Projected Shell Model Research Articles

Study of quasiparticle alignments and electromagnetic quantities in neutron-deficient even–even $$^{110-120}\hbox {Xe}$$ isotopes

The yrast states and quasiparticle alignments in even–even neutron-deficient xenon isotopes are studied in the projected shell model (PSM) framework. A new set of Nilsson parameters are proposed to reproduce the proton alignments in these isotopes. The B(E2) transition probabilities along the yrast line are obtained from the PSM wave functions and compared with the available experimental data. By using the same wave functions, g-factors are also predicted.

Changes of deformed shell gaps at N ∼ 100 in light rare-earth, neutron-rich nuclei

There has been compelling evidence indicating that deformed single-particle (SP) states in neutron-rich regions are different from those in the stable region. It was pointed out that for the light rare-earth (60Nd, 62Sm, and 64Gd), neutron-rich (N = 98 − 102) nuclei, the Woods–Saxon potential, the Nilsson modified oscillator potential with ‘universal’ parameters, and the folded Yukawa potential all failed to describe the correct ordering of neutron SP states. The location and size of deformed shell gaps in this mass region are under current debate. We propose a modification for the ‘standard’ Nilsson parameters of Bengtsson and Ragnarsson introduced in 1985. The proposed N-dependent spin–orbit interaction causes directly changes in deformed shell gaps with neutron number and deformation. By applying the modified Nilsson parameters to generate deformed bases for the projected shell model, we demonstrate that the calculation can explain consistently the current experimental data, including the ground state configuration in odd-neutron nuclei, upbending of the yrast moment of inertia at higher spins and the energies of 2-quasineutron 6− and 4− isomers in even–even nuclei.

Microscopic study of ground state bands in N = 45 and 46 isotones in mass region A ∼ 70–80

Using the angular-momentum projected quasiparticle states and a simple quadrupole–quadrupole + monopole–pairing + quadrupole-pairing type Hamiltonian, Projected Shell Model calculations have been performed to study the nuclear structure properties of some N = 45, 46 isotones in the mass region A = 70–80. The results are obtained for the yrast energy spectrum, rotational alignments, back-bending and reduced transition probabilities [B(M1) and B(E2)]. Results of our calculations exhibit satisfactory agreement with experimental data. Signature splitting and signature inversion which are the characteristics of this mass region have also been reproduced and discussed for these nuclei.

Two quasiparticle wobbling in the even-even nucleus 130Ba

Two newly observed bands built on a two-quasiparticle configuration in 130Ba have been investigated for the first time with the microscopic projected shell model. The experimental energy spectra and the available electromagnetic transition probabilities are well reproduced. The wobbling character of the higher band is revealed by the angular momentum projected wavefunctions via the K plot and the azimuthal plot. This provides the first strong microscopic evidence for wobbling motion based on a two-quasiparticle configuration in even-even nuclei.

Study of Back Bending in 157,158Er Isotopes by Using Electromagnetic Reduced Transition Probabilities

Electromagnetic reduced transition probabilities B(E2) and B(M1) calculated by using the PSM (Projected Shell Model) code are used to study the back-bending phenomenon in 157,158Er isotopes. As a result, two great drops in the B(E2)/B(M1) ratio were observed in spins 12 ħ and (37/2) ħ for the 158Er and the 157Er isotopes, respectively, which coincident exactly with the inertia change versus rotational frequency in the usual back-bending plots for these isotopes.

Study of multi-quasiparticle energy bands in neutron-deficient 117,119,121Cs

The projected shell model is employed to interpret multi-quasiparticle energy bands in odd-mass neutron-deficient 117–121Cs isotopes. The experimentally known energy bands and their configurations are reproduced well by employing this approach. The analysis of theoretical results predicts the low-lying spin states to arise from the single quasiparticle band albeit the high-spin states are seen to arise from the superposition of three quasiparticle bands. In addition, the experimentally known signature partner bands and band head energies are also predicted in these isotopes which can serve as a clue for planning new experiments.

High spin structure of 82Sr using triaxial projected shell model

Study of N≈Z nuclei in mass A ̴80 region, which is in the transitional region, is of interest due to the existence of abundant nuclear structure phenomena and this mass region is often characterized by shape co-existence. In present work, the multi-quasiparticle triaxial projected shell model (TPSM) approach is employed to study the high spin structures and to depict γ deformation in 82Sr. TPSM results for the yrast band and γ band-energies are compared with known experimental energies. The possibility of a 2γ-band is also predicted. The change in staggering phases for the γ band as a result of configuration mixing is also discussed.

