Abstract
Magic numbers are the backbone of the nuclear structure, serving as the basis for shell-model truncations, leading to the prediction of the island of stability, and linking to peaks in the solar-system abundance curve. Canonical nuclear magic numbers include 2, 8, 20, 28, 50, 82, and 126. It is now well established that these magic numbers are not universal over the nuclear landscape. This paper presents a brief review of recent highlights on the migration of magic numbers in neutron-rich nuclei, with particular emphasis on results obtained from knockout reactions. We focus on two key regions: 1) the loss of magicity at = 20 and 28, and 2) the emergence of new magic numbers at = 32, 34. Prospects for future measurements in these regions, enabled by new detection systems at upgraded and new facilities, are also discussed.
1 Introduction
Atomic nuclei with specific numbers of neutrons and/or protons exhibit particular stability. These numbers correspond to fully occupied nuclear shells separated by large energy gaps and define so-called ‘magic’ numbers. Canonical nuclear magic numbers found in stable nuclei include 2, 8, 20, 28, 50, 82, and 126. Nuclear shells and the associated magic numbers present the backbone of nuclear structure. Doubly magic nuclei provide a reasonable scheme for nuclear shell models to truncate the configuration space by assuming an inert doubly magic core. The existence of superheavy nuclei with Z > 104 is due to shell effects [1]. An “island of stability”, centering on the next potential doubly magic nucleus beyond 208Pb—with proton number = 114 and neutron number = 184 [2, 3], is predicted, stimulating ever-increasing experimental efforts to synthesize new superheavy elements. In addition, the shell effects at = 50, 82, and 126 are closely linked to the peaks in the solar-system abundance curve [4, 5].
Nuclear shell gaps are not observable but are inferred from a set of sensitive measurable quantities, such as nuclear masses, charge radii, energies, B (E2) transition probabilities, and spectroscopic factors from direct reactions. Magic nuclei typically exhibit high energies, suppressed B (E2) strengths, and discontinuities in separation energies and charge radii, reflecting enhanced rigidity associated with closed shells, while spectroscopic factors indicate near-full occupancy ( 2J + 1) below and almost empty ( 0) above the shell gap, consistent with the well-defined shell closure in the independent-particle model (IPM). Together, these complementary observables provide a coherent and robust picture of nuclear magicity. Over the past 4 decades, significant progress has been made in understanding the evolution of nuclear shell structure in light-to medium-mass nuclei, driven by advances in radioactive beam technology. Owing to the absence of repulsive Coulomb interactions, the neutron dripline lies much further from the -stability line than the proton dripline. To date, the proton dripline has been crossed up to 83 [6], whereas the neutron dripline is confirmed only up to = 10 [7]. Studies of neutron-rich systems with extremely unbalanced proton-to-neutron ratios have revealed a variety of exotic structures and dynamics, including the shell migration. Notable examples include the quenching of the canonical = 8, 20, 28 shell gaps [8–11] and the emergence of new subshell gaps at = 16, 32 and 34 [12–18]. These observations greatly stimulate the development of nuclear structure theories, although the underlying driving mechanism remains not fully understood. Tensor terms of the nucleon-nucleon interactions are suggested to play decisive roles in the shell evolutions [19], while three-body forces are found to be repulsive and help preventing overbinding in nuclei [20, 21].
It should be noted that strong shell reordering or new shell closures have been observed and predicted in nuclei with large proton-to-neutron asymmetry. To access spectroscopic information of such exotic nuclei with low production yields, it is essential to select proper reaction probes and develop novel detection systems. In this mini review, we focus on recent progress in understanding the disapperance of magic numbers at = 20 and 28, and the emergence of new magic numbers at 32 and 34 in neutron-rich nuclei, with particular emphasis on results obtained via the in-beam technique from knockout reactions.
2 In-beam -ray spectroscopy from knockout reactions
In-beam -ray spectroscopy following knockout reactions of fast beams often provides the first observable accessible to experiments to characterize the shell effects of exotic nuclei. In such reactions, projectiles with energies above 50 MeV/u impinge on a thick target to compensate for the low production yields of rare isotopes. By knocking out one or two nucleons, the residues will be populated in excited states, whose excitation energies can be determined from prompt de-excitation rays. One-nucleon knockout reactions have been widely used to probe single-particle structure of nuclei [22]. The momentum distributions of the residues are sensitive to the orbital angular momentum l of the removed nucleon, constraining spin-parity assignments. The measured partial cross sections, in comparison with single-particle cross sections calculated using reaction theory, yield the spectroscopic factors , which quantify the occupancy of a given orbital, albeit with some model dependence. Note that the center-of-mass motion correction is included in . The uncertainties arising from the choice of reaction frameworks, optical potentials, and nuclear structure inputs are significant and have been reviewed in Ref. [23]. A consistent treatment of the reaction calculation is thus essential when discussing spectroscopic strengths extracted from knockout reactions. When considering the IPM occupation, the reduction factor is defined as , which represents the ratio of the experimental cross section to the calculated single-particle cross section normalized to the full occupancy 2. For magic nuclei, is expected to be close to unity for orbitals below the Fermi surface and strongly suppressed for those above the shell gap.
