Triaxial Strongly Deformed Research Articles

How deformed are the TSD bands in odd Lu isotopes?

The experimental fingerprints for large deformation in the triaxial strongly deformed (TSD) bands of 163,165,167Lu are discussed. It is argued that these fingerprints are not very convincing. On the contrary, especially the fact that there exist strong interactions between the TSD bands and normal-deformed (ND) bands indicates that the deformation of the TSD bands cannot be very different from that of the ND bands. The need for detailed new experimental data is underlined.

Recent Results at Ultrahigh Spin: Terminating States and Beyond in Mass 160 Rare-earth Nuclei

A classic region of band termination at high spin occurs in rare-earth nuclei with around ten valence nucleons above the Gd-146 closed core. Results are presented here for such non-collective oblate (gamma = 60 degrees) terminating states in odd-Z Ho-155, odd-odd Ho-156, and even-even Er-156, where they are compared with neighbouring nuclei. In addition to these particularly favoured states, the occurrence of collective triaxial strongly deformed (TSD) bands, bypassing the terminating states and extending to over 65 (h) over bar, is reviewed.

Nuclear triaxiality in the A ∼ 160–170 mass region: the story so far

Research in nuclear triaxial deformation has revealed many exciting facts and figures over the last one and a half-decades. Although wobbling motion of nuclei was experimentally discovered at the beginning of the last decade, after almost 25 years of its prediction by Bohr and Mottelson, efforts are still being put to understand this rare nuclear phenomenon in greater detail. The concept of transverse wobbling is one such recent attempt which successfully explains the evolution of experimentally observed wobbling frequency with spin. The population of triaxial strongly deformed (TSD) bands in the A ∼160–170 region is favoured for which neutron number (N=92 or 94) is a topic of current debate. Experimental efforts are being put following Bengtsson’s calculations which indicate that the elevated yrast lines for N=92 isotones favour TSD population. In A ∼170 mass region, the ambiguity over the real character of certain strongly deformed bands has recently been removed by extensive experimental and theoretical efforts, and the bands have now been firmly established as either enhanced deformed (ED) or superdeformed (SD).

Interpretation of the high spin states inLu161: A paired and unpaired study

A paired cranked Nilsson-Strutinsky-Bogoliubov (CNSB) model is presented, which employs the same method to calculate the liquid-drop energy and moment of inertia as the unpaired cranked Nilsson-Strutinsky (CNS) model. In the CNSB model, the energy minimization is carried out in the mesh of pairing gaps $\ensuremath{\Delta}$ and Fermi levels $\ensuremath{\lambda}$ as well as deformation parameters. The high spin states in $^{161}\mathrm{Lu}$ are then investigated with the CNSB and CNS models. The terminating structure shows a striking similarity with these two models. Combining the CNSB and CNS models, a complete understanding of high spin structures, including the normal deformed (ND) and triaxial strongly deformed (TSD) bands and observed side bands in $^{161}\mathrm{Lu}$, is achieved. It appears that the only important paired crossings are the first ${i}_{13/2}$ neutron crossing and the first ${h}_{11/2}$ proton crossing. For the description of the unpaired high spin crossings, it is important to be able to distinguish between the pseudospin partners in the proton $\mathcal{N}=4$ shell, $({d}_{5/2}$,${g}_{7/2}$) and $({d}_{3/2}$,${s}_{1/2}$). The yrast bands are predicted to terminate, which explains the structure of a TSD-like band X2. A band crossing at $I\ensuremath{\approx}36.5$ for the TSD band in $^{161}\mathrm{Lu}$, unique within the chain of even-$N$ Lu isotopes, is well described by the CNSB model.

Triaxial strongly deformed structures in the even-even Hf nuclei

Two rotational bands of distinct character have been identified in 164 Hf from a recent experiment at Gammasphere. They are suggested to correspond to the long- anticipated triaxial strongly deformed (TSD) bands predicted by theoretical studies. The bands are substantially stronger in intensity and are located at lower spins than the pre- viously observed TSD bands in 168 Hf, and have been linked to the known states, hereby making 164 Hf the best even-even system for the study of TSD structures in the A ∼ 160 mass region. Cranking calculations based on the modified-oscillator model suggest that the bands are associated with four-quasiparticle configurations involving high-j intruder (i13/2) 2 proton orbitals. Wobbling mode has not been observed in 164 Hf and the possible reasons are discussed.