Microscopic study of electromagnetic properties and band spectra of neutron deficient 133,135,137Sm

A microscopic high spin study of neutron deficient and normally deformed 133,135,137Sm has been carried out in projected shell model framework. The theoretical results have been obtained for the spins, parities and energy values of yrast and excited bands. Besides this, the band spectra, band head energies, moment of inertia and electromagnetic transition strengths are also predicted in these isotopes. The calculations successfully give a deeper understanding of the mechanism of the formation of yrast and excited bands from the single and multi-quasi particle configurations. The results on moment of inertia predict an alignment of a pair of protons in the proton (1h11/2)2 orbitals in the yrast ground state bands of 133-137Sm due to the crossing of one quasiparticle bands by multi-quasiparticle bands at higher spins. The discussion in the present work is based on the deformed single particle scheme. Any future experimental confirmation or refutation of our predictions will be a valuable information which can help to understand the deformed single particle structure in these odd mass neutron deficient 133-137Sm.

Fast-timing measurements in the ground-state band of Pd114

Using a hybrid Gammasphere array coupled to 25 LaBr3(Ce) detectors, the lifetimes of the first three levels of the yrast band in ¹¹⁴Pd populated via ²⁵²Cf decay, have been measured. The measured lifetimes are τ₂+=103(10)ps, τ₄+=22(13)ps, and τ₆+≤10ps for the 2⁺₁, 4⁺₁, and 6⁺₁ levels, respectively. Palladium-114 was predicted to be the most deformed isotope of its isotopic chain, and spectroscopic studies have suggested it might also be a candidate nucleus for low-spin stable triaxiality. From the lifetimes measured in this work, reduced transition probabilities B(E2;J→J−2) are calculated and compared with interacting boson model, projected shell model, and collective model calculations from the literature. The experimental ratio RB(E₂)=B(E2;4⁺₁→2⁺₁)/B(E2;2⁺₁→0⁺₁)=0.80(42) is measured for the first time in ¹¹⁴Pd and compared with the known values RB(E₂) in the palladium isotopic chain: the systematics suggest that, for N=68, a transition from γ-unstable to a more rigid γ-deformed nuclear shape occurs.

Rotational behavior of thermally excited 56Fe, 58Fe and 60Fe isotopes

Thermal and rotational behaviors of neutron rich fp-shell isotopes such as [Formula: see text], [Formula: see text] and [Formula: see text] were analyzed microscopically within the framework of statistical theory of hot rotating nuclei (STHRN) for the angular momentum range (0–15)[Formula: see text] at excitation energy above 4[Formula: see text]MeV. Pair-breaking phenomenon and band-crossing phenomenon of Fe isotopes were discussed with and without the inclusion of BCS pairing. The STHRN method with BCS effect was extended to the Fe isotopes to determine the critical temperature [Formula: see text], where the pair-breaking phenomenon takes place and it is found to occur below [Formula: see text][Formula: see text]MeV. The observation of band-crossing phenomenon above the [Formula: see text] was explained without considering the effect of BCS pairing. The single particle energy levels were engendered from triaxially deformed Nilsson Hamiltonian. The outcomes of the present investigation on moment of inertia (MOI), back-bending phenomenon, spin cutoff parameter show a strong evidence of band-crossing phenomenon and it eventually led to a shape transition from spherical to noncollective oblate. Moreover, attention on separation energy of protons and neutrons reveals that the neutron pairs are responsible for band-crossing. STHRN method was able to reproduce results in good agreement with experiments and comparable with other theories such as projected shellmodel (PSM) and shellmodel Monte Carlo (SMMC) method.

Observation of quasi-$ \gamma$ bands in Te nuclei

Excited states of 124Te have been studied via 122Sn(9Be,$ \alpha$3n)124Te reaction at 48MeV. The spin and parity of several states have been confirmed on the basis of results of angular correlation ( $ R_{DCO}$ and linear polarization asymmetry measurements. The positive parity band structures of 124Te and 126Te have been investigated and the quasi-$ \gamma$ bands have been proposed in these nuclei. The structures of these bands have also been discussed under the framework of triaxial projected shell model (TPSM) calculations. The TPSM calculations fairly supported the interpretation of quasi-$ \gamma$ bands in 124Te and 126Te. The staggering pattern of odd-even spin states of the quasi-$ \gamma$ band suggests the $ \gamma$-soft behaviour of 124Te and the typical values of staggering factors indicate the presence of the E(5)-critical point symmetry in 124Te. The potential energy surface calculations have also been carried out for 124Te and 126Te, which also supports the $ \gamma$-soft behaviour for these nuclei.

