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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.

The second-phonon ( n ω = 2 ) wobbling band has been established in the nucleus 135Pr. Conclusive evidence for its wobbling nature comes from the Δ I = 1 , E2 character of the transitions between the new band and the previously identified transverse wobbler band ( n ω = 1 ) in this nucleus. Theoretical calculations in the framework of the quasiparticle triaxial rotor and triaxial projected shell models are found to be in good agreement with the experimental results.

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.

Lifetime measurements have been carried out for the levels of the negative parity yrast sequence in $^{103}\mathrm{Pd}$ nucleus using the Doppler shift attenuation method. The levels were populated via $^{94}\mathrm{Zr}(^{13}\mathrm{C}, 4n\ensuremath{\gamma})^{103}\mathrm{Pd}$ fusion-evaporation reaction at a beam energy of 55 MeV. De-exciting $\ensuremath{\gamma}$ rays were detected by utilizing the Indian National Gamma Array. The extracted transition probabilities and other auxiliary observations indicate that the sequence may be resulting from the antimagnetic rotational (AMR) motion of valence nucleons. The key characteristic feature of the AMR motion is the steady decrease of the $B(E2)$ transition probability with spin, which is seen in the present measured transitions for $^{103}\mathrm{Pd}$. The experimental results are compared with the theoretical predictions of tilted axis cranked approach based on the covariant density functional theory. It is noted that the properties of the AMR band structure for $^{103}\mathrm{Pd}$ predicted in this model analysis are in good agreement with the present experimental findings. Further, semi-classical particle-rotor model has been employed to substantiate the AMR interpretation of the observed band structure in $^{103}\mathrm{Pd}$ and it is shown that results are similar to the band structures observed in the neighboring isotopes, which have also been considered as candidates for AMR motion.

High-spin excited states of $^{135}\mathrm{Pm}$ have been investigated using the $^{107}\mathrm{Ag}(^{32}\mathrm{S}$, $2p2n$) reaction at 145-MeV beam energy. Three negative-parity bands have been investigated up to $\ensuremath{\sim}9.0\ensuremath{-}$, $7.6\ensuremath{-}$, and $6.9\ensuremath{-}\mathrm{MeV}$ excitation energies with spin parities $(55/{2}^{\ensuremath{-}})$, $(49/{2}^{\ensuremath{-}})$, and $(47/{2}^{\ensuremath{-}})\phantom{\rule{4pt}{0ex}}\ensuremath{\hbar}$, respectively. The experimental results have been interpreted using the triaxial projected shell model. It is shown that the observation of the parallel band structures can be explained in terms of crossing of two $s$ bands with the yrast band, which is a rare phenomenon for an odd-$A$ nucleus. Further, the lifetimes of levels in yrast band have been extracted using Doppler shift attenuation method. The evaluated $B(E2)$ values of yrast states show the loss of collectivity at higher spin, attributed to the alignment process.

A systematic study of the positive as well as negative parity yrast structure for the odd-mass 145−149Pm isotopes is carried out by using the angular momentum projection technique implemented in the projected shell model (PSM). By assuming an axial symmetry in the deformed basis, we have been able to calculate the yrast states up to a maximum spin of 47/2 for both positive and negative parity, and as a result, a good agreement is obtained with available experimental data for all isotopes. The intrinsic structure of positive and negative parity yrast states has been found to be built on πg7/2 $$ \otimes $$ νh11/2 and πh11/2 $$ \otimes $$ νh11/2 configurations, respectively. The phenomenon of back-bending has also been discussed in the present work. Reduced electric and magnetic transitions probabilities [B(E2) and B(M1)] are also calculated using the PSM wave functions. Since experimental data on B(E2) and B(M1) are not available, these results can serve as motivation for experimentalists to look for these data. However, the theoretical values of transition strength ratio B(M1)/B(E2) have been found to be matching excellently with the experimental ones.

