Intrinsic generation of angular momenta and entanglement in fission

Using time-dependent density functional theory, this study demonstrates that incorporating non-axial deformations in the spontaneous fission of $^{252}$Cf enables axial tilting rotations and broadens spin distributions, thereby reducing spin-spin correlations along the fission axis compared to axially symmetric trajectories.

The Gist

Imagine a heavy, unstable atom like a giant, wobbling water balloon. When this atom splits in two (a process called nuclear fission), it doesn't just break apart cleanly; it spins, twists, and throws off energy. For decades, scientists have been arguing about how these two new pieces (fragments) start spinning. Do they spin because of a chaotic collision after they break? Or do they start spinning while they are still connected, like dancers holding hands before a split?

This paper uses a super-computer simulation to settle a specific part of that debate. Here is the story in simple terms:

The Setup: The Perfectly Symmetric Dance

The researchers simulated the splitting of a Californium-252 atom. They started by looking at the "old way" of thinking: what if the atom stays perfectly symmetrical as it splits?

Imagine two ice skaters holding hands, spinning perfectly in sync. If they stay perfectly symmetrical (like a mirror image of each other), physics rules say they can only do two things:

1. Twist: Spin in opposite directions along the line they are separating (like twisting a towel).
2. Wriggle/Bend: Spin together or apart in a way that keeps them balanced.

In this "perfectly symmetrical" world, the spins are locked in a strict, predictable dance. If one spins left, the other must spin right. They are perfectly correlated, like two people walking in step.

The Twist: Breaking the Mirror

The big discovery in this paper is what happens when you stop pretending the atom is perfectly symmetrical. In the real world, atoms are often lumpy and uneven (like a potato rather than a sphere). The researchers let their simulation include these "lumps" (non-axial deformations).

Imagine the two ice skaters again, but now one is wearing a heavy backpack and the other isn't, or they are holding hands at a weird angle. The perfect symmetry is broken.

What changed?

1. The Dance Gets Messy: When the symmetry breaks, the strict rules loosen. The fragments are no longer forced to spin in perfect opposition. They can now tilt and spin in directions they couldn't before.
2. The "Entanglement" Fades: In the symmetrical world, the two fragments were tightly linked (entangled) in their spinning. If you knew one was spinning left, you knew the other was spinning right. But when the shape gets lumpy, this link weakens. The fragments become more independent. Knowing how one spins tells you less about the other.
3. The Angle Changes: The researchers looked at the angle between the two spins. In the symmetrical case, the spins tended to point in very specific, predictable directions. When they broke the symmetry, the spins pointed in a much wider variety of directions, smoothing out the sharp peaks that were there before.

The Analogy: The Spinning Top

Think of the atom as a spinning top that is about to crack in half.

• Symmetrical Case: If the top is perfectly round, when it cracks, the two halves fly apart spinning in a very predictable, mirrored pattern. They are like twins.
• Non-Symmetrical Case: If the top is slightly squashed or lumpy, when it cracks, the two halves fly apart spinning in a more chaotic, less predictable way. They are no longer twins; they are just two separate pieces doing their own thing.

The Bottom Line

The paper claims that shape matters. By ignoring the "lumpy" shapes of the nucleus, previous models were missing a huge piece of the puzzle. When you include these irregular shapes:

• The fragments spin in more varied directions.
• The "connection" (entanglement) between their spins gets weaker.
• The spins are less predictable and more spread out.

The researchers conclude that to truly understand how atoms split and spin, we cannot assume they are perfect, symmetrical spheres. We have to account for their messy, real-world shapes. This helps explain why the spins of the fragments are so different from what older, simpler models predicted.

Technical Summary

Technical Summary: Intrinsic Generation of Angular Momenta and Entanglement in Fission

Problem Statement
The generation of intrinsic spins in fission fragments (FFs) and their subsequent de-excitation via neutron evaporation and gamma-ray emission remain subjects of active debate. While experimental data suggests a "sawtooth" pattern in spin magnitudes and a lack of correlation between complementary fragments, theoretical interpretations vary. Time-dependent density functional theory (TDDFT) studies indicate that spins are determined prior to scission through collective bending and twisting modes, oriented primarily perpendicular to the fission axis. However, a persistent limitation in most microscopic theoretical treatments is the assumption of axial symmetry in the fissioning system. In reality, non-axial deformations (triaxiality) are known to influence nuclear structure and dynamics, yet their specific impact on spin generation, spin-spin correlations, and entanglement in spontaneous fission has not been systematically investigated.

Methodology
This study employs time-dependent relativistic density functional theory (TDDFT) to investigate the spontaneous fission (SF) of $^{252}\text{Cf}$. The system is chosen because its initial ground state carries zero total spin, allowing for the isolation of symmetry-breaking dynamics in spin generation.