Worldwide, leading facilities such as RIKEN, GSI, and FRIB have extensively employed knockout reactions to study exotic nuclei. Notably, the intense beams at RIKEN-RIBF have facilitated significant studies on shell evolution around = 20, 28, 32, 34 [24], while new generation facilities, including HIAF at IMP [25] and FAIR at GSI [26], will further extend these studies to higher beam energies above 400 MeV/u. At such energies, the final-state interactions are expected to be reduced, as the nucleon-nucleon total cross sections reach the minimum around 300 MeV [27]. However, the high beam energies also induce significant Doppler shifts between the observed –ray energy and those in the rest-frame of the residues [28], requiring precise information on i) the ejectile velocity when emitting the -rays and ii) the -ray emission angle for the Doppler correction. This imposes constraints on the choice of target thickness, which has been largely overcome by the MINOS setup [29]. MINOS combines a thick liquid-hydrogen target with a cylindrical TPC proton tracker, allowing to reconstruct the reaction vertex with a resolution of 4 mm (FWHM) [30]. Compared with passive heavy-ion targets, MINOS provides nearly an-order-of-magnitude increase in luminosity, while preserving Doppler-correction accuracy. MINOS is designed to fit the NaI (Tl) spectrometer DALI2 at the RIBF and has been used to study more than 50 neutron-rich nuclei from 52Ar to 110Zr [31–38]. As an example, with a 150 mm thick , the Doppler-corrected -ray energy resolution for 1.2 MeV rays emitted by nuclei moving at 0.6c improves from 89 to 59 keV when MINOS vertex information is used.
3 Disappearance of = 20 and 28 magic numbers
The pioneering work on the disappearance of a canonical magic number in the Segré chart was carried out in the neutron-rich = 20 region. In 1975, mass measurements performed at CERN revealed an unexpected excess of binding in 31Na and 32Na [9], challenging existing theoretical models. Later in 1990, extensive shell-model calculations by Warbuton et al [44] suggested that in very neutron-rich Ne, Na, and Mg isotopes the = 20 shell gap was quenched, and -shell intruder configurations became energetically more favorable than the normal -shell configurations. Nuclei exhibiting these characteristics are described as lying within the “island of inversion” (IOI). Subsequent experimental observations, such as the low-lying states and enhanced B (E2) transition probabilities in 32Mg [11, 45] and 30Ne [46, 47], were in line with the prediction. In particular, for 32Mg, a low-lying state was observed, interpreted to be spherical and coexisting with the prolate-deformed ground state [48].
Until now, considerable experimental and theoretical efforts have been devoted to mapping the precise boundaries of the IOI and tracing the evolution of intruder strengths along the isotopic and isotonic chains. 34Si was found to be a doubly magic nucleus [49–52], establishing the northern border in of the IOI, and the southern shore of the IOI has been pushed down to 29F [53, 54] and 28O [55] based on the -ray and invariant-mass spectroscopy using knockout reactions. In line with these findings, the SDPF-U-MIX20 calculations that describe well the IOI showed that the = 20 gap diminished from 7 MeV at = 14 to 2 MeV at = 8 [35]. In-beam -ray spectroscopy following one-neutron knockout reactions enables the direct extraction of intruder strengths in the ground state of the projectile via a combined analysis of the momentum distributions and partial cross sections. For 32Mg, the extracted S for neutron removal from the 2 and 1 intruder orbitals were 0.59 (11) and 1.19 (36), respectively, supporting the dominant 2p-2h configurations in its ground state [56]. In Ne isotopes, the S for the -wave intruder components was found to be 0.9 (1) in 30Ne [57], 1.16 (5) in 29Ne [58], and 0.34 (2) in 28Ne [59]. However, the predicted significant -wave intruder strengths to the bound final states of 27-29Ne were not observed [57–59]. Recent experiments confirmed that the 7/2- states of 27,29Ne are both unbound, located at an excitation energy 1.4–1.5 MeV higher than the SDPF-M prediction [60, 61]. These discrepancies present an ongoing challenge for a complete theoretical description of the IOI.
