Exact Treatment of Continuum Couplings in Nuclear Optical Potentials via Feshbach Theory

In the subatomic world, atomic nuclei are not solid, unchanging spheres. Many of them, especially those that are unstable or "weakly bound," are more like loose clusters of particles held together by a fragile grip. When these fragile nuclei collide with other atoms, they do not simply bounce off like billiard balls. Instead, the collision can stretch them, tear them apart, or cause them to break into smaller pieces that fly off in different directions. This process, known as breakup, is a major factor in how these particles interact. To predict what happens in these collisions, physicists use mathematical maps called optical potentials. Think of these maps as a way to describe the invisible force field that guides a particle's path. For decades, scientists have struggled to draw these maps accurately for fragile nuclei because the standard methods often ignore the complex, messy reality of the breakup process. They have relied on simplified versions that treat the nucleus as a single, rigid object, missing the subtle ways that the possibility of breaking apart changes the force felt by the incoming particle.

A team of researchers at Tongji University in Shanghai has now developed a new way to draw these maps that captures the full complexity of the breakup process without relying on those simplifying shortcuts. They focused on a specific collision: a deuteron, which is a nucleus made of just one proton and one neutron, hitting a nickel-58 nucleus. By using a sophisticated computer calculation that tracks every possible way the deuteron can vibrate or break apart, they constructed a complete picture of the interaction. Their work reveals that the force guiding the deuteron is not a simple, local push or pull. Instead, it is a "non-local" force, meaning the particle's path at one point depends on what happened to it at a different point in space and time. This happens because the deuteron can briefly split into its proton and neutron components, travel a short distance, and then recombine. The researchers found that this fleeting breakup creates a specific, structured pattern in the force field that previous methods could not see.

The team tested their new method by applying it to the collision of deuterons with nickel-58 at various speeds, ranging from 20 to 80 million electron volts. They compared their results against a highly detailed, standard calculation that tracks all the breakup possibilities directly. The new method, which creates a simplified two-body map from the complex three-body reality, reproduced the results of the detailed calculation with remarkable precision. This agreement proves that their new map correctly captures the physics of the interaction. In contrast, older methods that ignored the connections between different breakup states, or those that tried to force the complex force into a simple, local shape, failed to match the detailed results. The researchers showed that the force field generated by the breakup process has a rich, internal structure that extends over a finite distance, rather than being a single point of influence.

One of the most significant findings is how this force changes as the speed of the incoming particle changes. The researchers measured how much of the incoming beam is lost to breakup and how much is lost to other forms of absorption, such as the fragments being captured by the target nucleus. They discovered that as the energy of the collision increases, the fraction of particles that break apart and fly away as separate pieces grows larger. However, the total amount of energy lost to the breakup process does not simply keep increasing. Instead, the loss of energy peaks at intermediate speeds and then drops slightly at the highest speeds. This means that at lower and higher energies, the breakup fragments are more likely to escape, but at middle energies, the fragments are more likely to be absorbed by the target nucleus. This distinction is crucial because it shows that breakup and absorption are not separate steps that happen one after another; they are deeply intertwined processes that happen simultaneously.

The study also clarifies a long-standing issue in how physicists interpret these collisions. For a long time, scientists assumed that the force field could be described by a simple, bell-shaped curve, a shape that was popularized in earlier decades. The new calculations show that the actual shape of the force field is much more complex and does not follow that simple curve. The researchers were able to see the "spatial memory" of the interaction: the force field retains a record of the particle's journey through the breakup state. This level of detail was previously impossible to obtain because the mathematical tools used to simplify the problem often introduced errors or singularities, essentially breaking down at certain points. By keeping the full mathematical structure intact, the team avoided these errors and produced a smooth, reliable map of the interaction.

This work provides a direct numerical check on the theory that links complex many-body interactions to simpler effective forces. By proving that their new map works for the deuteron and nickel-58 collision, the researchers have validated a method that can be applied to other, more exotic nuclei. This is particularly important for understanding how elements are created in stars and how medical isotopes are produced, as these processes often involve unstable, weakly bound nuclei. The ability to accurately model these interactions without needing to run the most expensive, time-consuming calculations every time opens the door to more precise predictions in nuclear physics. The researchers noted that while their current work focuses on the deuteron, the same approach could be extended to other types of nuclear reactions, such as those involving the transfer of particles between nuclei.

The study does not claim to have solved every problem in nuclear physics, but it has removed a significant barrier to understanding how fragile nuclei behave. By showing that the force field is non-local and that the breakup process is inextricably linked to absorption, the researchers have provided a clearer picture of the subatomic world. Their findings suggest that the old, simplified views of these collisions are insufficient for describing the reality of weakly bound systems. The new method offers a way to see the hidden structure of these interactions, revealing that the path of a particle is shaped by a complex history of splitting and recombining that happens in the blink of an eye. This deeper understanding is a necessary step toward mastering the forces that govern the universe at its smallest scales.