⚛️ nuclear experiments

Probing the Size of Neutron and Proton Single-Particle Orbitals from Nucleon Knockout Reactions

This study investigates the spatial sizes of neutron and proton single-particle orbitals in calcium and scandium isotopes using nucleon knockout reactions at 230 MeV/nucleon, revealing that the 1p neutron orbitals are consistently 0.48–0.78 fm larger than the 0f7/2 orbitals while the evolution of valence proton orbital sizes remains inconclusive due to statistical uncertainties.

Original authors: M. Enciu, A. Obertelli, P. Doornenbal, C. Barbieri, S. Brolli, M. Heinz, W. Horiuchi, T. Inakura, W. H. Long, T. Miyagi, F. Nowacki, K. Ogata, A. Poves, A. Schwenk, K. Yoshida, N. L. Achouri, H. Baba

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: M. Enciu, A. Obertelli, P. Doornenbal, C. Barbieri, S. Brolli, M. Heinz, W. Horiuchi, T. Inakura, W. H. Long, T. Miyagi, F. Nowacki, K. Ogata, A. Poves, A. Schwenk, K. Yoshida, N. L. Achouri, H. Baba, F. Browne, D. Calvet, F. Château, S. Chen, N. Chiga, A. Corsi, M. L. Cortés, A. Delbart, J-M. Gheller, A. Giganon, A. Gillibert, C. Hilaire, T. Isobe, T. Kobayashi, Y. Kubota, V. Lapoux, H. N. Liu, T. Motobayashi, I. Murray, H. Otsu, V. Panin, N. Paul, W. Rodriguez, H. Sakurai, M. Sasano, D. Steppenbeck, L. Stuhl, Y. L. Sun, Y. Togano, T. Uesaka, K. Wimmer, K. Yoneda, O. Aktas, T. Aumann, L. X. Chung, F. Flavigny, S. Franchoo, I. Gasparic, R. -B. Gerst, J. Gibelin, K. I. Hahn, D. Kim, Y. Kondo, P. Koseoglou, J. Lee, C. Lehr, P. J. Li, B. D. Linh, T. Lokotko, M. MacCormick, K. Moschner, T. Nakamura, S. Y. Park, D. Rossi, E. Sahin, P-A. Söderström, D. Sohler, S. Takeuchi, H. Toernqvist, V. Vaquero, V. Wagner, S. Wang, V. Werner, X. Xu, H. Yamada, D. Yan, Z. Yang, M. Yasuda, L. Zanetti

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

Atoms are built from a dense core called the nucleus, which is made of protons and neutrons. For decades, physicists have known that these tiny particles do not sit still; they move in specific regions or "orbitals" around the center, much like planets orbit a star, though the rules governing their movement are far more complex. Understanding the size of these orbitals is crucial because the arrangement of protons and neutrons determines the stability and behavior of every element in the universe. While scientists have long been able to measure the overall size of a nucleus with great precision, figuring out the exact size of the individual orbitals for just the neutrons or just the protons has remained a difficult challenge, especially for unstable, short-lived atoms that cannot be found in nature.

A team of researchers recently tackled this problem by studying a specific family of calcium atoms, some of which are heavy and unstable. They wanted to see how the space occupied by a single neutron or proton changes as more neutrons are added to the core. To do this, they used a technique called nucleon knockout, which is essentially a high-speed collision experiment. They fired a beam of these calcium atoms at a target made of liquid hydrogen. When a fast-moving calcium atom hit a hydrogen proton, the impact was strong enough to knock a single neutron or proton out of the calcium nucleus. By carefully measuring the speed and direction of the remaining piece of the atom after the collision, the scientists could work backward to determine how far out that knocked-out particle had been traveling before it was hit. This method allowed them to map the spatial size of the specific orbital the particle came from.

The researchers focused on calcium isotopes with mass numbers 52, 53, and 54, which contain more neutrons than the stable calcium found in everyday life. They looked at two main scenarios: knocking out a neutron and knocking out a proton. When they analyzed the data for the neutrons, they found a clear and consistent pattern. The neutrons living in the outermost, or "valence," orbitals were significantly larger than those in the deeper, core orbitals. Specifically, the outer neutrons were found to be between 0.48 and 0.78 femtometers larger than the inner ones. A femtometer is an incredibly small unit of distance, equal to one quadrillionth of a meter, but in the world of atomic nuclei, this difference is substantial. This finding confirms a theoretical idea that as you fill up these outer neutron shells, they push the entire core of the nucleus to swell slightly, expanding the space available for all the particles inside.

