Point-by-Point model calculation of the prompt neutron multiplicity distribution $\nu \left(A\right)$ for ${}^{238}\mathrm{U}(n,f)$ at incident neutron energies ranging from 1 MeV to 80 MeV

Prompt neutron multiplicity distributions ν($A$) are generally required for prompt emission correction of double energy ($2E$) measurements of fission fragments in order to determine pre-neutron fragment properties. The lack of experimental ν($A$) data especially at higher incident neutron energy imposes the use of prompt emission models to predict ν($A$). At incident neutron energies ($\mathit{En}$) where multiple fission chances are involved, the Point-by-Point (PbP) model of prompt emission is able to provide the individual ν($A$) of the compound nuclei of the main and secondary nucleus chains that are undergoing fission at a given $\mathit{En}$. The total ν($A$) are obtained by averaging these individual ν($A$) over the fission chance probabilities (expressed as total and partial fission cross-section ratios). An indirect validation of the total ν($A$) results is proposed. At high $\mathit{En}$ (above 70 MeV) the PbP results of individual ν($A$) of the first few nuclei of the main and secondary nucleus chains exhibit an almost linear increase. This shape is explained by the damping of shell effects entering the superfluid expression of the level-density parameters. They tend to approach the asymptotic values for a great part of the fragments. This fact leads to a smooth and almost linear increase of fragment excitation energy with the fragment mass number that is reflected in a smooth and almost linear behavior of individual ν($A$). The comparison of the present results with those of the GEF code reveals different shapes of ν($A$) as well as different total average neutron multiplicity as a function of the $\mathit{En}$. At high $\mathit{En}$ the PbP calculations definitely reflect the influence of the almost linear shape of individual ν($A$) of the first few nuclei of the U and Pa chains. The differences between the total ν($A$) obtained by averaging the PbP results of individual ν($A$) over fission cross-section ratios of different evaluations are insignificant.

Figure 1

ν($A$) results of PbP (black stars) in comparison with the ν($A$) results of GEF-2014 (open blue circles) and GEF-2015 (red circles) at incident energies where only the first fission chance is involved. Examples of ν($A$) given by the scaling method, used to obtain the preliminary recovered pre-neutron fragment distributions reported in Ref. [1], are given with continuous dark-yellow lines.

Figure 2

PbP results of ν($A$) at $\mathit{En}$ ranging from 1 to 5 MeV. The corresponding ${\langle \nu \rangle }_{\mathrm{tot}}$ values obtained by averaging ν($A$) over the experimental $Y$($A$) data of JRC-IRMM are indicated in the figure too.

Figure 3

Comparison of the $Y$($A$) distributions of GEF-2015 (red circles) with the experimental data of JRC-IRMM (black and gray symbols) at $\mathit{En}$ where only the first fission chance is involved.

Figure 4

${\langle \nu \rangle }_{\mathrm{tot}}$ in the $\mathit{En}$ range where only the first fission chance is involved: experimental data (different open symbols) and results obtained by averaging the prompt neutron distribution ν($A$) of PbP and GEF-2015 over the experimental $Y$($A$) data of JRC-IRMM and the $Y$($A$) results of GEF-2015 as follows: ν($A$) of PbP averaged over $Y$($A$) of JRC-IRMM with red circles; ν($A$) of GEF-2015 averaged over $Y$($A$) of GEF-2015 with blue diamonds; ν($A$) of PbP averaged over $Y$($A$) of GEF-2015 with orange up-triangles; ν($A$) of GEF-2015 averaged over $Y$($A$) of JRC-IRMM with green down-triangles.

Figure 5

Examples of ν($A$) and ${\nu }_{\mathrm{pair}}\left(A\right)$ results at incident neutron energies where multiple fission chances are involved. The individual ν($A$) and ${\nu }_{\mathrm{pair}}\left(A\right)$ are plotted with open symbols for U nuclei and solid lines for Pa nuclei. The total ν($A$) and ${\nu }_{\mathrm{pair}}\left(A\right)$ are plotted with full black squares. Examples of ${\nu }_{\mathrm{pair}}\left(A\right)$ of the Wahl systematic [31] are given with gray stars.

Figure 6

PbP results of total ν($A$) at incident neutron energies were multiple fission chances are involved.

Figure 7

Left upper part: fission cross-section ratios of the ENDF/B-VII (red lines), JENDL4 (green lines), and BRC (black lines) evaluations. RF of the first fission chance are plotted with thick solid lines, RF of the second chance are plotted with dashed lines, RF of the third chance are plotted with dashed-dotted lines, RF of the fourth chance are plotted with this solid lines, and RF of the 5th chance are plotted with thin dashed lines. Others: examples of total ν($A$) obtained by averaging the PbP results of individual ν($A$) over RF of ENDF/B-VII (red stars), JENDL4 (green triangles), and BRC (black circles).

