Nonadiabatic quasiparticle approach for rotation-particle coupling in triaxial odd-$A$ nuclei

We discuss the formulation of a nonadiabatic approach to study the rotational states in triaxially deformed odd-$A$ nuclei. The rotation-particle coupling is treated microscopically by coupling the triaxial rotor states of the even-even core with the states of the valence particle in order to obtain the matrix elements of the odd-$A$ system. We arrive at a nonadiabatic quasiparticle approach where the rotational states can have contributions from various quasiparticle states near the Fermi level. We bring out the advantages of this approach over the conventional particle rotor model with a fixed or variable moment of inertia. One clear evidence favoring our approach is the rotation alignment phenomenon which is demonstrated in the case of ${}^{137}\mathrm{Pm}$. We discuss our results for ${}^{136}\mathrm{Nd}$ and ${}^{137}\mathrm{Pm}$, and justify that this approach is suitable also for studying nuclei away from stability.

Figure 1

The VMI parameters $C$ and ${\mathcal{I}}_0$ for the isotones of $N=76$ by using methods 1, 2, and 3. In (a) and (b) $\gamma$ is treated as a free parameter and in (c) and (d) $\gamma$ is fixed as the value obtained in macroscopic-microscopic calculations [37].

Figure 2

The softness parameters $\sigma$ and $b$ obtained from the methods 1, 2, 3, and 4 are shown for the isotones of $N=76$. This plot is made with Table 2. The $\gamma$ deformation is fixed as the value obtained in macroscopic-microscopic calculations [37].

Figure 3

Rotational spectra of ${}^{136}\mathrm{Nd}$ for the ground ($g$) and $\gamma$ bands. (a), (b), (c), and (d) represent the calculations with methods 1, 2, 3, and 4 for the triaxial rotor as described in Sec. 2a. The $g$ band contains only positive parity even angular momentum states, while the $\gamma$ band contains both even and odd angular momentum states with positive parity. Grey lines correspond to the experimental spectra [40] which are used in the fit. The solid lines represent our calculations with best fitting values of $\gamma$ and VMI parameters $C$ and ${\mathcal{I}}_0$ for methods 1, 2, and 3 and parameter $b$ for method 4.

Figure 4

The $\gamma$ band of ${}^{136}\mathrm{Nd}$ calculated with (a) method 2 (Sec. 2a2) and (b) method 4 (Sec. 2a4). Grey lines correspond to the experimental spectra [40].

Figure 5

Energy spectra of ${}^{136}\mathrm{Nd}$ calculated with different values of VMI parameter $b$ (17).

Figure 6

(a) Single-particle and (b) quasiparticle energies of ${}^{137}\mathrm{Pm}$ as a function of ${\beta }_2,\left({\beta }_4\right)$. (c) Quasiparticle energies of ${}^{137}\mathrm{Pm}$ as a function of $\gamma$. These energies are calculated with universal set of parameters [31]. The choice of ${\beta }_2$ and ${\beta }_4$ are consistent with the finite range droplet model calculations [39] and $\gamma$ is adjusted to have best fit for rotational spectra of ${}^{136}\mathrm{Nd}$, which is the core for ${}^{137}\mathrm{Pm}$. The solid and dashed lines correspond to the positive and negative parity states, respectively. The green dots represent the states of valence particle and the yellow line represents the chemical potential from BCS calculations. At zero deformation the degenerate states are labeled by $l_j$.

Figure 7

Spectra showing rotation alignment in ${}^{137}\mathrm{Pm}$ at $\gamma =0^{\circ }$ calculated using different methods with the BCS chemical potential $\lambda =-6$ MeV: (a) PRM with fixed moment of inertia (rigid core). (b) PRM with VMI (17) defined through the parameter $b$. (c) PRM with VMI (25) defined through the parameter $C$. (d) MPRM utilizing the experimental energies of the core ${}^{136}\mathrm{Nd}$.

Figure 8

Similar to Fig. 7, but at $\gamma ={25}^{\circ }$.

Figure 9

Similar to Fig. 8, but with the BCS chemical potential calculated at every deformation.

Figure 10

The ground band of ${}^{137}\mathrm{Pm}$ calculated with MPRM is plotted with respect to the nonaxial deformation $\gamma$. Experimental energies represented by the grey lines are taken from [41, 42].

Figure 11

The ground band along with the states with $I^{\pi }=7/2^{-}$ and $9/2^{-}$, of ${}^{137}\mathrm{Pm}$, calculated with MPRM is presented as a function of the axial deformation ${\beta }_2\left({\beta }_4\right)$ where $\gamma ={25}^{\circ }$. The thicker lines in grey and yellow colours correspond to the experimental energies of ground and the side bands [41, 42], respectively.

Figure 12

Ground band of ${}^{137}\mathrm{Pm}$ calculated within PRM, for different combinations of VMI parameter $b,$ and Coriolis attenuation coefficient $\rho$. Experimental energies represented by the grey lines are taken from [41, 42].

Figure 13

Low-lying negative parity bands of the odd-$A$ nucleus ${}^{137}\mathrm{Pm}$ calculated with deformation parameters (${\beta }_2$, ${\beta }_4$, $\gamma$) as mentioned in the inset. Experimental energies represented by the grey lines are taken from [41, 42].