Microscopic description of the odd-even effect in cold fission

The time-dependent equations of motion for the pair-breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two-fission fragment. The single-particle level scheme was calculated with the Woods–Saxon superasymmetric two-center shell model. This model provides a continuous variation of the energies from one nucleus up to two separated fragments. The dissipated energy resorts from the time-dependent pairing equations. A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones. This odd-even effect for cold fission was explained microscopically.

Figure 1

Ideal avoided-crossing regions between two adiabatic single- particle levels ${\epsilon }_i$ and ${\epsilon }_j$ characterized by the same good quantum numbers. Three possible transitions between configurations in an avoided-crossing region in the superfluid model are displayed: (a) The configuration remains unchanged after the passage through the avoided-crossing region. (b) A pair is broken. (c) A pair is created.

Figure 2

Nuclear shape parametrization. The elongation is defined as $R=z_2-z_1$. The curvature of the neck parameter is $C=S/R_3$, where $S=1$ for necked shapes in the median surface and $S=-1$ otherwise. The eccentricities of the fragments are ${\epsilon }_i=\sqrt{1-{(b_i/a_i)}^2}$ ($i=1,2$). The mass asymmetry parameter can be defined as $\eta =a_1{b}_1^2/\left(a_2{b}_2^2\right)$.

Figure 3

(a) The fission barrier is plotted with a full line. The lower single-particle excited seniority-two state for neutrons is plotted with a dot-dashed line while the first one for protons is plotted with a dashed line. The avoided-level-crossing regions are marked with arrows. The turning points $r_a$ and $r_b$ are defined for a given collective potential energy $U$. (b) The inertia in the nonadiabatic cranking approximation is plotted with a full line, the inertia within the Gaussian overlap approximation is plotted with a dashed line, and the inertia in the cranking approximation is plotted with a dot-dashed line.

Figure 4

Neutron single-particle-level scheme. The levels with spin projection $\Omega =5/2$ that give the lower-energy configuration for the unpaired fragments are plotted with a full line. The Fermi energy of the compound nucleus is displayed with a thick dashed line. Four avoided-level-crossing regions were identified for $R\approx 4.9$, 15.3, 17.1, and 19.6 fm.

Figure 5

Proton single-particle-level scheme. The levels with spin projection $\Omega =3/2$ that give the lower-energy configuration for the unpaired fragments are plotted with a full line. The Fermi energy of the compound nucleus is displayed with a thick dashed line. Four avoided-level-crossing regions were identified for $R\approx 9.1$, 14.7, 16, and 20 fm.

Figure 6

(a) Total dissipation energy after the scission $E^{*}$ as function of the internuclear velocity $v$ for the seniority-zero configuration. The neutron and proton components of the dissipated energy are plotted with a dot-dashed and a dashed line, respectively. (b) Total dissipation for the seniority-two configuration with lower single-particle excitation. The neutron and proton components are displayed with the same line types as in panel (a).

Figure 7

(a) Probability of realization of the adiabatic seniority-zero configuration at scission as function of the internuclear velocity for neutrons (full line) and protons (dot-dashed line). (b) Probability of the realization of the seniority-two configuration at scission with the lower single-particle excitation for neutrons (full line) and protons (dot-dashed line). The upper axis corresponds to the saddle-to-scission time $\tau$.

Figure 8

Full line: dependence of even even yield in arbitrary units $Y_0$ as function of the final excitation $E_x^{*}$ of the fission fragments. Dashed line: dependence of the odd-odd yields $Y_2$ as function of the excitation energy. Panel (a) corresponds to the cranking model, panel (b) is obtained with the nonadiabatic cranking approach, and panel (c) is obtained with the Gaussian overlap approximation.