Superdeformed Bands Research Articles

Nuclear softness in the variable moment of inertia model and its application to superdeformed bands in the mass region A ≈ 60 - 90

For superdeformed (SD) bands 58Ni (b1), 58Cu, 59Cu (b1), 61Zn, 62Zn, 65Zn, 68Zn, 84Zr, 86Zr (b1), 88Mo (b1, b2, b3) and 89Tc in the A ≈ 60 - 90 mass region, the nuclear softness (NS) parameter, σ, has been calculated using the VMINS3 model. The SD bands 58Ni (b1), 58Cu, 59Cu (b1), 62Zn, 65Zn, and 88Mo (b2, b3) have NS parameter values that are greater than those of the normal deformed bands, indicating smaller rigidity. The fluctuation of the NS parameter versus the gamma energy ratio, R, of SD bands in the A ≈ 60 - 90 mass region is one of the study's findings. The ratio of transition energies was used to calculate the band head spin, I0, by the Descartes method (the greatest technique to solve the quartic equation based on an auxiliary cubic equation) which was then confirmed by root mean square deviations. The estimated and observed transition energies are in good agreement.

Search for Super-Deformed Identical Bands in the A = 190 Mass Region on the Basis of Level Spin

The systematic study of Super-Deformed (SD) bands in the A=190 mass region has been performed. We observed a large number of pairs of SD bands, with different mass numbers, having transition energies nearly equal (within 3 keV) and having identical dynamic moments of inertia. The bands having nearly equal transitions energies and other parameters are called identical bands. We have performed detailed analysis and found 16 pairs of Super-Deformed Identical Bands (SDIBs) whose Eγ energies and moment of inertia are in good agreement with each other. The modified Variable Moment of Inertia (VMI) model is applied to 16 pairs of SDIBs to estimate the band spins by fitting the two parameters J0 and C. We found that out of 16 pairs, the band-head spin is consistent with moment of inertia and transition energies for four pairs. For another seven pairs, the transition energies and moment of inertia are identical, but originate from levels with different spins. The remaining five pairs have the identical energy but spins are either increasing or decreasing by one unit in the pair. Secondly, the NpNn scheme is applied to verify the existence of SDIBs. The parameters deduced from the NpNn scheme are also in good agreement for the mentioned case. The study indicates that each pair of conjugate nuclei have nearly identical spin, moment of inertia (dynamic) and gamma transition energy.

Analysis of super-deformed bands in A∼150 mass region: The principal-axis cranking versus the variable moment of inertia

A systematic study of the super-deformed (SD) bands in [Formula: see text] mass region is carried out by using two different approaches, namely, the Principal-axis Cranking (PAC) and the Variable Moment of Inertia (VMI). Both these approaches not only support each other’s results but also their hand-in-hand calculations help us in establishing the prominent features, like the band-head spin, parity, quadrupole moment and excitation energy, of the observed SD bands. Since the rotational frequency in the PAC model depends on angular momentum, parity, and quadrupole moment, by identifying the most common rotational frequency between the calculations and the experimentally observed ones, the band-head spin, and parity are likely to be extracted. On the other hand, the VMI model calculations are quite simple and are capable of predicting the threshold limits of band-head spin, the quadrupole moment, and the excitation energy, which in turn are significantly useful for deciding an appropriate configuration assignment in the PAC model calculations of the existing as well as upcoming SD bands. Further, the VMI model calculations involve the variation of the moment of inertia with angular momentum, which in turn is quite helpful in understanding the mechanism of the observed identical SD bands between a pair of neighboring nuclei.

Study of the High Rotational Bands of Moderately Heavy Nuclei

Abstract: This paper provides a study of the rotational properties of heavy and medium nuclei, particularly the paired nuclei existing in the rare-earth, including Gd, Er, …, first to have a good representation of the intrinsic prolate fundamentals of the considered nuclei. The most important residual nuclear interaction is the pairing force which makes it possible to couple the nucleons in pairs. To take it into account, we introduce the Bardeen-Cooper-Shrie formalism (BCS), developed to describe the phenomenon of superconductivity. The test wave function is then more elaborate than that of Hartree-Fock and corresponds to a state no longer of independent particles, but of independent quasi-particles. A quasi-particle state (qp) is a linear combination of particles and holes. The Routhian Hartree-Fock model through the analysis of the experimental spectra of rotation of the deformed nucleus was usewd. Knowing that this was originally expanded by Bohr-Mottelson by applying I(I+1) expansion, we modified an existing fixed code (HF) with axial symmetry, which extended in a way that allows us to add constraints on the angular momentum and kelvin rotation to the Hamiltonian known as Cr.HF (cranking version of this formalism), initially studied by P. Quentin. This modification led to good results, especially the spectra of rotation and the angular velocities as a function of the angular momentum. Besides, it led to a decrease in the moment of inertia after it was large in some models, such as in the Hartree-Fock-Bogoliubov (HFB) model. The rotational properties and the moments of inertia of the super-deformed bands of some deformed nuclei have been studied as well as in the mass regions A=190; A=160. The results were compared with experimental results which gave good agreement. This work will offer an interesting perspective necessary for certain improvements or extensions of the Cr.HF. Keywords: Microscopic mean field, Collective nuclear rotation, Angular momentum Routhian Hartree-Fock (RHF) model, Inertia moments, Angular velocity.

