some data. {copyright} {ital 1997} {ital The American Physical Society}

I = 2 Staggering in Ground Rotational Bands and A = 2 Mass Staggering in the Deformed Nuclei

We find a ΔI = 2 staggering or ΔI = 4 bifurcation phenomenon in many even-even mass ground rotational bands in rare-earth and actinide nuclei by performing staggering parameter analysis. We take the Interacting Boson Model and reproduce the ΔI = 4 bifurcation by slightly modifying the SU(3) limit of the IBM wave function. The IBM then predicts ΔA = 2 mass staggering in the deformed nuclei, which is also found experimentally.

Precision fit of nuclear rotational energies

A five-parameter I(I+1) expansion is presented and is used to analyse systematically the experimentally known level energies of the ground rotational bands in doubly-even deformed nuclei up to very high spin. Better and excellent agreement is obtained with experimental data. The expansion coefficients obtained by a least-squares fitting procedure are compared with those deduced from the previous two-parameter formulae showing closer results to the Holmberg - Lipas formula, as well as the Wu - Zeng expression.

A Step toward Large-Amplitude Description of the Ground Band around Band Crossing

The possible occurrence of large-amplitude motion in the band-crossing region is shown numeri­ cally for the ground rotational band. Calculations are based on the formalism by means of the self-consistent collective coordinate method. The phenomena in which the energy curve of the ground-state band crosses that of the rotation-aligned band (so-called s-band) have been mostly analyzed in the framework of mean-field theory. In particular, the roles of residual interactions have not yet been investigated in the band-crossing region, where the excitation energy of an aligned two-quasiparticle state is very low. The ground-state band (g-band) is not expected to be purely rotational around the band crossing, where it is thought to begin generating a new intrinsic mean field. Residual interaction is considered to play an essential role in this dynamics. For example, in 164Er the g-band and the s-band cross around the angular momentum 1=15. Just after the crossing, the yrast 1=16 state decays to two 1=14 states with nearly the same B(E2) value. l ) This fact implies that the structures of these states are not simple, though they are usually considered to be rotational states built on some intrinsic state. It is an important subject to study the dynamics of the transition (reorganization) process from the intrinsic state of the g-band to that of the s-band. As it is our aim to describe the transition process as large-amplitude motion, we first extract the relevant subspace connecting old and new intrinsic states, in which the large­ amplitude motion is properly described. Marumori and his collaborators attempted to define such a relevant subspace of the large-amplitude motion by making use of the selfconsistent collective coordinate (SeC) method. 2 ).3) In this paper we report the results of numerical application of the description given in the appendix of Ref. 3), which is now extended to the fourth power of an amplitude. 4 ) As we do not attempt to reproduce experimental values in this report, we adopt the model Hamiltonian of the pairing-plus-quadrupole force and the model space of two major shells for proton and neutron, respectively. In this model the g-band and the s-band of 164Er cross around 1=7. The main purpose of this paper is to study the dynamical roles of residual interaction at 1=6. In order to take new degrees of freedom for large-amplitude motion into account, we follow Ref. 3) and express a state of the g-band in the form

SEMICLASSICAL DESCRIPTION OF SPIN EXCITATIONS OF THE PARTICLE-CORE INTERACTION SYSTEM

A model Hamiltonian describing a nucleon moving in a shifted oscillator well and inter-acting with a harmonic core through a quadrupole-quadrupole term is semiclassically treated. The solutions of the linearised equations of motion are quantised by the Bohr-Sommerfeld procedure. Among the four RPA modes, there is one which corresponds to the harmonic vibration of the spin degree of freedom. This mode is analogous to the spin wave mode describing a system of interacting spins placed in a magnetic field. The state describes a harmonic wobbling motion around a stationary state belonging to the ground rotational band.

Coulomb excitation of 168Er

The nucleus 168Er has been Coulomb excited using beams of 40Ca, 58Ni and 208Pb. In addition lifetimes of eight states have been measured by the recoil distance method. About 50 E2 matrix elements in the ground and gamma rotational bands have been determined. The ground-band E2 matrix elements follow the predictions of a quadrupole-deformed rotor model. In the gamma band the measured E2 matrix elements exhibit a reduction of collectivity above spin 6.

Relations for the coefficients in the I(I+1) expansion for rotational spectra.

All the data now available for the ground rotational bands of actinide even-even nuclei and of rare-earth even-even nuclei (with bandcrossing angular momentum {ital I}{sub {ital c}}{ge}16) are analyzed with the four-parameter {ital I}({ital I}+1) expansion. The parameters obtained by the least-squares fitting definitely deviate from the relations derived from the Harris two-parameter {omega}{sup 2} expansion, but are closer to the relations deduced from the closed two-parameter expression by Wu and Zeng.