With the proton number just below the magic number Z = 50, the palladium isotopes are known to exhibit rich nuclear structure phenomena. Inspired by the extensive experimental as well as theoretical structural investigations of Pd isotopes, we have performed the detailed study of band structures of 108–118Pd isotopes using triaxial projected shell model approach employing the multi-quasiparticle configuration space. As the mass region A ~ 110 depicts well-developed yrast, γ- and 2γ-bands, the present work demonstrates the reliable description of these bands by performing exact three-dimensional angular-momentum projection technique. The results obtained are also consistent with the global predictions of non-axial shapes for these isotopes. Besides this, the present study predicts the involvement of 2-quasiparticle neutron bands in describing various structural properties of these isotopes up to high spin states and has also achieved an excellent agreement of theoretical results with the known experimental data.

Aiming to understand the role of triaxiality and the evolution of the ground state nuclear shapes, we have carried out a microscopic study for a series of chains of Pd, Xe, Ba, Nd, Sm, Gd, and Dy isotopes. This is done within the self-consistent Relativistic-Hartree-Bogoliubov (RHB) formalism, and supported by the Triaxial Projected Shell Model (TPSM) approach. Pairing interaction separable in the momentum space with DD-ME2 force parameter is used to generate the potential energy surfaces(PESs) under the axial and triaxial symmetry. Shape evolution manifest themself in very clear manner in almost all the isotopic chains. Properties of the global mimima have been found to be in good agreement with the available experimental data. Relatively flat PESs, and $\gamma$-soft nature, have been suggested $^{108,110}$Pd, $^{132,134}$Xe and $^{134}$Ba as good candidates for E(5) symmetry, while $^{102}$Pd is not found suitable for E(5) symmetry. The PESs with a bump, and rigidity against triaxial variable($\gamma$) suggested $^{150}$Nd, $^{152}$Sm and $^{154}$Gd to be good candidates while $^{150}$Sm and $^{156}$Dy are poor candidates of X(5) critical-point symmetry. The findings of the present RHB calculations supported by TPSM are qualitatively in good agreement with the experimental and other theoretical calculations., Comment: accepted in Nucl. Phys. A

Theoretical investigation of positive parity yrast band of odd-odd 130Pr nucleus is performed by applying the projected shell model. The present study is undertaken to investigate and verify the very recently observed side band in 130Pr theoretically in terms of quasi-particle (qp) configuration. From the analysis of band diagram, the yrast as well as side band are found to arise from two-qp configuration πh 11/2 ⊗ νh 11/2. The present calculations are viewed to have qualitatively reproduced the known experimental data for yrast states, transition energies, and B(M1) / B(E2) ratios of this nucleus. The recently observed positive parity side band is also reproduced by the present calculations. The energy states of the side band are predicted up to spin 25+, which is far above the known experimental spin of 18+ and this could serve as a motivational factor for future experiments. In addition, the reduced transition probability B(E2) for interband transitions has also been calculated for the first time in projected shell model, which would serve as an encouragement for other research groups in the future.

The basis space in the triaxial projected shell model (TPSM) approach is generalized for odd-odd nuclei to include two-neutron and two-proton configurations on the basic one-neutron coupled to one-proton quasiparticle state. The generalization allows to investigate odd-odd nuclei beyond the band crossing region and as a first application of this development, high-spin band structures recently observed in odd-odd $^{194-200}$Tl isotopes are investigated. In some of these isotopes, the doublet band structures observed after the band crossing have been conjectured to arise from the spontaneous breaking of the chiral symmetry. The driving configuration of the chiral symmetry in these odd-odd isotopes is one-proton and three-neutrons rather than the basic one-proton and one-neutron as already observed in many other nuclei. It is demonstrated using the TPSM approach that energy differences of the doublet bands in $^{194}$Tl and $^{198}$Tl are, indeed, small. However, the differences in the calculated transition probabilities are somewhat larger than what is expected in the chiral symmetry limit. Experimental data on the transition probabilities is needed to shed light on the chiral nature of the doublet bands., Comment: 11 pages, 17 figures, to appear in EPJA

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

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.

By using the phenomenological approach of Projected Shell Model (PSM), the positive and negative-parity band structures of odd mass neutron-rich 119–127 I nuclei have been studied with the deformed single-particle states generated by the standard Nilsson potential. For these isotopes, the band structures have been analyzed in terms of quasi-particles configurations. The phenomenon of backbending in moment of inertia is also studied in the present work. Besides this, the reduced transition probabilities, i.e. B ( E 2 ) and B ( M 1 ) , are obtained from the PSM wavefunction for the first time for yrast bands of these isotopes.

