Nuclear isomer is a current research focus. To describe isomers, we present a method based on the Projected Shell Model. Two kinds of isomers, K-isomers and shape isomers, are discussed. For the K-isomer treatment, K-mixing is properly implemented in the model. It is found however that in order to describe the strong K-violation more efficiently, it may be necessary to further introduce triaxiality into the shell model basis. To treat shape isomers, a scheme is outlined which allows mixing those configurations belonging to different shapes.
In the present work, we have done a systematic shell model study of $N=82$ and $N=126$ isotones. For the $N=82$ isotones, we have performed calculations using SN100PN interaction, while for $N=126$ isotones, we have used KHPE interaction. Similarities between these two isotonic chains have been reported, using the strong resemblance between the high-$j$ orbitals. Apart from the nuclear spectroscopic properties, we have also explained different isomeric states in these two regions. In the $N$ =82 region, we have mainly discussed the properties of the $6^+$ and $17/2^+$ isomers, while in the $N=…
Using a shell model which is capable of describing the spectra of upper $g_{9/2}$-shell nuclei close to the $N=Z$ line, we study the structure of two isomeric states $7^{+}$ and $21^{+}$ in the odd-odd $N=Z$ nucleus $^{94}$Ag. It is found that both isomeric states exhibit a large collectivity. The $7^{+}$ state is oblately deformed, and is suggested to be a shape isomer in nature. The $21^{+}$ state becomes isomeric because of level inversion of the $19^{+}$ and $21^{+}$ states due to core excitations across the $N=Z=50$ shell gap. Calculation of spectroscopic quadrupole moment indicates clear…
We probe the $N=82$ nuclear shell closure by mass measurements of neutron-rich cadmium isotopes with the ISOLTRAP spectrometer at ISOLDE-CERN. The new mass of $^{132}$Cd offers the first value of the $N=82$, two-neutron shell gap below $Z=50$ and confirms the phenomenon of mutually enhanced magicity at $^{132}$Sn. Using the recently implemented phase-imaging ion-cyclotron-resonance method, the ordering of the low-lying isomers in $^{129}$Cd and their energies are determined. The new experimental findings are used to test large-scale shell-model, mean-field and beyond-mean-field calculations, a…
The projected shell model is used to study the multi-quasiparticle and collective excitations of 178Hf. With an axially symmetric basis, the spin-16 isomer at 2.4 MeV appears to be well separated in energy/spin space from other configurations. However, projected energy surface calculations suggest that 178Hf has significant softness to axially asymmetric shapes, which can strongly modify the level distribution. The implications for photodeexitation of the isomer are discussed.
We present a comparative study of the high-spin ${11/2}^-$, ${10}^+$ and ${27/2}^-$ nuclear isomers, observed commonly in the full semi-magic $Z=50$ isotopic and $N=82$ isotonic chains, which are found to display nearly identical systematics in excitation energy and half-lives. A constant and particle number independent energy gap of $\sim$ 4 MeV between the $0^+$ ground states and ${10}^+$ isomers and, the ${11/2}^-$ and ${27/2}^-$ isomers exists before the mid-shell, which declines near the mid-shell, and then becomes constant at $\sim$ 3 MeV after the mid-shell. Large scale shell model calc…
The puzzle of finding consistent nuclear configurations for explaining both the decay probabilities and moments of the ${9/2}^-$, ${8}^+$ and ${21/2}^-$ isomers in and around $N=126$ closed shell has been approached in the generalized seniority scheme. Though $h_{9/2}$ is the dominant orbital near Fermi energy, the role of configuration mixing from the surrounding $f_{7/2}$ and $i_{13/2}$ orbitals is found to be very important for the consistent explanation of all the isomeric properties such as the $B(E2)$ rates, $Q-$moments and $g-$factors. The structural behavior of the closed shell $N=126$…
Collinear laser spectroscopy has been performed on the $^{79}_{30}$Zn$_{49}$ isotope at ISOLDE-CERN. The existence of a long-lived isomer with a few hundred milliseconds half-life was confirmed, and the nuclear spins and moments of the ground and isomeric states in $^{79}$Zn as well as the isomer shift were measured. From the observed hyperfine structures, spins $I = 9/2$ and $I = 1/2$ are firmly assigned to the ground and isomeric states. The magnetic moment $μ$ ($^{79}$Zn) = $-$1.1866(10) $μ_{\rm{N}}$, confirms the spin-parity $9/2^{+}$ with a $νg_{9/2}^{-1}$ shell-model configuration, in ex…
