Extended Exponential Model with Pairing Attenuation and Investigation of Energy Staggering and Identical Bands Effects in Superdeformed Thallium Nuclei

Abstract

A proposed extended exponential model with pairing attenuation (EEMPA) is applied to inspect and exhibit the $\Delta I=2$, $\Delta I=1$ staggering effects and the identical bands in transition energies $E_{\gamma}(I)$ of twelve superdeformed rotational bands (SDRB${}^{\mathrm{,}}$s) in Thallium nuclei. A computer simulated fitting search program has been written to determine the unknown spins and the values of the model parameters of the considered SDRB${}^{\mathrm{,}}$s so as to get a minimum root-mean-square deviation of the calculated $E_{\gamma}(I)$ and the observed ones. The adopted parameters are used to study the rotational frequency $\hbar\omega$, the dynamic $J^{(2)}$ and kinematic $J^{(1)}$ moments of inertia. The calculated results agree very well with the empirical ones. The behavior of $J^{(1)}$ and $J^{(2)}$ versus $\hbar\omega$ is discussed. To show the $\Delta I=2$ staggering in transition energies $E_{\gamma}(I)$ for ${}^{192}$Tl(SD2) and ${}^{194}$Tl(SD1, SD3, SD5), a staggering function which represents the finite difference approximation to the fourth-order derivative is used (Cederwall five-point formula). The $\Delta I=1$ staggering in signature partners of Tl nuclei: ${}^{191}$Tl(SD1, SD2), ${}^{192}$Tl(SD1, SD2), ${}^{193}$Tl(SD1, SD2), ${}^{194}$Tl(SD1, SD2), and ${}^{195}$Tl(SD1, SD2) are investigated by using a staggering function representing the differences between the average transitions $E_{\gamma}(I+2\to I)$, $E_{\gamma}(I\to I-2)$ in one band and the transition $E_{\gamma}(I+1\to I-1)$ in its signature partner. The difference in $E_{\gamma}(I)$ between transitions in SDRB${}^{\mathrm{,}}$s of the odd–odd ${}^{194}$Tl nucleus and its neighbors odd-$A$ nuclei ${}^{193,195}$Tl were small to large range of $\hbar\omega$, which indicates that these SDRB${}^{\mathrm{,}}$s have been considered as identical bands (IB${}^{\mathrm{,}}$s).