Weak-coupling caluclation on the deformed regions of Zr and Mo isotopes

A neutro-proteon weak-coupling model calculation is performed on the deformed regions of Zr and Mo isotopes. The simultaneous occupation of neutrons and protons of spin-orbit partner orbitals plays a crucial role in determining the onset of deformation. It is shown that the ground rotational band can be produced qualitatively by the isoscalar component of the n-p interaction in the100, 102Zr and102–106Mo isotopes.

Resonating-group-method study of alpha +40Ca elastic scattering and 44Ti structure.

The \ensuremath{\alpha}${+}^{40}$Ca elastic scattering cross sections in the energy range from 29.0 to 61.0 MeV are shown to be well reproduced by the resonating-group method by introducing a phenomenological imaginary potential. The local potential equivalent to the nonlocal potential of the resonating-group method is found to be very close to the optical potential of Delbar et al. which fits the scattering data well in a wide energy range. Since the optical potential of Delbar et al. is the so-called unique optical potential which is free from the discrete ambiguity, this result means that the resonating-group method is powerful and reliable for the study of the internucleus interaction. By calculating the bound and quasibound level spectra by the same \ensuremath{\alpha}${+}^{40}$Ca resonating-group method, it is found that the lowest positive parity rotational band is located near the observed ground rotational band of $^{44}\mathrm{Ti}$. In addition, when we use other effective two-nucleon forces so as to locate the calculated lowest ${0}^{+}$ state near the observed excited ${0}^{+}$ state with the excitation energy 8.54 or 11.2 MeV, which has large \ensuremath{\alpha} strength, the fitting of the elastic scattering data by the resonating-group method is found to be very bad. These results force us to regard that the structure of the $^{44}\mathrm{Ti}$ ground band as containing a large amount of \ensuremath{\alpha}${+}^{40}$Ca clustering component. However our present resonating-group method predicts the appearance of the negative parity rotational band with its bandhead ${1}^{\mathrm{\ensuremath{-}}}$ state below the excitation energy 10 MeV although there have been reported no such experimental indications at all of this.

Study of form factors of elastic and inelastic electron scattering by 20Ne

The elastic electron scattering form factor and also the form factor of inelastic scattering accompanied by nuclear transitions to excited states of the ground rotational band are calculated within the Sp z (2, R) model reproducing the dynamics of quadrupole nonsphericity of 20Ne. The Sp z (2, R) model is shown to describe correctly the experimental dependence of the elastic and inelastic (for the 0 + to 2 + state transition) form factors of the momentum transferred and, consequently, to remove some difficulties experienced by other dynamic models.

Microscopic Cluster Model for the FormulaBe Hypernucleus

The microscopic α + α + Λ cluster model is applied to describe the low-lying positive and negative parity states of 9ΛBe. The ground rotational band shows a weak coupling behavior between Λ and 8Be, while the negative parity states show a strong-coupling type band structure with K = 0− and K = 1−. The Λ binding energy in the ground state agrees very well with the observed value. The K = 0− and K = 1− band-head energies are in good correspondence to the 9Be(K−, π−)9ΛBe reaction data. Spectroscopic factors for the Λ + 8Be and α + 5ΛHe channels are evaluated. Electric quadrupole transition probabilities B(E2) are predicted.

On the properties of the extended ground rotational bands in actinide even-even nuclei

The energies of the yrast levels extending up to I = 28 in even-even actinide nuclei were analyzed in terms of an angular velocity expansion E rot( I) = ∑ i=1 n α i ω I 2 i with n = 2−7 to test the application of this expansion to high spin. A strong ω 2-dependence is observed for the Variable Moment of Inertia model (VMI) with two parameters α i ( ω 2), particularly at the higher spins. There is a marked difference in the ω 2-dependence, of these parameters between the N = 142 and other nuclei studied in this region to indicate structure effects are present at the higher spin.

Double Crossing of the Ground Rotational Band and Super Band

We suggest that the crossing of the ground rotational band by a second rotational-aligned band may in fact be a double crossing. From extrapolations of the rotational energy formula, we show that one could expect such a second crossing around I approx. = 30 in /sup 158/Er. This provides an alternative explanation in terms of a double band crossing for the second discontinuity recently observed in the moment of inertia about I = 28 in /sup 158/Er.

Microscopic approach to coupled modes in nuclei

A microscopic theory of low-lying collective states in even-even nuclei is developed. It treats rotations, vibrations and their coupling on the same footing. It is based on elementary transition operators, which are an extension of the phonon concept, on a density-matrix expansion in terms of these operators, and on equations of motion for the density matrix. The example of ground, beta and gamma rotational bands is treated up to second order of the density-matrix expansion. The cranking-model results for rotations and the random-phase approximation results for vibrations are reproduced, both for energies and E2 transition rates, by the same formalism in first order. For E2 transition rates the effects of rotation-quasiparticle, rotation-vibration and vibration-quasiparticle coupling are described in second order. This covers non-adiabatic corrections to the Alaga rules inside the bands and between the ground, beta and gamma bands, deviations in the intrinsic moments of the beta and gamma bands, and the beta-gamma transitional moment.