Origin of the dramatic change of fission mode in fermium isotopes investigated using Langevin equations

The fission of even-even fermium nuclides ${}^{250-260}\mathrm{Fm}$ at low excitation energy was studied using Langevin equations of three-dimensional nuclear-shape parametrization. The mass distributions of fission fragments show a dramatic change from an asymmetric shape for the lighter fermium isotopes to sharp symmetric fission for the heavier isotopes. The time evolution of the nuclear shape on the potential surface reveals that the lighter fermium isotopes showing asymmetric fission are trapped in the second minimum for a substantial length of time before overcoming the second saddle point. This behavior changes dramatically for the compact symmetric fission found in the heavier neutron-rich fermium nuclei that disintegrate immediately after overcoming the first saddle point, without feeling the second barrier, resulting in a fission time two orders of magnitude shorter.

Figure 1

Fission-fragment mass distributions of several actinide nuclides in the excitation-energy range $E^{*}=10-20\mathrm{MeV}$ (solid circles), obtained via multinucleon-transfer-induced fission [47, 51]. Results of Langevin calculations with four selected neck parameters $\epsilon$ are shown.

Figure 2

(a) Fission-fragment mass distributions, (b) total kinetic energy (TKE) distributions, and (c) mass-TKE distributions calculated for six fermium isotopes, ${}^{250-260}\mathrm{Fm}$. Literature data of spontaneous fission for FFMDs (${}^{254}\mathrm{Fm}$ [57], ${}^{256}\mathrm{Fm}$ [58], and ${}^{258}\mathrm{Fm}$ [6]) and TKE distributions (${}^{254}\mathrm{Fm}$ [57], ${}^{256}\mathrm{Fm}$ [59], and ${}^{258}\mathrm{Fm}$ [6]) are shown by black curves connected by solid circles. Solid curves in (c) correspond to the total $Q$ value in fission, $Q_{\mathrm{tot}}\left(A\right)$ (see text for more details).

Figure 3

Potential energy on the $z\text{−}\delta$ plane for (a) ${}^{254}\mathrm{Fm}$ and (b) ${}^{258}\mathrm{Fm}$, obtained at a fixed mass asymmetry $\alpha =0$. A sample shape trajectory is shown for each nucleus, selected from those leading to the dominant fission modes, i.e., elongated asymmetric fission (eAF) for ${}^{254}\mathrm{Fm}$ and compact symmetric fission (cSF) for ${}^{258}\mathrm{Fm}$. The trajectories are also shown on the $z\text{−}\alpha$ plane for (c) ${}^{254}\mathrm{Fm}$ and (d) ${}^{258}\mathrm{Fm}$ as well as the potential energy at a fixed $\delta$ value, 0.16 for ${}^{254}\mathrm{Fm}$ and $-0.08$ for ${}^{258}\mathrm{Fm}$, respectively. Nuclear shapes in each stage of fission trajectory are shown. The ground state ($•$) and the second minimum ($\blacksquare$) as well as the first (+) and the second saddle points ($\times$) are shown. The probability distributions of $\delta$ [(a) and (b)] and $\alpha$ [(c) and (d)] values at the scission point are shown for each nucleus, obtained by accumulating all the trajectories from the Langevin calculation.

Figure 4

Evolution of three shape parameters $z$ (top), $\delta$ (middle), and $\alpha$ (bottom) with time. The several points indicated by upper-case letters mean: (A for ${}^{254}\mathrm{Fm}$ and A' for ${}^{258}\mathrm{Fm}$) passing over the fist saddle point, (B for ${}^{254}\mathrm{Fm}$) trapping period in the second minimum, and (C for ${}^{254}\mathrm{Fm}$) overcoming the 2nd fission barrier. Scission points are shown by the solid diamonds. Selected nuclear shapes at each stage are shown.