In this paper, we investigate the generalized seniority scheme and the validity of Generalized Seniority Schmidt Model in and around the $Z=82$ semi-magic region. A consistently same multi-j configuration is used to explain all the nuclear spectroscopic properties such as $g-$factors, $Q-$moments and $B(E2)$ trends for the ${13/2}^+$, ${12}^+$ and ${33/2}^+$ isomers in all the three Hg, Pb and Po isotopic chains. The inverted parabolic $B(E2)$ trends for the first $2^+$ states in Hg, Pb and Po isotopes are also explained using the generalized seniority scheme. A comparison with the experimenta…
The close similarity between the shell structures in the 132Sn and 208Pb regions is a well known phenomenon. Thus, using the correspondence between the high-j orbits located above the Z=50 and Z=82 shell gaps, we discuss the evolutions of the fully aligned states with one broken proton pair in the N=82 and N=126 isotones. A long-lived isomeric state was discovered in 217Pa more than thirty years ago and despite two other experiments giving new experimental results, the discussions on its main properties (spin, parity, configuration) remained inconclusive. Then, using the comparison with the I^…
Contemporary data are analyzed concerning the half-lives of the $^{229m}$Th isomer in neutral atoms and various ions. It is explicitly shown that the isomer lifetime may strongly depend on the plain environmental physical conditions like pressure and temperature. Calculation is performed on the united platform of interplay of traditional and subthreshold resonance conversion. General agreement with experiment is obtained in the cases of Th$^I$ and Th$^{III}$, a prediction is made concerning Th$^{IV}$. Disagreement with the experiment in the case of Th$^{II}$ may be a consequence of a random ve…
In the present work available experimental data up to high-spin states of $^{119-126}$Sn isotopes with different seniority ($v$), including $v$ = 4, 5, 6, and 7 have been interpreted with shell model, by performing full-fledged shell model calculations in the 50-82 valence shell composed of $1g_{7/2}$, $2d_{5/2}$, $1h_{11/2}$, $3s_{1/2}$, and $2d_{3/2}$ orbitals. The results have been compared with the available experimental data. These states are described in terms of broken neutron pairs occupying the $h_{11/2}$ orbital. Possible configurations of seniority isomers in these nuclei are discus…
Recent experimental data are analyzed, concerning the half-lives of the 229mTh isomer in neutral atoms and various ions. Calculation is performed on the united platform of interplay of traditional and subthreshold resonance conversion. General agreement with experiment is obtained in the cases of Th I and Th III, a prediction is made concerning half-life in Th IV. Most critical is the case of Th II, where experimental data can be explained by interplay of various factors. A new physics is proposed, based on dependence of the nuclear lifetime on the ambient conditions, such as atmospheric press…
The neutron-rich nuclei 130Cd and 206Hg, so important in the astrophysical processes, may also be useful in tracking the evolution of nuclear shell gaps as one traverses the neutron-rich region. The high spin 8+ isomer in 130Cd and the 10+ isomer in 206Hg turn out to be the lampposts, which may shed light on the shell gaps and validity of the seniority scheme in the neutron-rich systems. We explore the robustness of the N=82 and N=126 magic numbers in the neutron-rich 130Cd and 206Hg nuclides, respectively. A parallel between the two nuclides in terms of the high-spin isomers allows us to inve…
A systematic study of 2-quasiparticle bands of the proton-rich odd-odd nuclei in the mass A ~ 70-80 region is performed using the projected shell model approach. The study includes Br-, Rb-, and Y-isotopes with N = Z+2, and Z+4. We describe the energy spectra and electromagnetic transition strengths in terms of the configuration mixing of the angular-momentum projected multi-quasiparticle states. Signature splitting and signature inversion in the rotational bands are discussed and are shown to be well described. A preliminary study of the odd-odd N = Z nucleus, 74Rb using the concept of sponta…
