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Register a new user$g$-factor and static quadrupole moment for the wobbling mode in $^{133}$La
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The $g$-factor and static quadrupole moment for the wobbling mode in the nuclide $^{133}$La are investigated as functions of the spin $I$by employing the particle rotor model. The model can reproduce the available experimental data of $g$-factor and static quadrupole moment. The properties of the $g$-factor and static quadrupole moment as functions of $I$ are interpreted by analyzing the angular momentum geometry of the collective rotor, proton-particle, and total nuclear system. It is demonstrated that the experimental value of the $g$-factor at the bandhead of the yrast band leads to the conclusion that the rotor angular momentum is $Rsimeq 2$. Furthermore, the variation of the $g$-factor with the spin $I$ yields the information that the angular momenta of the proton-particle and total nuclear system are oriented parallel to each other. The negative values of the static quadrupole moment over the entire spin region are caused by an alignment of the total angular momentum mainly along the short axis. Static quadrupole moment differences between the wobbling and yrast band originate from a wobbling excitation with respect to the short axis.
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The $g$-factor and static quadrupole moment of the nuclides $^{135}$Pr, $^{105}$Pd, and $^{187}$Au in the wobbling motion are investigated in the particle-rotor model as functions of the total spin $I$. The $g$-factor of $^{105}mathrm{Pd}$ increases with increasing $I$, due to the negative gyromagnetic ratio of a neutron valence-neutron. This behavior is in contrast to the decreasing $g$-factor of the other two nuclides, $^{135}$Pr and $^{187}$Au, which feature a valence-proton. The static quadrupole moment $Q$ depends on all three expectation values of the total angular momentum. It is smaller in the yrast band than in the wobbling band for the transverse wobblers $^{135}$Pr and $^{105}$Pd, while larger for the longitudinal wobbler $^{187}$Au.
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.
The static quadrupole moments (SQMs) of nuclear chiral doublet bands are investigated for the first time taking the particle-hole configuration $pi(1h_{11/2}) otimes u(1h_{11/2})^{-1}$ with triaxial deformation parameters in the range $260^circ leq gamma leq 270^circ$ as examples. The behavior of the SQM as a function of spin $I$ is illustrated by analyzing the components of the total angular momentum. It is found that in the region of chiral vibrations the SQMs of doublet bands are strongly varying with $I$, whereas in the region of static chirality the SQMs of doublet bands are almost constant. Hence, the measurement of SQMs provides a new criterion for distinguishing the modes of nuclear chirality. Moreover, in the high-spin region the SQMs can be approximated by an analytic formula with a proportionality to $cosgamma$ for both doublet bands. This provides a way to extract experimentally the triaxial deformation parameter $gamma$ for chiral bands from the measured SQMs.
In [S. Biswas et al., Eur. Phys. J. A 55, 159 (2019)] a longitudinal wobbling band was reported in $^{133}$La. The critical experimental proof for this assignment is the E2 dominated linking transitions between the wobbling and normal bands, which are supported by angular distribution and linear polarization measurements. However, severe problems are found in the reported experimental information, indicating that the assignment of wobbling band was not firmly established.
The dynamics of nuclear collective motion is investigated in the case of reflection-asymmetric shapes. The model is based on a new parameterization of the octupole and quadrupole degrees of freedom, valid for nuclei close to the axial symmetry. Amplitudes of oscillation in other degrees of freedom different from the axial ones are assumed to be small, but not frozen to zero. The case of nuclei which already possess a permanent quadrupole deformation is discussed in some more detail and a simple solution is obtained at the critical point of the phase transition between harmonic octupole oscillation and a permanent asymmetric shape. The results are compared with experimental data of the Thorium isotopic chain. The isotope Th-226 is found to be close to the critical point.
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