Shell effects in quasi-fission for calcium induced reactions forming thorium isotopes

Using time-dependent Hartree-Fock calculations for calcium-induced reactions on ytterbium, this study reveals that while quantum shell effects drive asymmetric quasi-fission toward a heavy fragment with Z~54, they do not induce a transition to symmetric modes in neutron-deficient systems, unlike in the fission of thorium compound nuclei.

The Gist

Imagine two giant, sticky balls of dough (atomic nuclei) smashing into each other at incredible speeds. Usually, when they hit, they might stick together to form one giant, wobbling blob (fusion). But often, they smash, swirl around each other, and then bounce apart again without ever fully merging. This "almost-sticking" event is called quasi-fission.

This paper is like a high-speed, microscopic movie camera that watches these collisions to see how the dough splits apart. The scientists wanted to know: Does the internal "recipe" of the dough (specifically, the arrangement of its particles) change how it splits?

Here is the breakdown of their discovery using simple analogies:

1. The Players: Calcium and Ytterbium

The researchers simulated collisions between Calcium (a small, round ball) and Ytterbium (a larger, slightly squashed ball). They used different "flavors" of Calcium, ranging from light to heavy, to see how the extra weight changed the outcome.

When these two smash together, they form a temporary, super-heavy "Thorium" blob. In the real world, Thorium usually splits in two. Sometimes it splits evenly (like cutting a sandwich in half), and sometimes it splits unevenly (like cutting a slice off the side).

2. The Mystery: The "Shell" Effect

Inside every atom, protons and neutrons like to sit in specific, organized layers, kind of like shells on an egg or seats in a theater. These are called quantum shells. When a shell is full, the atom is very stable and happy.

• The Old Rule: In normal nuclear fission (splitting), if you have a lot of neutrons, the atom prefers to split unevenly. It likes to keep one piece heavy (around 54 protons) and one piece light. This is because the heavy piece hits a "comfort zone" (a full shell) that makes it stable.
• The Change: Scientists knew that if you take away neutrons (making the atom "neutron-deficient"), this preference for uneven splitting disappears. The atom starts splitting evenly instead. It's like the "comfort zone" vanishes, so the dough has no reason to favor one side over the other.

3. The Big Surprise: Quasi-fission is Stubborn

The scientists asked: "Does this rule apply to quasi-fission too?"

They ran thousands of computer simulations (using a method called Time-Dependent Hartree-Fock, which is basically a super-accurate physics video game).

The Result:

• In normal fission: As they removed neutrons, the splitting changed from "uneven" to "even."
• In quasi-fission: Nothing changed! No matter how many neutrons they removed, the collision always stopped at the same point: a heavy piece with about 54 protons and a light piece with about 36 protons.

It's as if the dough had a magnetic memory. Even when the "comfort zone" (the shell) became weak or disappeared in the final product, the collision process itself was so fast and chaotic that it got stuck in the "uneven" pattern anyway. The system just couldn't break the habit.

4. Why Did This Happen? (The Landscape Analogy)

To explain this, the scientists looked at the Potential Energy Surface (PES). Think of this as a topographic map of a mountain range.

• The Valley: There is a deep valley on the map representing the "uneven split" (the stable 54-proton piece).
• The Hill: There is a high hill blocking the path to the "even split" valley.

In normal fission, the atom has time to roll around. If the neutrons are removed, the "uneven valley" gets shallower, and the "even valley" becomes easier to reach. The atom rolls over the hill and splits evenly.

In quasi-fission, the collision is like a fast-moving car driving down the mountain. It hits the "uneven valley" first. Because it's moving so fast and the "hill" blocking the even split is so high, the car never has time to climb over to the other side. It gets stuck in the uneven valley, even if that valley isn't as deep as it used to be.

The Takeaway

This paper tells us that quasi-fission is a different beast than normal fission.

