Reconciliation of wobbling motion with rotational alignment in odd mass nuclei

The transverse wobbling motion in odd-$A$ nuclei is investigated by means of a semiclassical treatment applied to a triaxial rotor Hamiltonian with a rigidly aligned high-$j$ quasiparticle. An additional spin-spin interaction which accounts for the rotational alignment mechanism is used to generalize the quasiparticle-rotor coupling. Its effect on the rotation dynamics is investigated in a classical mainframe. The quantum realization of the excitations associated to the transverse wobbling regime in the presence of additional alignment is achieved in a harmonic approximation. The quality of the approximation is investigated in a general theoretical context and particularly when applied to transverse wobbling excitations observed in five $A\approx 160$ nuclei and suggested for ${}^{183}\mathrm{Au}, {}^{135}\mathrm{Pr}$, and ${}^{105}\mathrm{Pd}$.

Figure 1

The dependence on canonical coordinates of the classical energy function for $\gamma =-{20}^{\circ }, j=13/2$, and a few half-integer values of $I$ in the absence of additional alignment $C=0$ (first row) and considering $C=1/{\mathcal{J}}_0$ (second row). The difference between two contour lines is scaled to have 20 contour lines in the range of energy for each case in part.

Figure 2

The critical angular momentum $I_c$ as a function of triaxiality $\gamma$ and for different values of alignment strength $C$ given in $1/{\mathcal{J}}_0$ units. The interval of $\gamma$ corresponds to the transition between the first and the second dynamical phases when $j=13/2$ and hydrodynamic MOI are used. The area below the curves corresponds to the transverse rotational regime.

Figure 3

$x=0$ profiles of the $j=13/2$ classical energy function for successive half-integer values of angular momentum and triaxiality $\gamma$ fixed at $-{20}^{\circ }$, without alignment (a) and with an alignment strength $C=1/{\mathcal{J}}_0$ (b). The dashed line in panel (a) represents the profile of the harmonic approximation.

Figure 4

The evolution of one phonon and two phonon ratio $I/I_{\text{min}}$ with angular momentum for $C=0,1,2{\mathcal{J}}_0^{-1}$ when (a) $j=13/2$ and $\gamma =-{20}^{\circ }$, (b) $j=13/2$ and $\gamma =-{25}^{\circ }$, and (c) $j=11/2$ and $\gamma =-{25}^{\circ }$.

Figure 5

The comparison of the experimental [4, 5, 6, 7, 8, 23, 59] one phonon and two phonon wobbling excitation energies for ${}^{161,163,165,167}\mathrm{Lu}$ and ${}^{167}\mathrm{Ta}$, with corresponding theoretical results.

Figure 6

Available experimental $\mathrm{\Delta }I=1$ out of band $E2$ transition probabilities normalized to in band $\mathrm{\Delta }I=2$ transition probabilities for ${}^{163,165,167}\mathrm{Lu}$ and ${}^{167}\mathrm{Ta}$ [4, 6, 7, 8, 23, 59], compared to theoretical predictions. Theoretical results for ${}^{167}\mathrm{Ta}$ can be considered as valid reference also for the case of ${}^{161}\mathrm{Lu}$, given the similar values of relevant parameters.

Figure 7

Experimental and theoretical $B(M1,I\to I-1)$ values connecting one phonon and yrast bands of ${}^{163}\mathrm{Lu}$. For theoretical calculation a quenching factor of 0.47 was used for $g_j$.

Figure 8

Comparison between experimental and theoretical wobbling energies (a) and $\mathrm{\Delta }I=1$ out of band $E2$ transition probabilities normalized to in band $\mathrm{\Delta }I=2$ transitions (b) for ${}^{135}\mathrm{Pr}$ [12, 13].

Figure 9

Comparison between experimental and theoretical wobbling energies (a) and $\mathrm{\Delta }I=1$ out of band $E2$ transition probabilities normalized to in band $\mathrm{\Delta }I=2$ transitions (b) for ${}^{105}\mathrm{Pd}$ [14].

Figure 10

Comparison between experimental and theoretical wobbling energies (a) and $\mathrm{\Delta }I=1$ out of band $E2$ transition probabilities normalized to in band $\mathrm{\Delta }I=2$ transitions (b) for ${}^{183}\mathrm{Au}$ [15].

Figure 11

The linear correlation between $C/A_3$ normalized to the mass number $A$ as a function of the quasiparticle spin.