⚛️ nuclear theory

Contribution of incomplete fusion to the evaporation residue cross sections in the $^{30}$ Si $+^{186}$ W reaction

This study employs the dinuclear system model to demonstrate that while complete fusion dominates evaporation residue formation in the $^{30}$ Si $+^{186}$ W reaction at lower energies, incomplete fusion via $\alpha$ -particle emission becomes the primary source of Rn isotopes at higher energies ( $E_{c.m.} > 128$ MeV), thereby successfully explaining the measured total evaporation residue cross sections.

Original authors: Bakhodir Kayumov, Avazbek Nasirov

Published 2026-10-14

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CC BY 4.0

Original authors: Bakhodir Kayumov, Avazbek Nasirov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, where matter is built from protons and neutrons, scientists often try to build heavier elements by smashing two lighter atomic nuclei together at incredible speeds. When these two nuclei collide, they can stick together to form a single, larger, and highly excited "compound nucleus." This new, unstable giant then cools down by shooting out tiny particles like neutrons or alpha particles, eventually settling into a new, heavy element. However, this process is fraught with difficulty. Often, instead of merging into one, the two colliding nuclei glance off each other or break apart into two different fragments before they ever fully unite. This failure to fuse is a major obstacle in the quest to create the heaviest elements in the universe.

To understand why these collisions succeed or fail, researchers look at the shape and orientation of the nuclei involved. Many atomic nuclei are not perfect spheres; some are flattened like a pancake, while others are stretched like a rugby ball. When these oddly shaped nuclei collide, the angle at which they hit each other matters immensely. Furthermore, at very high speeds, a nucleus might not just break apart into two equal halves, but might shed a small piece, like an alpha particle, while the rest of it fuses with the target. This phenomenon, known as incomplete fusion, creates a different set of heavy elements than the standard, complete fusion process. Understanding the balance between these different outcomes is crucial for predicting which heavy elements can be made and how much of them can be produced.

A team of researchers has recently taken a closer look at a specific collision between silicon-30 and tungsten-186 to see how these different fusion paths play out. By using a sophisticated computer model that treats the colliding nuclei as a temporary pair of dancing partners, they simulated what happens when these two atoms crash into one another. Their goal was to calculate how often the nuclei stick together completely, how often they break apart, and how often they undergo incomplete fusion, where a small piece is shed before the main bodies merge. They focused on a range of collision energies, from 107.6 to 172.2 million electron volts, a scale that represents the intense energy required to force these atomic cores together.

The researchers found that at lower collision energies, the primary way to create the desired heavy residue is through complete fusion, where the two nuclei merge into a single unit called radium-216. However, as the collision energy increases beyond approximately 128 MeV, the story changes dramatically. In this higher energy range, the incomplete fusion process becomes the dominant source of the final heavy elements. In this scenario, the colliding system breaks apart in a way that releases an alpha particle, leaving behind a slightly lighter nucleus called radon-212. The study shows that above this energy threshold, these radon isotopes produced by incomplete fusion contribute more than half of all the heavy residues observed.

This shift in dominance explains a long-standing puzzle in the experimental data. When scientists measure the total amount of heavy material produced in these collisions, the number does not drop off as quickly as one might expect when the energy gets very high. If only the complete fusion process were at work, the production of radium isotopes would plummet by a factor of ten as the energy rises. Yet, the total yield remains relatively steady. The researchers' calculations show that the incomplete fusion channel fills this gap. While the production of radium drops sharply, the production of radon via incomplete fusion rises to compensate, keeping the total number of heavy residues from collapsing.

The study also clarified the nature of the particles being produced. The researchers calculated that it is virtually impossible for the heavy radon isotopes, specifically radon-212 and radon-211, to be formed through the standard complete fusion route at these energies. The energy required to shed an alpha particle from the fully merged nucleus is simply too high. Instead, these specific isotopes are almost exclusively the result of the incomplete fusion process, where the alpha particle is shed early in the collision, leaving a "colder" and more stable radon nucleus behind. This distinction is so sharp that detecting these specific radon isotopes in an experiment would serve as a clear signature that incomplete fusion is occurring.

When the researchers compared their simulation results with actual experimental data, they found a strong agreement in the overall trends. The model successfully reproduced the slow decrease in the total yield of heavy elements at high energies, a feature that previous models focusing only on complete fusion could not explain. The calculated probability of the nuclei fusing completely, when considering only the most effective collision angles, was found to be about 0.75. This is slightly higher than the value of 0.57 derived from earlier experimental comparisons, suggesting that the theoretical model captures the physics of the collision with good precision. The work confirms that incomplete fusion is not just a minor side effect but a major player in the formation of heavy elements in this specific reaction, fundamentally shaping the landscape of what can be created in the laboratory.

