Energy dependence of fission-fragment mass distributions from strongly damped shape evolution

The recently developed treatment of Brownian shape evolution is refined to take account of the gradual decrease in microscopic effects as the nuclear excitation energy is raised. We construct effective potential-energy surfaces by multiplying the shell-plus-pairing correction term by a suppression factor that depends on the local excitation energy. While this approach is equivalent to the modification of the Fermi-gas level density parameter suggested by Ignatyuk et al. [Sov. J. Nucl. Phys. 29, 450 (1979)], we adopt a more general functional form for the suppression factor, which is adjusted to measured charge yields for ${}^{234}\mathrm{U}(E^{*}\approx 11\mathrm{MeV})$. The resulting model is benchmarked by comparison with 70 measured yields.

Figure 1

Schematic illustration of the construction of the effective potential $U_E\left(\mathbf{\chi }\right)$. Panel (a): The smooth macroscopic energy $U_{\mathrm{macro}}$ (solid curve) and the undulating shell-plus-pairing correction $U_{\mathrm{sh}+\mathrm{pair}}$ (dashed curve). Panel (b): The total potential $U=U_{\mathrm{macro}}+U_{\mathrm{sh}+\mathrm{pair}}$ (solid curve); for a given shape, the excitation energy is given by $E^{*}=E-U$, where $E$ is the specified total energy (dashed line), and the effective potential is given by $U_E=U_{\mathrm{macro}}+\mathcal{S}\left(E^{*}\right)U_{\mathrm{sh}+\mathrm{pair}}$ (dashed line) and thus approaches $E_{\mathrm{macro}}$ for large $E^{*}$.

Figure 2

The relative fragment charge yield for fission of ${}^{234}\mathrm{U}$ at $E^{*}=11\mathrm{MeV}$ as calculated with a purely exponential suppression function for various values of the damping energy $E_0$ (indicated in MeV). The experimental data for ${}^{234}\mathrm{U}(\gamma ,f)$ are from Ref. [17]. [The curve labeled “$E_0=0$” was obtained for a purely macroscopic potential, $U=U_{\mathrm{macro}}$.]

Figure 3

The suppression factor given in Eq. (19) with $E_0=15$ MeV and $E_1=20\mathrm{MeV}$ together with two purely exponential forms, $\mathcal{S}\left(E^{*}\right)=\exp (-E^{*}/E_0)$, that have either $E_0=15$ MeV or $E_0=60\mathrm{MeV}$.

Figure 4

Similar to Fig. 2, but the charge yield has been calculated with either the energy-independent potential $U_{T=0}=U_{\mathrm{macro}}+U_{\mathrm{sh}+\mathrm{pair}}$ (corresponding to $E_0=\infty$ in Fig. 2) or the purely macroscopic potential $U_{\mathrm{macro}}$ ($E_0=0$ in Fig. 2) as well as with the effective potential $U_E$ (4) using the suppression function (19) with $E_0=15$ MeV and $E_1=20\mathrm{MeV}$.

Figure 5

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 1–10.

Figure 6

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 11–20.

Figure 7

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 21–30.

Figure 8

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 31–40.

Figure 9

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al. [17] for cases 41–50.

Figure 10

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 51–60.

Figure 11

Comparison of the calculated fission-fragment charge yields with the GSI data by Schmidt et al.[17] for cases 61–70.

Figure 12

Comparison of the calculated fission-fragment charge yields with those measured by Schmidt et al.[17] for the elements that have $Z=85–90$ (lower panel) and $Z=89–92$ (upper panel), using the same layout as Fig. 20 in that paper. The subplots have horizontal axes covering $Z=20–70$, with tics for every ten units, and the vertical axes cover 0%–20%; the $\beta$-stable nuclei have gray backgrounds.