Stability of the wobbling motion in an odd-mass nucleus and the analysis of ${}^{135}\mathrm{Pr}$

We apply alternative representations of Holstein-Primakoff boson expansion to the particle-rotor model as useful probes to test the stability and the physical contents of the exact solution. The diagonal representations with the total and single-particle angular momenta along the same axis (longitudinal case) and along two perpendicular axes (transverse case) are employed according to the system with rigid or hydrodynamical moments of inertia (MoI). The longitudinal case gives the normal wobbling mode for rigid MoI, but does not give a stable solution for hydrodynamical MoI. The transverse case applied for hydrodynamical MoI describes a few low spin states with reduced alignment of the total spin in the parallel direction to the single-particle spin, but these states differ from the wobbling mode which involves a quantized unit of rotational angular momentum. Employing the angular-momentum-dependent rigid MoI which is derived from the self-consistent Hartree-Fock-Bogoliubov equation together with angular momentum and particle-number constraints, we obtain good fitting not only to the energy-level scheme, but also to the electromagnetic transition rates and the mixing ratio for ${}^{135}\mathrm{Pr}$.

Figure 1

The energy levels given by Eq. (33) for the rigid (rig) MoI at $\gamma ={30}^{\circ }$ as functions of $I$. The solid lines correspond to the levels with $I-j=$ even, while the dashed lines and the lowest solid lines correspond to the ones with $I-j=$ odd. A numeral beside each line denotes the value of $R$. A pair of quantum numbers $(n_{\alpha },n_{\beta })$ next to the bottom is assigned to the yrast for each $I$. The lowest solid lines represent (1,0) for the yrast levels of $I-j=$ odd for $I\ge 17/2$. The parameters are ${\mathcal{J}}_0=25$ ${\mathrm{MeV}}^{-1}$ and ${\beta }_2=0.18$.

Figure 2

The energy levels given by Eq. (38) for the hydrodynamical (hyd) MoI at $\gamma ={30}^{\circ }$ as functions of $I$. The solid lines correspond to the levels with $I-j=$ even, while the dashed lines and the second solid lines correspond to the ones with $I-j=$ odd. A numeral beside each line denotes the value of $R$. A pair of quantum numbers $(n_{\alpha },n_{\beta })$ next to the bottom is assigned to the yrast for each $I$. The solid lines for the yrare levels with $I\ge 17/2$ represent (1,0). The parameter ${\mathcal{J}}_0=25$ ${\mathrm{MeV}}^{-1}$ is the same as in Fig. 1.

Figure 3

The energy levels given by Eq. (41) for the hydrodynamical (hyd) MoI at $\gamma =6^{\circ }$ as functions of $I$. The solid lines correspond to the levels with $I-j=$ even, while the dashed lines correspond to the levels with $I-j=$ odd. A numeral beside each line denotes the value of $R$. A pair of quantum numbers $(R,R_z)$ next to the bottom is assigned to the yrast for each $I$. The parameter set is the same as in Fig. 2.

Figure 4

The energy levels given by Eq. (41) for the rigid (rig) MoI at $\gamma =6^{\circ }$ as functions of $I$. The solid lines correspond to the levels with $I-j=$ even, while the dashed lines correspond to the levels with $I-j=$ odd. A numeral beside each line denotes the value of $R$. A pair of quantum numbers $(R,R_z)$ next to the bottom is assigned to the yrast for each $I$. The parameter set is the same as in Fig. 1.

Figure 5

Alignments of ${\langle I_x^2\rangle }^{1/2}$ and ${\langle j_x^2\rangle }^{1/2}$ for rigid (rig) MoI as functions of $I$. The solid circles connected by solid lines show exact results for $I-j=$ even and the ones connected by dashed lines for $I-j=$ odd. The open circles connected by solid and dashed lines show the corresponding results obtained from the leading order approximation. The parameter set is the same as in Fig. 1.

Figure 6

Alignments of ${\langle I_y^2\rangle }^{1/2}$ and ${\langle j_y^2\rangle }^{1/2}$ for hydrodynamical (hyd) MoI as functions of $I$. The solid circles connected by solid lines show exact results for $I-j=$ even and the ones connected by dashed lines for $I-j=$ odd. The open circles connected by solid and dashed lines show the corresponding results obtained from the leading order approximation. The parameter set is the same as in Fig. 2.

Figure 7

Stability domain in the $\gamma \text{−}V$ plane for hydrodynamical (hyd) MoI with diagonal representation of $I_x$ and $j_x$ in the next-to-leading order approximation. The numerals denote the values of $I$. The solid lines are for the levels with $I-j=$ even, while the dashed lines are for those with $I-j=$ odd. The values of $(n_{\alpha },n_{\beta })$ are shown for each $I$. The arrows attached to the lines indicate the region where the stability conditions are satisfied.

Figure 8

Stability domain in the $\gamma \text{−}V$ plane for hydrodynamical (hyd) MoI with diagonal representation of $I_y$ and $j_y$ in the next-to-leading order approximation. The solid lines are for the levels with $I-j=$ even, while the dashed lines are for those with $I-j=$ odd. The arrows attached to the lines indicate the region where the stability conditions are satisfied.

