Cranking inertia of odd nuclei from time-dependent pairing equations: Application to Th cold fission

A new formalism for the nonadiabatic collective inertia from time-dependent pairing equations for odd numbers of nucleons is described in detail. The effective masses and the moments of inertia result from the occurrence of matrix elements of the time derivatives and of the angular couplings, respectively, between states of different seniority configurations. For low collective velocities, the formulas for the inertia reduce to the known cranking expressions available for quasitationary states. The effective mass and the perpendicular moment of inertia are evaluated for the ${}^{230–232}\mathrm{Th}$ parent nuclei along the fission path. The inertia for the even-odd system is larger than those of the even-even ones and exhibits fluctuations due to the intrinsic nuclear structure. The ground state theoretical value of the moment of inertia for ${}^{231}\mathrm{Th}$ agrees well with the evaluated experimental data if the blocking effect is neglected.

Figure 1

The deformation energy $V$ of the ${}^{230–232}\mathrm{Th}$ parent nuclei is represented as function of the distance between the centers of the spheroids associated with the nascent fragments denoted $R$. Panel (a) corresponds to ${}^{230}\mathrm{Th}$, panel (b) corresponds to ${}^{231}\mathrm{Th}$, and panel (c) is for ${}^{232}\mathrm{Th}$. The deformation energy obtained with high values of the pairing interaction $G$ is plotted with a full line. The dashed line is used for low values of $G$.

Figure 2

The pairing gap parameters $\mathrm{\Delta }$ and the pairing interaction $G$ as function of the elongation $R$ are plotted in panels (a) and (b), respectively. The full line corresponds to high values of $G$ obtained with the formalism of [64], while the dashed line is calculated with the formalism in [60], giving low $G$ values.

Figure 3

The neutron single-particle level scheme $\epsilon$ in the region of the Fermi energy is represented as function of the distance between the centers of the spheroids associated with the nascent fragments $R$. The spherical orbitals are marked with their spectroscopic notations on the left, for $R=0$, where a spherical nuclear shape is considered. From $R\approx 19.5\mathrm{fm}$, scission has occurred, and the single-particle energies remain unmodified. The ground state is located around a deformation of $R\approx 4.3\mathrm{fm}$. The levels with spin projection $\mathrm{\Omega }=1/2$ are plotted with the green curves, those with $\mathrm{\Omega }=3/2$ are plotted with the dark blue ones, while those with $\mathrm{\Omega }=5/2$ and $\mathrm{\Omega }=7/2$ are plotted with violet and blue, respectively. The Fermi single-particle level is identified with red points that are superimposed on the curves.

Figure 4

The effective masses $B$ along the fission trajectory displayed as function of the distance between the centers of the spheroids associated with the two fragments $R$. The parent nuclei are represented in the same order as in Fig. 1 and the meanings of the lines types are the same. In panel (b), the inertia is represented up to a maximal value of $19{\hslash }^2/\left(\mathrm{MeV}{\mathrm{fm}}^2\right)$.

Figure 5

The moment of inertia $I$ perpendicular to the axis of symmetry along the fission trajectory as function of the distance between the centers of the spheroids associated with the two fragments $R$. The parent nuclei are represented in the same order as in Fig. 1 and the meanings of the lines types are the same. In panel (b), the moments of inertia are represented up to a maximal value of $410{\hslash }^2/\mathrm{MeV}$.

Figure 6

The effective masses $B$ are displayed in panel (a) and the perpendicular moments of inertia $I$ in panel (b) as function of the distance between the centers of the spheroids associated with the two fragments $R$ for ${}^{231}\mathrm{Th}$. The blocking effect is taken into account. The full curve indicates high values of the pairing interaction $G$ while the dashed curve is for low values.