The results for the protons were less definitive. The team measured the size of the proton orbitals in the same calcium atoms, but the data was much noisier. Because the reaction that knocks out a proton happens less frequently than the one for neutrons, the researchers had fewer events to analyze, leading to larger statistical uncertainties. While the data suggested that the proton orbitals might also be getting slightly larger as more neutrons are added, the evidence was not strong enough to draw a firm conclusion. The measurements were consistent with the idea of expansion, but the margin of error was too wide to say for certain how much the protons had moved. This highlights a limitation in the current experiment: the method works very well for neutrons but requires more precise equipment and more data to be equally reliable for protons.

To ensure their measurements were accurate, the team compared their experimental results with advanced computer simulations based on the fundamental laws of physics. These simulations, which used different mathematical approaches to model the forces between particles, predicted the same large difference in size between the inner and outer neutron orbitals that the experiment observed. This agreement between the real-world data and the theoretical models gives scientists confidence that they are correctly understanding how these nuclear shells behave. The study also revisited older measurements of stable calcium atoms to see how the orbitals changed as the nucleus grew from its stable form to these heavier, unstable versions. The data suggested a trend where the proton orbitals expand as the nucleus gets heavier, but again, the uncertainty in the new measurements meant this trend could not be confirmed with high precision.

The work represents a significant step forward in mapping the internal structure of atomic nuclei. By successfully measuring the size of specific neutron orbitals in unstable isotopes, the researchers have provided a new tool for testing our understanding of nuclear forces. The clear difference found between the inner and outer neutron shells helps explain why the overall size of calcium nuclei increases so sharply after a certain number of neutrons is reached. However, the study also points out where more work is needed. To fully understand the behavior of protons in these heavy nuclei, future experiments will need to improve their ability to detect the rare proton-knockout events with greater precision. Until then, the picture of how protons and neutrons share space inside the heaviest calcium atoms remains partially incomplete, waiting for the next generation of experiments to fill in the gaps.

Technical Summary: Probing the Size of Neutron and Proton Single-Particle Orbitals from Nucleon Knockout Reactions

Problem and Motivation
Understanding the spatial size of atomic nuclei, particularly the matter, charge, and neutron radii, is fundamental to unraveling the complex interplay of forces within nuclear matter. While charge radii of stable and unstable nuclei have been measured with high precision via optical probes, neutron and matter radii for unstable isotopes remain largely inaccessible or lack high precision due to the limitations of elastic scattering and interaction cross-section measurements. Specifically, the behavior of neutron radii and neutron skin thickness is missing for most unstable isotopes. Furthermore, microscopic calculations have struggled to reproduce the steep increase in charge radii observed in calcium isotopes after the $N=28$ shell closure. This work addresses the need to probe the size of specific valence neutron and proton single-particle orbitals to understand relative changes between neighboring nuclei and the effect of adding one nucleon type on the other.

Methodology
The study investigates the size of single-particle orbitals in $^{52,53,54}$ Ca and $^{55}$ Sc using exclusive nucleon knockout reactions at approximately 230 MeV/nucleon. The experiment was conducted at the RIKEN Radioactive Isotope Beam Factory using a secondary beam of $^{52}$ Ca, $^{53}$ Ca, $^{54}$ Ca, and $^{55}$ Sc. The reactions of interest were quasi-free scattering processes: $(p, pn)$ for neutron knockout and $(p, 2p)$ for proton knockout.

The core methodology relies on measuring the momentum distributions of the reaction fragments. These distributions are interpreted within the Distorted Wave Impulse Approximation (DWIA) framework. The spatial extension of the knocked-out nucleon's wave function is shown to be sensitive to the shape of the measured momentum distribution.

