Spontaneous fission of the superheavy nucleus ${}^{286}\mathrm{Fl}$

The decimal logarithm of spontaneous fission half-life of the superheavy nucleus ${}^{286}\mathrm{Fl}$ experimentally determined is ${log}_{10}{T}_f^{exp}\left(s\right)=-0.632$. We present a method to calculate the half-life based on the cranking inertia and the deformation energy, functions of two independent surface coordinates, using the best asymmetric two center shell model. Spherical shapes are assumed. In the first stage we study the statics. At a given mass asymmetry up to about $\eta =0.5$ the potential barrier has a two hump shape, but for larger $\eta$ it has only one hump. The touching point deformation energy versus mass asymmetry shows the three minima, produced by shell effects, corresponding to three decay modes: spontaneous fission, cluster decay, and $\alpha$ decay. The least action trajectory is determined in the plane $(R,\eta )$, where $R$ is the separation distance of the fission fragments and $\eta$ is the mass asymmetry. We may find a sequence of several trajectories one of which gives the least action. The parametrization with two deformation coordinates $(R,\eta )$ and the radius of the light fragment, $R_2$, exponentially or linearly decreasing with $R$ is compared with the simpler one, in which $R_2=\mathrm{constant}$ and with a linearly decreasing or linearly increasing $R_2$. The latter is closer to the reality and reminds us about the $\alpha$ or cluster preformation at the nuclear surface.

Figure 1

PES of ${}^{286}\mathrm{Fl}$ vs $(R-R_i)/(R_t-R_i)\ge 0$ and $\eta =(A_1-A_2)/(A_1+A_2)$. Y+EM (bottom), Shell + Pairing corrections (center), and total deformation energy (top).

Figure 2

Deformation energy of ${}^{286}\mathrm{Fl}$ symmetrical fission. Important characteristics of the two humped barrier: first and second minima, $E_{m1},E_{m2}$, first and second barrier height, $B_1,B_2$, and the two turning points, $x_i,x_{\mathrm{exit}}$. The difference in energy from the exit dashed line and the deepest minimum, $E_{m1}$, is the zero-point vibration energy, $E_v$.

Figure 3

Contour plot of deformation energy of ${}^{286}\mathrm{Fl}$ shown as a PES in the upper panel of Fig. 1. The first and second minima of deformation energy at every value of mass asymmetry are plotted with dashed and dotted white lines.

Figure 4

The touching point deformation energy, $E_t$ (b), its macroscopic part, $E_{\mathrm{Y}+\mathrm{E}}=E_t-Q_{\mathrm{sp}}$ (c), and the contribution of shell and pairing corrections, $Q_{\mathrm{sp}}$ (a), for SF of ${}^{286}\mathrm{Fl}$, versus mass asymmetry.

Figure 5

Cranking inertia with proton and neutron contributions. Top: fission of ${}^{286}\mathrm{Fl}$ with ${}^{132}\mathrm{Sn}$ light fragment; $R_2=\mathrm{constant}$. (a) neutrons contributions; (b) protons contributions; (c) total. Bottom: symmetrical spontaneous fission of ${}^{286}\mathrm{Fl}$; exponential decrease of $R_2$. (d) total; (e) protons contributions; (f) neutrons contributions. $(R-R_i)/(R_t-R_i)$ and $B/m$ are dimensionless quantities.

Figure 6

Cranking inertia components for symmetrical fission of ${}^{286}\mathrm{Fl}$. Two independent deformation coordinates $(R,R_2)$. $R_2$ decreases exponentially with $R$. Top: Three components and the total: (a) $B_{RR}=B_{11}$; (b) $|B_{12}|$ ($B_{12}$ is a negative quantity); (c) $B_{22}$; (d) $B$. $B_{11}=B_{RR},B_{12}=2B_{R{R}_2}\frac{d{R}_2}{dR},B_{22}=B_{R_2{R}_2}{\left(\frac{d{R}_2}{dR}\right)}^2$. Bottom: Three components and the total: (e) $B_{RR}$; (f) $B_{R2R}$; (g) $B_{R2R2}$; (h) $B$. $(R-R_i)/(R_t-R_i)$ and $B/m$ are dimensionless quantities.

Figure 7

Decimal logarithm of nuclear inertia, ${log}_{10}(B/m)$, for fission of ${}^{286}\mathrm{Fl}$. $B/m$, $x$, and $\eta$ are dimensionless quantities.

Figure 8

Three least action trajectories on the contour plot of deformation energy of ${}^{286}\mathrm{Fl}$: (a) yellow dotted-line for variable $R_2$; (b) black solid line for $R_2=\mathrm{constant}$, and (c) cyan dashed-line for linearly increasing $R_2$. $x$ and $\eta$ are dimensionless quantities.

Figure 9

Fission barriers for ten different combinations of fragments. The light fragments are the following: (a) ${}^{134}\mathrm{Sn}$; (b) ${}^{136}\mathrm{Xe}$; (c) ${}^{132}\mathrm{Sn}$; (d) ${}^{134}\mathrm{Te}$; (e) ${}^{130}\mathrm{Te}$; (f) ${}^{130}\mathrm{Sn}$; (g) ${}^{143}\mathrm{La}$; (h) ${}^{128}\mathrm{Sn}$; (i) ${}^{126}\mathrm{Sn}$; (j) ${}^{124}\mathrm{Sn}$. Spontaneous fission of ${}^{286}\mathrm{Fl}$.