Isomeric states in the nuclei along the rapid proton capture process path are studied by the projected shell model. Emphasis is given to two waiting point nuclei Se-68 and Kr-72 that are characterized by shape coexistence. Energy surface calculations indicate that the ground state of these nuclei corresponds to an oblate-deformed minimum, while the lowest state at the prolate-deformed minimum can be considered as a shape isomer. Due to occupation of the orbitals with large K-components, states built upon two-quasiparticle excitations at the oblate-deformed minimum may form high $K$-isomers. Th…
Excited states in neutron-rich nuclei located south-east of $^{132}$Sn are investigated by shell-model calculations. A new shell-model Hamiltonian is constructed for the present study. The proton-proton and neutron-neutron interactions of the Hamiltonian are obtained through the existing CD-Bonn $G$ matrix results, while the proton-neutron interaction across two major shells is derived from the monopole based universal interaction plus the M3Y spin-orbit force. The present Hamiltonian can reproduce well the experimental data available in this region, including one-neutron separation energies, …
Deformation-energy surfaces of 54 even-even isotopes of Pt, Hg and Pb nuclei with neutron numbers up to 126 are investigated within a macroscopic-microscopic model based on the Lublin-Strasbourg-Drop macroscopic energy and shell plus pairing-energy corrections obtained from a Yukawa-folded mean-field potential at the desired deformation. A new, rapidly converging Fourier shape parametrization is used to describe nuclear shapes. The stability of shape isomeric states with respect to non-axial and higher-order deformations is investigated.
Isomeric studies in neutron-rich nuclei present a powerful tool to explore the structure at the nuclear extremes. We recently used the shell model calculations with Renormalized Charge Depen- dent Bonn (RCDB) effective interaction to calculate the properties of the $6^+$ seniority isomers in $^{134-138}$Sn in an attempt to resolve the anomalous B(E2) behavior of the $6^+$ isomer in $^{136}$Sn [Phys. Rev. C 91, 024321 (2015)]. We further explore these isomers by using the generalized seniority scheme for multi-j orbitals recently presented by us [Phys. Lett. B 753, 122 (2016)]; the B(E2) values…
A proper treatment of K-mixing is the key to understanding K-isomers. Here, we present a method based on the projected shell model. This method differs from the usual description of multi-quasiparticle states by introducing a transformation to the laboratory frame and a subsequent configuration mixing in that frame. It allows a quantitative study on the degree of K-violation through direct calculations of electromagnetic transitions.
To discuss whether an enhanced isomer decay is a preferred process in a plasma environment it is required to know the structure of the isomer as well as the nearby states. The spin-21/2, 6.85-hour high-spin isomer in 93Mo is investigated within a shell model which well describes nuclei in this mass region. By using the obtained wave-functions which correctly reproduce the observed B(E2), B(E4), and B(M1) transitions, characteristics of the isomer are shown in comparison with the isomeric states in neighboring nuclei. Calculations suggest that these high-spin isomers are formed with almost pure…
Isomeric states in the nuclei along the rapid proton capture process path are studied by the projected shell model. Emphasis is given to two waiting point nuclei 68Se and 72Kr that are characterized by shape coexistence. Energy surface calculations indicate that the ground state of these nuclei corresponds to an oblate-deformed minimum, while the lowest state at the prolate-deformed minimum can be considered as a shape isomer. The impact of these isomer states on isotopic abundance in x-ray bursts is studied in a multi-mass-zone x-ray burst model by assuming an upper-lower limit approach.
Shell model calculations have been done for interpreting two representative isomeric states in neutron rich $^{132}Sn$ region using SMN and SMPN Hamiltonians. They are, (i) 2.91 min isomer in $^{138}Cs$ and (ii) the 0.57 $μs$ isomer in $^{136}$Sb nuclei. The results are compared with those obtained with KH5082 and CW5082 Hamiltonians. How the results clearly distinguish the most appropriate interaction has been discussed.