Microscopic insight into the quasi-particle structure of odd-mass terbium isotopes

The present work has been focussed on the application of Projected Shell Model to study the negative parity rotational band structure up to the high spin states of the neutron-rich odd mass Terbium isotopes with neutron numbers ranging from 90 to 98. The structure evolution in these nuclei has been made on the basis of various nuclear structure properties like energy spectra, transition energies, wave functions and electromagnetic transition probabilities (B(E2) and B(M1)). Besides, the occurrence of back bending has also been reported in the present work.

Projected shell model analysis of structural evolution and chaoticity in fast-rotating nuclei

Rotationally-induced chaotic motion in fast-rotating nuclei is investigated using the projected shell model with an aggressively extended multi-quasiparticle configuration space. The two-body Hamiltonian in separable forms with quadrupole–quadrupole plus pairing forces is diagonalized in superimposed states with more than 2000 angular momentum projected configurations. By taking 164Yb as an example, it is shown that the experimental energies and moment of inertia (MoI) for the yrast band can be clearly described. The variations in MoI are explained in terms of band crossings among the 2-quasiparticle (qp), 4-qp, 6-qp, and 8-qp bands, corresponding to successive pair-breakings caused by rotation. Moreover, the chaotic motion in different spin and excitation regions is studied by quantitatively analyzing the branching number. It is found that the degree of chaoticity increases monotonically with spin and excitation energy. At the highest spins, bands built by different numbers of quasiparticles, with their own characteristic rotational behavior originally at low spins, tend to rotate uniformly with the same rotational frequency, indicating an emergent nuclear collectivity appearing from an extremely chaotic system. The important role played by nucleon pairing in increasing the chaoticity is emphasized.

Configuration-constrained calculations of high-<italic>K </italic>rotational bands in even-even isotones around <italic>N</italic>=100

Nuclear rotational spectroscopy has been extensively investigated in experimental and theoretical studies. In the collective model, a nucleus can be excited in two ways. One is by breaking pairs of nucleons and exciting nucleons to a higher level. An excited state with a higher K -value sometimes has a long lifetime. Usually those states with lifetimes longer than 1 ns are called isomeric state because of forbidden transitions from high K to low K state. The other way is by collective motions, like rotation and vibration. A rotational band built on the multi-particle state is called a sideband. In general, a nucleus can have more than one sideband. Investigations of sidebands can provide rich information about the nucleus structure, like single-particle level, nuclear shape, and electromagnetic transitions and so on. The traditional total Routhian surface method, which is a powerful tool for investigating nuclear rotation, has been successfully used to describe yrast bands. However, it often encounters non-convergence problems when calculating sidebands. To overcome this problem, we recently developed a configuration-constrained cranking Skyrme Hartree-Fock mean-field calculation with pairing treated by the particle-number-conserving (PNC) method for sidebands. The PNC pairing method follows the technique of a “standard” shell model, which diagonalizes the Hamiltonian in a truncated model space and guarantees the conservation of the particle number. The rare-earth ( Z =57−71) neutron-rich nuclides are good candidates for investigating the high- K rotational bands. In this mass region, nuclei with N ≥90 are known to be well deformed and many high- j orbitals exist near the neutron and proton Fermi surfaces. These orbitals can form various multi-particle configurations and lead to high- K rotational sidebands. We perform configuration-constrained cranking Skyrme Hartree-Fock mean-field calculations with pairing treated using the PNC method for the high- K rotational bands in even-even isotones near N =100. The K π=6− rotational bands in 160Sm, 164Dy, 166Er, 168Yb, 170Er, and 172Yb and the K π=4− rotational bands in 168Er and 170Yb have been observed during experiments. K π=6− isomers were first observed in very neutron-rich nuclei, i.e., 158Nd, 164Sm, and 166Gd. Further, K π=4− isomers were first observed in neutron-rich N =100 isotones, i.e., 160Nd, 162Sm, and 164Gd. Theoretically, several models have been used to study these neutron-rich nuclei, including the cranked shell model, projected shell model, deformed Hartree-Fock method, cranking PNC method, and quasiparticle random phase approximation. We systematically investigated these high- K rotational bands and the ground state bands in N =98, 100, and 102 isotones using the configuration-constrained cranking Skyrme Hartree-Fock mean-field calculation with PNC pairing. The Skyrme force of SKM* is adopted. The pairing strength is determined using the odd-even mass difference and a three-point formula, which includes the mean-field and blocking effects. The calculations reproduce the experimental moments of inertia of both the ground state and sidebands well. Configurations are assigned to the observed high- K rotational bands. Further, we predict possible high- K rotational bands in 158Nd, 162Gd, 160Nd, 162Sm, 164Gd, and 166Dy. This study on high- K rotational bands can provide new insights into the structure of the states, and the predictions may provide useful information for future experiments.

