Tilted-axis wobbling in odd-mass nuclei

A triaxial rotor Hamiltonian with a rigidly aligned high-$j$ quasiparticle is treated by a time-dependent variational principle, using angular momentum coherent states. The resulting classical energy function has three unique critical points in a space of generalized conjugate coordinates, which can minimize the energy for specific ordering of the inertial parameters and a fixed angular momentum state. Because of the symmetry of the problem, there are only two unique solutions, corresponding to wobbling motion around a principal axis and, respectively, a tilted axis. The wobbling frequencies are obtained after a quantization procedure and then used to calculate $E2$ and $M1$ transition probabilities. The analytical results are employed in the study of the wobbling excitations of ${}^{135}\mathrm{Pr}$ nucleus, which is found to undergo a transition from low angular momentum transverse wobbling around a principal axis toward a tilted-axis wobbling at higher angular momentum.

Figure 1

Wobbling phase diagram for independent rigid MOI represented for $I=35/2$ and $j=11/2$. In the region near the origin bounded by $S_{13/2,11/2}$ the wobbling modes do not exist. The transversal regime of the first wobbling mode is denoted with $\left(t1\right)$. The two closed curves show the relationship among the rigid MOI parametrized as in (3.1) for $\beta =0.2$ and $\beta =0.3$, when the asymmetry ${\gamma }_{rig}$ is varied.

Figure 2

Evolution of the separatrix $S_{Ij}$ as a function of angular momentum for $j=11/2$ and $13/2$.

Figure 3

Wobbling phase diagram for hydrodynamic MOI represented in the Cartesian coordinates $x=2Icos\gamma$ and $y=2Isin\gamma$. In the middle exclusion region the wobbling modes not exist. The transversal regime of the first wobbling mode is denoted with $\left(t1\right)$.

Figure 4

Evolution of wobbling frequency given in units of ${\mathcal{J}}_0^{-1}$ as a function of $2I$ from the transversal regime of the wobbling mode 1 to the tilted-axis wobbling mode 2. The curves correspond to $\gamma =-{10}^{\circ },-{15}^{\circ }$, and $-{20}^{\circ }$ with associated critical spins around $I=21/2,25/2$, and $33/2$.

Figure 5

Tilting angle ${\alpha }_2$ defined by (2.19) is given as a function of $2I$ for few $\gamma$ values from the phase space of the wobbling mode 2 with hydrodynamic MOI.

Figure 6

Experimental excitation energies [26] of the wobbling band in ${}^{135}\mathrm{Pr}$ compared with the theoretical wobbling energy.

Figure 7

Comparison of $n=0$ and $n=1$ energy levels between theoretical results and experimental data [26] for ${}^{135}\mathrm{Pr}$.