Angular correlation between the fission fragment intrinsic spins

The generation of fission fragment (FF) spins and of their relative orbital angular momentum has been debated for more than six decades, and no consensus has been achieved so far. The interpretation of recent experimental results of Wilson et al., Nature (London) 590, 566 (2021) has been challenged by several recent theoretical studies, which are not in agreement with one another. According to the interpretation by Wilson et al., Nature (London) 590, 566 (2021), the FF spins emerge very long after scission occurrs. Randrup and Vogt Phys. Rev. Lett. 127, 062502 (2021), while agreeing that the FF spins are uncorrelated, conclude on the basis of a phenomenological model that these spins are uncorrelated already before scission. Bulgac et al., Phys. Rev. Lett. 128, 022501 (2022) in a fully microscopic study demonstrate that the primordial FF spin final values are defined before the emission of prompt neutrons and statistical gammas and are strongly correlated with a relative angle between spins close to $2\pi /3$, a result in full agreement with the present independent analysis. The prompt neutrons and statistical gammas carry a significant amount of angular momentum according to the study of Stetcu et al., Phys. Rev. Lett. 127, 222502 (2021), which can lead to a decorrelation of the FF bandhead spins of the yrast lines measured by Wilson et al., Nature (London) 590, 566 (2021), and which also provides arguments why these measured spins are so different from the primordial FF spins evaluated by Bulgac et al., Phys. Rev. Lett. 128, 022501 (2022). Here, I show that the unexpected character of the angular correlation between the primordial FF intrinsic spins, recently evaluated by Bulgac et al., Phys. Rev. Lett. 128, 022501 (2022), which favors FF intrinsic spins pointing predominantly in opposite directions, can be understood by using simple general phase space arguments. The observation by Wilson et al., Nature (London) 590, 566 (2021) that the FF spins are uncorrelated follows from both the results of the microscopic calculation of Bulgac et al., Phys. Rev. Lett. 128, 022501 (2022) and the present analysis of the full correlated probability distribution of the FF spins together with the relative orbital angular momentum of the primordial FFs. These arguments may apply also to heavy-ion collisions and, since there is no use of specifics of the particle interactions, the present results might apply to atomic and molecular systems as well.

Figure 1

In this figure the bullets fill out a three-face pyramid with the apex at (0, 0, 0). The green bullets show the triplets $(S^L,S^H,\mathrm{\Lambda })$ for which $cos{\varphi }^{LH}<0$, the blue bullets show those for $cos{\varphi }^{LH}=0$, and the red bullets are for $cos{\varphi }^{LH}>0$, when $S_{\text{max}}=3$. The green bullets correspond to ${\varphi }^{LH}>\pi /2$ and the red bullets correspond to ${\varphi }^{LH}<\pi /2$. The ratio of red to green bullets for any $S_{\text{max}}$ value is always close to 0.5, which means that the number of configurations in which the FF intrinsic spins point in opposite direction is dominant. The black lines are the intersections of the pairs of planes $(S^L=S^H,\mathrm{\Lambda }=0)$, $(S^L=\mathrm{\Lambda },S^H=0)$, and they cross at (0, 0, 0). The points on each face are joined by thin lines.

Figure 2

The uniform angular momentum distribution (8) was obtained with an $S_{\text{max}}=30$ [see Eq. (8)] and is represented here as a histogram with 20 bins. The statistical distribution was obtained with ${\sigma }^L=108.2,{\sigma }^H=44.8$, and ${\sigma }^{\mathrm{\Lambda }}=161.3$, which closely reproduce the corresponding distributions obtained in Refs. [20, 22] for ${}^{240}\mathrm{Pu}$. The freya distribution was obtained with typical parameters from [17, 18]. The $\hslash \to 0$ classical limit was obtained by taking $S_{\text{max}}\to \infty$ for the uniform distribution. The results obtained with a modified freya prescription $P_{\text{model}\text{F}}$ introduced here are shown with magenta circles. The average and variance (in parentheses) of ${\varphi }^{LH}/\pi$ for each distribution are displayed next to each label. The lines connecting the markers are only for guiding the eye. For visual purposes the distribution displayed here differs by a factor from Eq. (16), $P\left({\varphi }^{LH}\right)=\pi p\left({\varphi }^{LH}\right).$ The freya results from Ref. [18] and reproduced here are markedly different from the distributions derived here and from the microscopically evaluated distribution $p\left({\varphi }^{LH}\right)$ in Ref. [22]. In the space $(S^L,S^H,\mathrm{\Lambda })$ the allowed configurations with ${\varphi }^{LH}=\pi$ correspond to the lateral lower faces of the pyramid in Fig. 1 [see Eqs. (6b) and (6c)], when $\mathrm{\Lambda }=|S^L-S^H|$. The allowed configurations with ${\varphi }^{LH}=0$ correspond to upper face of the pyramid $\mathrm{\Lambda }=S^L+S^H$ in Fig. 1 [see Eq. (6a)]. All faces have dimension 2, as opposed to the dimension 3 of the space of the rest of allowed configurations $(S^L,S^H,\mathrm{\Lambda })$, which makes obvious the presence of the strong suppression at ${\varphi }^{LH}=0,\pi$. This clear discrepancy between the predictions made in Refs. [17, 18] and reproduced here and the predictions made in the present work and in Ref. [22] remains to be resolved in future experiments, see also Sobotka's talk at the Workshop of Fission Fragment Angular Momenta [41].

Figure 3

The distributions $P_1\left(S^{L,H}\right)$ (9a) along with the distributions $\tilde{P}\left(S^{L,H}\right)$ and $P\left(\mathrm{\Lambda }\right)$ [see Eqs. (28) and (29)], obtained from the distribution ${\overline{P}}_3(S^L,S^H,\mathrm{\Lambda })$ (27) by summing over the rest of angular momenta, are shown for ${}^{252}\mathrm{Cf}$. The FF intrinsic spin distributions $P_1\left(S^F\right)\approx \tilde{P}\left(S^F\right)$ and the relative orbital angular momentum distribution $P\left(\mathrm{\Lambda }\right)$ obtained from Eq. (29) are very similar in shape to the statistical distribution $P_1\left(\mathrm{\Lambda }\right)$ from Eq. (9b).

Figure 4

The distribution $P\left({\varphi }^{LH}\right)$ (normalized as in Fig. 2) obtained using the distributions ${\overline{P}}_3(S^L,S^H,\mathrm{\Lambda })$ using Eq. (12) (Statistical) and ${\tilde{P}}_3(S^L,S^H,\mathrm{\Lambda })$ using Eq. (27) (Clebsch-Gordan). Both these distributions show a preference for angles ${\varphi }^{LH}>\pi /2$ and both of them show a clear suppression of the FF intrinsics spins either parallel or antiparallel to each other. The statistical distribution here is slightly different from the one shown in Fig. 2, since in that case I used $P_1\left(\mathrm{\Lambda }\right)$ from Eq. (9b) while here I used $P\left(\mathrm{\Lambda }\right)$ from Eq. (29).