The description of nuclear superdeformed rotational bands in A∼190 region with pq formula

The nuclear superdeformed (SD) rotational bands in [Formula: see text] mass region were systematically analyzed with the two-parameter [Formula: see text] formula. The study of the observed transition energies by using the best-fitting method shows that the [Formula: see text] formula can provide excellent descriptions of the SD bands. The analysis of two types of moments of inertia ([Formula: see text] and [Formula: see text]) and bandhead moment of inertia ([Formula: see text]) also demonstrates the reliability of the [Formula: see text] formula in describing the SD bands. Finally, by using the [Formula: see text] formula, the unobserved transition energies for the lower and higher spin regions were predicted herein.

Energy Staggering and Identical Superdeformed Bands in $${}^{{191{-}195}}$$Tl Nuclei within Modified Perturbed $${SU(3)}$$ Limit of The sdg IBM

Energy Staggering and Identical Superdeformed Bands in $${}^{{191{-}195}}$$Tl Nuclei within Modified Perturbed $${SU(3)}$$ Limit of The sdg IBM

Super- and hyperdeformation inZn60, Zn62, andGe64at high spins

Background: The observation of the superdeformed (SD) bands in $^{60,62}\mathrm{Zn}$ indicates that the particle number 30 is a magic particle number, where two- and four-neutron single particles are considered to be promoted to the intruder $1{g}_{9/2}$ shell. However, the SD-yrast band in $^{62}\mathrm{Zn}$ is assigned negative parity.Purpose: I investigate various SD configurations in the rapidly rotating $^{60,62}\mathrm{Zn}$ and $^{64}\mathrm{Ge}$, and attempt elucidating the different roles of the energy gaps at particle numbers 30 and 32.Method: I employ a nuclear energy-density functional method: the configuration-constrained cranked Skyrme-Kohn-Sham approach is used to describe the rotational bands near the yrast line.Results: The negative-parity SD bands appear higher in energy than the positive-parity SD-yrast band in $^{60}\mathrm{Zn}$ by about 4 MeV, which is indicative of the SD doubly magic nucleus. However, the energy gap in $^{64}\mathrm{Ge}$ is smaller $\ensuremath{\approx}2--3$ MeV, though the quadrupole deformation of the SD states in $^{64}\mathrm{Ge}$ is greater than that of $^{60}\mathrm{Zn}$. The present calculation predicts the occurrence of the hyperdeformed state in $^{60}\mathrm{Zn}$ and $^{64}\mathrm{Ge}$ at a high rotational frequency $\ensuremath{\approx}2.0\phantom{\rule{4pt}{0ex}}\mathrm{MeV}/\ensuremath{\hbar}$ due to the occupation of the ${h}_{11/2}$ shell.Conclusions: An SD-shell gap at particle number 30 and 32 appears at different deformations and the energy gap at particle number 32 is low, which make the SD structures of $^{62}\mathrm{Zn}$ unique, where the negative-parity SD states appear lower in energy than the positive-parity one.

Cranking covariant density functional theory with a shell-model-like approach for the superdeformed band of 42Ca

The superdeformed rotational band in 42Ca is studied with the cranking covariant density functional theory complemented by a shell-model-like approach for treating the pairing correlations. The microscopic and self-consistent description of the superdeformed rotational band is obtained. The calculated energy surfaces show local minimums at [Formula: see text] from rotational frequency [Formula: see text] [Formula: see text] to [Formula: see text][Formula: see text]MeV. The shape coexistence of spherical, normal deformation and superdeformation is found at [Formula: see text][Formula: see text]MeV. The single-particle levels and configurations are analyzed in details with the deformation. The configuration of the superdeformed band is figured out as [Formula: see text]. The single-particle Routhians indicate that the neutrons configuration plays a key role in the formation of the superdeformed band, and the change of the protons configuration at [Formula: see text][Formula: see text]MeV terminates the superdeformed band. The importance of pairing correlation to the superdeformed band is also studied in terms of the moments of inertia and the angular momentum.

