Dynamical interpretation of average fission-fragment kinetic energy systematics and nuclear scission

A dynamical interpretation of the well-known systematics for average total kinetic energy of fission fragments $\langle E_K\rangle$ over a wide range of the Coulomb parameter ($600<Z^2/A^{1/3}<2200$) is given. Different scission criteria traditionally employed in fission theory—at zero neck radius and at finite neck radius—have been applied in dynamical calculations. Both have resulted in a fairly good description of the dependence of $\langle E_K\rangle$ on the Coulomb parameter. The results of dynamical calculations of $\langle E_K\rangle$ within three-dimensional Langevin dynamics show that the mean distance between the centers of mass of nascent fragments at the scission configuration increases linearly with the parameter $Z^2/A^{1/3}$. This distance changes approximately from $2.35R_0$ for ${}^{119}\mathrm{Xe}$ to $2.6R_0$ for ${}^{256}\mathrm{Fm}$. In spite of this increase in mean distance between future fragments at scission, the linear dependence of $\langle E_K\rangle$ on the parameter $Z^2/A^{1/3}$ remains approximately valid over a wide range of the Coulomb parameter $Z^2/A^{1/3}$.

Figure 1

The mean trajectories for the ${}^{256}\mathrm{Fm},{}^{224}\mathrm{Th}$, and ${}^{142}\mathrm{Ce}$ fissioning nuclei in the collective coordinates $q_1$ and $q_2$ ($q_3=0$). The dashed curve corresponds approximately to the bottom of the LDM fission valley and lies at $h=\alpha =0$. The examples of nuclear shapes are presented for the appropriate values of the parameters $q_1$ and $q_2$ ($q_3=0$).

Figure 2

The distance between mass centers of nascent fragments $\langle D\rangle$ of the mean scission deformations for the different scission criteria. The circles correspond to the zero-neck scission criterion. The triangles correspond to the scission criterion at finite neck radius $R_N=0.3R_0$. The squares correspond to the hydrodynamic scission criterion [6, 7]. The solid lines connect symbols that correspond to the calculations along the mean trajectories. The dotted lines correspond to the calculations along the bottom of the LDM fission valley ($h=\alpha =0$). The value $\langle D\rangle =2.62$ is obtained for the zero-neck-radius scission criterion, $\langle D\rangle =2.33$ for scission criterion $R_N=0.3R_0$, and $\langle D\rangle =2.22$ for hydrodynamic scission. The dashed line corresponds to the calculations of the critical deformation $D^{\mathrm{crit}}$ performed in Ref. 1.

Figure 3

Plot of $\langle E_K\rangle ,\langle E_{\mathit{ps}}\rangle$, and $\langle V_n\rangle$ values as a function of the parameter $Z^2/A^{1/3}$ for different fissioning nuclei. The solid line represents the $\langle E_K\rangle$ systematics proposed in Refs. 8, 49, Eq. (21); the dashed line represents Viola's systematics [48], Eq. (20). The filled squares are experimental data of $\langle E_K\rangle$. The open triangles are results of $\langle E_K\rangle$ calculations with scission criterion at the finite neck radius $R_N=0.3R_0$. The open circles are results of calculations with the zero-neck scission criterion. Open squares are results of $\langle E_{\mathit{ps}}\rangle$ calculations and inverted triangles are results of $\langle V_n\rangle$ calculations with scission criterion at the finite neck radius $R_N=0.3R_0$.

Figure 4

The sum $\langle E_{\mathit{ps}}\rangle +\langle V_n\rangle$ calculated for different scission criteria as a function of the parameter $Z^2/A^{1/3}$. Filled circles are results of calculations with the zero-neck scission criterion. Filled triangles are results of calculations with scission criterion at the finite neck radius $R_N=0.3R_0$. Filled squares are results of calculations with the hydrodynamic scission criterion.

Figure 5

The mean scission deformation parameters $\langle {\beta }_{E_K}\rangle$ and $\langle {\beta }_D\rangle$ vs the parameter $Z^2/A^{1/3}$. The filled circles, triangles, and squares are $\langle {\beta }_D\rangle$ values obtained from calculations with the zero neck radius $R_N=0$, finite neck radius $R_N=0.3R_0$, and hydrodynamic scission criterion, respectively. The dashed lines are obtained by the least-squares method on the basis of circles, triangles, and squares. The dotted curve is the $\langle {\beta }_{E_K}\rangle$ dependence given by Eq. (25) with the coefficients $C_1$ and $C_2$ from the Viola's systematics [48] [Eq. (20)]. The dashed-dotted curve is the $\langle {\beta }_{E_K}\rangle$ dependence given by Eq. (25) with the coefficients $C_1$ and $C_2$ from the $\langle E_K\rangle$ systematics proposed in Refs. 8, 49 [Eq. (21)].

Figure 6

Comparison of experimental average fission-fragment kinetic energies and the calculated values of the Coulomb interaction energies at scission as a function of the parameter $Z^2/A^{1/3}$. The solid line is Viola's systematics [48] for $\langle E_K\rangle$. The dotted curve is the calculated dependence of $\langle V_C\rangle$ given by Eq. (27) with the coefficients obtained from the criterion at finite neck radius $R_N=0.3R_0$.