• Theoretical Framework: The evolution of the quasiparticle vacuum is modeled using the time-dependent BCS approximation with the relativistic density functional PC-PK1 and a monopole pairing interaction. While the fully dynamical Hartree-Fock-Bogoliubov (HFB) theory is not used due to computational constraints regarding non-axial symmetry breaking, the BCS approach allows for large-amplitude collective motion driven by nucleon density evolution, with the pairing gap adapting instantaneously.
• Initial Conditions: The study initializes TDDFT evolution at exit points beyond the fission barrier, following quantum tunneling from the ground state. Four distinct trajectories are analyzed:
  1. Trajectory 1: Axially symmetric and reflection symmetric ($\beta_{30}=0$).
  2. Trajectory 2: Axially symmetric but reflection asymmetric (finite $\beta_{30}$).
  3. Trajectory 3 & 4: Non-axial (triaxial) trajectories with finite $\beta_{31}$ deformations, breaking axial symmetry.
• Analysis Techniques:
  • Angular Momentum Projection (AMP): Used to obtain probability distributions $P(I, I_z)$ for spin magnitude $I$ and its projection $I_z$ on the computational lattice's z-axis.
  • Joint Distributions: Computed for spin components along the y-axis (perpendicular to the reaction plane) and z-axis (fission axis) to identify rotational modes (wriggling, bending, twisting, and axial/tilting).
  • Mutual Information: Calculated to quantify spin-spin correlations and entanglement between heavy and light fragments.
  • Opening Angle: The distribution of the angle $\phi_{HL}$ between the intrinsic spins of the two fragments is evaluated.

Key Results

1. Spin Magnitudes and Mass Dependence:

   • A systematic mass dependence is observed where the lighter fragment consistently exhibits higher spin values due to greater deformation, except in the reflection-symmetric case (Trajectory 1).
   • Breaking reflection symmetry (Trajectory 2) enhances the probability of higher spin projections ($|I_z|$) for the heavy fragment compared to the symmetric case.
   • Non-axial trajectories (3 and 4) show significantly greater dispersion in spin projections ($I_z$) compared to axial cases.

2. Rotational Modes and Symmetry Constraints:

   • Axial Symmetry (Trajectories 1 & 2): Strict constraints enforce conservation of the total spin projection along the fission axis ($I^H_z + I^L_z = 0$). This restricts dynamics exclusively to the twisting mode (counter-rotation) along the z-axis. Along the y-axis, bending and wriggling modes occur with equal probability (25% per quadrant), reflecting underlying symmetry.
   • Non-Axial Symmetry (Trajectories 3 & 4): Breaking axial symmetry relaxes the constraint $I^H_z + I^L_z = 0$. This allows for the emergence of axial (tilting) collective rotations (cooperative rotation) alongside twisting modes. The joint probability distributions $P(I^H_y, I^L_y)$ become asymmetric, showing unequal probabilities for bending versus wriggling modes. The spin projection $I_z$ is no longer a conserved quantum number for the total system.

3. Correlations and Entanglement:

   • Mutual information analysis reveals that breaking axial symmetry significantly reduces spin-spin correlations along the fission axis (z-axis). The correlation indicator $C(H_z, L_z)$ for non-axial trajectories is only $1/6$ to $1/5$ of that for axially symmetric cases.
   • Correlations perpendicular to the fission axis (y-axis) are more resilient to symmetry breaking, though they still exhibit reductions in the axial cases compared to non-axial ones.

4. Opening Angle Distribution:

   • For axially symmetric trajectories, the opening angle distribution $P(\phi_{HL})$ reproduces previous microscopic trends: vanishing at small angles ($<30^\circ$) and $180^\circ$, with a pronounced peak at $\approx 30^\circ$.
   • Including triaxial degrees of freedom (Trajectories 3 & 4) considerably reduces the main peak at $30^\circ$, shifts the distribution toward larger angles, and smooths out the secondary peak near $165^\circ$.

Significance and Claims
The paper claims to present the first investigation of the influence of non-axial trajectories on fission fragment spins and their correlations using time-dependent relativistic density functional theory. The primary significance lies in demonstrating that triaxial degrees of freedom are not merely structural details but fundamentally alter the dynamics of spin generation:

• They broaden the distribution of spin projections on the fission axis.
• They enable collective tilting rotations that are strictly forbidden in axially symmetric trajectories.
• They reduce spin-spin correlations (entanglement) along the fission axis, as quantified by mutual information.
• They significantly alter the predicted opening angle distribution between fragment spins.

The authors note that while their approach improves upon axially symmetric models, it relies on the BCS approximation, which neglects dynamical aspects of the pairing field and quantum fluctuations in collective coordinates. Consequently, the study describes individual scission events via single trajectories rather than a full ensemble of fluctuating paths. However, the results underscore the necessity of including triaxial degrees of freedom in microscopic models to accurately describe the complex nature of fragment spin generation and entanglement in nuclear fission.