Unified description of fission in fusion and spallation reactions

We present a statistical-model description of fission, in the framework of compound-nucleus decay, which is found to simultaneously reproduce data from both heavy-ion-induced fusion reactions and proton-induced spallation reactions at around $1$ GeV. For the spallation reactions, the initial compound-nucleus population is predicted by the Liège intranuclear cascade model. We are able to reproduce experimental fission probabilities and fission-fragment mass distributions in both reactions types with the same parameter sets. However, no unique parameter set was obtained for the fission probability. The introduction of fission transients can be offset by an increase of the ratio of level-density parameters for the saddle-point and ground-state configurations. Changes to the finite-range fission barriers could be offset by a scaling of the Bohr-Wheeler decay width as predicted by Kramers. The parameter sets presented allow accurate prediction of fission probabilities for excitation energies up to $300$ MeV and spins up to $60$ $\hslash$.

Figure 1

Comparison of the distributions of excitation energy and spin populated in the ${}^{19}\mathrm{F}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{181}$$\mathrm{Ta}\to {}^{200}$Pb fusion reaction for $E^{*}=90$ and 150 MeV (horizontal lines) with the INCL4.5 prediction for the 1-GeV $p+{}^{208}$Pb spallation reaction (contours, logarithmically spaced). Inset: Distribution of fissioning remnants for the spallation reaction (same contour levels). The dashed line is the macroscopic yrast line from Sierk [25].

Figure 2

Comparison of GEMINI$++$ predictions to the experimental evaporation-residue (ER) and fission (fis) excitation functions for the ${}^{19}\mathrm{F}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{181}$Ta reaction.

Figure 3

Comparison of experimental and calculated residual mass distributions for the 1-GeV $p+{}^{208}$Pb reaction. Predictions of the INCL4.5-GEMINI$++$ code are shown for different adjustments of the fission width. Experimental data are from Ref. [43].

Figure 4

Comparison of experimental and calculated fission (fis) and evaporation-residue (ER) excitation functions for the indicated reactions. Solid lines: Bohr-Wheeler fission width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.036$, no fission delay. Dashed lines: Lestone fission width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.057$, 1-zs fission delay. Dotted lines: Bohr-Wheeler fission width, energy-dependent effective $a_{\mathrm{f}}/a_{\mathrm{n}}$ ratio with $r=1.0747$.

Figure 5

Same as Fig. 4.

Figure 6

Same as Fig. 4.

Figure 7

Same as Fig. 4.

Figure 8

Residue-mass distribution for the $p+{}^{197}$Au reaction at 1 GeV. Predictions of the INCL4.5-GEMINI$++$ code are shown for different adjustments of the fission width. Experimental data are from Ref. [42].

Figure 9

Same as Fig. 8, but for $p+{}^{208}$Pb at 500 MeV. Experimental data from Refs. [45] and [51]. Experimental fission cross sections have been multiplied by a factor of 146/232 (see text for details).

Figure 10

Same as Fig. 8, but for $p+{}^{238}$U at 1 GeV. Experimental data are from Refs. [44], [52], and [53].

Figure 11

Excitation curve for the fission cross section in $p+{}^{181}$Ta. Predictions of the INCL4.5-GEMINI$++$ code are shown for different adjustments of the fission width. Experimental data are from Ref. [50].

Figure 12

Comparison of experimental (data points) and simulated (curves) distributions of the primary fission-fragment masses for the ${}^{198}\mathrm{Pb}$ compound nucleus formed in the ${}^{16}\mathrm{O}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{182}$W reaction at the two indicated excitation energies. The total fission cross section has been calculated using the Bohr-Wheeler formalism, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.036$, without any fission delay. The dotted curve was obtained using the Moretto formalism with Sierk’s conditional barriers to define the mass distributions. Solid and dashed curves were obtained using the Rusanov systematics with the saddle- and scission-point temperatures, respectively.

Figure 13

Comparison of experimental (data points) and simulated (curves) distributions of the primary fission-fragment masses for the ${}^{216}\mathrm{Ra}$ compound nucleus at $E^{*}~60$ MeV formed in the ${}^{12}\mathrm{C}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{204}$Pb, ${}^{19}\mathrm{F}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{197}$Au, and ${}^{30}\mathrm{Si}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{186}$W reactions. The total fission cross section has been calculated using the Bohr-Wheeler formalism, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.036$, without any fission delay. To aid viewing, the data and curves have been scaled by the indicated amounts.

Figure 14

Comparison of experimental (data points) and simulated (curves) distributions of the fission-fragment masses for the 1-GeV $p+\mathrm{Pb}$ spallation reaction. The dotted (blue) curve was obtained using the Moretto formalism with Sierk’s conditional barriers to define the mass distributions. The solid (black) and dashed (red) curves were obtained using the Rusanov systematics with the saddle- and scission-point temperatures, respectively.

Figure 15

Comparison of asymmetry dependences of conditional barriers for ${}^{149}\mathrm{Tb}$ and ${}^{194}\mathrm{Hg}$ nuclei at $J=0$. Solid curves are predictions from Sierk’s finite-range calculations. Dashed curves are parabolic functions with curvatures taken from the Rusanov systematics and with symmetric fission barriers taken from the finite-range calculations. The mass-asymmetry coordinate is defined in terms of $A_1$ and $A_2$, the two masses following binary division.

Figure 16

Same as Fig. 3, but for the Bohr-Wheeler fission width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.036,$ and three values of fission delay.

Figure 17

Same as Fig. 3, but for the Lestone fission width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.057$, and three values of fission delay.

Figure 18

Comparison of GEMINI$++$ predictions using (a) the Bohr-Wheeler and (b) the Lestone fission formalism for the experimental evaporation-residue and fission excitation functions for the ${}^{19}\mathrm{F}\mathrm{\hspace{0.5em}}+\mathrm{\hspace{0.5em}}{}^{181}$Ta reaction. Curves are labeled by the $t_{\mathrm{delay}}$ and $a_{\mathrm{f}}/a_{\mathrm{n}}$ values obtained from fitting the fission cross section for the 1-GeV $p+\mathrm{Pb}$ reaction.

Figure 19

Fission probability as a function of the excitation energy of remnants in 2.5-GeV $p+\mathrm{Au}$, Bi, and U reactions. Solid lines: Bohr-Wheeler width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.036$. Dashed lines: Lestone, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.057$, $t_{\mathrm{delay}}=1$ zs. Dotted line: Bohr-Wheeler width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.02$. Dash-dotted line: Bohr-Wheeler width, $a_{\mathrm{f}}/a_{\mathrm{n}}=1.00$.

Figure 20

Same as Fig. 19, for a Bohr-Wheeler width (solid lines) or a Lestone width with a 1-zs fission delay (dashed lines) and an energy-dependent effective $a_{\mathrm{f}}/a_{\mathrm{n}}$ ratio, Eq. (4).