Fragment Intrinsic Spins and Fragments’ Relative Orbital Angular Momentum in Nuclear Fission

While nuclear fission has been studied for more than eight decades Hahn and Strassmann (1939), a complete microscopic description based on quantum many-body theory is still lacking. Typical microscopic approaches rely on unverified assumptions and/or strong restrictions, thus rendering the treatment incomplete. Phenomenological models are based on the imagination of their creators, rather than rigorous quantum mechanics or direct experimental information. Meitner and Frisch (1939) correctly identified the main driver of nuclear fission: namely, the competition between the Coulomb energy and the surface potential energy. The formation of the compound nucleus and its extremely slow shape evolution toward the outer fission barrier is correctly encapsulated by Bohr’s compound nucleus concept Bohr (1936); Bohr and Wheeler (1939). The saturation properties of nuclei along with the symmetry energy constrain the flow of the nuclear fluid from the moment the compound nucleus is formed until scission, which evolves like an incompressible liquid drop of almost constant local proton-neutron mixture. The spin-orbit interaction and pairing correlations control the finer details on how the emerging fission fragments (FFs) are formed, favoring asymmetric fission yields at low excitation energies Strutinsky (1967); Brack et al. (1972); Bertsch (1980); Bertsch and Bulgac (1997). The critical theoretical ingredients are thus well-known: the incompressibility of nuclear matter, the symmetry energy strength, the surface tension and the proton charge, the spin-orbit and the pairing correlations strengths. Only recently, a well-founded formalism free of restrictions that incorporates all of these features has been implemented and the nonequilibrium character of the nuclear large amplitude collective motion, particularly from the outer saddle to the scission configuration and the excitation energy sharing mechanism between FFs have been unambiguously proven microscopically Bulgac et al. (2016, 2019, 2020).

The FFs’ intrinsic spins have been the subject of old and renewed experimental and theoretical investigations Strutinsky (1960); Ericson (1960); Nix and Swiatecki (1965); Moretto and Schmitt (1980); Wilson et al. (2021); Bulgac et al. (2021); Marević et al. (2021); Randrup and Vogt (2021). In the 1960s, it was conjectured that the emerging FFs acquire intrinsic spins due to the existence of several collective FF spin modes: the double-degenerate transversal modes, wriggling and bending, and the longitudinal modes, twisting and tilting. The origin of the relative orbital angular momentum between fragments has never been elucidated within a fully microscopic framework. Consider the clean case of spontaneous fission of ²⁵²Cf from its ground state with $S_{0}^{\pi}=0^{+}$. The final three angular momenta satisfy the conservation law

where ${\bm{S}}^{\text{L,H}}$ are the FF intrinsic spins and ${\Lambda}$ is the FFs’ relative orbital angular momentum, which is an integer. Classically, these three vectors lie in a plane and ${\bm{\Lambda}}={\bm{R}}\times{\bm{P}}$, is perpendicular to the fission direction, where ${\bm{R}},{\bm{P}}$ are the FFs’ relative separation and momentum. On its way to scission this nucleus elongates along a spontaneously broken symmetry direction and the fledging FFs emerge. The longer the nuclear elongation the larger the moment of inertia of the entire nuclear system is and the overall rotational frequency controlled by ${\bm{\Lambda}}$ is slower. As FFs emerge, being by nature nonspherical, they rotate with intrinsic spins ${\bm{S}}^{\text{L}}$ and ${\bm{S}}^{\text{H}}$, while at the same time they also rotate as a dumbbell around their common center of mass with the angular momentum ${\bm{\Lambda}}$. Until scission, these three angular momenta can vary, subject to restriction Eq. (1). After scission, when the mass and energy exchange between emerging FFs stops, these angular momenta cease to evolve in time (apart from small effects of the Coulomb interaction between FFs Strutinsky (1960); Bulgac (2020)). Before scission the FF identities are not well-defined, because matter, momentum, and energy are flowing between them. The FF intrinsic spins and ${\bm{\Lambda}}$ are well-defined only at a sufficiently relative large separation. Even though the initial nuclear system ²⁵²Cf has a vanishing initial spin $S^{\pi}_{0}=0^{+}$, the FFs emerge as wave packets of deformed nuclei, characterized by rotation and vibrational bands. Similar to the well-known bicycle wheel classroom physics demos Phy , the dynamics of a spontaneously fissioning ²⁵²Cf resembles the dynamics of an instructor on a freely rotating stand (${\bm{\Lambda}}$) holding two bicycle wheels (${\bm{S}}^{\text{L,H}}$), and is nothing like a “snapping rubber band” Wilson et al. (2021), which does not rotate.

