Analysis of the total kinetic energy of fission fragments with the Langevin equation

We analyzed the total kinetic energy (TKE) of fission fragments with three-dimensional Langevin calculations for a series of actinides and Fm isotopes at various excitation energies. This allowed us to establish systematic trends of TKE with $Z^2/A^{1/3}$ of the fissioning system and as a function of excitation energy. In the mass-energy distributions of fission fragments we see the contributions from the standard, super-long, and super-short (in the case of ${}^{258}\mathrm{Fm})$ fission modes. For the fission fragments mass distribution of ${}^{258}\mathrm{Fm}$ we obtained a single peak mass distribution. The decomposition of TKE into the prescission kinetic energy and Coulomb repulsion showed that decrease of TKE with growing excitation energy is accompanied by a decrease of prescission kinetic energy. It was also found that transport coefficients (friction and inertia tensors) calculated by a microscopic model and by macroscopic models give drastically different behaviors of TKE as a function of excitation energy. The results obtained with microscopic transport coefficients are much closer to experimental data than those calculated with macroscopic ones.

Figure 1

The two-center shell model shape profile.

Figure 2

The nuclear potential (2) of the two-center shell model at $\rho =0$.

Figure 3

Fission fragment yield of ${}^{236}\mathrm{U}$. Top: at $E_x=6.545$ MeV; bottom: at $E_x=20$ MeV. The black histogram shows our calculation with pre-neutron (pre-$n)$ emission for $E_x=6.545$ MeV and flat-neutron (flat-$n)$ emission for $E_x=20$ MeV. The green histogram shows our calculation with neutron emission calculated from [41] (top) and neutron emission calculated using gef [42] (bottom). Our calculations (denoted by red circles) are compared with evaluated post-neutron (post-$n)$ distributions stored in jendl. We also compare them with (a) our previous data [3] for single-chance, flat-neutron (flat-$n)$ emission calculations using microscopic transport coefficients.

Figure 4

Fission fragment yield for ${}^{258}\mathrm{Fm}$ at $E_x=3.34$ MeV in comparison with experiment [45].

Figure 5

Fission fragments mass systematics (solid line) obtained by fitting the average mass for light and heavy fission fragments (filled circles). These mass averages are calculated from the fission fragment mass yield of JENDL 14 MeV incident neutron data. We also plot ${}^{256,258}\mathrm{Fm}$ experimental results from [44] (open squares) and [45] (filled squares). We compare them with (a) our previous data [46, 47] (blue inverted triangles) for compound nucleus at $E_x=20$ MeV.

Figure 6

The minimized-in-$\delta$ potential energy surface of ${}^{236}\mathrm{U}$ at $T=0$ calculated within TCSM. The white line shows the position of zero neck radius for $\delta =0$ (solid) and $\delta =0.1$ (dash).

Figure 7

The calculated distribution of fission events of ${}^{236}\mathrm{U}$ in kinetic energy and fragment mass at $E_x=6.545$ MeV. The red curve denotes the kinematically allowed maximal value of TKE, namely, $Q+E_x$.

Figure 8

The average value of TKE (top) and average elongation $〈R_{12}〉$ of fissioning nucleus at the scission point for $E_x=6.545$ MeV (red) and $E_x=20$ MeV (blue). The experimental data are taken from (a) [49] $\left(E_n\approx 0\right)$ and (b) [50] $(E_n=15.5$ MeV), correspondingly.

Figure 9

The minimized-in-$\delta$ potential energy surface of ${}^{258}\mathrm{Fm}$ at $T=0$ calculated within TCSM. The white line shows the position of zero neck radius for $\delta =-0.2$ (dash) and $\delta =0.15$ (solid).

Figure 10

The calculated mass-energy distribution of fission events of ${}^{258}\mathrm{Fm}$ at $E_x=3.34$ MeV. The red curve denotes kinematically allowed maximum values of TKE, namely, $Q+E_x$.

Figure 11

Systematical trends of average TKE of fission fragments as a function of $Z^2/A^{1/3}$ of the fissioning system. We compare our present results with (a) [52] and (b) [54] linear least squares of the TKE systematics as a function of the fissioning system. Included are (c) evaluated data from JENDL [43] and experimental spontaneous TKE from (d) [55] and (e) [56]. We compare also with several other nuclides using our previous results at $E_x=20$ MeV [46, 47] for (f) microscopic and (g) macroscopic transport coefficients.

Figure 12

The TKE as a function of neutron energy $E_n$ calculated for compound nucleus ${}^{236}\mathrm{U}$. The red lines are the present results for pre-neutron (pre-$n)$ emission, single chance fission TKE, ${\mathrm{KE}}_{\mathrm{pre}}$, and ${\mathrm{KE}}_{\mathrm{Coul}}$. We compare ${}^{236}\mathrm{U}$ with (a) our previous data [46] calculations with macroscopic transport coefficients without using the effective temperature description in Eq. (10), (b) experimental data from [50], and (c) experimental data from [58].

Figure 13

The TKE as a function of neutron energy $E_n$ calculated for three compound nuclei: ${}^{239}\mathrm{U},{}^{240}\mathrm{Pu}$, and ${}^{232}\mathrm{Pa}$ on the top, middle, and bottom panels respectively. The red lines are the present results for pre-neutron emission (pre-$n)$, single chance fission for the corresponding compound nucleus TKE. Our calculated TKE's for ${}^{240}\mathrm{Pu}$ are compared with experimental data from (a) [61], (b) [62], and (c) [63]. We compare the ${}^{239}\mathrm{U}$ fission TKE from post-neutron (post-$n)$ and pre-neutron emission with (d) experimental data from [60]. ${}^{232}\mathrm{Pa}$ TKE's are compared with experimental data for thermal neutrons in (e)[64].

Figure 14

The dependence of average ${\mathrm{KE}}_{\mathrm{pre}}$ on the fragment mass. ${\mathrm{KE}}_{\mathrm{pre}}$ was calculated for $E_x=6.545$ MeV (black), $E_x=20$ MeV (blue), and $E_x=30$ MeV (red).