Nuclear moment of inertia as an indicator of the phase transition in a finite system

The purpose of this article is to derive the analytic expression for the angular-momentum dependence ($I$ dependence) of the moment of inertia in microscopic mean-field theory for both even-even and odd-mass nuclei. Based on the constrained Hartree-Fock-Bogoliubov theory, the Coriolis antipairing effect is taken into account as the second-order perturbation to the BCS basis together with the blocking effect. Instead of integration, an asymptotic series expansion is applied to the quantity in which finiteness of the nuclear system becomes tangible in the high-spin region, where the gap parameter $\Delta$ becomes much smaller than the average single-particle level distance $d$. As a result, $\Delta$ keeps a small but finite value even for high-spin states, showing that there is no sharp phase transition in the nucleus. Analytic formulas are derived for the $I$ dependence of the moment of inertia for different regions of $\Delta \ge d/2$ and $\Delta <d/2$.

Figure 1

Behavior of $g\left(x\right)$ in Eq. (14) as a function of $x(={\varepsilon }_{\alpha }-\lambda$) for $\xi (=2\Delta /\delta )=0.2$.

Figure 2

Comparison of the MoI as functions of $\xi$ for three cases with different ${\langle g\rangle }_{\mathrm{av}}$ and a common ${\mathcal{J}}_x^{\mathrm{rig}}=68$ ${\mathrm{MeV}}^{-1}$. Equation (17) is represented by BH (dotted line), Eq. (16) by BM (dashed line), and Eq. (15) by the solid line in the middle of BH and BM.

Figure 3

Behavior of $f\left(x\right)$ in Eq. (21) as a function of $x(={\varepsilon }_{\alpha }-\lambda )$ with $\Delta =0.1$ MeV and $\delta =1$ MeV.

Figure 4

The angular momentum $I$ measured from band head $I_0$ calculated from Eq. (26) as a function of $\xi (=2\Delta /\delta )$. The parameters are given in the text.

Figure 5

Behavior of $N\left(x\right)$ in Eq. (28) as a function of $x(={\varepsilon }_{\alpha }-\lambda )$ with $\Delta =0.1$ MeV and $\delta =1$ MeV.

Figure 6

The gap $\Delta$ as a function of the angular-momentum difference $I-I_0$ for an even-even nucleus. The dashed line is based on Eq. (25) with Eqs. (15) and (23), while the solid line is based on Eq. (25) with Eqs. (30) and (31). The starting gap parameter is ${\Delta }_0=0.8$ MeV at $I=I_0$, and the other parameters are given in the text.

Figure 7

The moment of inertia (MoI) ${\mathcal{J}}_x$ as a function of the angular-momentum difference $I-I_0$ for an even-even nucleus. The dashed line is based on Eq. (26), and the solid line on Eq. (32). The parameters are the same as those in Fig. 7.

Figure 8

Comparison of gap $\Delta$ between the even-even case (${\Delta }^{\left(\mathrm{even}\right)}$ with dashed line) and the odd-mass case (${\Delta }^{\left(\mathrm{odd}\right)}$ with solid line). The parameters for the odd-mass case are $j_{>}^2=12,j_{<}^2=10,{\xi }_0=0.6$, and $\eta =0.6$. Lines for the even-even case are the same as those in Fig. 6.

Figure 9

Comparison of the MoI between the even-even case (dashed line) and the odd-mass case (solid line). The parameters are the same as those in Fig. 8.

Figure 10

Comparison of ${\mathcal{J}}_x/{\mathcal{J}}_x^{\mathrm{rig}}$ in the approximate sum form between the even-even case (dashed line) and the odd-mass case (solid line) in the high-spin region. In both cases, the parameters are the same as those in Figs. 8 and 9.

Figure 11

Comparison of the MoI between two cases with $j_{>}^2=18$ (dashed line) and $j_{>}^2=6$ (solid line) for the odd-mass nucleus. In both cases, the other parameters are common, i.e., $j_{<}^2=j_{>}^2-2,{\xi }_0=0.7$, and $\eta =0.6$.

Figure 12

Comparison of the MoI for different $\eta$ values in the odd-mass nucleus as functions of $I-I_0$. The solid line corresponds to $\eta =0.6$ and the dashed line to $\eta =1.2$. The other parameters are common in both cases, i.e., ${\xi }_0=0.6,j_{>}^2=12$, and $j_{<}^2=10$.

Figure 13

Difference between the function $g\left(x\right)+g(x-\delta )$ given by Eq. (14) (exact) and the approximate one given by Eq. (A2) (approx.) within the interval $-\delta /2\le x={\varepsilon }_{\alpha }-\lambda \le \delta /2$.

Figure 14

Comparison between the functions $F\left(x\right)$ and $G\left(x\right)$ as functions of $x(={\varepsilon }_{\alpha }-\lambda )$ for the case of $\Delta =0.1$ MeV and $\delta =1$ MeV.

Figure 15

Difference $F\left(x\right)-G\left(x\right)$ as a function of $x(={\varepsilon }_{\alpha }-\lambda )$ for the case of $\Delta =0.1$ MeV and $\delta =1$ MeV.