Features of Triaxial Superdeformed Bands in Odd Mass Nuclei Using the Cranked Nilsson–Strutinsky Model

Features of Triaxial Superdeformed Bands in Odd Mass Nuclei Using the Cranked Nilsson–Strutinsky Model

Parity partner bands in 163Lu: A novel approach for describing the negative parity states from a triaxial super-deformed band

The wobbling spectrum of [Formula: see text]Lu is described through a novel approach, starting from a triaxial rotor model within a semi-classical picture, and obtaining a new set of equations for all four rotational bands that have wobbling character. Redefining the band structure in the present model is done by adopting the concepts of Signature Partner Bands and Parity Partner Bands. Indeed, describing a wobbling spectrum in an even–odd nucleus through signature and parity quantum numbers is an inedited interpretation of the triaxial super-deformed bands.

Properties of superdeformed bands in A < 100 mass region within the variable moment of inertia model

The variable moment of inertia (VMI) model has been used to study the properties of superdeformed (SD) bands in A < 100 mass region. The experimental γ-ray (E γ ) transitions energies are least square fitted to obtain the model parameters such as band head moment of inertia, (J 0), and stiffness constant, (C). The observed E γ transition energies obtained using the VMI formula are in excellent agreement with the experimental values. In the paper, we have studied the dynamic moment of inertia (J (2)) and its variation with rotational frequency (ℏ ω). We also studied the ΔI = 2 staggering phenomena in SD bands of 61Zn(SD), 62Zn(SD1), 80Sr(SD1), 81Sr(SD1), 82Sr(SD), 83Sr(SD), 84Zr(SD), 86Zr(SD1), 86Zr(SD4), 87Nb(SD2), 89Tc(SD) and 91Tc(SD) nuclei. The VMI model brings very comprehensive interpretation for spin assignment and to study the properties of SD bands in light mass region nuclei.

Description of Superdeformed Bands of the Isotones $$\boldsymbol{N=113}$$ for Nuclear Mass Region $$\boldsymbol{A\sim 190}$$

Electric quadrupole $$\gamma$$ -ray transition energies $$E_{\gamma}$$ , rotational frequencies $$\hbar\omega$$ and dynamic $$J^{(2)}$$ and kinematic $$J^{(1)}$$ moments of inertia have been calculated for yrast and excited superdeformed (SD) bands in Hg, Tl, Pb isotonic ( $$N=113$$ ) nuclei by using two-parameter power-law formula. The model parameters and the bandhead spins were determined using a $$\chi^{2}$$ fit by random minimization via Monte Carlo simulation technique with a uniform distribution of random numbers. The calculated results of $$E_{\gamma}$$ agreed with the experimental data published in National Nuclear Data Center (nndc) very well, and present assigned spins of the studied superdeformed rotational bands (SDRB’s) are consistent with the spins suggested by other previous formulae. The majority of the studied SD bands displayed the same large, smooth increase in $$J^{(2)}$$ by increasing $$\hbar\omega$$ . A saturation of $$J^{(2)}$$ at higher values of $$\hbar\omega$$ was observed for the two bands (1, 2) in $${}^{195}$$ Pb due to the Pauli blocking of the quasineutron alignment in the presence of quasiproton alignment. The studied SD bands exhibited staggering in their transition energies. Five pairs of signature partner SDRBs exhibited an $$\Delta I=1$$ staggering effect in their transition energies, namely $${}^{193}$$ Hg (SD1, SD2), $${}^{193}$$ Hg (SD3, SD4), $${}^{194}$$ Tl (SD1, SD2), $${}^{195}$$ Pb (SD1, SD2), and $${}^{195}$$ Pb (SD3, SD4). To investigate this $$\Delta I=1$$ staggering, we extracted the difference between the average transition $$I+2\to I$$ and $$I\to I-2$$ energies in one band and the transition $$I+1\to I-1$$ energies in the signature partner. Also, a $$\Delta I=2$$ staggering exists in three SD bands of odd–odd $${}^{194}$$ Tl. This $$\Delta I=2$$ staggering effect was determined by calculating the energy difference between two consecutive $$\gamma$$ -ray transitions as a function of rotational frequency after subtracting a smooth reference expressed in the notation of Cederwall.