Identification of triaxial strongly deformed bands in164Hf

Two new rotational bands of distinct character have been identified in ${}^{164}\mathrm{Hf}$. They are suggested to correspond to the long-anticipated triaxial strongly deformed (TSD) bands predicted by theoretical studies. The bands have been linked to known states, and the level spins and energies could be determined. The bands are also substantially stronger in intensity and are located at lower spins than the previously observed TSD bands in ${}^{168}\mathrm{Hf}$, hereby making ${}^{164}\mathrm{Hf}$ the best even-even system so far for the study of TSD structures in the $A\ensuremath{\sim}160$ mass region. Cranking calculations based on the modified-oscillator model suggest that the bands are associated with four-quasiparticle configurations that involve high-$j$ intruder ${({i}_{13/2})}^{2}$ proton orbitals.

Quadrupole moments of coexisting collective shapes at high spin in154Er

Four high-spin collective bands have been populated in Er-154(68)86 via the Pd-110(Ti-48, (4)n gamma)Er-154 reaction. Average transition quadrupole moments Q(t) have been measured for three of the bands by using the Doppler-shift attenuation method. The strongest band has a value of Q(t) = 11.0 +/- 1.0 e b, similar to values found recently for four triaxial strongly deformed (TSD) bands in Er-157,Er-158. The second band has a value of Q(t) = 19.5 +/- 3.2 e b, consistent with a predicted axially symmetric superdeformed (SD) shape, similar in deformation to the Dy-152 isotone, and is used as a calibration point. The third, new band has a value of Q(t) = 9.9 +/- 2.2 e b. The results confirm the unexpectedly large Q(t) moments for the favored TSD bands in light erbium isotopes. (Less)

New Studies on the Aspects of Nuclear Shapes

Using the pairing-deformation-frequency self-consistent total-Routhiansurface and configuration-constrained potential-energy-surface calculations, we have studied nuclear deformation and its effect on the structure of nuclei. It was found that the high-order multipolarity-six (β6) deformation plays a significant role in superheavy nuclei. Possible non-collective highspin isomeric states which locate in the second well of actinide nuclei have been investigated with the predictions of excitation energies and configurations. High-spin isomers can extend shape coexistence in A ∼ 190 neutrondeficient nuclei. Triaxiality with γ 30 is found in the ground and excited rotational states of the A ∼ 70 germanium isotopes. Octupole correlations have also been discussed in different mass regions. In recent experiments, the textbook nucleus 158Er has been reached at ultrahigh spins around 65∼. We have studied 158Er ultrahigh-spin states by means of the self-consistent tilted-axis-cranking method based on the Nilsson shell correction and the Skyrme-Hartree-Fock model. The calculation with a ≈ 12 ° triaxialstrongly-deformed (TSD) excited configuration can well reproduce the observed large transitional quadrupole moment. It is demonstrated that the TSD minimum at negative γdeformation which appears in the principalaxis-cranking approach is a saddle point if allowing the rotational axis to change direction.

Relative quadrupole moments of exotic shapes at ultrahigh spin in 154Er: calibrating the TSD/SD puzzle

Transition quadrupole moments, Qt, of two ultrahigh-spin, collective structures in 154Er have been measured for the first time using the Doppler Shift Attenuation Method (DSAM). Data were acquired at the ATLAS accelerator facility of Argonne National Laboratory, using the Gammasphere detector array. A thick, gold-backed 110Pd foil was bombarded by a beam of 48Ti ions at 215 MeV. The Qt for each band was determined from the Doppler shift of gamma rays emitted by the resulting recoil nuclei. The extracted transition quadrupole moments are significantly different in magnitude, suggesting the two structures in 154Er represent distinct exotic nuclear shapes, namely axial superdeformed (SD) with Qt ≈ 20 eb, and triaxial strongly deformed (TSD) with Qt ≈ 11 eb. Indeed, the results calibrate the quadrupole moments of TSD bands recently measured in light erbium nuclei, 157,158Er.

