Fission-fragment charge yields: Variation of odd-even staggering with element number, energy, and charge asymmetry

Background: Fission-fragment charge-yield distributions exhibit a pronounced odd-even staggering. For actinide nuclei the staggering decreases with increasing proton number and with increasing excitation energy. In our calculations of fission yields [Phys. Rev. Lett. 106, 132503 (2011)] we obtained charge-yield distributions for a number of actinide nuclides by means of random walks on tabulated five-dimensional potential-energy surfaces. However, because the potential-energy model treats the system as a single, compound system during all stages of the fission process, in which individual fragment properties do not appear, no odd-even staggering appeared in the calculated yield curves.

Purpose: We have recently become aware that in the experimental data displayed in Fig. 1 in the above paper, there is a remarkable similarity in the odd-even staggering in fission of ${}^{240}\mathrm{Pu}$ at thermal neutron energy and fission of ${}^{234}\mathrm{U}$ in photon-induced fission at around 11 MeV. We discuss how this similarity and how the variation in the magnitude of the odd-even staggering for three Th isotopes with charge asymmetry and isotope can be qualitatively understood based on strongly damped shape evolution on our calculated five-dimensional potential-energy surfaces.

Methods: We conduct random walks on our tabulated five-dimensional potential-energy surfaces and study the difference between the total compound-nucleus energy and the potential energy for the different systems from saddle to scission. Under the strong-damping assumption this difference is the internal excitation energy. We also determine this quantity for different charge splits, symmetric and asymmetric.

Results: We find that the magnitude of the odd-even staggering in the charge distribution in the several cases studied here correlates well, inversely, with the excitation energy above the potential-energy surface in the postsaddle region.

Conclusions: Because the observed magnitude of the odd-even staggering correlates well with excitation energy over the region where the individual character of the fission fragments emerges, the Brownian shape-motion method can be expected to reproduce this feature, provided a potential-energy model is developed that accounts for how the nascent fragment properties are expressed in the calculated potential-energy surfaces.

Figure 1

Measured charge yields in ($\gamma$,f) reactions for ${}^{228}\mathrm{Th}$ (a), ${}^{226}\mathrm{Th}$ (b), and ${}^{222}\mathrm{Th}$ (c), from [6], leading to $E^{*}\approx 8$–14 MeV; they include contamination from fission after 1n ($\approx 15\%$) and 2n ($\approx 5\%$) emission. The average excitation energies $\langle E_{\mathrm{ex}}\left(\mathrm{a}\right)\rangle$ and $\langle E_{\mathrm{ex}}\left(\mathrm{s}\right)\rangle$ at the end of the asymmetric and symmetric tracks are also given, see text for definitions.

Figure 2

Measured fission-fragment charge yields. The data in (a) and (c) are for $(n_{\mathrm{th}},\mathrm{f})$ reactions leading to $E^{*}=6.54$ MeV and $E^{*}=6.84$ MeV for ${}^{240}\mathrm{Pu}$ and ${}^{234}\mathrm{U}$, respectively, [7] in which some original data are from [8], while the data in (b) are for ($\gamma$,f) reactions leading to $E^{*}\approx 8$–14 MeV; they include contamination from fission after 1n ($\approx 15\%$) and 2n ($\approx 5\%$) emission [6]. The average excitation energies $\langle E_{\mathrm{ex}}\left(\mathrm{a}\right)\rangle$ at the end of the asymmetric tracks are also given, see text for definitions.

Figure 3

Calculated potential energy $U$ along the optimum path to scission for ${}^{234}\mathrm{U}$ (top panel) and ${}^{240}\mathrm{Pu}$ (bottom panel). Plotted for ${}^{234}\mathrm{U}$ are the total energies of the compound nuclei following capture of a thermal neutron or an 11-MeV photon (these total energies are conserved during the evolution so they appear as horizontal lines as functions of deformation), while for ${}^{240}\mathrm{Pu}$ only the total energy for thermal fission is plotted. Because all energies are given relative to the macroscopic energy of the spherical shape, the excitation energy at a specific deformation is the difference between the total energy and the potential-energy curve, $E_{\mathrm{ex}}=E_{\mathrm{tot}}-U$. The energies tabulated in Table 1 are based on ground-state energies that are the sum of the potential energy at the ground-state minimum and a zero-point energy, which is around 0.9 MeV for the nuclei studied here. The ground-state energy has been indicated by a short horizontal line about 0.9 MeV above the potential minimum.