Band structures of the neutron-rich Mo- and Ru-isotopes around A $\sim $ 110 are investigated using the triaxial projected shell model (TPSM) approach employing multi-quasiparticle configuration space. The mass region under investigation depicts a rich variety of band structures with well developed $\gamma$- and $\gamma\gamma$-bands, and quasiparticle excitations based on them. It is demonstrated that TPSM provides a reasonable description of most of the observed properties, in particular, detailed structure variations observed in Mo-isotopes are well reproduced in the present work., Comment: 19 pages, 7 figures, to appear in Nucl. Phys. A

Inspired by the availability of recent experimental as well as theoretical data on the energy levels of odd-mass 151-161Pm and odd-odd 154,156Pm, we applied the theoretical framework of the projected shell model to further understand the nuclear structure of these nuclei. The calculations closely reproduced the experimental data reported for the yrast bands of these isotopes by assuming an axial (prolate) deformation of ~0.3. Other properties along the yrast line, such as transition energies and transition probabilities, have also been discussed. Band diagrams are plotted to understand their intrinsic multi-quasiparticle structure, which turn out to be dominated by 1-quasiparticle bands for the odd-mass Pm isotopes and 2-quasiparticle bands for the doubly-odd Pm isotopes under study. The present study not only confirms the recently reported experimental/theoretical data, but also extends the already available information on the energy levels and adds new information regarding the reduced transition probabilities.

In this study, the multi-quasiparticle triaxial projected shell model (TPSM) is applied to investigate -vibrational bands in transitional nuclei of . We report that each triaxial intrinsic state has a -band built on it. The TPSM approach is evaluated by the comparison of TPSM results with available experimental data, which shows a satisfactory agreement. The energy ratios, B(E2) transition rates, and signature splitting of the -vibrational band are calculated.

The present work focuses on the successful application of the Projected Shell Model (PSM) in studying the yrast bands of neutron-rich odd-mass 111-119In nuclei. For these isotopes, the band structures have been analyzed in terms of quasi-particles configurations. The phenomenon of rotational alignment and back bending in moment of inertia is also studied in the present work. The intrinsic structure of 111-119In nuclei in the present study has also been described as arising due to an interplay between proton and neutron orbitals belonging to 1- and 3-qp configurations. Besides this, the reduced transition probabilities, B(E2) and B(M1) , are also obtained from the PSM wave functions.

Excited band structures recently observed in $^{156}\mathrm{Dy}$ are investigated using the microscopic triaxial projected shell model (TPSM) approach and the quasiparticle random phase approximation (QRPA) based on the rotating mean field. It is demonstrated that new observed excited bands, tracking the ground-state band, are the $\ensuremath{\gamma}$ bands based on the excited two-quasineutron configurations as conjectured in the experimental work.

The positive and negative parity rotational band structure of the neutron rich odd mass Eu isotopes with neutron numbers ranging from 90 to 96 are investigated up to the high angular momentum. In the theoretical analysis of energy spectra, transition energies and electromagnetic transition probabilities we employ the projected shell model. The calculations successfully describe the formation of the ground and excited band structures from the single particle and multi quasiparticle configurations. Calculated excitation energy spectra, transition energies, exact quantum mechanically calculated B(E2) and B(M1) transition probabilities are compared with experimental data wherever available and a reasonably good agreement is obtained with the observed data. The change in deformation in the ground state band with the increase in angular momentum and the increase in neutron number has also been established.

A systematic study of the yrast band structure for the neutron-rich odd-mass 103-111Tc nuclei is carried out using Projected Shell Model. The rotational band structure has been studied up to a maximum spin of $ 59/2^{+}$ . Excellent agreement with available experimental data for all isotopes is obtained. The energy spectra and electromagnetic transition strengths in terms of the configuration mixing of the angular-momentum projected multi-quasiparticle states are studied in detail. Signature splitting in the yrast rotational band is well described in the perspective of nuclear structure physics. The back-bending phenomenon is also well described for these nuclei in the present work.