The rotational properties of the neutron rich Nd and Sm isotopes with mass number $A\approx150$ are systematically investigated using the cranked shell model with pairing correlations treated by a particle-number conserving method, in which the Pauli blocking effects are taken into account exactly. The 2-quasiparticle states in even-even Nd and Sm isotopes with excitation energies lower than 2.5~MeV are systematically calculated. The available data can be well reproduced and some possible 2 and 4-quasiparticle isomers are also suggested for future experiments. The experimentally observed rotat…
The $10^+$ and $27/2^-$ isomers of the $Z>64$, $N=82$ nuclei are investigated in the shell model framework. We derive an extended seniority reduction formula for the relevant $E2$ transition strengths. Based on the extended formula, as well as on the approximate degeneracy among the $0h_{11/2}$, $2s_{1/2}$ and $1d_{3/2}$ orbits, we argue that the $B(E2)$ data require the $^{146}$Gd core excitation. The energy levels of both parities and the $B(E2)$ values are simultaneously reproduced by a multi-$j$ shell model calculation with the MSDI, if the excitations from $(0g_{7/2}1d_{5/2})$ to $(0h_{11…
Inspired by the newly discovered isomeric states in the rare-earth neutron-rich nuclei, high-$K$ isomeric states in neutron-rich samarium and gadolinium isotopes are investigated within the framework of the cranked shell model (CSM) with pairing correlation treated by a particle-number-conserving (PNC) method. The experimental multi-particle state energies and moments of inertia are reproduced quite well by the PNC-CSM calculations. A remarkable effect from the high-order deformation $\varepsilon_{6}$ is demonstrated. Based on the occupation probabilities, the configurations are assigned to th…
Study of the N ~ Z nuclei in the mass-80 region is not only interesting due to the existence of abundant nuclear structure phenomena, but also important in understanding the nucleosynthesis in the rp-process. It is not feasible to apply a conventional shell model due to the necessary involvement of the g_{9/2} sub-shell. In this paper, the projected shell model is introduced to this study. Calculations are systematically performed for the collective levels as well as the quasi-particle excitations. It is demonstrated that calculations with this truncation scheme can achieve a comparable qualit…
Triggering of long-lived nuclear isomeric states via coupling to the atomic shells in the process of nuclear excitation by electron capture (NEEC) is studied. NEEC occurring in highly-charged ions can excite the isomeric state to a triggering level that subsequently decays to the ground state. We present total cross sections for NEEC isomer triggering considering experimentally confirmed low-lying triggering levels and reaction rates based on realistic experimental parameters in ion storage rings. A comparison with other isomer triggering mechanisms shows that, among these, NEEC is the most ef…
Strong residual correlations between neutrons and protons in N ~ Z systems can lead to unusual structure. Using the spherical shell model, we show that a low-excitation shape isomer can occur in the odd-odd N=Z nucleus 66As. This extends the picture of shape coexistence beyond even-even nuclei. Furthermore, it is demonstrated that in 66As and in the N=Z+1 nucleus 67As, a new type of isomer, which we term j-isomer, can be formed. The underlying mechanism for the isomerism formation is structure change in the isomeric states, which involves either an alignment of a neutron-proton pair from the h…
We present a fully quantum-mechanical, microscopic, unified treatment of ground-state band and multi-phonon $γ$-vibrational bands using shell model diagonalization with the triaxial projected shell model. The results agree very well with data on the g- and $γ$-band spectra in $^{156-170}$Er, as well as with recently measured $4^+$ 2-phonon $γ$-bandhead energies in $^{166}$Er and $^{168}$Er. Multi-phonon $γ$-excitation energies are predicted.
The projected shell model implements shell model configuration mixing in the projected deformed basis. Our analysis on the recently observed superdeformed band in $^{36}$Ar suggests that the neutron and proton 2-quasiparticle and the 4-quasiparticle bands cross the superdeformed ground band at the same angular momentum. This constitutes a picture of band disturbance in which the first and the second band-crossing, commonly seen at separate rotation frequencies in heavy nuclei, occur simultaneously. We also attempt to understand the assumptions of two previous theoretical calculations which int…
New shell model Hamiltonians are derived for the T=1 part of the residual interaction in the f5/2 p3/2 p1/2 g9/2 model space based on the analysis and fit of the available experimental data for 57Ni-78Ni isotopes and 77Cu-100Sn isotones. The fit procedure, properties of the determined effective interaction as well as new results for valence-mirror symmetry and seniority isomers for nuclei near 78Ni and 100Sn are discussed.