Figure 8

Comparison of ν($A$) results of PbP (black stars) with those of GEF-2015 (full red circles) and GEF-2014 (open blue circles) at ten incident neutron energies ranging from 6 MeV to 80 MeV. Results of the scaling method are exemplified with a continuous dark-yellow line at $En=15$ MeV and 20 MeV.

Figure 9

Total average prompt neutron multiplicity ${\langle \nu \rangle }_{\mathrm{tot}}$ (full circles) as a sum of the average number of pre-fission neutrons ${\langle \nu \rangle }_{\mathrm{prefiss}}$ (open circles) and the average number of neutrons emitted by fragments ${\langle \nu \rangle }_{\mathrm{FF}}$ (stars) in comparison with experimental data (open black symbols). The results of the GEF version of 2014 are plotted in blue, and those of the GEF version of 2015 are plotted in green. Our results are plotted in red.

Figure 10

${\langle \nu \rangle }_{\mathrm{tot}}$ (full circles), ${\langle \nu \rangle }_{\mathrm{FF}}$ (stars), and ${\langle \nu \rangle }_{\mathrm{prefiss}}$ (open circles) of GEF-2015 (olive) and those reported in Tudora et al. [11] (red). ${\langle \nu \rangle }_{\mathrm{FF}}$ obtained by averaging ν($A$) of PbP over $Y$($A$) of GEF are plotted with wine stars. The corresponding ${\langle \nu \rangle }_{\mathrm{tot}}$ (as a sum of this ${\langle \nu \rangle }_{\mathrm{FF}}$ and ${\langle \nu \rangle }_{\mathrm{prefiss}}$ of BRC) are plotted with wine circles with a cross inside.

Figure 11

$Y$($A$) distributions used to compare ${\langle \nu \rangle }_{\mathrm{FF}}$ based on ν($A$) distributions of PbP, Wahl, and GEF: $Y$($A$) of GEF at $En=6\mathrm{MeV}$ with red circles, experimental $Y$($A$) data of JRC-IRMM at $En=5.8\mathrm{MeV}$ with black squares, and $Y$($A$) of GEF at $En=25\mathrm{MeV}$ with open green circles.

Figure 12

Level-density parameter of the fragments at scission for the main fissioning nucleus ${}^{239}\mathrm{U}$ in the $n+{}^{238}\mathrm{U}$ reaction at $En=80\mathrm{MeV}$ (red circles) and $En=2\mathrm{MeV}$ (blue squares). The asymptotic level-density parameter is plotted with a black dashed line.

Figure 13

Excitation energy of fully accelerated fragments of the main fissioning nucleus ${}^{239}\mathrm{U}$ in the $n+{}^{238}\mathrm{U}$ reaction at an incident neutron energy of 80 MeV.

Figure 14

A schematic representation of the systematic behavior of the ratio ${\nu }_H/{\nu }_{\mathrm{pair}}$ or $E_H^{*}/\mathrm{TXE}$ as a function of $A_H$ at different excitation energies of the fissioning nucleus: going from excitation energies corresponding to $\mathit{En}$ values below the second fission chance threshold (solid red line and wine dashed line) up to very high $\mathit{En}$ (black line) when $E^{*}\left(A\right)$ and ν($A$) exhibit a linear increase.

Figure 15

The individual ν($A$) of ${}^{239}\mathrm{U}$ at $En=80\mathrm{MeV}$ is plotted with open black squares and its linear fit with a red line. The individual ${\nu }_{\mathrm{pair}}\left(A\right)$ is plotted with open gray circles. The constant ${\nu }_{\mathrm{pair}}$ value obtained by considering the linear fit of ν($A$) is plotted with a dashed blue line.

Figure 16

Level-density parameter of the superfluid model (using the Ignatiuk parametrizations of $\tilde{a}$ and γ [17]) as a function of fragment energy for five representative fragmentations of the fissioning nucleus ${}^{239}\mathrm{U}$. The level-density parameters of heavy fragments are plotted with full red circles, and those of the complementary light fragments are plotted with open blue circles. The asymptotic level-density parameters of heavy and light fragments are plotted with red and blue lines, respectively. The values of the level-density parameters and excitation energies at scission (for the case of $En=80\mathrm{MeV}$) are given in the figure legends. The shell corrections of Möller and Nix [18] (entering the superfluid formula) are given in the lower right part.

Figure 17

Excitation energy at scission (resulting from the partition at scission under the condition of statistic equilibrium, i.e., equal nuclear temperature of complementary fragments) as a function of fragment mass for the fragments of five representative fragmentations of ${}^{239}\mathrm{U}$ at $En=80\mathrm{MeV}$. $ZA=1000Z+A$ of the fragments are indicated in the figure.