Correlations Between Fission Fragment and Neutron Anisotropies in Neutron-Induced Fission

Being able to accurately and reliably model the fission process and the resulting observables is important for a variety of applications, including security, non-proliferation, energy, and fundamental science. While microscopic models for this process, beginning from fundamental nucleon-nucleon interactions, are available Schunck et al. (2014); Bulgac et al. (2016), more often these models are constructed based on a combination of macroscopic and microscopic theories Randrup and Möller (2011); Möller and Ichikawa (2015). Still, at the moment, these methods are limited; they are computationally expensive, and the range of observables that can be calculated generally ends at the post-scission fission fragment yields. These results can then be used as input for codes that calculate the prompt and delayed neutron and $\gamma$-ray observables.

These codes (e.g, $\mathtt{CGMF}$ Talou et al. (in progress); Becker et al. (2013), $\mathtt{FREYA}$ Verbeke et al. (2015, 2018), $\mathtt{HF^{3}D}/\mathtt{BeOH}$ Okumura et al. (2018), $\mathtt{FIFRELIN}$ Litaize et al. (2015), $\mathtt{GEF}$ Schmidt et al. (2016)) employ a variety of physics models describing how the excitation and kinetic energies are shared between the light and heavy fission fragments, and determine neutron and $\gamma$-ray observables on either an event-by-event basis or as averages. Event-by-event calculations allow for the determination of all observables, including correlations, as consistently as possible, from scission to prompt particle emissions, conserving energy, spin, and parity at every step of the decay. One effect that has been studied over the years, but is not always fully included in these types of codes, is the anisotropic emission of fission fragments and neutrons during the decay. There are several sources of this anisotropy: fission fragment emission with respect to the incident beam, neutron emission from the fission fragments (including whether neutrons are emitted from the fully accelerated fragments), along with the emission of scission, multi-chance, and pre-equilibrium neutrons before fission.

Although, there is no reason to expect that fission fragments should be emitted anisotropically with respect to the incoming neutron beam from the compound nucleus, many experimental studies from the 1950’s through today have shown that there can be a large degree of anisotropy - defined as the ratio of the cross section at $0^{\circ}$ to the cross section at $90^{\circ}$. The enhancement with respect to the beam axis can be as much as 30% to 60%. There have, in particular, been several measurements of the anisotropy of actinides, including ²³⁵U Vorobyev et al. (2015); Ahmad et al. (1979); Brolley et al. (1955); Simmons and Henkel (1960); Geppert-Kleinrath et al. (2019), ²³⁸U Vorobyev et al. (2015); Henkel and Brolley (1956); Brolley and Dickinson (1954); Birgersson et al. (2009); Ryzhov et al. (2005); Vivès et al. (2000); Simmons and Henkel (1960), and ²³⁹Pu Leachman and Blumberg (1965); Blumberg and Leachman (1959); Simmons and Henkel (1960).

This anisotropy was postulated, by A. Bohr Bohr (1956), to stem from the populated transition states at the fission barrier of the parent nucleus, which would account for the changing anisotropy as a function of incident neutron energy. At low incident energies, few states are populated, resulting in an anisotropic distribution of fission fragments depending on the quantum numbers of the states that are populated. As the incident energy increases, more states are available in the parent nucleus (and the angular distribution becomes more isotropic) until the second-chance fission channel opens up, at which point the $A$ nucleus is fissioning instead of the $A+1$ system, lowering the number of available states, increasing the anisotropy again. Of course, the picture is more complicated than this since the opening of the second-chance fission channel leads to a possibility of fission from either the $A+1$ or $A$ compound nucleus. A recent measurement of the anisotropy of ²³⁹Pu confirmed that the changes in anisotropy track the shape of the fission cross section Geppert-Kleinrath et al. (2019), in agreement with many previous measurements that observed the same trends, e.g. Leachman and Blumberg (1965); Blumberg and Leachman (1959); Simmons and Henkel (1960).