Shape coexistence in neutron-deficient Hg188 investigated via lifetime measurements

Shape coexistence in the $Z \approx 82$ region has been established in mercury, lead and polonium isotopes. Even-even mercury isotopes with $100 \leq N \leq 106$ present multiple fingerprints of this phenomenon, which seems to be no longer present for $N \geq 110$. According to a number of theoretical calculations, shape coexistence is predicted in the $^{188}$Hg isotope. The $^{188}$Hg nucleus was populated using two different fusion-evaporation reactions with two targets, $^{158}$Gd and $^{160}$Gd, and a beam of $^{34}$S, provided by the Tandem-ALPI accelerators complex at the Laboratori Nazionali di Legnaro. The channels of interest were selected using the information from the Neutron Wall array, while the $\gamma$ rays were detected using the GALILEO $\gamma$-ray array. The lifetimes of the excited states were determined using the Recoil Distance Doppler-Shift method, employing the dedicated GALILEO plunger device. Using the two-bands mixing and rotational models, the deformation of the pure configurations was obtained from the experimental results. The extracted transition strengths were compared with those calculated with the state-of-the-art symmetry-conserving configuration-mixing (SCCM) and five-dimentional collective Hamiltonian (5DCH) approaches in order to shed light on the nature of the observed structures in the $^{188}$Hg nucleus. An oblate, a normal- and a super-deformed prolate bands were predicted and their underlying shell structure was also discussed.

Signature-dependent triaxiality for shape evolution from superdeformation in rapidly rotating 40Ca and 41Ca

Abstract We investigate the possible occurrence of highly elongated shapes near the yrast line in $^{40}$Ca and $^{41}$Ca at high spins on the basis of the nuclear energy-density functional method. Both the superdeformed (SD) yrast configuration and the yrare configurations on top of the SD band are described by solving the cranked Skyme–Kohn–Sham equation in the three-dimensional coordinate space representation. It is suggested that some of the excited SD bands undergo band crossings and develop to hyperdeformation (HD) beyond $J \simeq 25 \hbar$ in $^{40}$Ca. We find that the change of triaxiality in response to rotation plays a decisive role in the shape evolution towards HD, and that this is governed by the signature quantum number of the last occupied orbital at low spins. This mechanism can be verified in an experimental observation of the positive-parity SD yrast signature-partner bands in $^{41}$Ca, one of which ($\alpha=+1/2$) undergoes crossings with the HD band, while the other ($\alpha=-1/2$) shows smooth evolution from collective rotation at low spins to non-collective rotation with an oblate shape at termination.

Moments of Inertia of SDIBs in A = 190 Mass Region using VMI Model

A lot of identical bands are known at present in the Normal Deformed (ND) region. In our study of the occurrence and properties of identical bands in Super-Deformed (SD) nuclei we first applied the modified Variable Moment of Inertia (VMI) model to extract the band-head spin of Super-Deformed bands. The calculated transition energies, level spins and dynamic moment of inertia are systematically examined. Then, in the framework of theoretical model several identical bands are identified. The kinematic and dynamic moment of inertia have been calculated for the six pairs of Super-Deformed Identical Bands (SDIBs) which was not reported earlier in the literature. Thus, the results are significant. In all the cases J(2) is significantly higher than J(1) over a large range of frequency.

On the question of spin assignment and quantized alignment in the A~ 100-140 superdeformed mass region

The structure of the superdeformed (SD) mass region, A~ 100-140 is re-studied by using a direct approach based on the rotational energy formula E (I, K). To determine the lowest spin (If) and the K-value of the considered SD bands; namely, 104Pd(b1), 132Ce(b1), 134Nd(b1), 136Nd(b2), 142Sm(b1), 148Eu(b1, b2), 131Ce(b1, b2), 133Ce(b1, b2), 133Pr(b1, b3), 137Sm(b1) and 143Eu(b1), we study the dependence of the dynamic, (2) and static, (1) , moments of inertia on the rotational frequency (ℏω). Using a search method, the higher order inertial parameters A and B of such method were determined by fitting the proposed transition energies with their observed ones. The calculated transition energies are strongly dependent on the identified spins. It was found that, when an accurate bandhead spin (I0) is assigned, a good agreement is achieved between the calculated and observed transition energies. The results for the spin are in satisfactory agreement with other theoretical results and available experimental data. A good agreement between the calculated and corresponding experimental transition energies of the SD bands is obtained which supports our methodology. The incremental alignment and also the angular momentum alignment, for the identical SD bands exist among the considered ones, have been studied and the results verified well Stephens’s law of alignment.