Diversity of shapes and rotations in the γ-soft 130Ba nucleus: First observation of a t-band in the A = 130 mass region

Several new bands have been identified in 130Ba, among which there is one with band-head spin 8+. Its properties are in agreement with the Fermi-aligned νh11/22, 7/2−[523]⊗9/2−[514] Nilsson configuration. This is the first observation of a two-quasiparticle t-band in the A=130 mass region. The t-band is fed by a dipole band involving two additional h11/2 protons. The odd-spin partners of the proton and neutron S-bands and the ground-state band at high spins are also newly identified. The observed bands are discussed using several theoretical models, total Routhians surfaces (TRS), tilted axis cranking (TAC), particle rotor model (PRM) and projected shell model (PSM), which strongly suggest the coexistence of prolate and oblate shapes polarized by rotation aligned two-proton and two-neutron configurations, as well as prolate collective rotations around axes with different orientations. With the new results, 130Ba presents one of the best and most complete sets of collective excitations that a γ-soft nucleus can manifest at medium and high spins, revealing a diversity of shapes and rotations for the nuclei in the A=130 mass region.

Multichiral facets in symmetry restored states: Five chiral doublet candidates in the even-even nucleus Nd136

A triaxial projected shell model including configurations with more than four quasiparticles in the configuration space is developed and applied to investigate the recently reported five chiral doublets candidates in a single even-even nucleus $^{136}\mathrm{Nd}$. The energy spectra and transition probability ratios $B(M1)/B(E2)$ are reproduced satisfactorily. The configuration mixing along the rotational bands is studied by analyzing the intrinsic composition of the eigenfunctions. The chiral geometry of these nearly degenerate bands is examined by the K plot and the azimuthal plot, and the evolution from the chiral vibration to the static chirality with spin is clearly demonstrated for four pairs of partner bands. For bands D3 and D3-C, due to strong configuration mixing, their azimuthal plots show features of planar rotation and it is difficult to interpret them as chiral partners.

Microscopic insight into the nuclear structure properties of odd-mass 101−109Cd isotopes

The purpose of the present work is to study the negative parity yrast bands in odd-mass 101−109Cd nuclei in the framework of Projected Shell Model (PSM). The results are obtained for the energy spectrum, reduced transition probabilities [B(E2)], rotational alignment etc. The yrast and excited band spectra of these isotopes are found to arise from the quasiparticle states around the Fermi level. The yrast and excited band spectra are found to depend sensitively on the different single particle and multi-quasiparticle configurations at low- and high-spins respectively. This work also substantiates the observed presence of anti-magnetic rotation in 105,107,109Cd nuclei.

$\gamma$γ-vibration in 198Hg

Low lying states of $^{198}$Hg have been investigated via $^{197}$Au($^{7}$Li, $ \alpha $2n$ \gamma $)$ ^{198} $Hg reaction at E$ _{\text{beam}} $ = 33 MeV and 38 MeV and the members of $ \gamma $-vibrational band have been identified. Results are compared with the systematic of this mass region and found in agreement. Observed band structures have been interpreted using the theoretical framework of microscopic triaxial projected shell model (TPSM) approach and it is shown that TPSM results are in fair agreement with the observed energies.

Superdeformed band in the N = Z + 4 nucleus 40Ar : A projected shell model analysis

40Ar is among the few sd -shell nuclei known to exhibit superdeformation. An interesting question is whether the experimentally identified superdeformed rotational band in 40Ar has an axially or triaxially deformed shape. Projected shell model calculations assuming an axially deformed basis are performed up to high spins for this nucleus. Our detailed analysis of the wave functions reveals that the high-spin states are dominated by mixed 0-, 2-, and 4-quasiparticle configurations with different K-components. The calculated electric quadrupole transition probabilities reproduce well the known experimental data and suggest a reduced but still significant collectivity in the high spin region. Due to the mixing of various multi-quasiparticle K-configurations, axial asymmetry emerges in the calculation. Throughout the entire rotational band, the extracted triaxial deformation parameters are small but non-zero. Relations of triaxiality and unnatural spin-parity states of the superdeformed side-band are discussed.

Systematic study of two-quasiparticle structure of the neutron-rich odd-odd rubidium nuclei

A theoretical study of the nuclear structure of odd-odd 88–96Rb nucleus in the A∼100 mass region is carried out by using the angular-momentum-projection technique implemented in the Projected Shell Model (PSM). For these isotopes, PSM calculations are performed to obtain the yrast spectra and the description of the formation of the yrast level structure from multi-quasi-particle configurations is also presented. The back-bending in moment of inertia and transition energies have also been calculated and compared with the available experimental data. In addition, the reduced transition probabilities, i.e. B(E2) and B(M1), are also obtained for the yrast band of these isotopes for the first time by using PSM wave-functions.