We use the time-dependent density functional theory (TDDFT) extended to superfluid systems (see recent reviews Bulgac (2013, 2019) and Refs. Bulgac et al. (2016, 2019, 2020, 2021)) to determine the triple probability distribution $P(S^{\text{L}},S^{\text{H}},\Lambda)$,   $\sum_{S^{\text{L,H}},\Lambda}P(\Lambda,S^{\text{L}},S^{\text{H}})=1$, by performing a triple angular momenta projection of the overlap Bulgac (2021)

where $Oz$ the fission axis and the magnitudes of the angular momenta satisfy the triangle restriction

and $|\Phi\rangle$ is the fissioning nucleus intrinsic wave function. In case of ²³⁶U^∗ and ²⁴⁰Pu^∗ the initial spin $S_{0}\neq 0$ and then $|{\bm{\Lambda}}-{\bm{S}}_{0}|=|{\bm{S}}^{\text{L}}+{\bm{S}}^{\text{H}}|$ and since $S_{0}\ll\langle\Lambda\rangle$ then $\Lambda\approx|{\bm{S}}^{\text{L}}+{\bm{S}}^{\text{H}}|$ with good accuracy. We determined the probability distribution $p(\cos\phi^{\text{LH}})$, where $\phi^{\text{LH}}$ is the angle between $S^{\text{L}}$ and $S^{\text{H}}$ by constructing a histogram of the expectation of the cosine between

where we used the Langer correction Langer (1937) in the denominator. Note that the relative angle $\phi^{\text{LH}}$ does not depend on a lab or body reference frame. Optimally, one should consider also an additional projection to enforce the value of total angular momentum ${\bm{S}}_{0}$, with the rotation operator $P_{0}=e^{i\gamma(J_{x}^{\text{L}}+J_{x}^{\text{H}}+\Lambda_{x})}$, where $\Lambda_{x}$ rotates the entire system around its center of mass, a procedure that is expected to lead only to minor corrections Bulgac et al. (2021). We replaced this projection with the equivalent triangle restriction

We performed TDDFT fission calculations of ²³⁶U, ²⁴⁰Pu, and ²⁵²Cf using two different nuclear energy density functionals (NEDFs), SkM^∗ Bartel et al. (1982) and SeaLL1 Bulgac et al. (2018), in simulation boxes $30^{2}\times 60$ with a lattice constant and $l=1$ fm and a corresponding momentum cutoff $p_{\text{cut}}=\pi\hbar/l\approx 600$ MeV/c, and using the LISE package as described in Refs. Bulgac et al. (2016); Jin et al. (2021); Bulgac et al. (2020). The excitation energies for ²³⁶U and ²⁴⁰Pu were chosen close to the neutron threshold, thus emulating the reactions ²³⁵U(n${}_{\text{th}}$,f) and ²³⁹Pu(n${}_{\text{th}}$,f). The initial nuclear wave function $|\Phi\rangle$ was evolved in time from various initial deformations $Q_{20}$ and $Q_{30}$ of the mother nucleus near the outer saddle until the FFs were separated by more than 30 fm as in Refs Bulgac et al. (2019, 2020, 2021) and their shapes relaxed. In the case of ²⁵²Cf(sf) we started the simulation outside the barrier for energies close to the ground state energy. The current implementation of the TDDFT framework Bulgac (2013, 2019) has proven capable of providing answers to a wide number of problems in cold atom physics, quantum turbulence in fermionic superfluids, vortex dynamics in neutron star crust, nuclear fission, and reactions. Density Functional Theory and Schrödinger descriptions are mathematically identical quantum many-body frameworks for one-body densities Dreizler and Gross (1990); Marques et al. (2006, 2012), with the proviso that in nuclear physics neither NEDF nor the internucleon forces are known with sufficient accuracy Salvioni et al. (2020).

The distributions of the FFs’ orbital angular momenta, see Fig. 1, are the first unrestricted microscopic extractions of these quantities. As the masses of ²³⁶U, ²⁴⁰Pu, and ²⁵²Cf are close to one another, the $\Lambda$ distributions obtained by performing a single angular projection of the overlap $\langle\Phi|\Phi(\beta_{0})\rangle=\langle\Phi|e^{i\beta_{0}(J_{x}^{\text{L}}+J_{x}^{\text{H}})}|\Phi\rangle$, as in Ref. Bulgac et al. (2021), are very similar. Such individual intrinsic spin distributions can be recovered independently from our triple projection results from $P(\Lambda,S^{\text{L}},S^{\text{H}})$ as follows

and a comparison between results using the single and the triple projections in case of induced fission of ²⁵²Cf are shown in Fig. 2. The more precise triple projection leads to larger FF intrinsic spins by about $2\ldots 3\,\hbar$, while the average orbital angular momentum $\Lambda$ decreases by about 1 $\hbar.$ (Similar corrections to the FF intrinsic spins would be required for the estimates presented in Ref. Marević et al. (2021).) As demonstrated in Ref. Stetcu et al. (2021), the emission of neutrons and statistical gammas reduces the FF spins by $\approx 3.5-5\,\hbar$ by the time the FF decay reaches the yrast bandhead, corresponding to the FF spin values measured by Wilson et al. (2021). The sum of the yrast bandhead spins Wilson et al. (2021) for ²⁵²Cf, $6.85\,\hbar$ for the heavy FF and 6.44 $\hbar$ for the light FF respectively (averaged over all measured FFs) with the angular momentum loss to decay $\approx 3.5-5\,\hbar$ estimated in Ref. Stetcu et al. (2021), using standard phenomenological inputs however, agree reasonably well with our estimates of the average intrinsic FF spins in Fig. 2.