Importance of multicranked configuration mixing for angular-momentum-projection calculations: Study of superdeformed rotational bands in Dy152 and Hg194

Recently we investigated an effective method of multicranked configuration mixing for angular-momentum-projection calculations, where several cranked mean-field states are coupled after projection: The basic idea was originally proposed by Peierls and Thouless more than fifty years ago. With this method a good description of the rotational band has been achieved in a fully microscopic manner. In the present work, we apply the method to the high-spin superdeformed band, for which a long rotational sequence is observed, and study how a good description is obtained for the rotational spectrum as well as the ${\mathcal{J}}^{(1)}$ and ${\mathcal{J}}^{(2)}$ moments of inertia as functions of angular momentum. The Gogny D1S force is employed as an effective interaction, and the yrast superdeformed bands in $^{152}\mathrm{Dy}$ and $^{194}\mathrm{Hg}$ are taken as typical examples in the $A\ensuremath{\approx}150$ and $A\ensuremath{\approx}190$ regions, respectively. The effect of pairing correlations is examined by the method of variation after particle-number projection to understand the different behaviors of ${\mathcal{J}}^{(2)}$ moments of inertia observed in these two nuclei. The particle-number projection on top of the angular-momentum projection has been performed for the first time with the multicranked configuration mixing.

α and triton clustering in Cl35

Coupling of cluster and deformed structures are important for dynamics of nuclear structure. Threshold energy has been discussed to explain cluster structures coupling to deformed states but relation between threshold energy and excitation energy has open problems. Negative-parity superdeformed (SD) states were observed by a $\gamma$-spectroscopy experiment in $^{35}$Cl but its detailed structure is unclear. By analyzing coupling of cluster structures in deformed states and high-lying cluster states in $^{35}$Cl, cluster structures coupling to deformed states and excitation energy of high-lying cluster states are investigated. The antisymmetrized molecular dynamics (AMD) and the generator coordinate method (GCM) are used. An AMD wave function is a Slater determinant of Gaussian wave packets. By energy variational calculations with constraints on deformation and clustering, wave functions of deformed structures and $\alpha$- and $t$-cluster structures are obtained. Adopting those wave functions as GCM basis, wave functions of ground and excited states are calculated. Various deformed bands are obtained and predicted. A $K^\pi = \frac{1}{2}^-$ deformed band, which corresponds to the observed SD band, dominates deformed structure and compact $\alpha$- and $t$-cluster structure components. Particle-hole configurations of the dominant components with deformed and cluster structures are similar. In high-lying states, almost pure $\alpha$- and $t$-cluster states are obtained in negative-parity states, and excitation energies of the $t$-cluster states are higher than those of $\alpha$-cluster states. In conclusions, particle-hole configurations of cluster structure with small intercluster distance are important for coupling to low-energy deformed states. Threshold energies reflect to excitation energies of high-lying almost pure cluster states.

Empirical evidence of a superrigid structure of “flat” superdeformed bands

For the first time, we present the empirical evidence of superrigid character of the ``flat'' superdeformed bands in the Tl and Pb isotopes. For this purpose, we have used various rotational energy formulas. The free parameters of superdeformed (SD) bands in the Tl and Pb isotopes have been extracted and systematically studied to obtain the proposed empirical evidence. In particular, the intraband transition energies of the SD bands have been split into rotational and shape fluctuation energies. The role of the vibrational factor in the evolution of dynamic moment of inertia, softness parameter, alignment, and effective pairing gap parameter of the flat SD bands has been studied. Using these models, two distinct natures have been identified for the SD bands in the Tl and Pb isotopes. Our study establishes the role of shape fluctuations, vibrational effect, deformation, and pairing correlations for the unusual behavior of the dynamic moment of inertia in the flat SD bands of the $A\ensuremath{\approx}190$ mass region.

Prediction of band-head spin of triaxial super-deformed bands using the modified VMI model

The band-head spin $ (I_{0})$ of triaxial super-deformed (TSD) bands recently available in the rare-earth region are evaluated by using the modified variable moment of inertia (VMI) model. Quantitatively good results of the $ \gamma$ -ray energies and the spins of the TSD bands for Tm, Lu and Hf isotopes are obtained. The ratio of transition energies over spin $ E_{\gamma}/2I$ is used to evaluate the correct spin of the band-head by the VMI equation. The calculated and observed transition energies agree quite well when the correct band-head spin $ (I_{0})$ is obtained. This method provides consistently good spin assignment of TSD bands which could help in designing future experiments for these bands. In the present paper, we have reported the band-head spin for 20 triaxial super-deformed bands known experimentally. Out of 20 bands, the band-head spin is assigned correctly for 16 TSD bands which match with the suggested values and for the rest of the 4 TSD cases the spin predicted is either higher or lower than the experimentally suggested value.