There have also been several studies to assess whether or not the prompt neutrons are emitted isotropically from the fission fragments, along with whether the neutrons are emitted from the fully accelerated fragments, both of which are commonly assumed in the fission fragment deexcitation codes mentioned above. Often, differences between the measured and calculated neutron angular distributions are attributed to scission neutrons, those neutrons that are emitted during the rupture of the neck of the compound nucleus. However, this interpretation does not take into account the assumptions in the model that could change the resulting calculations (such as that all prompt neutrons are emitted from the fully accelerated fragments). The percentage of neutrons that are not emitted from fully accelerated fission fragments is somewhat of a controversy Bowman et al. (1962, 1963); Kapoor et al. (1963); Skarsvåg and Bergheim (1963); Carjan and Rizea (2019).

As the incident neutron energy increases, reaction channels beyond first-chance fission open. Besides the scission neutrons that may be emitted, other pre-fission neutrons can be emitted before the compound nucleus fissions. Most often, these pre-fission neutrons are emitted from the fully equilibrated compound nucleus. However, for incident neutrons with high enough energy, $\gtrsim 12$ MeV, a neutron can be emitted with or without the compound nucleus forming. When the compound nucleus does not form, the emitted neutrons are considered to be pre-equilibrium. These pre-equilibrium neutrons have a different energy spectrum from the pre-fission neutrons coming from the compound system Gadioli and Hodgson (1992) and have a forward-peaked angular distribution, more akin to the angular distribution of an inelastically scattered neutron than the isotropic distributions of the pre-fission neutrons emitted from the compound system Feshbach et al. (1980). Pre-equilibrium neutrons have been definitively observed and their angular distributions measured for the first time for neutron-induced fission of ²³⁹Pu Kelly et al. (2019).

Recently, the fission fragment anisotropy and pre-equilibrium angular distributions have been implemented into the Monte Carlo Hauser-Feshbach code $\mathtt{CGMF}$ Talou et al. (in progress); Becker et al. (2013). In this work, we describe the implementation and study of the correlations between the fission fragment and neutron directions. This paper is organized in the following manner: in Section II, we discuss the underlying physics of the angular distributions as well as the $\mathtt{CGMF}$ code and the implementation of these sources of anisotropy; in Section III, we present our results and compared to experimental data; and finally, we conclude in Section IV.

There are several physical processes that could lead to an anisotropic distribution of the prompt neutrons emitted during the fission process. The following sections describe these processes along with the Monte Carlo Hauser-Feshbach code $\mathtt{CGMF}$, used here to calculate the decay of the fission fragments by prompt neutron and $\gamma$-ray emissions.

We now present results for the case of the neutron-induced fission on ²³⁹Pu for incident energies up to 20 MeV. Because the sources of anisotropy can become entangled and more difficult to understand when we consider them together, we first examine, in sequence, the compounding effects due to the recoil of the target, the anisotropic distribution of the fission fragments, and then the angular sampling of the pre-equilibrium neutrons. Figure 2(a) shows the compounding effect on the fission fragment angular distribution when each of these three effects is added. The red dotted line shows the effect due to the recoil of the target from momentum conservation of the interaction with the beam, which can be up to 5% at 20 MeV. To this, the fission fragment anisotropy due to the transition states is added (black solid line). Finally, the angular sampling of the pre-fission neutrons is included (blue dashed line). As expected, there is very little change in the fragment angular distribution when the pre-equilibrium neutrons are sampled from an anisotropic distribution compared to the isotropic one.

The angular anisotropy of the emitted neutrons is lower compared to the fission fragment anisotropy. The same additions as in Fig. 2(a) are shown in panel (b) but for the angular distributions of the neutrons emitted from the fission fragments, without pre-fission neutrons included. Because the neutrons are emitted isotropically in the center-of-mass frame of each fission fragment, the values for the anisotropy are reduced. Although the pre-equilibrium neutrons amount to less than 20% of the pre-fission neutrons, there is a noticeable boost to the angular distribution at forward angles when these events are included. The forward-peaked angular distribution translates to an increase in the anisotropy, as seen in the difference between the black solid and blue dashed lines in Fig. 2(c).