Figure 9

The alignments of ${\langle I_x^2\rangle }^{1/2},{\langle I_y^2\rangle }^{1/2},{\langle j_x^2\rangle }^{1/2}$, and ${\langle j_y^2\rangle }^{1/2}$ for the rigid (rig) MoI as functions of $I$. The solid circles connected by solid lines show ${\langle I_x^2\rangle }^{1/2}$ and ${\langle I_y^2\rangle }^{1/2}$ for the levels with $I-j=$ even, while open circles connected by dashed lines show the corresponding results for $I-j=$ odd. The closed triangles connected by solid lines show ${\langle j_x^2\rangle }^{1/2}$ and ${\langle j_y^2\rangle }^{1/2}$ for the levels with $I-j=$ even, while the open triangles connected by dashed lines show the corresponding results for $I-j=$ odd. The parameters are ${\mathcal{J}}_0=25$ ${\mathrm{MeV}}^{-1},V=1.6$ MeV, ${\beta }_2=0.18$, and $\gamma ={26}^{\circ }$.

Figure 10

The alignments of ${\langle I_x^2\rangle }^{1/2},{\langle I_y^2\rangle }^{1/2},{\langle j_x^2\rangle }^{1/2}$, and ${\langle j_y^2\rangle }^{1/2}$ for the hydrodynamical (hyd) MoI as functions of $I$. The solid circles connected by solid lines show ${\langle I_x^2\rangle }^{1/2}$ and ${\langle I_y^2\rangle }^{1/2}$ for the levels with $I-j=$ even, while the open circles connected by dashed lines show the corresponding results for $I-j=$ odd. The closed triangles connected by solid lines show ${\langle j_x^2\rangle }^{1/2}$ and ${\langle j_y^2\rangle }^{1/2}$ for the levels with $I-j=$ even, while the open triangles connected by dashed lines show the corresponding results for $I-j=$ odd. The parameter set is the same as in Fig. 9.

Figure 11

Stability domain in the $\gamma \text{−}V$ plane for hydrodynamical MoI with diagonal representation of $I_y$ and $j_x$ in the next-to-leading order approximation. The solid lines correspond to the yrast levels of $I=11/2$ and 31/$2(I-j=$ even), while the dashed lines correspond to those of $I=13/2$ and 33/$2(I-j=$ odd). The domain borders for the other levels are between these lines. The arrows attached to the lines indicate the region where stability conditions are satisfied.

Figure 12

Comparison of the squared absolute amplitudes in Eq. (27) in the next-to-leading order approximation as functions of $I$. The solid circles correspond to $|u_{\pm }{|}^2$, solid triangles to $|v_{\pm }{|}^2$, open circles to $|w_{\pm }{|}^2$, and open triangles to $|t_{\pm }{|}^2$. Both $|w_{-}{|}^2$ and $|t_{-}{|}^2$ are quite small and lie along the abscissa. The parameter set is the same as in Fig. 9.

Figure 13

The square of shift parameters $r^2$ and $q^2$ as functions of $I$ in the next-to-leading order approximation. The solid circles correspond to $r^2$, and open circles to $q^2$. The parameter set is the same as in Fig. 9.

Figure 14

The alignments of ${\langle R_x^2\rangle }^{1/2},{\langle R_y^2\rangle }^{1/2}$, and ${\langle R_z^2\rangle }^{1/2}$ for the hydrodynamical (hyd) MoI as functions of $I$. The solid lines are for the levels with $I-j=$ even, while the dashed lines for those with $I-j=$ odd. The parameter set is the same as in Fig. 9.

Figure 15

The alignments of ${\langle R_x^2\rangle }^{1/2},{\langle R_y^2\rangle }^{1/2}$, and ${\langle R_z^2\rangle }^{1/2}$ for the rigid (rig) MoI as functions of $I$. The solid and open circles correspond to ${\langle R_x^2\rangle }^{1/2}$ and ${\langle R_y^2\rangle }^{1/2}$, while solid and open triangles correspond to ${\langle R_z^2\rangle }^{1/2}$. The solid lines are for the levels with $I-j=$ even, while the dashed lines for those with $I-j=$ odd. The parameter set is the same as in Fig. 9.

Figure 16

Behavior of three angular momenta $\vec{I},\vec{j}$, and $\vec{R}(\equiv \vec{I}-\vec{j})$ for the hydrodynamical MoI. (a) The case of the lowest spin state of $I=j=11/2$, and (b) the one of the high spin state of $I=31/2$. Both drawings are designed in consistence with the exact results of the root mean squares of angular-momentum components. Further details are presented in the text.

Figure 17

Comparison of $E\left(I\right)-0.02I(I+1)$ between theoretical results and experimental data in Refs. [13, 19]. Theoretical values are shown by solid triangles for band 1 and solid circles for bands 2 and 4, while experimental data for band 1 are shown by open triangles and bands 2 and 4 by open circles. Band 2 is from $I=13/2$ to 25/2, while band 4 is from $I=17/2$ to 33/2.

Figure 18

Comparison between theory and experimental data in the backbending plot for bands 1, 2, and 4. Theoretical values are shown by solid triangles for band 1 and by solid circles for bands 2 and 4, and experimental values are shown by open triangles for band 1, and by open circles for bands 2 and 4.

Figure 19

Comparison between theory and experimental data in the backbending plot for band 1. Theoretical values are shown by solid triangles for $I\le 31/2$ and by solid circles for $I\ge 35/2$. Experimental values are shown by open circles. Solid circles are normalized at $I=35/2$.