Fission-fragment mass distributions from strongly damped shape evolution

Random walks on five-dimensional potential-energy surfaces were recently found to yield fission-fragment mass distributions that are in remarkable agreement with experimental data. Within the framework of the Smoluchowski equation of motion, which is appropriate for highly dissipative evolutions, we discuss the physical justification for that treatment and investigate the sensitivity of the resulting mass yields to a variety of model ingredients, including in particular the dimensionality and discretization of the shape space and the structure of the dissipation tensor. The mass yields are found to be relatively robust, suggesting that the simple random walk presents a useful calculational tool. Quantitatively refined results can be obtained by including physically plausible forms of the dissipation, which amounts to simulating the Brownian shape motion in an anisotropic medium.

Figure 1

The effect of the discretization of the shape space on the calculated fragment charge distribution is illustrated by comparing the results of the Metropolis walk introduced in Ref. [1] with those of the corresponding continuous process; the experimental data [27, 28] are also shown. Throughout, as in Ref. [1], the calculated mass yields $Y\left(A_{\mathrm{f}}\right)$ have been transformed to charge yields by a simple scaling, $Y\left(Z_{\mathrm{f}}\right)=(A_0/Z_0)Y\left(A_{\mathrm{f}}\right)$, where $Z_0$ and $A_0$ are the charge and mass numbers of the fissioning nucleus. The odd-even staggering seen in the data is due to pairing and this effect is not present in the potential-energy surfaces because existing pairing models treat the fissioning nucleus as a single system, even near scission.

Figure 2

The effect of changing the lattice spacing for the Metropolis walk: The standard lattice [2] is modified in the $Q_2$ direction by introducing two additional lattice sites between the original lattice sites, thus tripling the density of lattice sites in that direction; subsequently, the likelihood for considering the $Q_2$ direction as a candidate for the next step is enhanced by a factor of nine (solid curve), which is seen to compensate exactly for the reduced lattice spacing. Also shown are the experimental data [27, 28].

Figure 3

The charge yields for neutron-induced fission of ${}^{240}$Pu and ${}^{236,234}$U calculated with increasingly isotropic mobility tensors, as obtained by using Eq. (27) with $f=0.2,1,\infty$ (the latter is fully isotropic and has been indicated as $f\gg 1$), together with the experimental data [27, 35]. Those in the top three panels are for (n${}_{\mathrm{th}}$,f) reactions [27], while the data in the bottom panel is for ($\gamma$,f) reactions leading to $E^{*}\approx$ 8–14 MeV; they include contamination from multichance fission [28].

Figure 4

The charge yield for thorium isotopes calculated with increasingly isotropic mobility tensors, as obtained by using Eq. (27) with $f=0.2,1,\infty$ (the latter being fully isotropic), together with the experimental data obtained with ($\gamma$,f) reactions leading to $E^{*}\approx$ 8–14 MeV; they include contamination from multichance fission [28].

Figure 5

Charge yields resulting from Metropolis walks on the 5D potential-energy surfaces associated with the 3QS shape family, together with the corresponding results obtained with 3D surfaces generated by minimizing the 5D surfaces with respect to the individual fragment deformations, ${\epsilon }_{\mathrm{f}1}$ and ${\epsilon }_{\mathrm{f}2}$. Also shown are the experimental data [27, 28].

Figure 6

Similar to Fig. 5, but for fission of the isotopes ${}^{222,224,226,228}$Th; the data are from Ref. [28].

Figure 7

Three shapes relevant for fission of ${}^{222}$Th: the outer potential-energy minimum (which is reflection symmetric) where the fission isomeric state resides (top), the outer saddle which is asymmetric (middle), and the most probable fragment division which is symmetric (bottom).

Figure 8

The evolution of the mass-asymmetry distribution from the region of the second saddle ($I=20$) toward scission is illustrated by the charge distribution $P(Z_{\mathrm{f}};I)$ for increasing values of $I$ (the lattice index giving the quadrupole moment $Q_2$ of the nuclear shape), as obtained for an ensemble of 10 000 Metropolis walks.

Figure 9

The sensitivity of the charge yields to the shape dependence of the Wigner term is illustrated for ${}^{226}$Th, using either the standard potential-energy surface [2], in which the Wigner term changes gradually as the shape evolves, or a modified energy surface obtained with a Wigner term that remains constant until scission occurs; the three panels show results calculated with increasingly isotropic mobility tensors as obtained by using Eq. (27) with $f=0.2,1,\infty$; the experimental data are also shown [28].

Figure 10

The charge yield for the ${}^{234}$U($\gamma$,f) reaction extracted from Metropolis walks using either the macroscopic expression for the level density, $a_A={\tilde{a}}_A\equiv A/e_0$, with the standard value $e_0=8$ MeV and also $e_0=10$ MeV, or the microscopic expression (28); the experimental data are included [28].

Figure 11

Results of our scission model (see text) for ${}^{240}$Pu and ${}^{222,224,228}$Th for the scission neck radii $c_{\mathrm{sc}}=2.5,2.0,1.5\mathrm{fm}$, together with the corresponding experimental data [27, 28].

Figure 12

Results of the Metropolis walk [1] for the cases shown in Fig. 11, obtained with the same values of the critical neck radius $c_0$, together with the corresponding experimental data [27, 28].