We can define a lower bound on the outgoing neutron energy, the neutron threshold energy, $E_{\mathrm{thres}}$. As $E_{\mathrm{thres}}$ is raised, the magnitudes of the neutron anisotropies are enhanced. In Fig. 3, we show the neutron anisotropy as a function of $E_{\mathrm{thres}}$ for several incident neutron energies, indicated by the different colored symbols. Panel (a) shows the neutron anisotropy when pre-fission neutrons are removed from the analysis, and the neutron anisotropy when they are included is shown in panel (b). In both panels, the hashed areas give the uncertainties coming from the fit to Legendre polynomials, as described in Sec. II.4.3. When pre-fission neutrons are not considered, as in Fig. 3(a), we see that the neutron anisotropy trends toward the fission fragment anisotropy as $E_{\mathrm{thres}}$ increases. Although this picture is complicated by the addition of pre-fission neutrons, there appear to be strong correlations between the neutron anisotropy and $E_{\mathrm{thres}}$ that could be used to extract information about the fission fragment anisotropy.

As shown in Fig. 2, the emission of prompt neutrons dilutes the signature of the anisotropy coming from the fission fragments, as the neutrons are emitted isotropically in the center-of-mass frame of the fission fragments. However, as seen in Fig. 3(a), as the outgoing neutron energy increases, the neutron anisotropy approaches the fission fragment anisotropy. Therefore, if neutrons with a high enough energy are measured, the fission fragment anisotropy could be extracted from the neutron anisotropy. Of course, pre-fission neutrons complicate this picture, as in Fig. 3(b), since they are emitted isotropically in the rest frame of the compound nucleus. A hint of this difference when the pre-fission neutrons are included is seen in Fig. 4 which shows the anisotropy of the fission fragments in black, prompt neutrons from the fragments in blue, and all neutrons in red, compared to the experimental data for the fission fragments, light grey circles. Around $E_{\mathrm{inc}}=5$ MeV, where the second chance fission channel opens, very slight differences can be seen between the calculations of the neutron anisotropy with and without pre-fission neutrons, but these differences are primarily within the uncertainty bands from each calculation. When the pre-equilibrium neutrons are first emitted, around $E_{\mathrm{inc}}=12$ MeV, the difference between the two neutron anisotropies becomes more visible, as some of the pre-fission neutrons are now emitted preferentially in the direction of the incident neutron beam.

In most cases, the most energetic neutrons are those that are emitted along the direction of the fission fragments, as they gain the strongest kinematic boost from the motion of the fragments. Therefore, as can be seen in the differences between panels (a), (b), and (c) in Fig. 5, as the energy threshold for the neutrons increases, the neutron anisotropy is more aligned with the anisotropy of the fission fragments. When $E_{\mathrm{thres}}=7$ MeV, Fig. 5(b), the neutron anisotropy and associated uncertainties from the fit to the Legendre polynomials covers the spread in the experimental data, with 62% of the data falling within the confidence bands for the neutron anisotropy when experimental uncertainty is taken into account. We do see that the fission fragment anisotropy extracted from $\mathtt{CGMF}$, black solid lines in Fig. 5, is somewhat higher than the experimental anisotropy, due to the kinematic boost from the incident neutron beam. The correction for the neutrons from the recoil of the target is less than 4%.