• Normal Fission is like a slow, careful hiker who changes their path based on the terrain.
• Quasi-fission is like a speeding bullet that follows a specific trajectory regardless of how the terrain changes slightly.

The "shell effects" (the quantum rules that make certain atoms stable) act like a strong magnet that pulls the collision into an uneven split, and this pull is so strong that even when the atom becomes "neutron-deficient," the collision refuses to switch to a symmetric (even) split.

In short: The universe has a "habit" of making uneven splits in these heavy collisions, and it's very hard to break that habit, even when the physics suggests it should change.

Technical Summary

1. Problem Statement

The paper addresses the role of quantum shell effects in quasi-fission, a process in heavy-ion collisions where significant mass transfer occurs between colliding nuclei without the formation of a fully equilibrated compound nucleus.

• Context: In the fission of actinide compound nuclei, shell effects drive an asymmetric mass split (producing fragments around $Z \approx 54$). However, as thorium isotopes become more neutron-deficient ($A < 226$), experimental fission data shows a transition to symmetric fission.
• The Question: Does a similar transition from asymmetric to symmetric modes occur in quasi-fission when the neutron number of the reactants is varied? Specifically, do neutron-deficient systems in quasi-fission lose the asymmetric character driven by shell effects, or do they persist?
• Gap: Previous studies often focused on single compound nuclei or relied on static mean-field calculations. There was a need for a microscopic, dynamical study comparing a range of isotopes to disentangle shell effects from sequential fission or other experimental artifacts.

2. Methodology

The authors employed the Time-Dependent Hartree-Fock (TDHF) theory, a microscopic approach that describes nuclear structure and dynamics without free parameters, relying solely on the Skyrme energy density functional (EDF).

• Reaction System: The study simulated collisions between Calcium isotopes ($^{40-56}\text{Ca}$) and Ytterbium-176 ($^{176}\text{Yb}$).
  • Compound Nuclei: These reactions form Thorium isotopes ranging from $^{216}\text{Th}$ to $^{232}\text{Th}$, spanning the region where the fission mode transitions from symmetric to asymmetric.
  • Energy: Calculations were performed at center-of-mass energies ($E_{c.m.}$) of 172 MeV (10% above the Coulomb barrier) and 180 MeV (15% above) for select cases.
• Simulation Parameters:
  • EDF: The SLy4d parametrization was used.
  • Geometry: The $^{176}\text{Yb}$ target was treated as prolate ($\beta_2 \approx 0.33$), while Ca isotopes were spherical. Four initial orientations of the target deformation axis (0°, 45°, 90°, 135°) were considered.
  • Dynamics: The simulations tracked the evolution of the system until two separate fragments formed (separated by 26 fm) or fusion occurred.
  • Contact Time ($\tau$): Defined as the duration the neck density exceeds $0.08 \text{ fm}^{-3}$.
  • Data Volume: A total of 480 TDHF calculations were performed.
• Complementary Analysis: To interpret the dynamical results, the authors computed Potential Energy Surfaces (PES) for the resulting Thorium isotopes ($^{216-232}\text{Th}$) using the constrained Hartree-Fock + BCS method (SKYAX code), analyzing the energy landscape in terms of quadrupole ($Q_{20}$) and octupole ($Q_{30}$) deformations.

3. Key Results

A. Mass Equilibration and Fragment Formation

• Universal Asymmetry: Contrary to the behavior observed in the fission of neutron-deficient thorium isotopes, all simulated quasi-fission reactions ($^{40-56}\text{Ca} + ^{176}\text{Yb}$) exhibited a strong preference for asymmetric mass splits.
• Stopping Point: Mass transfer consistently halted when a heavy fragment with $Z \approx 54$ protons was formed. This corresponds to a light fragment with $Z \approx 36$.
• Neutron Distribution: While proton numbers ($Z$) were tightly clustered around these shell-stabilized values, neutron numbers ($N$) showed a broader distribution, suggesting that proton shell effects are the primary driver of the final asymmetry in these reactions.
• Energy Independence: The asymmetric behavior persisted even when the collision energy was increased from 10% to 15% above the Coulomb barrier.