Self-Consistent Tilted-Axis-Cranking Study of Triaxial Strongly Deformed Bands inEr158at Ultrahigh Spin

Stimulated by recent experimental discoveries, triaxial strongly deformed (TSD) states in (158)Er at ultrahigh spins have been studied by means of the Skyrme-Hartree-Fock model and the tilted-axis-cranking method. Restricting the rotational axis to one of the principal axes--as done in previous cranking calculations--two well-defined TSD minima in the total Routhian surface are found for a given configuration: one with positive and another with negative triaxial deformation γ. By allowing the rotational axis to change direction, the higher-energy minimum is shown to be a saddle point. This resolves the long-standing question of the physical interpretation of the two triaxial minima at a very similar quadrupole shape obtained in the principal-axis-cranking approach. Several TSD configurations have been predicted, including a highly deformed band, which is a candidate for the structure observed in experiment.

Properties of triaxial, strongly deformed bands inTa167andLu167and the top-on-top model

Based on the particle-rotor model with one particle coupled to a triaxially deformed rotor, the experimental excitation energy relative to a reference ${E}^{*}\ensuremath{-}\mathit{aI}(I+1)$ and the ratio between interband and intraband electromagnetic transitions are well reproduced for $^{167}\mathrm{Ta}$ with $\ensuremath{\gamma}={19}^{\ifmmode^\circ\else\textdegree\fi{}}$. The same parameter set for the angular-momentum-dependent rigid-body moments of inertia attains good agreement with experimental data for the positive-parity triaxial, strongly deformed (TSD) band levels in $^{167}\mathrm{Lu}$. An attempt is made to investigate the negative-parity TSD band in $^{167}\mathrm{Lu}$.

Description of the Triaxial Strongly Deformed Bands in 160,161Tm and 163Tm

Properties of the triaxial strongly deformed (TSD) bands of 160,161Tm and 163Tm are investigated systematically within the supersymmetry scheme including many-body interactions and a perturbation possessing SO(5)(or SU(5)) symmetry on the rotational symmetry. Quantitatively good results of the γ-ray energies, the dynamical moments of inertia and the spin of the TSD bands in 160,161Tm and 163Tm are obtained. The calculation shows that competition between the pairing and anti-pairing effects exist in these TSD bands. Meanwhile, the SU(3) symmetry in TSD bands is broken more seriously than that in superdeformed bands.

Identification of triaxial strongly deformed bands inHf168

Two new rotational bands of distinct character have been identified in 164 Hf. They are suggested to correspond to the long-anticipated triaxial strongly deformed (TSD) bands predicted by theoretical studies. The bands have been linked to known states, and the level spins and energies could be determined. The bands are also substantially stronger in intensity and are located at lower spins than the previously observed TSD bands in 168 Hf, hereby making 164 Hf the best even-even system so far for the study of TSD structures in the A ∼ 160 mass region. Cranking calculations based on the modified-oscillator model suggest that the bands are associated with four-quasiparticle configurations that involve high-j intruder (i13/2) 2 proton orbitals.

Nuclear shapes of highly deformed bands inHf171,172and neighboring Hf isotopes

A Gammasphere experiment was carried out to search for triaxial strongly deformed (TSD) structures in $^{171,172}\mathrm{Hf}$ and the wobbling mode, a unique signature of nuclei with stable triaxiality. Three strongly deformed bands in $^{172}\mathrm{Hf}$ and one in $^{171}\mathrm{Hf}$ were identified through $^{48}\mathrm{Ca}$($^{128}\mathrm{Te}$, xn) reactions. Linking transitions were established for the band in $^{171}\mathrm{Hf}$ and, consequently, its excitation energies and spins (up to $111/2\ensuremath{\hbar}$) were firmly established. However, none of the $^{172}\mathrm{Hf}$ sequences were linked to known structures. Experimental evidence of triaxiality was not observed in these bands. The new bands are compared with other known strongly deformed bands in neighboring Hf isotopes. Theoretical investigations within various models have been performed. Cranking calculations with the Ultimate Cranker code suggest that the band in $^{171}\mathrm{Hf}$ and two previously proposed TSD candidates in $^{170}\mathrm{Hf}$ and $^{175}\mathrm{Hf}$ are built on proton $({i}_{13/2}{h}_{9/2})$ configurations, associated with near-prolate shapes and deformations enhanced with respect to the normal deformed bands. Cranked relativistic mean-field calculations suggest that band 2 in $^{175}\mathrm{Hf}$ has most likely a near-prolate superdeformed shape involving the $\ensuremath{\pi}{i}_{13/2}\ensuremath{\bigotimes}\ensuremath{\nu}{j}_{15/2}$ high-$j$ intruder orbitals. It is quite likely that the bands in $^{172}\mathrm{Hf}$ are similar in character to this band.