Technical Summary: Contribution of Incomplete Fusion to Evaporation Residue Cross Sections in the $^{30}\text{Si} + ^{186}\text{W}$ Reaction

Problem Statement
The formation of evaporation residues (ER) in heavy-ion collisions is a complex process competing with quasifission. While it was historically believed that the hindrance to complete fusion becomes significant only in reactions with a product of charge numbers $Z_1Z_2 > 1600$ , experimental data for the $^{30}\text{Si} + ^{186}\text{W}$ reaction ( $Z_1Z_2 = 1036$ ) suggested otherwise. Previous studies indicated that the yield of Ra isotopes (formed via complete fusion) decreases significantly at higher excitation energies, yet the total ER yield (including Rn and Fr isotopes) remains relatively high. This discrepancy suggested the presence of a mechanism beyond complete fusion and standard quasifission. Specifically, Berriman et al. noted that the non-Ra residues could be affected by the breakup of the projectile into $\alpha$ -particle clusters at large orbital angular momenta ( $\ell$ ), leading to incomplete fusion (ICF). The theoretical explanation for this large yield of non-Ra residues, particularly Rn isotopes, was the primary subject of this investigation.

Methodology
The authors analyzed the $^{30}\text{Si} + ^{186}\text{W}$ reaction within the framework of the Dinuclear System (DNS) model, incorporating both complete fusion and ICF mechanisms.

Capture and Dynamics: The capture cross section was calculated by solving dynamical equations of motion for the relative distance and orbital angular momentum, averaging over the orientation angles of the deformed $^{30}\text{Si}$ (oblate) and $^{186}\text{W}$ (prolate) nuclei.
Potential Energy Surface (PES): The evolution of the DNS was governed by a PES including shell effects. The model considered the competition between nucleon transfer leading to complete fusion and the breakup of the DNS via quasifission.
Incomplete Fusion Mechanism: ICF was treated not as a pre-capture breakup, but as the breakup of the DNS from a very mass-asymmetric configuration ( $\alpha + ^{212}\text{Rn}$ ) populated by nucleon transfer at large $\ell$ . The probability of $\alpha$ -particle emission was calculated based on the population of this specific DNS configuration and its breakup rate through the quasifission barrier.
Statistical De-excitation: The survival probabilities of the excited compound nucleus $^{216}\text{Ra}^*$ (from complete fusion) and the conjugate nucleus $^{212}\text{Rn}^*$ (formed after $\alpha$ emission in ICF) were calculated using a statistical model. This included neutron, proton, and $\alpha$ emission widths, fission barriers (corrected by Kramers and Strutinsky factors), and level densities with shell damping.

Key Contributions and Results

Mechanism of ICF: The study confirms that ICF in this reaction proceeds via the formation of a mass-asymmetric DNS configuration ( $\alpha + ^{212}\text{Rn}$ ) followed by the emission of an $\alpha$ particle. This process is dominant at large orbital angular momenta ( $\ell \approx 20\text{--}60\hbar$ ).
Cross Section Contributions:
- At lower collision energies ( $E_{c.m.} \lesssim 125$ MeV), complete fusion is the primary source of evaporation residues (Ra isotopes).
- Above $E_{c.m.} \approx 128$ MeV, the contribution of incomplete fusion becomes dominant, accounting for more than half of the total calculated ER cross section.
- The ICF cross section increases by two orders of magnitude from the lowest energies to $E_{c.m.} \approx 120$ MeV, reaching approximately 30 mb at higher energies.
Reproduction of Experimental Data:
- The sum of complete fusion (Ra isotopes) and ICF (Rn isotopes) cross sections successfully reproduces the slow decrease of the total measured ER cross section at high energies.
- In contrast, the yield of Ra isotopes alone decreases by an order of magnitude in this energy range.
- The calculated total ER cross section agrees with experimental data within 17% for $E_{c.m.} = 120\text{--}151$ MeV.
Fusion Probability: The fusion probability calculated for partial waves contributing to ER formation ( $\ell \leq 30\hbar$ ) is approximately 0.75. This exceeds the value of $0.57 \pm 0.08$ extracted from experimental data by Berriman et al., suggesting that quasifission hinders complete fusion in this reaction even at $Z_1Z_2 = 1036$ .
Isotopic Signatures: The calculation of de-excitation channels reveals that the complete fusion contribution to the formation of $^{212}\text{Rn}$ and $^{211}\text{Rn}$ is negligible (cross sections $< 10^{-3}$ mb). These isotopes are formed almost exclusively via ICF, with cross sections in the ICF channel being more than three orders of magnitude larger.

Significance
The paper demonstrates that incomplete fusion, driven by the breakup of a mass-asymmetric DNS configuration at large orbital angular momenta, is a critical mechanism for explaining the total evaporation residue yield in the $^{30}\text{Si} + ^{186}\text{W}$ reaction. The inclusion of the ICF channel resolves the discrepancy between the rapid drop in Ra isotope yields and the slower decline of the total ER yield observed experimentally. Furthermore, the study identifies the production of $^{212}\text{Rn}$ and $^{211}\text{Rn}$ isotopes, coincident with $\alpha$ particles, as a distinct signature of the ICF mechanism in this reaction, providing a potential test for the DNS model of incomplete fusion. The results also reinforce the conclusion that quasifission significantly hinders complete fusion in reactions with $Z_1Z_2$ well below the traditionally accepted threshold of 1600.

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