Scission-point configuration within the two-center shell model shape parameterization

Within the two-center shell model parameterization we defined the optimal shape that fissioning nuclei attain just before the scission and calculated the total deformation energy (liquid-drop part plus the shell correction) as a function of the mass asymmetry and elongation at the scission point. The three minima corresponding to one mass-symmetric and two mass-asymmetric peaks in the mass distribution of fission fragments are found in the deformation energy at the scission point. The calculated deformation energy is used in a quasistatic approximation for the estimation of the total kinetic and excitation energies of fission fragments and the total number of emitted prompt neutrons. The calculated results reproduce rather well the experimental data on the position of the peaks in the mass distribution of fission fragments, and the total kinetic and excitation energies of fission fragments. The calculated value of neutron multiplicity is somewhat larger than experimental results.

Figure 1

Solutions of Eq. (3) for a few values of Lagrange multiplier ${\lambda }_2$.

Figure 2

Liquid-drop deformation energy (5) as a function of parameter $R_{12}$ for a few values of the fissility parameter $x_{\mathrm{LD}}$.

Figure 3

The mean field potential of the two-center shell model.

Figure 4

The liquid-drop energy (2) calculated with the optimal shapes (3) (solid line) and with the two-center shell model parameterization (dashed line) minimized with respect to $z_0,\epsilon$ and the fragments deformation ${\delta }_1,{\delta }_2$ for the mass asymmetry $\alpha =0$ and $\alpha =0.2$ as a function of the distance $R_{12}$ (2) between centers of mass of left and right parts of the drop for the fissility parameter $x_{\text{LD}}=0.75$. The dotted line shows the result of minimization within TCSM parameterization with the ${\delta }_1={\delta }_2$ restriction.

Figure 5

The change of the shape of the drop around the scission point.

Figure 6

The relative Coulomb and surface energies (7) as a function of the distance $R_{12}$ (2) between centers of mass of left and right parts of the system calculated for the shapes shown in Fig. 5 (solid lines) and for the two separated spheres (dashed lines).

Figure 7

The dependence of the liquid-drop energy (5) on the distance $R_{12}$ between centers of mass of left and right parts of the nucleus.

Figure 8

The liquid-drop deformation energy of ${}^{236}\text{U}$ as a function of the elongation $R_{12}$ (2) and the mass number $A_H$ of a heavy fragment.

Figure 9

The total (liquid-drop plus shell correction) deformation energy of ${}^{236}\text{U}$ as a function of the elongation $R_{12}$ (2) and the mass number $A_H$ of a heavy fragment.

Figure 10

The total deformation energy at the maximal elongation $R_{12}=R_{12}^{\left(\text{crit}\right)}$ for the asymmetry corresponding to heavy fragment mass $A_H=140$ calculated with the solutions of Eq. (20) as a function of Lagrange multipliers ${\lambda }_5$ and ${\lambda }_6$; see Eq. (20). The yellow contour lines show the value of $R_{12}^{\left(\text{crit}\right)}$.

Figure 11

The shell component of the scission point deformation energy of ${}^{236}\text{U}$ as a function of the heavy fragment mass number $A_H$ and the maximal elongation $R_{12}^{\left(\text{crit}\right)}$. The mean value (21) of $R_{12}^{\left(\text{crit}\right)}$ is shown by the thick line.

Figure 12

Total energy (liquid drop plus shell correction) for ${}^{236}\text{U}$ at the scission point as a function of the heavy fragment mass number $A_H$ and the maximal elongation $R_{12}^{\left(\text{crit}\right)}$.

Figure 13

The experimental [25] and calculated values of the mass distribution of fission fragments in the reaction ${}^{235}\text{U}+n_{\text{th}}$.

Figure 14

The total kinetic energy (24) calculated with (solid lines) and without (dashed lines) account of fragments' prescission kinetic energy. The experimental results (solid circles) are taken from Refs. [26, 27, 28, 29, 30, 31]. The $Q$ values for fission of ${}^{232}\mathrm{Th},{}^{235}\text{U},{}^{239}\mathrm{Pu}$, and ${}^{245}\mathrm{Cm}$ are shown by dotted lines.

Figure 15

The total excitation energy (29) (solid line) and the jump of liquid-drop energy during the neck rupture (8) (dashed line).

Figure 16

(a) The deformation energy of ${}^{236}\text{U}$ just before scission (solid line) and immediately after scission (dashed line) as a function of fragment mass number $A$; (b) the primary excitation energy $E_{\text{def}}^{\text{jbs}}-E_{\text{def}}^{\text{ias}}$; (c) the deformation energy at spherical shape (dashed line) and at the ground state (solid line); (d) the extra deformation energy (8) of fission fragments; and (e) the excitation energy (33) of fission fragments. All energies in this figure are given in MeV.

Figure 17

The excitation energy (33) available for prompt neutron emission (open circles) and the experimental results for neutron multiplicity [37, 38] multiplied by half of the two-neutron separation energy.

Figure 18

The calculated value of total neutron multiplicity (35) (open squares) and the experimental results [39] (solid squares). The dashed line shows the estimate (30).