Angular momentum effects in fission

Background: The role of angular momentum in fission has long been discussed but the observable effects are difficult to quantify.

Purpose: We discuss a variety of effects associated with angular momentum in fission and present quantitative illustrations.

Methods: We employ the fission simulation model freya, which is well suited for this purpose because it obeys all conservation laws, including linear and angular momentum conservation at each step of the process. We first discuss the implementation of angular momentum in freya and then assess particular observables, including various correlated observables. We also study potential effects of neutron-induced fission of the low-lying isomeric state of ${}^{235}\mathrm{U}$ relative to the ground state.

Results: The fluctuations inherent in the fission process ensure that the spin of the initial compound nucleus has only a small influence on the fragment spins, which are therefore nearly uncorrelated. There is a marked correlation between the spin magnitude of the fission fragments and the photon multiplicity. We also consider the dynamical anisotropy caused by the rotation of an evaporating fragment and study especially the distribution of the projected neutron-neutron opening angles, showing that while it is dominated by the effect of the evaporation recoils, it is possible to extract the signal of the dynamical anisotropy by means of a Fourier decomposition. Finally, we note that the use of an isomeric target, ${}^{235\mathrm{m}}\mathrm{U}$($n_{\mathrm{th}}$,f), may enhance the symmetric yields and can thus result in higher neutron multiplicities for low total fragment kinetic energies.

Conclusions: While the initial angular momentum of the fissioning nucleus tends to have little effect on the observables, those of the produced fragments influence the emitted neutrons and photons in a significant and correlated manner which may be exploited experimentally to elucidate the fission process.

Figure 1

The distribution of the opening angle ${\varphi }_{\mathrm{LH}}$ between the spins of the two primary fission fragments from ${}^{235}\mathrm{U}(n$,f) for two neutron energies: thermal neutrons and $E_n=20$ MeV. Also shown is the lowest-order Fourier approximation, $P\left({\varphi }_{\mathrm{LH}}\right)=1+f_{2}cos{\varphi }_{\mathrm{LH}}$.

Figure 2

The angular distribution of the evaporated neutrons relative to the spin direction of the emitting fragment, $d{N}_n/d{\mathrm{\Omega }}_{nS}$, averaged over the evaporation chains for the entire fragment yield distribution from ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f). In addition to the freya simulation results (filled red circles), the lowest two Legendre approximations are shown.

Figure 3

The mean number of photons emitted from ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) (filled blue circles) as a function of the sum of the two primary fragment spin magnitudes, $S=S_L+S_H$. The associated event-by-event photon multiplicity dispersion is indicated by the vertical bars. Also shown are the mean multiplicities for $E_n=20$ MeV (open red squares).

Figure 4

A contour plot of the velocity distribution of neutrons evaporated following ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) is shown. The $z$ direction is chosen to be along the motion of the light primary fragment in each event. Red circles have radii corresponding to constant neutron kinetic energies of 1 MeV (solid) and 2 MeV (dashed).

Figure 5

The laboratory angular distribution of the evaporated neutrons from ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) relative to the direction of the light primary fragment, $d{N}_n/dcos{\theta }_n$, is shown for both the standard freya scenario when the neutrons acquire a rotational boost (curves) and a modified scenario when this boost is omitted (symbols). In addition to considering neutrons of all energies (black), the figure also shows the result of including either only neutrons with energies above 2 MeV or below 1 MeV.

Figure 6

Two-neutron angular correlations for ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f), gating on either the total spin magnitude $S=S_L+S_H$ (a) or the total photon multiplicity $N_{\gamma }$ (b). The angular correlation functions $P\left({\varphi }_{nn}\right)$ for either “soft” neutrons ($E_n<2$ MeV) or “hard” neutrons ($E_n>2$ MeV) are presented in (a) for events with either a low or a high spin magnitude ($S\le 7\hslash$ or $S\ge 8\hslash$, respectively) and in (b) for events with either a low or a high total photon multiplicity ($N_{\gamma }\le 7$ or $N_{\gamma }\ge 8$, respectively).

Figure 7

The distribution of the neutron-neutron opening angle in the plane transverse to the direction of the light product nucleus, ${\varphi }_{nnL}$, is shown for ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) as obtained from 40 million freya events (filled circles). The extracted first and second Fourier components are also shown.

Figure 8

The neutron multiplicity $\nu$ as a function of the TKE for ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) in two target scenarios: In the first scenario (filled blue squares), the target is the ground-state ${}^{235\mathrm{gs}}\mathrm{U}$ and the spin of the fissioning compound nucleus ${}^{236}\mathrm{U}$ is $S_0=4\hslash$. In the alternative scenario (open red circles), the target is the isomeric-state ${}^{235\mathrm{m}}\mathrm{U}$ and $S_0=0$. In both cases, the symbols indicate the average multiplicities at the particular TKE. The dispersion of the multiplicity distribution at a given TKE is indicated by the error bars (so these are not uncertainties on the calculated results, which are based on $4\times {10}^6$ freya events). The experimental data (filled circles) are from Göök et al. [30].

Figure 9

The neutron multiplicity distribution $P\left(\nu \right)$ for the two cases of ${}^{235}\mathrm{U}(n_{\mathrm{th}}$,f) shown in Fig. 8: ${}^{235\mathrm{gs}}\mathrm{U}$ (filled blue squares) and ${}^{235\mathrm{m}}\mathrm{U}$ (open red circles), shown on a logarithmic scale in order to better bring out the enhancement of high-multiplicity events.