Unlike the = 20 magic number, the = 28 shell gap originates from the spin-orbit splitting between 1 and 1. Figure 1 shows the measured E() systematics along different isotopic chains [39]. The removal of only two protons from the doubly magic 48Ca nucleus leads to a 2,255 (1) keV reduction in E() for 46Ar [62, 63]. In contrast to = 20 nuclei 36S and 34Si, E() of 44S [64] and 42Si [65] become much lower at = 28, supporting the collapse of the = 28 shell gap and the formation of another IOI. 44S [66] exhibited shape coexistence with a deformed ground state, while 42Si [67] and 40Mg [68] were suggested to be a well-deformed oblate and prolate rotor, respectively. For 42Si, the level observed at 2,150 (13) keV was tentatively assigned as the state based on comparisons with calculations [69]. 40Mg is the most neutron-rich even-even = 28 isotone known to lie inside the dripline. However, its two-neutron separation energy = 670 (710) keV has a large uncertainty [6], leaving its weak binding effects poorly constrained. The low-lying excited states of 40Mg were first studied via knockout reactions at the RIBF [68]. The observed excitation spectrum revealed a pattern distinctly different from that of its neighbors 36,38Mg [70], indicating that weak-binding effects may play a critical role [71]. In particular, 40Mg was predicted to be a promising candidate for a halo nucleus with two neutrons occupying the 2 intruder orbital [72–74]. Further measurements, such as interaction cross sections of Mg isotopes, are highly desirable to clarify this intriguing phenomenon. The measured E() systematics along the Mg isotopic chain, shown in Figure 1, displays a striking long flat trend from = 20 to 28, supporting the merging of the = 20 and = 28 shell quenching [70].
FIGURE 1
4 Emergence of = 32 and 34 magic numbers
= 32 and 34 are two intimately connected new subshells emerging in the neutron-rich -shell nuclei. In the shell model framework, the weakening of the attractive - tensor interactions when removing protons from the orbital from 64Ni to 54Ca raises the energy of the orbitals with respect to and [17]. Consequently, a new = 34 subshell closure develops above the = 32 subshell.
The = 32 subshell gap is determined by the spin-orbit splitting of the 2 and 2 orbitals. As shown in Figure 1, the measured E() systematics along the Ca isotopic chain exhibit a pronounced peak at = 32, supporting its doubly magic nature [14]. This finding was further confirmed by the subsequent mass measurements. The neutron shell gaps extracted from masses indicate that the = 32 magicity is quite localized: it peaks at 52Ca [75], remains strong in 51K [76] and 53Sc [77, 78], weakens in 54Ti [79], and vanishes for 22 [80]. However, the laser spectroscopy revealed unexpected large charge radii for Ca isotopes up to = 32 [81] and for K isotopes up to = 33 [82], interpreted as challenging the = 32 closed-shell character. New insights to understand these seemingly conflicting interpretations have been obtained from a quasi-free 52Ca(p, pn) measurement with MINOS [83]. Using a consistent DWIA framework for reaction calculations, the measured partial cross sections from 48,52,54Ca(p, pn) were used to extract reduction factors for single-particle states below and above the Fermi surface. The 52Ca results displayed a pattern characteristic of magic nuclei, similar to 48,54Ca, with for orbitals below the Fermi level and very small values for those above, providing direct experimental evidence for its doubly magic nature. Furthermore, leveraging the Fourier transform relationship between momentum and spatial distributions, the measured momentum distributions were used to extract the root-mean-square (rms) radii of the and orbitals, with the latter exceeding the former by 0.61 (23) fm. The impact of the choice of optical potentials on momentum distributions was estimated to be within 4.5%. This result is consistent with the 0.7 fm difference predicted by modified-shell-model calculations [84], which proposed that the occupation of spatially extended orbitals beyond = 28 influenced the proton distributions and led to the large charge radii of K and Ca isotopes while preserving the = 32 magicity. To further test this interpretation, systematic (p, pn) studies to extract the rms radii of fp-shell orbitals in other isotopes such as 53,54Ca are highly desirable.
50Ar is the most neutron-rich even-even = 32 isotone for which -ray spectroscopy is available. Shell model calculations with the modified SDPF-MU interaction, describing well the measured E() systematics, predicted that the = 32 subshell gap in 50Ar [16] was as strong as in 52Ca. Nevertheless, the E() of 50Ar is lower than those at = 28 and 34. This behavior was attributed to the onset of collectivity associated with the open proton shell in Ar isotopes based on the 50Ar(p, pn) studies [85]. Very recently, ab initio calculations employing realistic nuclear forces suggested that the = 32 subshell was gradually quenched as protons are removed from = 20 [40]. In contrast, mean-field calculations predicted that this subshell persisted until = 16, with 48S being a doubly magic nucleus [86]. The measurement of E() in the more neutron-rich isotone, like 48S, via in-beam -ray spectroscopy following knockout reactions is thus crucial to discriminate between these two scenarios and clarify how the = 32 subshell evolves below = 18.
