$\beta$ decay of ${}^{252}\mathrm{Cf}$ in the transition from the exit point to scission

Upon increasing significantly the nuclear elongation, the $\beta$-decay energy grows. This paper investigates within a simple yet partly microscopic approach, the transition rate of the ${\beta }^{-}$ decay of the ${}^{252}\mathrm{Cf}$ nucleus on the way to scission from the exit point for a spontaneous fission process. A rather crude classical approximation is made for the corresponding damped collective motion assumed to be one dimensional. Given these assumptions, we only aim in this paper at providing the order of magnitudes of such a phenomenon. At each deformation the energy available for ${\beta }^{-}$ decay, is determined from such a dynamical treatment. Then, for a given elongation, transition rates for the allowed (Fermi) $0^{+}⟶0^{+} \beta$ decay are calculated from pair correlated wave functions obtained within a macroscopic-microscopic approach and then integrated over the time corresponding to the whole descent from exit to scission. The results are presented as a function of the damping factor (inverse of the characteristic damping time) in use in our classical dynamical approach. For instance, in the case of a descent time from the exit to the scission points of about ${10}^{-20}$ s, one finds a total rate of $\beta$ decay corresponding roughly to 20 events per year and per milligram of ${}^{252}\mathrm{Cf}$. The inclusion of pairing correlations does not affect much these results.

Figure 1

Mass excess of Pu-Ku nuclei with $A=252$ evaluated within the LSD model [1]. Each curve corresponds to different elongation of nuclei $\rho$ defined in the text. These values are reported on the r.h.s. side of the every second curve. The upper curve (labeled as “sph.”) presents the mass excesses calculated for spherical nuclei.

Figure 2

Proton (p) and neutron (n) single-particle levels of ${}^{252}\mathrm{Cf}$ in the ground state (l.h.s.) and at the scission point (r.h.s.).

Figure 3

Shell (top, l.h.s), pairing (top, r.h.s.), liquid drop (bottom, l.h.s.), and total deformation energy of ${}^{252}\mathrm{Cf}$ as function of the relative distance between the fission fragments $\left(\rho \right)$ and the neck parameter $\left(\kappa \right)$.

Figure 4

Macroscopic (solid line) and macroscopic-microscopic (dashed line) fission barriers and the $\beta$-decay energy $Q\left({\beta }^{-}\right)$ (dotted line) of ${}^{252}\mathrm{Cf}$ as a function of nuclear elongation.

Figure 5

Collective elongation $q=(R_{12}-R_{12}^{\mathrm{exit}})/R_0$ as function of time [Eq. (5)] for different values of the nuclear damping $k/\mu$.

Figure 6

Pre-scission kinetic energy for a few values of the reduced friction $\left(k/\mu \right)$ as function of time. The time at which scission configuration is reached for a given $k/\mu$ value is marked by a cross.

Figure 7

Number of $\beta$ decays per year for a 1 mg sample of ${}^{252}\mathrm{Cf}$ as a function of the descent time $t_{sc}$ from the exit to the scission points.