Excited 0+ states are studied in the framework of the projected shell model, aiming at understanding the nature of these states in deformed nuclei in general, and the recently observed 13 excited 0+ states in 158Gd in particular. The model, which contains projected two- and four-quasiparticle states as building blocks in the basis, is able to reproduce reasonably well the energies for all the observed 0+ states. The obtained B(E2) values however tend to suggest that these 0+ states might have a mixed nature of quasiparticle excitations coupled to collective vibrations.
Transition probabilities of the ground-state bands in $γ$-soft nuclei are studied for the first time using the triaxial projected shell model approach. It is observed that the angular-momentum dependence of the transition quadrupole moment $Q_t$ is related to the triaxial deformation of the nuclear mean-field potential. The introduction of the $γ$-degree of freedom in the shell model basis is shown to have a little influence on the {\it constant} behavior of the low-spin $Q_t$ in a well-deformed nucleus. However, the {\it increasing} collectivity with spin for the low-spin states in a $γ$-soft…
We perform particle-number projected mean-field study using the recently developed symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations. Realistic calculations have been performed in sd- and fp-shell nuclei using the shell model empirical intearctions, USD and GXPFIA. It is demonstrated that the mean-field results for energy surfaces, obtained with these shell model interactions, are quite similar to those obtained using the density functional approaches. Further, it is shown that particle-number projected results, for neutron rich isotopes, can lead to different ground-state shapes in c…
A shape phase transition is demonstrated to occur in 190W by applying the Projected Shell Model, which goes beyond the usual mean-field approximation. Rotation alignment of neutrons in the high-j, i_{13/2} orbital drives the yrast sequence of the system, changing suddenly from prolate to oblate shape at angular momentum 10$\hbar$. We propose observables to test the picture.
The systematics of g factor of first excited 2^+ state vs neutron number N is studied by the projected shell model. The study covers the even-even nuclei of all isotopic chains from Gd to Pt. g factors are calculated by using the many-body wavefunctions that reproduces well the energy levels and B(E2)'s of the ground-state bands. For Gd to W isotopes the characteristic feature of the g factor data along an isotopic chain is described by the present model. Deficiency of the model in the g factor description for the heavier Os and Pt isotopes is discussed.
The projected shell model analysis is carried out using the triaxial Nilsson+BCS basis. It is demonstrated that, for an accurate description of the moments of inertia in the transitional region, it is necessary to take the triaxiality into account and perform the three-dimensional angular-momentum projection from the triaxial Nilsson+BCS intrinsic wavefunction.
A systematic study of the yrast-band structure for the proton-rich, even-even mass-80 nuclei is carried out using projected shell model approach. We describe the the energy spectra, transition quadrupole moments and gyromagnetic factors. The observed variations in energy spectra and transition quadrupole moments in this mass region are discussed in terms of the configuration mixing of the projected deformed Nilsson states as a function of shell filling.