B. Total Kinetic Energy (TKE)

• The TKE of the outgoing fragments correlated strongly with the charge asymmetry. More symmetric splits resulted in lower TKE (indicating more elongated shapes at scission), while the asymmetric splits (around $Z \approx 54$) corresponded to higher TKE, implying a more compact scission configuration.
• This correlation was consistent across all isotopes, reinforcing that the exit channel shapes are similar regardless of the entrance channel neutron number.

C. Potential Energy Surfaces (PES) Interpretation

• Fission vs. Quasi-fission: The PES calculations revealed that for neutron-deficient thorium isotopes (e.g., $^{216}\text{Th}$), the asymmetric valley in the potential energy landscape becomes very shallow and is separated from the symmetric valley by a significant barrier.
  • In fission, the system must overcome this barrier to reach the asymmetric valley; for neutron-deficient isotopes, the barrier prevents population of the asymmetric mode, leading to symmetric fission.
  • In quasi-fission, the system enters the potential landscape with high excitation and angular momentum. The authors argue that the system falls directly into the asymmetric valley before the barrier can prevent it, or that the dynamical trajectory bypasses the barrier that restricts fission.
• Fragment Stability: PES analysis of the heavy fragments showed that nuclei near $Z=54$ and $N=84$ (deformed shell closures) have low deformation energy costs, making them energetically favorable products for the quasi-fission process. Magic numbers ($Z=50, N=82$) were avoided due to high deformation energy costs.

4. Key Contributions

1. Dissociation of Fission and Quasi-fission Trends: The study provides microscopic evidence that the transition from asymmetric to symmetric fission observed in neutron-deficient thorium isotopes does not apply to quasi-fission. Quasi-fission remains asymmetric across the entire isotopic chain studied.
2. Mechanism of Asymmetry: It clarifies that the persistence of asymmetry in quasi-fission is due to the system falling into the asymmetric valley of the PES during the dynamical evolution, whereas in fission, static barriers prevent this population in neutron-deficient systems.
3. Validation of Shell Effects: The results confirm that quantum shell effects (specifically the octupole-deformed shell gaps at $Z \approx 52-56$ and $Z \approx 36$) dominate the fragment mass distribution in slow quasi-fission, even in systems where the compound nucleus would otherwise fission symmetrically.
4. Microscopic Benchmark: The work provides a comprehensive dataset of 480 TDHF simulations, offering a rigorous benchmark for understanding mass equilibration timescales and the role of orientation in heavy-ion collisions.

5. Significance

• Nuclear Structure Physics: The findings deepen the understanding of how shell effects influence nuclear dynamics in non-equilibrium conditions. It highlights that the "memory" of shell closures persists in the exit channel of quasi-fission even when the compound nucleus properties suggest a different outcome.
• Superheavy Element Synthesis: Understanding the competition between fusion and quasi-fission is critical for synthesizing superheavy elements. Since quasi-fission is the primary barrier to fusion, knowing that it is driven by specific shell-stabilized fragments helps in selecting optimal projectile-target combinations to minimize quasi-fission yields or steer them toward desired outcomes.
• Theoretical Framework: The successful application of TDHF to reproduce these complex shell-driven phenomena validates the approach as a predictive tool for nuclear reaction dynamics, particularly in regimes where experimental data is scarce or difficult to interpret due to sequential fission effects.

In conclusion, the paper demonstrates that quasi-fission is robustly asymmetric for calcium-induced reactions forming thorium isotopes, driven by shell effects that stabilize fragments at $Z \approx 54$. This behavior contrasts sharply with the fission of the same compound nuclei, where neutron deficiency leads to symmetric fission, a difference explained by the distinct dynamical pathways and barrier structures in the potential energy surfaces.