Lifetime measurements of triaxial strongly deformed bands inTm163

With the Doppler Shift Attenuation Method, quadrupole transition moments Q{sub t} were determined for the two recently proposed triaxial strongly deformed (TSD) bands in {sup 163}Tm. The measured Q{sub t} values indicate that the deformation of these bands is larger than that of the yrast signature partners. However, the measured values are smaller than those predicted by theory. This observation appears to be valid for TSD bands in several nuclei of the region.

Quadrupole moment measurements of TSD1 and TSD2 bands in 167Lu

The triaxial strongly deformed (TSD) bands in167Lu were populated by the 123Sb(48Ca, 4n) reaction with a beam energy of 203 MeV. Gamma rays, requiring five fold or more in prompt coincidence, were detected with the Gammasphere spectrometer. Of particular interests are TSD bands 1 and 2 which have previously been interpreted as zero phonon and one phonon wobbling bands, respectively. Using the Doppler shift attenuation method (DSAM), a preliminary transition quadrupole moment of 6.9+0.3−0.3 eb was extracted for the TSD1 band. Data analysis continues for TSD2 which is considerably more weakly populated.

Wobbling excitations in strongly deformed Hf nuclei?

Two Gammasphere experiments have been performed in order to establish the possible triaxial nature of strongly deformed (SD) bands in 174Hf. A lifetime measurement, using the Doppler-shift attenuation method, confirmed the large deformation of the four previously observed bands in this nucleus with transition quadrupole moments ranging from 12.6 to 13.8 b. These values are significantly larger than those predicted for triaxial minima by ultimate cranker (UC) calculations. A thin-target, high-statistics experiment was also carried out to search for linking transitions between the SD bands. No such transitions, which represent an experimental signature for wobbling modes, were observed. Four additional SD bands were found in 174Hf together with a single SD band in 173Hf. These results indicate that the strongly deformed sequences of N≈102 Hf isotopes behave differently than the triaxial strongly deformed (TSD) bands found in Lu nuclei near N=92. The interpretation of these bands in terms of possible stable triaxial deformation is confronted with the experimental findings and UC predictions.

Quantum numbers for the triaxial strongly deformed bands in odd mass nuclei

A set of quantum numbers describing both precessions of total spin I⃗ and single particle spin j⃗ is defined by applying boson expansion method to the particle-rotor model with one nucleon coupled to a triaxially deformed core. The derived algebraic energy formula can be applied to classify the triaxial, strongly deformed (TSD) bands in Lu isotopes.

Coexisting wobbling and quasiparticle excitations in the triaxial potential well of 163 Lu

High-spin states of the nucleus 163Lu have been populated through the fusion-evaporation reaction 139La(29Si, 5n) with a beam energy of 157 MeV. In addition to the two lowest excited triaxial strongly deformed (TSD) bands, recently interpreted as one- and two-phonon wobbling excitations, a third excited TSD band has been firmly established decaying to the yrast TSD band. The assignment of this band as a three-quasiparticle band shows together with the normal deformed (ND) level scheme the presence not only of shape coexistence between ND and TSD structures, but also an interplay of wobbling and quasiparticle excitations in the triaxial strongly deformed potential well of 163Lu.

The wobbling mode in 167Lu

High spin states in 167Lu were populated through the 123Sb(48Ca,xn) reaction at 203 MeV. Four, presumably triaxial, strongly deformed (TSD) bands have been found in this nucleus. Several transitions linking an excited TSD band (TSD2) to the lowest (yrast) TSD band (TSD1) were observed. The electromagnetic properties of the connecting transitions have been investigated. Evidence for the assignment of TSD2 as a wobbling mode built on TSD1 is presented. This assignment is based on comparisons of the experimental data to theoretical calculations. The wobbling mode of excitation is an unambiguous signal of a stable triaxial deformation associated with these bands.