Systematic Study of Proton, Two-Proton, Alpha, and Cluster Radioactivity Half-Lives based on the Deformed Gamow-like Model and Tabular Prior-data Fitted Network ($\mathrm{TabPFN}$)

Atoms are the fundamental building blocks of the visible universe, yet their cores are not always stable. In the heart of an atom, protons and neutrons are held together by powerful forces, but when the balance tips, the nucleus can become unstable and shed particles to find a more comfortable state. This process, known as radioactive decay, is a natural clock that ticks away at the very edge of matter. Some atoms shed a single proton, others release a pair of protons, while many emit a tight bundle of two protons and two neutrons, known as an alpha particle. In rare cases, even larger clusters of particles break away. Predicting exactly how long these unstable atoms will last before they decay is a central challenge in nuclear physics, as it reveals the hidden rules governing the structure of matter and helps scientists understand the limits of the periodic table.

For nearly a century, physicists have relied on mathematical models to estimate these lifetimes, often treating the nucleus as a simple sphere and calculating how likely a particle is to tunnel through an energy barrier to escape. While these traditional models capture the basic physics, they often struggle to match the precise measurements made in laboratories, especially when dealing with complex, oddly shaped nuclei or rare decay types. To bridge this gap between theory and reality, a team of researchers has developed a new approach that combines a classic physical model with a modern artificial intelligence tool. By merging the established laws of nuclear physics with a machine learning system designed to learn from small datasets, they have created a method that predicts the lifetimes of decaying atoms with unprecedented accuracy.

The researchers focused on four distinct ways unstable atoms shed their excess energy: the emission of a single proton, the simultaneous release of two protons, the ejection of an alpha particle, and the emission of larger clusters of particles. They gathered data on 583 different radioactive nuclei, a collection that included rare examples of two-proton emitters, dozens of proton emitters, hundreds of alpha emitters, and a selection of cluster emitters. Their starting point was a well-known physical framework called the deformed Gamow-like model. This model treats the nucleus not as a perfect sphere, but as a shape that can be stretched or squashed, which affects how easily particles can escape. The model calculates the time it takes for a particle to tunnel through the nuclear barrier, but it relies on a few adjustable numbers to match reality, such as the effective size of the nucleus and a factor that accounts for how the internal arrangement of protons and neutrons might slow down the process.

To improve these calculations, the team introduced a sophisticated machine learning system known as a Tabular Prior-data Fitted Network. Unlike many artificial intelligence systems that require massive amounts of data to learn, this tool is specifically designed to make high-precision predictions from small datasets, making it ideal for nuclear physics where experimental data can be scarce. The researchers did not simply let the computer guess the answers; instead, they used the AI to learn the subtle patterns that the physical model missed. They fed the system information about the nucleus, such as the number of protons and neutrons, the energy released during decay, and the shape of the nucleus. The AI then learned to predict a specific correction factor that adjusts the theoretical calculation to match what is actually observed in experiments.

The results of this hybrid approach were striking. When the researchers tested their new method against the experimental data, the predictions became significantly more accurate across all four decay modes. For the emission of single protons, the error in the predictions dropped by more than 80 percent compared to the physical model alone. For alpha decay, which is the most common form of radioactive decay studied, the error decreased by nearly 88 percent. The most successful version of their model, which combined the optimized physical parameters with the AI corrections, achieved a level of precision that was roughly 82 percent better than the uncorrected physical model. This improvement was not just a statistical fluke; the model maintained its accuracy even when tested on data it had never seen before, proving that it had learned the underlying rules of decay rather than just memorizing the examples.

Beyond simply getting the numbers right, the new model successfully reproduced complex behaviors that are difficult to capture with simple formulas. For instance, it accurately described how the lifetimes of alpha-emitting atoms change as the number of neutrons increases, including the sudden jumps in stability that occur when the nucleus reaches a specific "magic" number of neutrons, such as 126. This ability to capture the shell-closure effect suggests that the model has implicitly learned the structural details of the nucleus, even though it was not explicitly programmed with those rules. The researchers also used their refined model to predict the lifetimes of atoms that have not yet been observed or measured in laboratories, including potential candidates for two-proton decay and superheavy elements at the very edge of the periodic table. These predictions provide a roadmap for future experiments, guiding scientists on where to look for new forms of matter.

The study demonstrates that the future of nuclear physics may lie in the careful integration of traditional physical theories with modern data-driven tools. By respecting the established laws of quantum mechanics while allowing machine learning to correct for the complexities of real-world nuclei, the researchers have created a powerful new instrument for exploring the atomic world. Their work shows that even in a field governed by fundamental constants, there is still room for refinement, and that combining human insight with artificial intelligence can reveal patterns that neither could find alone. This hybrid framework offers a reliable way to predict the behavior of unstable atoms, helping to map the boundaries of nuclear stability and deepening our understanding of the forces that hold the universe together.