Experimental Setup: The MINOS setup (liquid hydrogen target and Time Projection Chamber) was used for reaction vertex reconstruction. Fragment identification and momentum measurement were performed using the SAMURAI spectrometer and BigRIPS separator. Final states were tagged via $\gamma$ -ray spectroscopy using the DALI2+ array.
Analysis: Theoretical momentum distributions were calculated using DWIA, where the knocked-out nucleon's wave function was modeled as a bound state of a Woods-Saxon potential. The radial parameter ( $r_0$ ) and diffuseness parameter ( $a_0$ ) were varied to fit the experimental momentum distributions. The optimum $r_0$ was determined via $\chi^2$ minimization, from which the single-particle root-mean-square (rms) radius ( $R_{s.p.}$ ) was extracted.
Theoretical Comparison: The extracted radii were compared against state-of-the-art theoretical predictions, including:
- Ab initio calculations: In-Medium Similarity Renormalization Group (IMSRG) and Self-Consistent Green's Function (SCGF) methods based on chiral effective field theory interactions.
- Mean-field calculations: Skyrme-Hartree-Fock (HF) and Relativistic Hartree-Fock (RHF) using various energy-density functionals.

Key Contributions and Results

Extraction of Single-Particle Radii: This work presents the first extraction of single-particle orbital sizes for the $\nu 0f_{7/2}$ , $\nu 1p$ , $\pi 0d_{3/2}$ , and $\pi 1s_{1/2}$ orbitals in neutron-rich calcium isotopes using momentum distributions from knockout reactions.
Neutron Orbital Size Evolution: A systematic sensitivity study confirms that the momentum distribution method is robust. The results consistently show that the $1p$ neutron orbitals ( $\nu 1p_{3/2}$ and $\nu 1p_{1/2}$ ) in $^{52-54}$ Ca are significantly larger than the deeply bound $\nu 0f_{7/2}$ core orbital. The difference in rms radii is found to be between 0.48 and 0.78 fm. Additionally, the $\nu 1p_{1/2}$ orbitals are slightly larger than the $\nu 1p_{3/2}$ orbitals.
Proton Orbital Size Evolution: For proton orbitals, the size of the $\pi 0d_{3/2}$ orbital remains constant across $^{52-54}$ Ca, with extracted values of 3.98(18) fm for $^{52}$ Ca, 4.11(14) fm for $^{53}$ Ca, and 3.85(17) fm for $^{54}$ Ca. The $\pi 1s_{1/2}$ orbital shows a marginal increase toward $^{54}$ Ca, with values of 3.57(33) fm for $^{52}$ Ca, 3.72(26) fm for $^{53}$ Ca, and 4.25(31) fm for $^{54}$ Ca. However, the authors note that the statistical uncertainties for the proton channels are large due to lower cross-sections in $(p, 2p)$ reactions, rendering the evolution of valence proton orbitals inconclusive.
Comparison with Theory:
- The large difference between the $0f_{7/2}$ and $1p$ neutron orbital sizes is well reproduced by ab initio VS-IMSRG and SCGF calculations, as well as Relativistic Hartree-Fock (RHF) models.
- Standard Hartree-Fock calculations for $^{52}$ Ca agree with experimental values within $2\sigma$ .
- Regarding proton orbitals, theoretical models predict a slight shrinking of $sd$-shell orbitals from $^{40}$ Ca to $^{48}$ Ca followed by a steady expansion to $^{54}$ Ca. However, the magnitude of the increase observed experimentally (relative to $^{48}$ Ca values from $(e, e'p)$ data) is larger than what the theoretical models predict, though a $2\sigma$ agreement is maintained.
Quenching Factors: The study reports isospin-dependent quenching factors. Neutron knockout channels show no quenching (factors of 0.96 and 1.0), while proton knockout channels exhibit significant quenching (factors of 0.51–0.66), consistent with literature findings.

Significance and Claims
The paper claims that the study provides direct experimental evidence for a "core swelling" effect triggered by the filling of large, spatially extended $1p$ neutron orbitals beyond $N=28$ . This mechanism offers an explanation for the steep, non-linear increase in bulk charge radii observed in calcium isotopes, a phenomenon that has challenged traditional microscopic calculations.

The authors emphasize that while the neutron orbital size differences are robustly determined and consistent with ab initio predictions, the conclusions regarding proton orbital evolution are limited by statistical uncertainties inherent to the $(p, 2p)$ reaction channels. Consequently, the paper calls for future systematic experiments with improved statistics and momentum resolution to refine the understanding of neutron and matter radii. It also highlights the theoretical need to move beyond localized Woods-Saxon potential approximations by integrating ab initio transition amplitudes directly into reaction theory frameworks to achieve a coherent description of nuclear states.

More like this