Raising the neutron threshold energy pushes the effect of the pre-fission neutrons to higher incident energies. For $E_{\mathrm{thres}}=5$ MeV, as in Fig. 5(a), pre-fission neutrons only begin to have an effect when the incident neutron energy is above 10 MeV. As the threshold energy is increased, the pre-fission neutrons have an increasingly smaller effect on the overall neutron anisotropy; the difference in the anisotropy is only seen for higher incident neutron energies. The pre-fission neutron energy spectrum is typically peaked around 2 MeV and drops off at higher outgoing neutron energies. However, around $E_{\mathrm{inc}}\sim 9-10$ MeV, the pre-fission neutrons become more numerous than the neutrons emitted from the fission fragment and dominate the anisotropy when only higher neutron energies are considered. As the neutron threshold energy continues to increase, the emission of high-energy pre-fission neutrons would not leave the residual compound with enough energy to fission. Thus, the neutrons with the highest energies are emitted from the fission fragments, in the same direction as the fragments. The trade-off here is that fewer neutrons are emitted with these highest energies, which is clear from the growing uncertainty bands between each of these energy thresholds.

We can then calculate a correlation factor between the neutron anisotropy and the fission fragment anisotropy, defined as the ratio of the neutron anisotropy (including pre-fission neutrons) to the fission fragment anisotropy at every incident and threshold neutron energy. In Fig. 6, this ratio is plotted as a function of incident neutron energy and threshold neutron energy. Thus, knowing the incident and outgoing neutron energies and measuring the neutron anisotropy, the fission fragment anisotropy could be extracted with this correlation factor. It is worth noting that the correlation factor rises above one at the highest incident energies and neutron threshold energies due to the inclusion of the pre-fission neutrons, consistent with what is shown in Fig. 5 in red.

We also performed these studies for neutron-induced fission on ^{233,234,235,238}U, ²³⁷Np, and ²⁴¹Pu, all available in $\mathtt{CGMF}$. In all cases, we found very similar results to what was shown here for ²³⁹Pu(n,f). The most notable difference was for even isotopes of uranium, which were the only even-even targets studied here. Differences in the spin of the target and the available transition states can lead to large differences in the fission fragment anisotropy between even-even and even-odd target nuclei Simmons and Henkel (1960); Mueller et al. (2012). An increase of anisotropy is seen for all targets at the opening of the higher multi-chance fission channels, due to the population of few transition states in residual fissioning nucleus with less excitation energy. The density of states in the residual nucleus depends on the odd/even character of the nucleus. For these isotopes, the values of the anisotropy of the fission fragments could reach nearly to 2, significantly larger than any of the values shown here for ²³⁹Pu(n,f). When the fragment anisotropy was this large, the neutron anisotropy did not reach those values, regardless of $E_{\mathrm{thres}}$ (within the limits of the statistics of our calculations). When the fragment anisotropy is high, increasing the energy threshold for the neutrons has only a small effect on the resulting neutron anisotropy. For this case, the neutrons seem to already be boosted in the direction of the fission fragments. Increasing the neutron threshold energy would then have a smaller effect on the resulting neutron anisotropy, as fewer neutrons are being excluded based on these cuts. Further studies on this feature should be performed.

Recently, several measurements of the angles and energies of prompt neutrons from the fission of major actinides have been made with the Chi-Nu detector array at LANL. The Chi-Nu experimental setup consists of two detector arrays used in separate measurements: a 22-detector Li-glass array to measure outgoing neutrons with energy from 0.01 – 1.59 MeV and a 54-detector liquid scintillator array to measure outgoing neutrons from approximately 0.89 – 10 MeV. The spectra from these detectors are combined to form prompt fission neutron spectrum (PFNS) measurements from 10 keV to 10 MeV for incident neutron energies from 1 – 20 MeV. More details can be found in Devlin et al. (2018); Kelly et al. (2018a, b). Prompt neutrons from neutron-induced fission on ²³⁸U were measured with the Chi-Nu liquid scintillator array to study the PFNS up to 20 MeV incident neutron energy. The Chi-Nu liquid scintillator array covers nine outgoing neutron angles from 30 – 150^∘ in the lab frame, allowing for a comparison between the $\mathtt{CGMF}$ calculations of the neutron anisotropies to those measured by Chi-Nu.