Turning to the = 34 subshell closure, the first experimental evidence for its existence came from the measured E() of 54Ca ( = 20) from knockout reactions [17]. This finding was further supported by the S extracted from the 54Ca(p, pn) reaction [87] and the mass measurement [88]. For nuclei with 20, a rapid weakening of the = 34 subshell closure was observed, based on systematic measurements of the excitation energies [89, 90], B(E2) transition probabilities [91, 92], and masses [78]. However, the = 34 subshell gap was predicted to be enhanced below = 20 by the shell-model calculations employing both phenomenological [41] and chiral-effective-field-theory interactions [40], which described well available experimental data. Notably, 48Si was predicted to be a new doubly magic nucleus far away from stability with a 3.9 MeV = 34 shell gap [18]. Thanks to the intense beams and the MINOS setup at the RIBF, the first in-beam spectroscopy of 52Ar [18] was achieved and yielded a excitation energy of 1,656 (18) keV, which is the highest among the measured Ar isotopes with 20, as shown in Figure 1. The results supported the persistence of the = 34 subshell closure below = 20, in line with the predicted trend.
5 Outlook
Over the past two decades, studies of the migration of magic numbers have been significantly extended to extremely neutron-rich systems, primarily through the in-beam -ray spectroscopy following the knockout reactions. Neutron-rich N = 20 and 28 nuclei are flagship cases of the disappearance of canonical magic numbers in the Segré chart, forming an extended region of deformation by merging two IOIs. It should be emphasized that excited states are bandheads of unfavored configurations and are key for understanding the competition between shell gaps and many-body correlations. However, for many nuclei within the IOI, such as 30, 32Ne, 34,36,38Mg, the locations of the states remain unknown. This is primarily because these low-lying states often do not decay via transitions, requiring particle detection to determine their excitation energies. The STRASSE system [93], consisting of a thick target and a silicon tracker, has been developed to perform simultaneous in-beam and missing-mass spectroscopy at the RIBF, offering a unique opportunity to study these states. While the emergence of the new = 32 and 34 magic numbers is well established in the Ca region, their evolution below = 18 towards more neutron-rich systems remains an open question. The nucleus 48S (, ) involves two spin-orbit splitting subshells. Its first in-beam -ray spectroscopy would offer key information about the shell evolution in this region. Looking ahead, next-generation RI-beam facilities will further advance these studies. Following the planned upgrade of the RIBF, a new scintillator array, HYPATIA [94], comprising about one thousand HR-GAGG (Ce) and detectors, is under construction. With the 2 pA238U primary beam at the upgraded RIBF, HYPATIA will enable the -ray spectroscopy of extremely exotic nuclei such as 60Ca using (p, 2p) reactions with secondary-beam intensities at the level of 1 pps [95]. Notably, the HIAF facility in China is entering the commissioning phase. Its high beam energy, ranging from several hundred MeV/u to a few GeV/u, will increase the fraction of fully stripped heavy ions, offering an excellent platform to study shell evolution via knockout reactions in the heavier mass region, such as the structure of neutron-rich = 126 nuclei. A thick liquid hydrogen target dedicated to such studies at HIAF is now under construction. The stage is set for continued fruitful advances in understanding exotic nuclear structure.
Statements
Author contributions
HL: Writing – review and editing, Investigation, Writing – original draft, Software, Funding acquisition.
Funding
The author(s) declared that financial support was received for this work and/or its publication. HL thanks the International Union of Pure and Applied Physics (IUPAP) C12 Commission for this opportunity, and is honored to be the recipient of the 2025 IUPAP Early Career Scientist Prize in Nuclear Physics (C12). This work was supported by the National Key R&D Program of China (Grant No. 2023YFA1606403, No. 2024YFE0102800), the National Natural Science Foundation of China (Grant No. 12375111, No. 12361141822), and the Fundamental Research Funds for the Central Universities of China.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
knockout reactions, magic numbers, magicity loss at N = 20 and 28, new magic numbers at N = 32 and 34, shell evolution
Citation
Liu H (2026) Exploring the migration of magic numbers in neutron-rich systems via knockout reactions. Front. Phys. 14:1778872. doi: 10.3389/fphy.2026.1778872
Received
31 December 2025
Revised
25 March 2026
Accepted
26 March 2026
Published
12 May 2026
Volume
14 - 2026
Edited by
Chong Qi, Royal Institute of Technology, Sweden
Reviewed by
Jin Lei, Tongji University Press, China
Updates
Copyright
© 2026 Liu.
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*Correspondence: Hongna Liu, hongna.liu@bnu.edu.cn
ORCID: Michał Ciborowski, orcid.org/0000-0002-5881-8353
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