Low-energy collective structure of $^{92,94}$Zr and $^{94}$Mo from ($e,e^{\prime}$) and ($p,p^{\prime}$) scattering II. Signatures of mixed-symmetry states and origin of collectivity

Inside the heart of every atom lies a nucleus, a dense cluster of protons and neutrons bound together by forces so powerful they defy everyday intuition. While we often picture these particles as static building blocks, they are actually in a constant state of collective motion, vibrating and spinning in complex patterns. For decades, physicists have sought to understand the rules governing these movements, particularly in a class of atoms known as vibrational nuclei, where the entire core swells and shrinks like a breathing sphere. Among the most intriguing of these movements are states where protons and neutrons move in opposition to one another, a phenomenon called a mixed-symmetry state. Identifying these states is crucial because they act as fundamental building blocks for understanding how atomic nuclei organize themselves, yet they are notoriously difficult to spot because they often hide among other, more common vibrations.

A team of researchers has now taken a fresh look at three specific atoms—zirconium-92, zirconium-94, and molybdenum-94—to find a new way to recognize these elusive mixed-symmetry states. Instead of relying solely on the traditional method of measuring how strongly these states emit magnetic radiation, the scientists combined data from two different types of particle collisions: firing electrons at the nuclei and firing protons at them. By analyzing how the nuclei responded to these different projectiles, the team was able to map out the internal structure of the vibrations with unprecedented clarity. Their work confirms that these mixed-symmetry states exist in these atoms and reveals that their unique character is defined by a specific sign change in how protons and neutrons move relative to each other. This discovery provides a new, reliable signature for spotting these states, while also challenging previous assumptions about how to identify them and shedding light on the hidden sources of their strength.

To understand what the researchers were looking for, one must first grasp the basic nature of nuclear vibrations. In a perfectly symmetric vibration, protons and neutrons move in perfect unison, like a choir singing the same note in harmony. This is called a fully symmetric state. However, nature often allows for a more complex arrangement where protons and neutrons move in opposite directions, akin to two groups in a choir singing different notes that clash slightly. This is the mixed-symmetry state. For a long time, the primary way to find these states was to look for a strong magnetic transition, a specific type of energy release that happens when a mixed-symmetry state interacts with a fully symmetric one. While this method has worked in the past, it can be confusing because other, non-collective states can sometimes mimic this behavior, leading to ambiguity. The researchers in this study wanted to find a more direct way to see the internal structure of these vibrations, one that didn't rely on the strength of a magnetic signal but rather on the physical shape and size of the vibration itself.

The team focused their investigation on three isotopes: zirconium-92, zirconium-94, and molybdenum-94. These atoms are ideal for study because they are "vibrational," meaning their nuclei behave like liquid drops that can oscillate. The researchers used a powerful theoretical framework called the quasiparticle-phonon model to simulate the behavior of these nuclei. This model allowed them to calculate the expected energy levels, the shapes of the vibrations, and how the nuclei should react when hit by particles. They then compared these calculations against a vast amount of experimental data, including the results of electron scattering and proton scattering experiments. In electron scattering, a beam of electrons bounces off the nucleus, revealing details about the distribution of electric charge. In proton scattering, a beam of protons hits the nucleus, revealing details about the distribution of matter, which includes both protons and neutrons. By combining these two perspectives, the team could reconstruct a complete picture of the nuclear vibrations.

The results of this combined analysis were striking. The researchers found that the theoretical model agreed remarkably well with the experimental data across the board. It correctly predicted the energy levels of the excited states, the magnetic moments, and the probabilities of transitions between different states. Most importantly, the model successfully described the momentum transfer dependence in the scattering experiments. This is a technical way of saying that the way the nuclei scattered the particles depended on the angle and energy of the collision in a specific pattern that matched the theory. This agreement gave the team confidence that their model was accurately capturing the wave functions, or the mathematical descriptions, of the nuclear states. With this confidence established, they could look deeper into the nature of the mixed-symmetry states.

