Energy partition in low energy fission

The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time-dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the other separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The fission path is obtained in the frame of the macroscopic-microscopic model. The single-particle-level schemes are obtained within the two-center Woods-Saxon shell model. It is shown that the available intrinsic dissipated energy is not shared proportionally to the masses of the two fission fragments. If the heavy fragment possesses nucleon numbers close to the magic ones, the accumulated intrinsic excitation energy is lower than that of the light fragment.

Figure 1

Nuclear shape parametrization. Two ellipsoids of different eccentricities are smoothly joined with a third surface. Two cases are obtained: (a) the curvature of the neck is positive ($S=1$) and (b) the curvature is negative ($S=-1$).

Figure 2

(right) For the left axis, the lowest-energy Woods-Saxon wave function for the two center model ${\phi }_0(z)$ as a function of $z$ for different values of the distance between the centers is displayed with a thin line. The right axis corresponds to a plot with a thick curve of a section in the middle of the neutron Woods-Saxon potential $V_0(z)$ at the same values of the distance between centers. (left) A representation of the Woods-Saxon potential $V_0$ in the cylindrical $\rho$ and $z$ coordinates is made. The distances between centers $R$ are 0, 6, 12, 18, and 21 fm from top to bottom. The configurations displayed correspond to the minimal ${}^{234}\mathrm{U}$ fission trajectory.

Figure 3

${}^{234}\mathrm{U}$ fission barrier $V$ for a final partition ${}^{102}\text{Zr}+{}^{132}\text{Te}$ determined along the minimal action trajectory. Some particular shapes related to the ground state, the extremes of the barrier, the exit point, and the scission point are inserted in the plot. The distances for the elongation $R$ that characterizes the shapes are 4.17, 7.7, 10.43, 12.64, 15.53, 17.53, and 20.2 fm.

Figure 4

Neutron level diagram for the ${}^{234}\mathrm{U}$ fission with respect to the elongation $R$.

Figure 5

Same as Fig. 4 for the proton diagram.

Figure 6

Dissipation energy $E^{*}$ calculated for the proton level scheme (top) and for the neutron one (bottom) as a function of the elongation $R$. The calculations are made within relation (15) and Eqs. (14) without imposing any condition for fixing the number of of particles. The dashed line, the dot-dashed line, and the solid line correspond to internuclear velocities $\mathit{dR}/\mathit{dt}$ of ${10}^4$, ${10}^5$, and ${10}^6$ m/s, respectively.

Figure 7

(a) The TDPE dissipated energy is plotted with a solid thick line as a function of the elongation $R$ for the neutron working space for the collective velocity $\mathit{dR}/\mathit{dt}={10}^6$ m/s. The TDPEC dissipated energy (after imposing the condition for projecting the number of particles) is plotted with a thin line. The dissipated energy obtained by replacing the ground state of the parent nucleus within the ground states of the two nascent fragments is displayed with a dashed line. (b) The number of pairs $N_{p_1}$ and $N_{p_2}$ located on the levels that belong to the first and second well are plotted with thick lines in the framework of the TDPE. The number of pairs within the TDPEC are plotted with thin lines. $N_{p_2}$ corresponds to the heavy fragment and it is always larger than $N_{p_1}$.

Figure 8

Excitation energies for ${}^{102}\mathrm{Zr}$ (dashed line) and ${}^{132}\mathrm{Te}$ (solid line) as a function of the internuclear velocity $\mathit{dR}/\mathit{dt}$. The excitations after scission are displayed for the (a) neutron and (b) proton contributions. (c) The total dissipation energies accumulated in each fragment are presented with the same line types.