In order to exclude many sources of systematic error from the experiment, the neutron anisotropies from Chi-Nu are presented as a ratio of experimental data at 30^∘ and 90^∘, normalized to the lowest incident neutron energy range,

The same ratio can be calculated from the $\mathtt{CGMF}$ histories and compared to the measurement from Chi-Nu. This comparison is shown in Fig. 7. For the best trade-off between statistics and high incident neutron energies where the neutron anisotropy most closely tracks the fission fragment anisotropy, the sum in Eq. (7) is taken from $E_{i}=5-10$ MeV. The data from Chi-Nu are shown as the black filled stars, compared to the $\mathtt{CGMF}$ calculation of Eq. (7) as the red filled circles. The $\mathtt{CGMF}$ calculations are scaled such that $\rho(E_{\alpha})=1$ for thermal neutrons, and Chi-Nu was scaled to match the relative magnitude of the $\mathtt{CGMF}$ calculations to be able to compare the shapes of the two ratios.

The shapes of the distributions from Chi-Nu and $\mathtt{CGMF}$ are mostly consistent within the experimental uncertainties. The experimental uncertainties are large compared to those of the calculation, which are only statistical and do not account for model defects. In the calculated and measured ratios, there is a rise up to $E_{\mathrm{inc}}\sim 2$ MeV, followed by a flattening before the second-chance fission channel opens around incident energies of 6 MeV. Both ratios flatten off again, and then there is another increase in $\rho(E_{\alpha})$ where the third-chance fission channel opens, although there could be a disagreement between the incident neutron energy where this channel opens experimentally and in $\mathtt{CGMF}$. The Chi-Nu experiment was not optimized to extract the neutron anisotropy, and a different set-up could be put in place to more accurately measure this quantity. The $\mathtt{CGMF}$ calculations could then be used to convert the experimental values of $\rho(E_{\alpha})$ to fission fragment anisotropies and compare with data measured directly.

In summary, angular distribution sampling for both the fission fragment anisotropy and the pre-equilibrium neutrons were recently introduced into the Monte Carlo fission code $\mathtt{CGMF}$. Experimental values for the fission fragment anisotropy were fitted using cubic splines while the pre-equilibrium neutron angular distributions were calculated with the semi-direct model of Feshbach, Kerman, and Koonin and fitted as a function of neutron incident energy and compound excitation energy using Kalbach systematics. In itself, the inclusion of these effects is important to interpret experimental results from detector setups that measure neutron angles and energies, such as the Chi-Nu array. We also found that high-energy neutrons track the fission fragment anisotropy relatively well, and this congruence improves as the energy threshold of the neutrons is increased. This was studied for neutron-induced fission reactions on ^{233,234,235,238}U, ²³⁷Np, and ^239,241Pu.

Excluding pre-fission neutrons, for a threshold energy of 7 MeV for the outgoing neutrons, the uncertainty of the theory coming from the Legendre polynomial fit to the $\mathtt{CGMF}$ calculations is similar to the spread in the available data. When the pre-fission neutrons are included, the neutron anisotropy does not change until after the opening of third-chance fission. This difference in the neutron anisotropy when not including and including pre-fission neutrons is only seen at increasingly higher incident neutron energies if the neutron threshold energy is increased, but statistics become significantly worse. A correlation factor, such as the one calculated here with $\mathtt{CGMF}$, could be used to extract the fission fragment anisotropy from the neutron anisotropy or vise-versa depending on the incident and outgoing threshold neutron energies, even when pre-fission neutrons are included in the analysis. Here, we performed a preliminary comparison with ²³⁸U(n,f) data measured at Chi-Nu, using a ratio method which showed rather good agreement between the shapes of the $\mathtt{CGMF}$ calculations and experimental measurement.

There are still other effects that we have not taken into account in this study that could also contribute to neutron anisotropy, including the angular distribution of the neutrons in the center-of-mass frame of the fission fragments, the emission of neutrons from fission fragments that are not fully accelerated, and the emission of scission neutrons. These effects should be studied theoretically and experimentally and included in the model.