The band structures of $^{68,70}$Ge, $^{128,130,132,134}$Ce and $^{132,134,136,138}$Nd are investigated using the triaxial projected shell model (TPSM) approach. These nuclei depict forking of the ground-state band into several s-bands and in some cases, both the lowest two observed s-bands depict neutron or proton character. It was discussed in our earlier work that this anamoluos behaviour can be explained by considering $\gamma$-bands based on two-quasiparticle configurations. As the parent band and the $\gamma$-band built on it have the same intrinsic structure, g-factors of the two bands are expected to be similar. In the present work, we have undertaken a detailed investigation of g-factors for the excited band structures of the studied nuclei and the available data for a few high-spin states are shown to be in fair agreement with the predicted values., Comment: 29 pages, 20 figures. Nucl. Phys. A (2017)

Excited states in the $^{101}\mathrm{Pd}$ nucleus were investigated through the $^{75}\mathrm{As}(^{31}\mathrm{P},2p3n)$ fusion-evaporation reaction at ${E}_{\mathrm{lab}}=125$ MeV by using the Indian National Gamma Array spectrometer equipped with 21 clover Ge detectors. The level scheme is considerably extended for medium spin values. New positive-parity band structures in $^{101}\mathrm{Pd}$ have been studied within the framework of the projected shell model and are found to undergo transition from single quasiparticle to high-$K$ three quasiparticle configuration after band crossing, i.e., from principal-axis rotation to tilted-axis rotation. The negative-parity band structures are discussed in the framework of the hybrid version of tilted-axis cranking shell model calculations. The observed alignment gain in the lowest excited $\ensuremath{\nu}{h}_{11/2}$ negative-parity band results from successive ${(\ensuremath{\nu}{g}_{7/2})}^{2}$ and ${(\ensuremath{\pi}{g}_{9/2})}^{2}$ pair alignments. The higher excited negative-parity bands are reproduced for the $\ensuremath{\nu}[{h}_{11/2}{({g}_{7/2}/{d}_{5/2})}^{2}]$ and ${(\ensuremath{\pi}{g}_{9/2})}^{\ensuremath{-}2}\ensuremath{\bigotimes}\ensuremath{\nu}{h}_{11/2}$ configurations.

Rotational spectra of odd-odd Rh and Ag isotopes are investigated with the primary motivation to search for the spontaneous chiral symmetry breaking phenomenon in these nuclei. The experimental results obtained on the degenerate dipole bands of some of these isotopes using a large array of gamma detectors are discussed and studied using the triaxial projected shell (TPSM) approach. It is shown that, first of all, to reproduce the odd-even staggering of the known yrast bands of these nuclei, large triaxial deformation is needed. This large triaxial deformation also gives rise to doublet band structures in many of these studied nuclei. The observed doublet bands in these isotopes are shown to be reproduced reasonably well by the TPSM calculations. Further, the TPSM calculations for neutron-rich nuclei indicate that the ideal manifestation of the chirality can be realised in 106Rh and 112Ag , where the doublet bands have similar electromagnetic properties along with small differences in excitation energies.

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 (1h 11/2)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.

By employing a systematically parametrized Hamiltonian and the best fit of the various input parameters, high-spin yrast energy states for an isotopic chain of odd mass 59-69Cu nuclei have been investigated by using a novel computational quantum mechanical framework-projected shell model. Comparison of calculations and experiments yields good agreement. The present study of various intriguing nuclear structure properties along the yrast lines in these odd proton isotopes reflects some interesting informative nuclear physics results. The calculations successfully describe the formation of the yrast level structures from multi-quasi-particle configurations based on $\pi f \otimes \nu g$ bands for 59-69Cu isotopes. The present calculations indicate the evolution of the nuclear structure near the magic nuclei, Ni, and also provide an indication of coexistence of both, collective as well as single-particle, levels for 69Cu nucleus at N=40.

Excited states of $^{133}$La have been investigated to search for the wobbling excitation mode in the low-spin regime. Wobbling bands with $n_\omega$ = 0 and 1 are identified along with the interconnecting $\Delta I$ = 1, $E2$ transitions, which are regarded as one of the characteristic features of the wobbling motion. An increase in wobbling frequency with spin implies longitudinal wobbling for $^{133}$La, in contrast with the case of transverse wobbling observed in $^{135}$Pr. This is the first observation of a longitudinal wobbling band in nuclei. The experimental observations are accounted for by calculations using the quasiparticle-triaxial-rotor (QTR) model, which attribute the appearance of longitudinal wobbling to the early alignment of a $\pi=+$ proton pair., Comment: 9 pages, 6 figures, 1 table

The structure of $^{70}$Ge has been studied through in-beam gamma ray spectroscopy. A new band structure is identified that leads to forking of the ground-state band into two excited bands. Band structures have been investigated using the microscopic triaxial projected shell model approach. The observed forking is demonstrated to result from almost simultaneous band crossing of the two neutron aligned and the \gamma-band built on this two-quasiparticle configuration with the ground-state band.