In Fig. 3 we show the triple distribution $P(\Lambda,S^{\text{L}},S^{\text{H}})$ for odd values of $\Lambda$. The even values of $\Lambda$ fixes both FF parities to be identical, $\pi^{\text{L}}=\pi^{\text{H}}$, while in case of odd $\Lambda$ these parities are opposite, $\pi^{\text{L}}=-\pi^{\text{H}}$, since for ²⁵²Cf $S_{0}^{\pi}=0^{+}$. The distribution $P(\Lambda,S^{\text{L}},S^{\text{H}})$ is nonvanishing only in the region defined by Eq. (3).

The distribution of the angles between the intrinsic spins ${\bm{S}}^{\text{L}}$ and ${\bm{S}}^{\text{H}}$ is particularly instructive and qualitatively different from previous predictions. It was assumed a number of times in the literature, see Refs. Randrup and Vogt (2021); Vogt and Randrup (2021) and references therein, that the two intrinsic spins are very weakly correlated at most. In particular, this was one of the main interpretations of the experimental results recently published by Wilson et al. (2021). If that were case, the distribution $p(\phi^{\text{LH}})$ would basically be flat, similar to the predictions in Refs. Vogt and Randrup (2021); Randrup and Vogt (2021), with those results reproduced in this figure. In Fig. 4, the distribution $p(\phi^{\text{LH}})$ evaluated by us is clearly not a uniform distribution, with a prominent maximum at an angle $\phi^{\text{LH}}\approx 2\pi/3$ Bulgac . The probability of having angles $\phi^{\text{LH}}\geq\pi/2$ is $\approx 0.71/0.72$ (SeaLL1/SkM^∗), which means that the two FF intrinsic spins are predominantly pointing in opposite directions and that the the bending modes are predominantly favored over the wriggling modes. In Fig. 4, we used instead of the correlated evaluated distribution $P(\Lambda,S^{\text{L}},S^{\text{H}})$ the uncorrelated distribution $P(\Lambda)P(S^{\text{L}})P(S^{\text{H}})\triangle$ obtained using Eqs. (6-7), shown with triangles. The results appear very similar, even though $P(\Lambda,S^{\text{L}},S^{\text{H}})$ is drastically different from $P(\Lambda)P(S^{\text{L}})P(S^{\text{H}})\triangle$, where $P(\Lambda)P(S^{\text{L}})P(S^{\text{H}})$ is non-vanishing outside the region (3), the white regions in Fig. 3 and in evaluating the distribution $p(\theta_{\text{LH}})$ we have imposed the (3) restriction and renormalized the distribution $P(\Lambda)P(S^{\text{L}})P(S^{\text{H}})\triangle$ by a (not shown) factor $\approx 0.74$. Fig. 4 unfortunately does not reveal the large amount of FF intrinsic spin correlations, which are not merely geometrical in nature, since

when the geometrical constraint Eq. (5) is taken into account.

In Fig. 4 we plot the recent published results obtained with the phenomenological model FREYA, where Randrup and Vogt (2021) discussed the generation of the fragment angular momentum in fission. In Ref. Randrup and Vogt (2021) the claim is made that, unlike the conclusion reached by Wilson et al. (2021), i.e., that the FF intrinsic spins were formed after scission and are uncorrelated, the primordial intrinsic spins emerge uncorrelated before scission. This argument is based on the assumptions that the FF spins dynamics is governed by the rotational energy

where $I^{\text{L,H,R}}$ are the FFs and orbital moments of inertia, satisfying the relation $I^{\text{R}}\approx 10\,I^{\text{L,H}}$. The only correlation between $S^{\text{L,H}}$ is due to the third term, which is quantitatively small and which one can hardly quantify as highly correlated, is in stark contradistinction with our microscopic results in the same figure. While at first glance this assumption appears valid, see also Refs. Moretto and Schmitt (1980); Vogt and Randrup (2021), upon closer analysis it becomes clear that the most general form allowed by symmetry is

with a nondiagonal $3\times 3$ effective inertia tensor $\tensor{\bm{I}}$ in general.

The impact of the emission of neutrons and $\gamma$ rays on the spin of the FFs was discussed in Ref. Stetcu et al. (2021) within the Houser-Feshbach framework Hauser and Feshbach (1952), where it was demonstrated that the intrinsic FF spins can be changed on average by 3.5 - 5 $\hbar$, a process that leads to a strong decorrelation of the observed FF spins, a process strongly underestimated by the model of Ref. Wilson et al. (2021). The experimental data Wilson et al. (2021) characterizes only the yrast bandhead FF spins after a large amount of the internal FF excitation energy, $\approx$ 20 MeV per FF Bulgac et al. (2016, 2019, 2020); Schmidt and Jurado (2018); Talou et al. (2021); Randrup and Vogt (2009), was carried away by emitted particles. The work presented here can better guide phenomenological models Vogt and Randrup (2021); Randrup and Vogt (2021); Becker et al. (2013); Talou et al. (2021) and further extend the analysis in Ref. Stetcu et al. (2021), which all rely on a quite large number of fitting parameters.