The study confirmed that the mixed-symmetry states in these three nuclei are indeed characterized by a sign change between the leading proton and neutron configurations. In simpler terms, the dominant way the protons move is opposite to the dominant way the neutrons move. This sign change is the fingerprint of the mixed-symmetry state. The researchers discovered that this sign change has a direct and measurable consequence on the transition densities, which describe how the charge and matter are distributed during the vibration. Specifically, the transition density for the mixed-symmetry state has a different shape compared to the fully symmetric state. This difference manifests as a shift in the angular distribution of the scattered protons. When protons hit the mixed-symmetry state, they scatter at slightly different angles compared to when they hit the fully symmetric state, even if the energy of the collision is the same. This shift is a new, independent signature that allows scientists to identify mixed-symmetry states without needing to rely on the strength of magnetic transitions.

This new signature proved to be particularly useful because it helped distinguish the mixed-symmetry states from other, non-collective states that might have similar energies. In the past, scientists sometimes struggled to tell if a state was a true mixed-symmetry vibration or just a random collection of particles moving together. The shift in the scattering pattern provided a clear distinction. For instance, in zirconium-92 and molybdenum-94, the mixed-symmetry states showed a distinct shift in their scattering angles compared to all other low-lying states. This allowed the researchers to confidently identify the candidates for these states. However, the study also revealed that the situation is more complex for higher-energy vibrations. When looking at states with different spins, such as those involving octupole or hexadecapole vibrations, the identification became much harder. The researchers found that these higher-energy states often mix with other configurations, making it difficult to isolate a pure mixed-symmetry state based on magnetic transitions alone.

The paper also addressed a long-standing question about the origin of collectivity in these nuclei. Collectivity refers to the phenomenon where many particles move together in a coordinated way, creating a strong, unified vibration. The researchers found that this collectivity is not generated solely by the particles in the outermost shell of the nucleus, as was sometimes assumed. Instead, a significant portion of the strength comes from the coupling of these outer particles to very high-energy states deep within the nucleus, known as giant resonances. These giant resonances are like massive, collective vibrations of the entire nucleus that exist at much higher energies. The study showed that the interaction between the low-energy vibrations and these high-energy giant resonances is what gives the low-energy states their collective strength. This finding explains why the transition strengths are so large and why the simple models that only look at the outer shell are insufficient.

One of the most significant outcomes of this work is the re-evaluation of a proposed method for identifying mixed-symmetry states. Previously, some researchers suggested using a specific ratio of proton and neutron transition matrix elements to identify these states. The idea was that if the ratio was large, it indicated a mixed-symmetry state. However, the new analysis showed that this ratio is not a reliable indicator. The researchers demonstrated that the collectivity of these states is heavily influenced by the high-energy giant resonances, which dilute the ratio and make it difficult to use as a clear signature. In fact, for the mixed-symmetry states in these nuclei, the ratio turned out to be smaller than expected, sometimes even smaller than that of the fully symmetric states. This finding effectively rules out the use of that specific ratio as a standalone tool for identification, forcing the scientific community to rely on more robust methods like the transition density analysis used in this study.

The researchers also explored the possibility of finding mixed-symmetry states in other types of vibrations, such as octupole (three-dimensional shape changes) and hexadecapole (more complex shape changes). While the model predicted the existence of such states, the analysis suggested that they are likely to be highly fragmented. This means that instead of appearing as a single, clear state, the mixed-symmetry character is spread out over several different energy levels. This fragmentation makes it very difficult to identify them experimentally, especially since they often appear at energies where they mix with other two-phonon states. For the octupole states, the model predicted that the mixed-symmetry candidate would be located at much higher energies than previously thought, likely above four million electron volts, which is beyond the range of many previous experiments. This suggests that earlier claims of identifying these states might need to be revisited.

In the end, this work provides a comprehensive and detailed picture of the low-energy collective structure in zirconium and molybdenum nuclei. By combining electron and proton scattering data with advanced theoretical modeling, the researchers have established a new, reliable way to identify mixed-symmetry states based on the differences in how protons and neutrons move. This method bypasses the ambiguities of previous techniques and offers a clearer view of the nuclear landscape. The study confirms that the mixed-symmetry states in these nuclei are real and distinct, characterized by a specific sign change in their internal structure. It also highlights the critical role of high-energy giant resonances in generating the collective behavior of the nucleus. While the identification of these states in more complex vibrations remains challenging, the tools and insights developed in this paper provide a solid foundation for future exploration. The ability to distinguish these states with such precision opens the door to a deeper understanding of the fundamental forces that hold the atomic nucleus together and the complex dance of particles within it.