A method for calculation of Gamow-Teller transition rates is developed by using the concept of the Projected Shell Model (PSM). The shell model basis is constructed by superimposing angular-momentum-projected multi-quasiparticle configurations, and nuclear wave functions are obtained by digonalizing the two-body interactions in these projected states. Calculation of transition matrix elements in the PSM framework is discussed in detail, and the effects caused by the Gamow-Teller residual forces and by configuration-mixing are studied. With this method, it may become possible to perform a state…
Inspired by the recent experimental data (Phys. Lett. B {\bf 675} (2009) 420), we extend the triaxial projected shell model approach to study the $γ$-band structure in odd-mass nuclei. As a first application of the new development, the $γ$-vibrational structure of $^{103}$Nb is investigated. It is demonstrated that the model describes the ground-state band and multi-phonon $γ$-vibrations quite satisfactorily, supporting the interpretation of the data as one of the few experimentally-known examples of simultaneous occurrence of one- and two-$γ$-phonon vibrational bands. This generalizes the wel…
The Triaxial Projected Shell Model (TPSM) has been successful in providing a microscopic description of the energies of multi-phonon vibrational bands in deformed nuclei. We report here on an extension of the TPSM to allow, for the first time, calculations of B(E2) values connecting gamma- and gamma-gamma-vibrational bands and the ground state band. The method is applied to 166,168Er. It is shown that most of the existing B(E2) data can be reproduced rather well, thus strongly supporting the classification of these states as gamma-vibrational states. However, significant differences between th…
We expand the triaxial projected shell model basis to include triaxially-deformed multi-quasiparticle states. This allows us to study the yrast and gamma-vibrational bands up to high spins for both gamma-soft and well-deformed nuclei. As the first application, a systematic study of the high-spin states in Er-isotopes is performed. The calculated yrast and gamma-bands are compared with the known experimental data, and it is shown that the agreement between theory and experiment is quite satisfactory. The calculation leads to predictions for bands based on one- and two-gamma phonon where current…
The projected shell model with configuration mixing for nuclear chirality is developed and applied to the observed rotational bands in the chiral nucleus $^{130}$Cs. For the chiral bands, the energy spectra and electromagnetic transition probabilities are well reproduced. The chiral geometry illustrated in the $K~plot$ and the $azithumal~plot$ is confirmed to be stable against the configuration mixing. The other rotational bands are also described in the same framework.
Recent experimental data have demonstrated that $^{76}$Ge may be a rare example of a nucleus exhibiting rigid $γ$-deformation in the low-spin regime. In the present work, the experimental analysis is supported by microscopic calculations using the multi-quasiparticle triaxial projected shell model (TPSM) approach. It is shown that to best describe the data of both yrast and $γ$-vibrational bands in $^{76}$Ge, a rigid-triaxial deformation parameter $γ\approx 30^\circ$ is required. TPSM calculations are discussed in conjunction with the experimental observations and also with the published resul…
Chiral rotation observed in $^{128}$Cs is studied using the newly developed microscopic triaxial projected shell model (TPSM) approach. The observed energy levels and the electromagnetic transition probabilities of the nearly degenerate chiral dipole bands in this isotope are well reproduced by the present model. This demonstrates the broad applicability of the TPSM approach, based on a schematic interaction and angular-momentum projection technique, to explain a variety of low- and high-spin phenomena in triaxial rotating nuclei.
Total and parity-projected level densities of iron-region nuclei are calculated microscopically by using Monte Carlo methods for the nuclear shell model in the complete $(pf+0g_{9/2})$-shell. The calculated total level density is found to be in good agreement with the experimental level density. The Monte Carlo calculations offer a significant improvement over the thermal Hartree-Fock approximation. Contrary to the Fermi gas model, it is found that the level density has a significant parity-dependence in the neutron resonance region. The systematics of the level density parameters (including s…
The Projected Shell Model is a shell model theory built up over a deformed BCS mean field. Ground state and excited bands in even-even nuclei are obtained through diagonalization of a pairing plus quadrupole Hamiltonian in an angular momentum projected 0-, 2-, and 4-quasiparticle basis. The residual quadrupole-quadrupole interaction strength is fixed self-consistently with the deformed mean field and the pairing constants are the same used in constructing the quasiparticle basis. Taking $^{160}Dy$ as an example, we calculate low-lying states and compare them with experimental data. We exhibit …
Systematic study of E2 matrix elements in the framework of the Triaxial Projected Shell Model
University of Notre Dame（圣母大学）；Government Degree College Shopian（肖皮安政府学位学院）；Government Degree College Kulgam（库尔加姆政府学位学院）；Institute of Physics (IOP)（物理研究所）；Homi Bhabha National Institute（霍米·巴巴国家研究所）；University of Kashmir（克什米尔大学）；Islamic University of Science and Technology（伊斯兰科学技术大学）
A systematic study of the yrast band in 154-164 Dy isotopes using the Projected Shell Model is presented. It is shown that, in the context of the present model, enlarging the mean field deformation by about 20 % allows a very good description of the spectrum of yrast band in these isotopes. The dependence of the B(E2) values on angular momentum is also better described when larger deformations are used. The observed oscillation of g-factors at low spin states remains an open question for this model.