The projected shell model (PSM) with the deformed single-particle states, generated by the standard Nilsson potential, is applied to study the negative-parity high spin states of [Formula: see text] nuclei. The nuclear structure quantities like band structure and back-bending in moment of inertia have been calculated with PSM method and are compared with the available experimental data. In addition, the reduced transition probabilities, i.e., B[Formula: see text] and B[Formula: see text], are also obtained for the yrast band of these isotopes for the first time by using PSM wave function. A multi-quasiparticle structure has been predicted for [Formula: see text] isotopes by the present PSM calculations.

High-spin levels of [Formula: see text]Mo have been reinvestigated by analyzing the high statistics [Formula: see text]-[Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text] coincidence data from the spontaneous fission of [Formula: see text]Cf taken with the Gammasphere detector array. Two bands and 30 new transitions have been identified. A potential energy surface calculation has been performed. The calculation confirmed the 3/2[Formula: see text][411] configuration of the ground state band and 5/2[Formula: see text][532] for the 346[Formula: see text]keV excited band, as assigned in the previous work. The two newly established bands were proposed to be one- and two-phonon [Formula: see text] vibrational bands coupling to the 5/2[Formula: see text][532] Nilsson orbital, respectively. Triaxial projected shell model calculations have been applied to explain the level structure and are found in good agreement with experimental data.

A systematic investigation of the nuclear observables related to the triaxial degree of freedom is presented using the multi-quasiparticle triaxial projected shell model (TPSM) approach. These properties correspond to the observation of $\gamma$-bands, chiral doublet bands and the wobbling mode. In the TPSM approach, $\gamma$-bands are built on each quasiparticle configuration and it is demonstrated that some observations in high-spin spectroscopy that have remained unresolved for quite some time could be explained by considering $\gamma$-bands based on two-quasiparticle configurations. It is shown in some Ce-, Nd- and Ge-isotopes that the two observed aligned or s-bands originate from the same intrinsic configuration with one of them as the $\gamma$-band based on a two-quasiparticle configuration. In the present work, we have also performed a detailed study of $\gamma$-bands observed up to the highest spin in Dysposium, Hafnium, Mercury and Uranium isotopes. Furthermore, several measurements related to chiral symmetry breaking and wobbling motion have been reported recently. These phenomena, which are possible only for triaxial nuclei, have been investigated using the TPSM approach. It is shown that doublet bands observed in lighter odd-odd Cs-isotopes can be considered as candidates for chiral symmetry breaking. Transverse wobbling motion recently observed in $^{135}$Pr has also been investigated and it is shown that TPSM approach provides a reasonable description of the measured properties., Comment: Submitted to special Physica Scripta edition "Focus Issue on Nuclear Structure: Celebrating the 75 Nobel Prize" in Physics

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 data are still sparse. It is observed that gamma-bands for neutron-deficient isotopes of 156Er and 158Er are close to the yrast band, and further these bands are predicted to be nearly degenerate for high-spin states., 6 pages, 9 figures

The low-lying states of 108Ag near the I π = 6+ isomer have been investigated with the 11B + 100Mo reaction at 39 MeV beam energy. We aim at understanding the structure of the states near the isomer and identifying possible isomer depletion paths. From the γ–γ and γ–γ–γ coincidence analysis, spectroscopy of the excited states near the isomer has been carried out to establish new transitions and modification of the previously known level scheme. The present work suggests a total of three possible transitions at energies below 500 keV from the isomer to higher excited levels, whose subsequent decay can branch to the ground state. The spins and parities of these states have been established with angular correlation and polarization measurements. The branching ratios and multipolarities of the associated γ-rays were used in the estimation of the integral cross section for induced isomer depletion via these states. The experimental data have been compared to the results of projected Hartree–Fock calculations to understand the configurations of the levels.